Measuring Noise in the VHF Band and Its Affects on Low SNR Signal

Measuring Noise in the VHF Band and Its Eﬀect On Low SNR Signal Detection

Hunter A. DeJarnette III

Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulﬁllment of the requirements for the degree of

Master of Science in Electrical Engineering

R. Michael Buehrer, Chair Steven W. Ellingson Claudio R. C. M. da Silva

September 26, 2012 Blacksburg, Virginia

Hunter A. DeJarnette III

(ABSTRACT)

With the increasing demand for access to the crowded radio frequency spectrum, cognitive radios have been suggested as one solution. Cognitive radios would be frequency agile and able to sense their radio environment and opportunistically use empty spectrum. Spectrum sensing, monitoring a given band of spectrum to see if it is occupied, is an essential part of a cognitive radio. The preferred method of spectrum sensing is the energy detector, which does not require any a priori information about the signal to be detected and is computationally simple to implement. Man-made noise, impulsive in nature, has also become more prevalent with the widespread use of electronic devices. In this thesis, we took measurements of man- made impulsive noise in the broadcast digital television bands to measure its presence, power, and spatial correlation. The eﬀects of impulsive noise on the detection performance of an energy detector were analyzed. Lastly, a wideband RF receiver was designed, built, and tested on the Virginia Tech campus, which would be well suited both to spectrum sensing and taking measurements of impulsive noise. Dedication

To Leslie, you are the love of my life.

iii Acknowledgments

I would like to thank my advisor, Dr. Buehrer. I have learned so much working with him, and I appreciate all the time that he has invested in me to see this through to completion. It has really been a wonderful two years.

I would like to thank Dr. Ellingson and the students of ECE 4984 RF Spectrum Sensing: Bill Clark, Mike Hutton, Chris Jennette, Jarrette Keen, JB Calpo, and Neeraj Pramodkumar. I learned a lot both in this class and from working with such a great team of fellow students. I don’t think I would have been able to ﬁnish this thesis without the experience of this class.

I can’t forget the fellow students in Wireless@VT. Too many names to list here, but I have had a wonderful time working with you during my time at Virginia Tech. You have always been there to help answer my questions and keep me going.

iv Contents

1 Introduction1

1.1 Motivation...... 1

1.2 Contributions...... 3

1.3 Thesis Summary...... 4

2 Measurement Campaign8

2.1 Single Antenna...... 9

2.1.1 Measurement Setup...... 9

2.1.2 Time Resolution...... 12

2.1.3 Amplitude Distribution...... 13

2.1.4 Time Correlation...... 16

2.1.5 Average power vs. frequency at diﬀerent times of day...... 19

2.2 Multi Antenna...... 21

2.2.1 Measurement Setup...... 21

2.2.2 Measured Parameters...... 22

2.2.3 Veriﬁcation of the Measurement System...... 23

v 2.2.4 Results...... 25

2.3 Conclusions...... 26

3 Impulsive Noise Model 32

3.1 Introduction...... 32

3.2 Middleton’s Model...... 33

3.3 Simpliﬁcation of Middleton’s Models...... 36

3.3.1 Simpliﬁed Class B Model...... 37

3.3.2 Statistics for Gaussian Noise...... 38

3.3.3 Statistics for Impulsive Noise Using Weibull Distribution...... 39

3.3.4 Simpliﬁed Class A Model...... 41

3.4 Estimating the Parameters for the Two Models...... 41

3.5 Conclusion...... 48

4 Eﬀects of Impulsive Noise on the Energy Detector 49

4.1 System Model and Detector Statistics...... 50

4.1.1 System Model...... 50

4.1.2 Detector Statistics...... 51

4.2 SNR Walls for an Energy Detector...... 53

4.2.1 Reviewing the SNR Wall for Gaussian Noise of Uncertain Power... 53

4.2.2 SNR Wall for Impulsive Noise...... 56

4.3 Eﬀects of Impulsive Noise on the Sample Complexity of an Energy Detector 59

vi 4.4 Eﬀect on the Central Limit Theorem Convergence...... 61

4.5 Conclusions...... 64

5 Eﬀects of Spatially Correlated Impulsive Noise on the Energy Detector 68

5.1 System Model...... 69

5.1.1 Deﬁnition of Terms...... 70

5.2 Correlated Noise Measurements...... 71

5.2.1 Correlated Noise Model...... 74

5.3 Performance of an Energy Detector in Uncorrelated Impulsive Noise..... 78

5.3.1 Probabilities of False Alarm and Missed Detection for a Single Channel Realization...... 78

5.3.2 Average Probabilities of False Alarm and Missed Detection...... 80

5.4 Performance of an Energy Detector in Spatially Correlated Impulsive Noise. 80

5.4.1 Calculating the Correlation in Noise Power for Impulsive Noise.... 81

5.4.2 Probabilities of False Alarm and Missed Detection for Spatially Cor- related Noise...... 84

5.5 Numerical Simulations...... 86

5.6 Conclusions...... 89

6 Receiver Design 90

6.1 Overview of Project...... 90

6.2 Design Methodology...... 92

6.3 System Architecture...... 96

vii 6.3.1 Signal Flow and Design Decisions...... 96

6.3.2 GNI Analysis...... 98

6.4 Results...... 103

6.5 Conclusions...... 106

7 Conclusions 109

7.1 Summary...... 110

7.2 Future Work...... 111

Appendices 113

A GNI Analysis 113

B RF Board Details 115

B.1 Diplexer Design...... 116

B.2 Schematic, Layout and Bill of Materials...... 118

C Measurement Results 128

C.1 Near a Cubicle...... 136

C.2 Near an Elevator...... 143

C.3 Near a Refrigerator...... 150

C.4 Outdoors Near Construction...... 157

C.5 Average Power vs. Frequency for Diﬀerent Times of Day...... 164

Bibliography 166

viii List of Figures

2.1 Representative Amplitude Probability Distribution (APD) graph...... 16

2.2 Examples of periodic and aperiodic impulsive noise measurements...... 18

2.3 Examples of aperiodic and periodic impulsive noise pulse shapes...... 19

2.4 Representative autocorrelation functions for impulsive noise...... 20

2.5 Representative autocovariance functions...... 20

2.6 Average power in broadcast TV channels for VHF low band...... 21

2.7 Correlation coeﬃcient for Channels 3-5 vs. fractional wavelength...... 26

2.8 Correlation coeﬃcient for Channels 1-2 vs. fractional wavelength...... 29

2.9 External noise ﬁgure over the measured frequencies for a city environment using (2.11). The parameters c = 76.8 and d = 27.7 are for a city environment taken from [37]...... 31

3.1 Comparison of APD graph of Class A and Class B models at low probabilities with bandwidth B = 5 MHz...... 36

3.2 Comparison of the APD graph for Weibull and Gaussian distributed noise. 40

3.3 APD for measurements taken in an oﬃce outside of a cubicle along with Class A and B noise models with measurement bandwidth B = 5 MHz...... 46

ix 3.4 APD for measurements taken outdoors near a construction site along with Class A and B noise models with measurement bandwidth B = 5 MHz... 46

3.5 APD for measurements taken in an oﬃce outside of a cubicle along with Class A and B noise models with measurement bandwidth B = 5 MHz...... 47

3.6 APD for measurements taken near a refrigerator along with the Class B noise model with measurement bandwidth B = 5 MHz...... 47

4.1 SNR Wall for additive white Gaussian noise (AWGN) noise. The location of the SNR Wall is calculated using (4.16)...... 56

4.2 SNR Wall for -mixture noise with = .002 and κ = 26dB...... 59

4.3 Eﬀect of the values of κ and on the SNR Wall...... 60

4.4 Ratio of variance of δ for -mixture and Gaussian noise. The increased variance of the test statistic requires a proportionally longer sensing time to get the same performance as in the AWGN case...... 61

4.5 Receiver operating curve for N = 10, 000, = 0.01, and snr = −15 dB. Increasing values of κ negatively impact the ROC curve for GMM noise... 62

4.6 Comparing the convergence of the Central Limit Theorem (CLT) for -mixture noise and AWGN...... 63

4.7 Eﬀect of slow convergence of the CLT on the receiver operating curve.... 63

4.8 Simulation to demonstrate the SNR Wall. The noise power uncertainty was

set to 0.35 dB, PFA = 0.1. The SNR Wall was calculated using (4.20).... 66

4.9 Failure of signal detection when the noise power is underestimated. The de-

cision threshold was set using the estimated noise so that PMD = 0.1. The SNR Wall was calculated using (4.28)...... 67

x 4.10 Failure of signal detection when the noise power is overestimated. The decision

threshold was set using the estimated noise so that PFA = 0.1. The SNR Wall was calculated using (4.28)...... 67

5.1 Example of using a threshold to compare the correlation of the impulsive and nominal components of the noise. The threshold, τ, is set to 5 dB where the APD deviates from the Gaussian distribution. The measurement was taken at 79 MHz with an antenna separation of 3 m...... 75

5.2 Comparison of the crosscorrelation coeﬃcients of the impulsive and nominal components of the noise plotted against a chosen threshold. Measurements were taken at a center frequency of 79 MHz with an antenna separation of 3 m. 77

5.3 Simulating the correlation of the instantaneous power of a Gaussian mixture model based on the correlation of the impulsive component with = 0.01, κ = 100, and 1e6 samples...... 84

5.4 Comparing the eﬀect of sensing time on correlated impulsive noise that arrives at all antennas simultaneously vs. impulsive noise that arrives randomly. For

this simulation, PFA = 0.1, = 0.01, κ = 100...... 87

5.5 Comparing the eﬀect of additional antennas for correlated impulsive noise that arrives at all antennas simultaneously vs. arriving at diﬀerent antennas

randomly. For this simulation = 0.01, κ = 100, N = 20e3, PFA = 0.1.... 88

6.1 Full working RF spectrum sensing system...... 91

6.2 System architecture for the RF board...... 97

6.3 Final design for the RF board...... 104

6.4 System transfer function of the RF board for the 25-80 MHz band...... 105

6.5 System transfer function of the RF Board for the 116-174 MHz Band.... 106

xi 6.6 RF measurements taken in the 25-80 MHz band near the campus of Virginia Tech...... 107

6.7 RF measurements taken in the 116-174 MHz band near the campus of Virginia Tech...... 108

B.1 Diplexer design for the RF board...... 118

B.2 Measured transfer function for diplexer ﬁlters connected in parallel..... 119

B.3 RF board schematic...... 120

B.4 RF board schematic...... 121

B.5 RF board schematic...... 122

B.6 RF board layout...... 123

B.7 RF board layout...... 124

B.8 Bill of Materials...... 125

B.9 Bill of Materials...... 126

B.10 Bill of Materials...... 127

C.1 Measurements for the 1 dB compression point using 15 dB of attenuation.. 129

C.2 Measurements near a cubicle with a center frequency of 57 MHz and measurement bandwidth B = 5 MHz...... 136

C.3 Measurements near a cubicle with a center frequency of 63 MHz and measurement bandwidth B = 5 MHz...... 137

C.4 Measurements near a cubicle with a center frequency of 69 MHz and measurement bandwidth B = 5 MHz...... 137

xii C.5 Measurements near a cubicle with a center frequency of 79 MHz and measurement bandwidth B = 5 MHz...... 138

C.6 Measurements near a cubicle with a center frequency of 85 MHz and measurement bandwidth B = 5 MHz...... 138

C.7 Measurements near a cubicle with a center frequency of 177 MHz and measurement bandwidth B = 5 MHz...... 139

C.8 Measurements near a cubicle with a center frequency of 183 MHz and measurement bandwidth B = 5 MHz...... 139

C.9 Measurements near a cubicle with a center frequency of 189 MHz and measurement bandwidth B = 5 MHz...... 140

C.10 Measurements near a cubicle with a center frequency of 195 MHz and measurement bandwidth B = 5 MHz...... 140

C.11 Measurements near a cubicle with a center frequency of 201 MHz and measurement bandwidth B = 5 MHz...... 141

C.12 Measurements near a cubicle with a center frequency of 207 MHz and measurement bandwidth B = 5 MHz...... 141

C.13 Measurements near a cubicle with a center frequency of 213 Mhz and measurement bandwidth B = 5 MHz...... 142

C.14 Measurements near and elevator with a center frequency of 57 MH and measurement bandwidth B = 5 MHz...... 143

C.15 Measurements near and elevator with a center frequency of 63 MH and measurement bandwidth B = 5 MHz...... 144

C.16 Measurements near and elevator with a center frequency of 69 MH and measurement bandwidth B = 5 MHz...... 144

xiii C.17 Measurements near and elevator with a center frequency of 79 MH and measurement bandwidth B = 5 MHz...... 145

C.18 Measurements near and elevator with a center frequency of 85 MH and measurement bandwidth B = 5 MHz...... 145

C.19 Measurements near and elevator with a center frequency of 177 MH and measurement bandwidth B = 5 MHz...... 146

C.20 Measurements near and elevator with a center frequency of 183 MH and measurement bandwidth B = 5 MHz...... 146

C.21 Measurements near and elevator with a center frequency of 189 MH and measurement bandwidth B = 5 MHz...... 147

C.22 Measurements near and elevator with a center frequency of 195 MH and measurement bandwidth B = 5 MHz...... 147

C.23 Measurements near and elevator with a center frequency of 201 MH and measurement bandwidth B = 5 MHz...... 148

C.24 Measurements near and elevator with a center frequency of 207 MH and measurement bandwidth B = 5 MHz...... 148

C.25 Measurements near and elevator with a center frequency of 213 MH and measurement bandwidth B = 5 MHz...... 149

C.26 Measurements near a Microwave with a center frequency of 57 MH and measurement bandwidth B = 5 MHz...... 150

C.27 Measurements near a Microwave with a center frequency of 63 MH and measurement bandwidth B = 5 MHz...... 151

C.28 Measurements near a Microwave with a center frequency of 69 MH and measurement bandwidth B = 5 MHz...... 151

xiv C.29 Measurements near a Microwave with a center frequency of 79 MH and measurement bandwidth B = 5 MHz...... 152

C.30 Measurements near a Microwave with a center frequency of 85 MH and measurement bandwidth B = 5 MHz...... 152

C.31 Measurements near a Microwave with a center frequency of 177 MH and measurement bandwidth B = 5 MHz...... 153

C.32 Measurements near a Microwave with a center frequency of 183 MH and measurement bandwidth B = 5 MHz...... 153

C.33 Measurements near a Microwave with a center frequency of 189 MH and measurement bandwidth B = 5 MHz...... 154

C.34 Measurements near a Microwave with a center frequency of 195 MH and measurement bandwidth B = 5 MHz...... 154

C.35 Measurements near a Microwave with a center frequency of 201 MH and measurement bandwidth B = 5 MHz...... 155

C.36 Measurements near a Microwave with a center frequency of 207 MH and measurement bandwidth B = 5 MHz...... 155

C.37 Measurements near a Microwave with a center frequency of 213 MH and measurement bandwidth B = 5 MHz...... 156

C.38 Outdoor measurements taken near a construction site with a center frequency of 57 MHz...... 157

C.39 Outdoor measurements taken near a construction site with a center frequency of 63 MHz...... 158

C.40 Outdoor measurements taken near a construction site with a center frequency of 69 MHz...... 158

xv C.41 Outdoor measurements taken near a construction site with a center frequency of 79 MHz...... 159

C.42 Outdoor measurements taken near a construction site with a center frequency of 85 MHz...... 159

C.43 Outdoor measurements taken near a construction site with a center frequency of 177 MHz...... 160

C.44 Outdoor measurements taken near a construction site with a center frequency of 183 MHz...... 160

C.45 Outdoor measurements taken near a construction site with a center frequency of 189 MHz...... 161

C.46 Outdoor measurements taken near a construction site with a center frequency of 195 MHz...... 161

C.47 Outdoor measurements taken near a construction site with a center frequency of 201 MHz...... 162

C.48 Outdoor measurements taken near a construction site with a center frequency of 207 MHz...... 162

C.49 Outdoor measurements taken near a construction site with a center frequency of 213 MHz...... 163

C.50 Average Power in Broadcast TV Channels for VHF Low...... 164

C.51 Max Power in Broadcast TV Channels for VHF Low...... 165

xvi List of Tables

2.1 1 dB compression point input referred vs. attenuation. Measured bandwidth

B = 5 MHz and reference temperature To = 290 K...... 11

2.2 Broadcast television stations to be measured in the lower VHF...... 12

2.3 Broadcast television stations to be measured in the upper VHF...... 13

2.4 Verifying the multi-antenna setup...... 24

2.5 Multi-antenna measurements...... 27

2.6 Multi-antenna measurements...... 28

2.7 Multi-antenna measurements...... 29

3.1 Summary of models for man-made noise...... 36

3.2 Parameters used for the simulation...... 45

3.3 Kullback-Leibler divergence test...... 45

4.1 Single-antenna measurements...... 65

5.1 Correlation for impulsive and Gaussian components of noise...... 76

6.1 Design considerations for the RF board...... 96

xvii 6.2 Design implications for the RF board...... 96

6.3 GNI analysis for the 25-80 MHz band maximum gain...... 99

6.4 GNI analysis for the 25-80 MHz band minimum gain...... 100

6.5 GNI analysis for the 116-174 MHz band maximum gain...... 101

6.6 GNI analysis for the 116-174 MHz band minimum gain...... 102

B.1 Interconnects for the RF board...... 116

B.2 Values for the diplexer...... 117

C.1 Single-antenna measurements...... 129

C.2 Single-antenna measurements...... 130

C.3 Single-antenna measurements...... 131

C.4 Single-antenna measurements...... 132

C.5 Single-antenna measurements...... 133

C.6 Single-antenna measurements...... 134

C.7 Single-antenna measurements...... 135

xviii Chapter 1

Introduction

1.1 Motivation

This thesis is concerned with the measurement of impulsive noise and the analysis of its eﬀect on low signal to noise ratio (SNR) signal detection using an energy detector in the VHF broadcast TV bands. In this section, we motivate the choice to focus on each of these topics: spectrum sensing, energy detection, low SNR signal detection, impulsive noise, and the VHF TV bands.

Spectrum Sensing

Signal detection, or in this case spectrum sensing, refers to the act of determining whether a signal is present in a given band of the radio spectrum. Traditionally, governments have given exclusive use of bands of spectrum to speciﬁc licensed users. It has been noted that the demand for new spectrum is out pacing these traditional policies [1]. Cognitive radio is one suggested solution to the problem of spectrum scarcity [2]. Cognitive radios would be aware of and able to adapt to their environment. In particular, they would be able to access unused licensed spectrum on an ad hoc basis as long as they did not interfere with

1 2 the primary or licensed users. Measurement studies have shown that there is in fact a great deal of unused licensed spectrum that may be available for secondary use [3–6]. Spectrum sensing is a necessary step to enable cognitive radio and secondary spectrum reuse without causing interference to primary licensed users.

Energy Detector

The most common methods of spectrum sensing include a matched ﬁlter, energy detection, and cyclostationary based feature detection. Among these methods energy detection is preferred because it does not require any a priori knowledge about the signal to be detected and because of its computational simplicity [8]. The energy detector makes a decision about the presence of a signal by averaging the amount of energy in a band and comparing that to a given threshold [7]. The energy detector, however, is not without its limitations. Since the energy detector does not make use of any a priori knowledge about the signal to be detected, it is not able to distinguish between a primary user and noise [9]. It is also not well suited to the detection of spread spectrum signals [10]. In spite of these limitations, the energy detector is often the preferred solution since it does not put any burden on the primary user and it is simple to implement.

Low SNR Signal Detection

An energy detector requires knowledge of the noise power so that the threshold can be chosen appropriately. It has been shown that in the presence of uncertainty in the noise power there is an SNR below which the energy detector cannot detect the signal. This is called the SNR Wall [11]. Low SNR detection is important due to the hidden node problem [9], which is caused because the secondary user does not have access to the channel between the transmitter and the primary user. Since secondary use of the spectrum may face a low SNR environment and there is a SNR Wall associated with energy detectors, the performance of an energy detector in a realistic low SNR noise scenario is of interest. 3

Impulsive Noise

The interest in impulsive noise is predicated on the fact that these detection strategies have been optimized for an additive white Gaussian noise (AWGN) channel. In the presence of non-Gaussian noise, the performance of these detectors degrades. Furthermore, many studies have shown the presence of non-Gaussian man-made impulsive noise [12–15]. We are interested to see the eﬀect of this impulsive noise on the energy detector, especially in low SNR environments.

VHF Broadcast TV Bands

Finally, it was decided to take the impulsive noise measurements in the VHF digital TV bands for this thesis. The digital TV bands have been suggested as one place where spectrum could be used on a secondary basis [17, 18]. Therefore, as likely candidates for secondary spectrum reuse, these bands make for an interesting and relevant place to take impulsive noise measurements.

1.2 Contributions

There are four main contributions of this thesis: a measurement campaign of the VHF digital TV bands, an analysis of the performance of the energy detector in the presence of impulsive noise particularly the SNR Wall, analysis of the multi-antenna energy detector in the presence of spatially correlated impulsive noise, and building an RF board that can be used to take wideband impulsive noise measurements. While there have been many measurement campaigns that have shown the presence of impulsive noise, most of them have been at narrow bandwidths on the order of kHz. Two notable exceptions to this are measurement campaigns using 40 MHz in the ISM band [19] and 10 MHz in the UHF broadcast TV band [13]. Both of these campaigns sought to describe the statistics of the impulse waveforms. The 4 contribution of this measurement campaign was twofold: to add additional measurements of impulsive noise at lower frequencies and two take simultaneous measurements with two antennas to measure the correlation. The results of our measurements, primarily defined by the APD graph, were confirmed by comparing them to established models. We then analyzed the SNR Wall in the case of impulsive noise. After examining our data, we suggested a model for correlated impulsive noise in which impulses occur at all antennas at the same time. Using this model and previous work from [16], we analyzed the performance of an energy detector when impaired by correlated impulsive noise. The final contribution of this work was building and testing an RF receiver that could be used to make wideband noise measurements. This board also targeted the VHF band, though it measured frequencies generally below most but not all of the digital TV channels.

1.3 Thesis Summary

Problem Statement

In this thesis we sought to take measurements of man-made impulsive noise in the digital TV bands and match those results to established models. Once appropriate models were chosen, we sought to characterize how this impulsive noise would aﬀect an energy detector in a low SNR environment. In the rest of this section, we introduce chapter by chapter what work was done.

Chapter2: Measurement Campaign

A measurement campaign was conducted to measure the presence of impulsive man-made noise in the VHF digital broadcast TV bands. A measurement bandwidth of 5 MHz was chosen to approximate the width of a broadcast TV channel. Dual antenna measurements were taken at diﬀerent distances to measure the amount of spatial correlation in the impulsive 5 noise. All data was taken using real-time spectrum analyzers and stored as IQ data in Matlab ﬁles for later processing. The primary means of assessing the results was the amplitude probability distribution (APD) graph, which measures the percentage of time that a given amplitude is exceeded. Measurements were taken over the course of several days, and it was found that impulsive noise in excess of 35 dB greater than the nominal background noise was frequently present.

Chapter3: Noise Model

Two models from the work of [22] were reviewed. The first model was called Class A, where the receive bandwidth is wider than the noise impulses and thus they pass through unchanged. The second model was Class B, which describes the situation where the bandwidth of the noise impulses is larger than the receive bandwidth of the communication system and thus the pulses are smeared. Simplified versions of these models were compared to representative measurements from Chapter2. Using the Kullback-Leibler distance, it was found that the Class A and B models were a closer fit to the measurements than the AWGN model by as much as 18 dB. The Class B model was closer than the Class A model by 1 to 6 dB. Though the Class B model had a better fit, the Class A model was simpler to implement and analyze. We determined that the added performance of the Class B model did not out weigh these difficulties, and the Class A model was chosen for the analytical work in later chapters.

Chapter4: Eﬀects of Impulsive Noise on the Energy Detector

The effect of a simplified version of the Class A model from Chapter3 on the energy detector was analyzed. The two most important findings were that the presence of impulsive noise increases both the SNR Wall and the required sensing time to achieve a given performance requirement. The performance requirement is defined as achieving specified probabilities 6

of false alarm, PFA, and missed detection, PMD. In the case of AWGN, the author of [11] found that the SNR Wall arises from uncertainty in the average noise power. Since impulsive noise requires more parameters to describe and thus has more uncertainty, the SNR Wall increases. The increase of the SNR Wall is due to this increased uncertainty in its average power. Therefore, for the same uncertainty in average power impulsive noise and AWGN both have the same SNR Wall.

In this chapter we examined the eﬀect of these additional parameters on the uncertainty and the location of the SNR Wall. Sensing time is increased because the rare higher amplitude impulses cause the variance of the energy detector to increase, which requires a corresponding increase in sensing time to achieve the same performance.

Chapter5: Eﬀects of Spatially Correlated Impulsive Noise on the Energy Detector

The performance of an energy detector was analyzed in the presence of spatially correlated impulsive noise. Assumptions about the model for spatially correlated impulsive noise were made based on the measurement campaign in Chapter2. The most important assumption was that noise impulses were present at each antenna simultaneously. Intuitively, this makes sense since impulsive noise is generally from an external source such as car ignitions, ﬂuo- rescent lighting, etc. If it can be assumed that the signal to be detected is highly correlated across antennas, then it seems likely that the noise impulses would also be highly correlated across antennas. Additionally, it was assumed that only the noise impulses were correlated and that the nominal component of the noise remained uncorrelated. It was found that if noise impulses arrive at the antenna simultaneously, then the overall noise power will be highly correlated. Using our model for the correlated noise, we showed that even if the amplitudes of the noise impulses themselves were uncorrelated, the total noise power will still be highly correlated. This results in a signiﬁcant degradation in performance for a multi-antenna system. 7

Chapter6: Receiver Design

For this chapter we designed and built a RF receiver that could be used for wideband spectrum sensing measurements. When taking impulsive noise measurements, it is important to use as wide a bandwidth as possible [19]. This is because a wider bandwidth will have a better time resolution and be able to more accurately describe the statistics of the noise impulses. The RF board that we designed simultaneously receives two bands: 25-80 MHz and 116-174 MHz. The sensitivity of the RF board was designed to be limited by external noise in the lower band and variable attenuators were used in order to keep the board linear throughout a range of input signal powers. Tests were performed around the Virginia Tech campus in order to demonstrate the performance of the board. Chapter 2

Measurement Campaign

For this thesis, a measurement campaign was conducted on the Virginia Tech campus to measure the presence, level, and time domain statistics of man-made noise in the VHF digital television bands. The purpose of this campaign was to compare the measurements to established models, which will be discussed in Chapter3, and to examine the spatial correlation between two antennas in an indoor environment. Single antenna measurements were used to measure the presence and power of impulsive noise, and dual-antenna measurements were used to measure the spatial correlation. Each measurement was between 50,000 and 200,000 samples with a sampling rate of 7 Msps. Each of these measurements was given a numeric name beginning with 00001, and data such as time of day, sampling rate, and measured bandwidth were kept for each measurement. Real-time spectrum analyzers were used that have the capability to output the time domain samples in in-phase and quadrature format. In addition to these measurements, there was also a series of single antenna measurements taken indoors to measure the level of background noise at diﬀerent times of the day. For these measurements, the bands of interest were swept and the average and maximum values were recorded over about a two minute time period. The results of all these sets of measurements can be found below. In Section 2.1 the results for the single antenna measurements will be discussed including the measurement setup, amplitude distribution,

8 9 time correlation, and average power vs. frequency. Section 2.2 contains the results for the two antenna measurements.

2.1 Single Antenna

2.1.1 Measurement Setup

The goal of the single antenna measurements is to measure the amplitude probability distribution, time correlation and the maximum and average power levels in the VHF broadcast television bands at different times of day. The broadcast television stations that were measured can be found in Tables 2.2 and 2.3. Though each TV channel has a bandwidth of 6 MHz, all measurements were made using a bandwidth of 5 MHz. This was chosen because of the equipment used. The measurements were taken using a real-time spectrum analyzer connected to a Diamond D-130NJ Wideband Discone Antenna (25-1300 MHz) by a 20 foot low-loss coax cable. The spectrum analyzer performed the required amplification and mixing to baseband. The lower bound of the useful range of operation of the spectrum analyzer is described by the minimum detectable signal and the upper bound by the 1 dB compression point. The minimum detectable signal is the smallest signal power required over a given bandwidth to be recognizable over the internal noise of the receiver. The equipment used had a minimum detectable signal of about -163 dBm/Hz. Using a measurement bandwidth of 5 MHz and reference noise temperature, To, of 290 K, this comes out to -96 dBm or about 11 dB above the thermal noise floor. The input referred 1 dB compression point is the input power required for the device to measure 1 dB lower than it should. In order to maintain linear operation, it is important that the maximum power is below the 1 dB compression point. The spectrum analyzers used have an option to vary the attenuation, and the 1 dB compression point was measured at four different attenuation settings ranging from 0-15 dB. This was done by gradually increasing the input power until the measured power was 1 dB lower than expected if the equipment was in linear operation. Table 2.1 10

has the 1 dB compression points for each of the four attenuation settings. The ﬁrst column is the attenuation setting in dB, the second column is the 1 dB compression point in dBm, and the third column is the 1 dB compression point measured in dB above the thermal noise ﬂoor. An example of a measurement of the 1 dB compression point using 15 dB of attenuation can be found in Figure C.1 in AppendixC. The implication of the minimum detectable signal for the measurement campaign is that any man-made noise signals that

are lower than 11 dB over kToB at 5 MHz will be drowned out by the internal noise of the receiver. The implication of the 1 dB compression point is that, depending on attenuation

setting, external signals with power greater than 51.4 - 76.3 dB above kToB will appear to be lower than they actually are. Practically speaking, it is desirable to keep the maximum input power ﬁve to ten dB below the 1 dB compression point.

After noting this range of operation, it is useful to see if the actual measurements fall within it. With only two exceptions, the maximum input power of the measurements taken when

the attenuation was set to 0 dB ranged from -80.5 to -60.2 dBm (26.5 to 46.8 dB above kToB), which at the maximum was 4.6 dB below the 1 dB compression point. The two exceptions, with indexes 00298 and 00299, were both taken near a refrigerator and there were brief bursts

of noise that were -58.7 and -56.9 dBm (48.3 and 50.1 dB above kToB). These were 3.1 and 1.3 dB below the compression point, respectively. Plots of these measurements can be found in Figures C.34 and C.35 in AppendixC. There may be a small amount of compression of the highest amplitudes in these two measurements.

The maximum impulsive component of the power for the measurements taken with the

attenuation set to 5 dB ranged from -80 to -48.1 dBm (27 to 58.9 dB above kToB). In the worst case this is 7.6 dB below the 1 dB compression point and so should be in the linear range of operation. Similarly, the maximum power in the measurements taken using 10 dB

ranged from -69.1 to -44.9 dBm (37.9 to 62.1 dB above kToB), which falls at least 8.9 dB below the 1 dB compression point. There was only a single measurement taken with the attenuation set to 15 dB with a maximum input power of -67 dBm (40 dB above kToB), which is well below the 1 dB compression point. 11

Table 2.1: 1 dB compression point input referred vs. attenuation. Measured bandwidth B

= 5 MHz and reference temperature To = 290 K

Attenuation 1 dB Compression Point [dBm] dB above kToB 0 -55.6 dBm 51.4 5 -40.5 dBm 66.5 10 -36.0 dBm 71.0 15 -30.7 dBm 76.3

There were also a few measurements taken with the attenuation set to 20 dB. The maximum input power ranged from -55.9 to -34 dBm (51.1 to 73.0 dB above kToB). While we did not measure the 1 dB compression point with the attenuation set to 20 dB, the highest power measurements were 3.3 dB below the 1dB compression point when the attenuation is set to 15 dB. The 1 dB compression point with the attenuation set to 20 dB will be higher. Therefore, this measurement should fall within the linear range of operation of the spectrum analyzer.

All of these maximum powers are well above the minimum detectable signal. Examining the average power of these measurements shows that the nominal component of the noise for many of the outdoor measurements was equivalent to the internal noise of the receiver. This is not a problem since our primary interest is in the impulsive component of the noise which, as can be seen above, was much higher than the minimum detectable signal. The indoor measurements, on the other hand, had higher average power and were usually about 10 dB or more above the internal noise of the receiver.

The amount of attenuation was varied from measurement to measurement in order to get the maximum sensitivity while keeping the spectrum analyzer in a linear state of operation. The output was in the form of complex baseband samples, which could be analyzed afterwards using a software package such as Matlab. For each broadcast television station, 5 MHz of 12

Table 2.2: Broadcast television stations to be measured in the lower VHF

Broadcast Television Lower VHF Channel Start Freq [MHz] Center Freq [MHz] Stop Freq [MHz] 2 54 57 60 3 60 63 66 4 66 69 72 5 76 79 83 6 82 85 88

bandwidth was measured using a sample rate of 7 Msps for a duration of 28 ms. There were a total of four measurement locations, three indoors and one outdoors. The indoor measurements were all taken in an oﬃce environment. The locations were just outside a cubicle, next to a refrigerator and microwave, and next to an elevator. The outdoor measurements were taken just outside of the building adjacent to a parking lot and a construction site. The results of the measurements will be discussed in terms of amplitude distribution, time correlation, and average and max power vs. frequency for diﬀerent times of day.

2.1.2 Time Resolution

The choice of a measurement bandwidth of 5 MHz, which is close to that used by digital television, limits the period of a resolvable pulse. The system will be able to resolve pulses that have a duration longer than approximately 400 ns, which is twice the reciprocal of the measurement bandwidth. Pulses that are narrower than this will be smeared in time by the receiving ﬁlter. Extensive impulsive noise measurement campaigns have been done at a variety of frequencies and bandwidths. Blackard [19] performed indoor impulsive noise measurements in the ISM band using a measurement bandwidth of 40 MHz. It was found 13

Table 2.3: Broadcast television stations to be measured in the upper VHF

Broadcast Television Upper VHF Channel Start Freq [MHz] Center Freq [MHz] Stop Freq [MHz] 7 174 177 180 8 180 183 186 9 186 189 192 10 192 195 198 11 198 201 204 12 204 207 210 13 210 213 216

that sources of impulsive noise such as fluorescent lights had an average pulse duration of about 150 ns. Similarly, Sanchez et al. [13] performed impulsive noise measurements in a digital TV UHF channel using a measurement bandwidth of 10 MHz. The measurements were taken both outdoors in an urban area as well as indoors near a fluorescent light. It was found that in all cases the average pulse duration was between 210 and 240 ns. It is to be expected that using a measurement bandwidth of 5 MHz will mean that at least some of the noise pulses will be shorter than the time resolution of our measurement system. However, since the primary concern is on the effect of the impulsive noise on a receiver operating in the digital TV band, not on the statistics of the noise pulse shapes themselves, we believe that the information is still useful.

2.1.3 Amplitude Distribution

The amplitude probability distribution (APD) graph is commonly used to describe man- made noise [14, 15]. The APD graph is a way of plotting the complementary cumulative 14

distribution function (CCDF) of the instantaneous amplitude of a signal in such a way that is useful for radio engineers. The square of the instantaneous amplitude is plotted on the y-axis in decibels. The probability of exceeding that amplitude is plotted on the x-axis. Although it is called an amplitude probability distribution graph, it should be noted that the y-axis is actually scaled such that it is the instantaneous power that is plotted. The x-axis is scaled such that the amplitude of a complex Gaussian random variable, which is Rayleigh distributed, will appear as a straight line. This is useful for impulsive noise measurements because background noise is well modeled as a complex Gaussian random variable. The amplitude of such noise will appear as a straight line on the APD graph. If the slope of this line becomes sharper, that indicates the presence of impulsive noise. The cumulative distribution function is deﬁned as the probability that a random variable is less than a given value [20]

Z x FX (x) = P [X ≤ x] = p(u)du, (2.1) −∞

where FX (x) is the cumulative distribution function, P [X ≤ x] is the probability that the random variable, X is less than a given value x, and p(x) is the probability density function for the random variable X. The complementary cumulative distribution function is the opposite, in that it is the probability that a random variable will be greater than a given value

Z ∞ 0 FX (x) = 1 − FX (x) = P [X > x] = 1 − FX (x) = p(u)du, (2.2) x

0 where FX is the CCDF. The scaling of the x, and y-axes is important. The x-axis is Rayleigh 1 scaled which means that 1−log ln 0 is plotted on the x-axis. The y-axis is scaled in dB FX (x) −23 greater than kToB, where k is Boltzmann’s constant 1.3807 ∗ 10 J/K, To is the reference temperature taken to be 290 K [36], and B is the bandwidth of the measurement in Hz. All measurements were taken with a bandwidth of 5 MHz. 15

A representative APD graph from the single antenna measurements can be found in Figure 2.1. Two measurements are plotted. The dashed line is the measurement taken with the antenna removed, and a 50 ohm load connected to the terminals of the spectrum analyzer. The solid line is one of the noise measurements taken with the antenna attached that shows the presence of impulsive noise. The dashed line, the noise from the 50 ohm load, is a straight line across the graph. This is because the measurement is well modeled as a complex Gaussian random variable whose amplitude is Rayleigh distributed. The slope of the APD graph for a Rayleigh distributed random variable is fixed. Increasing the variance of the complex Gaussian random variable, or increasing the power of the noise, will cause the line to shift upward on the APD graph, but the slope will not change. The measurement using the antenna shows man-made noise which is made up of two components. On the right hand side of the graph the nominal, or Gaussian, component of the man-made noise can be seen. It has the same slope as that of the 50 ohm load measurement but is shifted higher on the y-axis. This reflects the higher power delivered by the antenna vs. the 50 ohm load. On the left hand side of the graph, the slope increases at low probabilities indicating the departure from a complex Gaussian distribution. This change in the distribution, describing rare high amplitude events, is caused by the presence of impulsive noise. On the far left-hand side of the graph, the slope levels off. This is due to low-probability high amplitude impulsive noise. In Chapter3 the statistics of both Gaussian noise and impulsive noise will be examined in greater detail along with 1st order statistical models.

The measurement campaign found that impulsive noise was common in all of the measured bands in all four locations. The indoor noise showed much higher average power than the outdoor measurements. It was noticed that the impulsive noise, as plotted on the APD graph, could often change signiﬁcantly from measurement to measurement. One complicating factor to measuring the impulsive noise was the presence of coherent signals in our bands of interest. This was most pronounced in that there was a strong signal in the 63 MHz channel throughout many of the indoor measurements. APD graphs as well as tables listing each measurement, average power, and time it was taken for all the measurements in all four 16 locations can be found in AppendixC.

60 Impulsive Noise Antenna 50 50 Ohm Load

40 Nominal Noise 30

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure 2.1: Representative Amplitude Probability Distribution (APD) graph

2.1.4 Time Correlation

The APD graph only describes the 1st order statistics of the instantaneous power of the noise. It ignores any time correlation. In this section, the time correlation of the noise samples will be examined. Speciﬁcally, the autocovariance of the magnitude of the noise is estimated for each of the noise samples. The phase is ignored for purposes of measuring the autocovariance because the focus of this thesis is addressing the eﬀects of impulsive noise on energy detectors which ignore phase.

The covariance can be calculated from the autocorrelation of a stochastic process. The autocorrelation function of a stochastic process is [20]

RX (t1, t2) = E[x(t1)¯x(t2)], (2.3) where E[·] refers to the expected value, x(t) is a stochastic process,x ¯(t) is the complex con- 17

jugate of x(t), and t1 and t2 are two points in time. In order to estimate the autocorrelation, two important assumptions were made about the data: that it was ergodic in both the mean and autocorrelation and that it was wide-sense stationary over the duration of the measurement. If a stochastic process is ergodic in the mean, then the time-average will equal the ensemble average for a single inﬁnitely long realization of the stochastic process [21]. Similarly, if a stochastic process is ergodic in the autocorrelation, then for a single inﬁnitely long realization the autocorrelation will converge to the true value. A wide-sense stationary stochastic process has a mean that is constant over time, as well as an autocorrelation func-

tion that is only a function of the delay between the two time samples, τ = t2 − t1. Hence, the deﬁnition of the autocorrelation function can be rewritten as

RX (τ) = E[x(t)¯x(t + τ)], (2.4)

Since the instantaneous power of the measured signals will have a nonzero mean, the autocovariance is used. The autocovariance is calculated by subtracting the means from the autocorrelation and is given by

2 CX (τ) = RX (τ) − µX , (2.5)

where CX (τ) is the autocovariance for a given delay, τ, and µX is the mean of the stochastic process x(t). Finally, in order to calculate the autocovariance between signals of varying power it is helpful to normalize it. This can be done by using the correlation coeﬃcient

CX (τ) ρX (τ) = 2 , (2.6) σX

2 where ρX (τ) is the correlation coeﬃcient at a delay of τ, and σX is the variance of the

random process x(t). The correlation coeﬃcient is on the interval 0 ≤ |ρX (τ)| ≤ 1, where values close to 0 indicate that there is very little correlation, and values close to 1 indicate a high degree of correlation. 18

After examining the impulsive noise captured in the measurements, two types of noise stand out: periodic and aperiodic. A time domain picture comparing examples of the magnitude of these two can be found in Figure 2.2. The left is a plot of periodic impulsive noise measured indoors near a cubicle at 57 MHz. The pulses occur approximately once every 108 samples, which is 15 µs. On the right is an example of aperiodic impulsive noise measured outdoors near a construction site where the pulses are occurring randomly in time. A close up of an individual pulse from each measurement can be found in Figure 2.3.

Periodic Impulsive Noise Aperiodic Impulsive Noise 0.8 Periodic Aperiodic 0.7

0.6 0.1 0.5

0.4

0.3 Magnitude [mV] Magnitude [mV] 0.05 0.2

0.1

0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Time [ms] Time [ms]

Figure 2.2: Examples of periodic and aperiodic impulsive noise measurements

The period of the periodic impulsive noise can be seen clearly when examining the correlation coefficient at various delays as shown in Figure 2.4. The figure confirms that the periodicity occurs approximately once every 108 samples for the periodic impulsive noise while the correlation coefficient for the aperiodic noise is approximately zero for τ > 0. Figure 2.5 shows a close up of the correlation coefficient for the periodic, aperiodic, and 50 Ohm load noise samples. The close up shows that the correlation coefficients for the aperiodic and 50 Ohm load noise are nearly identical. For τ > 0 it can be seen that the coefficient of both is approximately less than or equal to 0.1 and smoothly goes to zero as τ is increased. The correlation coefficient of the periodic impulsive noise on the other hand, is approximately less 19

Normalized Periodic Noise Pulse Normalized Aperiodic Impulsive Noise 1 1 Periodic Aperiodic 0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

Magnitude 0.4 Magnitude 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Time [µs] Time [µs]

Figure 2.3: Examples of aperiodic and periodic impulsive noise pulse shapes.

than or equal to 0.2 for τ > 0. Instead of smoothly going to zero, the correlation coeﬃcient has a ripple.

2.1.5 Average power vs. frequency at diﬀerent times of day

In order to measure average power vs. frequency at different times of day, the same measurement setup was used. However, instead of measuring 5 MHz of data at a time, the two clusters of broadcast television stations were measured separately by sweeping the spectrum analyzer across each band. The average and max-hold functions were used on the spectrum analyzer with a measuring time of approximately two minutes. From this data, the power in each television channel could be measured and plotted against the time of day. These measurements were made over the course of three days at different times taking measurements approximately every 15 minutes from 8:30 am until 8:30 pm. While there is not enough data to draw many conclusions about the RF environment, it is clear that the level of noise can vary significantly throughout the day. Figure 2.6 shows that noise power in a 5 MHz channel can vary by nearly 15 dB throughout the day. The power in the 5 MHz band around 63 20

Periodic Impulsive Noise Aperiodic Impulsive Noise 1 1 Periodic Aperiodic 0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

Correlation Coefficient 0.3 Correlation Coefficient 0.3

0.2 0.2

0.1 0.1

0 0 −200 −100 0 100 200 −200 −100 0 100 200 Delay in Samples Delay in Samples

Figure 2.4: Representative autocorrelation functions for impulsive noise

Examples of Autocovariance Functions for Impulsive Noise 1 Periodic 0.9 Aperiodic 0.8 50−Ohm

0.7

0.6

0.5

0.4

Correlation Coefficient 0.3

0.2

0.1

0 −10 −5 0 5 10 Delay in Samples

Figure 2.5: Representative autocovariance functions 21

MHz is generally much stronger than that of the other frequencies measured. This is because during the measurement time,there was a strong carrier in the channel centered at 63 MHz.

Average Power in TV Broadcast Channels −55 57 MHz 63 MHz −60 69 MHz 79 MHz 85 MHz −65

−70

−75

−80 Power in 5 MHz Channel [dBm/5MHz]

−85 8 10 12 2 PM 4 6 8 Time of Day

Figure 2.6: Average power in broadcast TV channels for VHF low band

2.2 Multi Antenna

2.2.1 Measurement Setup

For the multi-antenna measurements, a second antenna and spectrum analyzer were used. The two antennas were identical wideband discone antennas. The second real-time spectrum analyzer was the same model as the first. In order to make sure the measurements were locked in time, an external voltage trigger was used to start the measurements. Furthermore, to eliminate, as much as possible, any local oscillator offset between the spectrum analyzers, an external reference frequency of 10 MHz was used. In order to verify the measurement setup, a few tests were performed, which will be discussed in Section 2.2.3. Measurements were taken with the two antennas separated from 1m to 6m in five MHz bandwidths in the 5 broadcast channels in the lower VHF band. No measurements were taken for the upper 22

VHF channels. The sampling rate was 7 Msps, and the total number of samples for each measurement was 50,000 for a recording time of approximately 7.2 ms.

2.2.2 Measured Parameters

While the single antenna measurements were used to measure the presence of impulsive noise, the dual antenna measurements were used to measure the time and spatial correlation between the two antennas. Specifically, the time and spatial correlation of the complex baseband measurements were calculated. For each set of measurements, two parameters were calculated: the crosscorrelation coefficient and the average power. Again it was assumed that the measured data was both ergodic in the mean and autocorrelation, as well as wide- sense stationary over the measurement interval. Calculating the crosscorrelation coefficient for two different stochastic processes is very similar to the process used for calculating the autocorrelation with a single process. First the crosscorrelation, RXY (τ), which is similar to (2.3), was calculated as

RXY (τ) = E[x(t), y¯(t + τ)], (2.7) where instead of a single stochastic process there are now two, x(t) and y(t). The crosscovariance can be calculated just like in (2.5)

CXY (τ) = RXY (τ) − µX µ¯Y , (2.8)

where CXY (τ) is the crosscovariance, µX is the mean of x(t) andµ ¯Y is the complex conjugate of the mean of y(t). Finally, the correlation coeﬃcient between two stochastic processes can be calculated from the crosscovariance

CXY (τ) ρXY (τ) = , (2.9) σX σY 23

where σx and σy are the standard deviations of the stochastic processes x(t) and y(t). Gen- erally, the correlation coefficient will take on different values as the delay, τ, is changed. Since the spectrum analyzers were externally triggered and frequency locked, the correlation coefficient should be highest when τ = 0. However, just in case one of the spectrum analyzers was off by a sample or more, the correlation coefficient was calculated for several different values of τ and the largest value was used. For the two antenna measurements, the correlation coefficient was calculated for the complex series, magnitude, and the phase separately. The average power is defined

2 wo = E[w(t)] = E[|x(t)| ], (2.10)

where wo is the average power, w(t) is the instantaneous power, and x(t) is the complex baseband representation of the signal.

2.2.3 Veriﬁcation of the Measurement System

In order to test the measurement system, both correlated and uncorrelated control signals were measured to see how it performed. First, both spectrum analyzers’ antennas were replaced by a 50 ohm load and simultaneous measurements were taken. The correlation coefficient for the complex baseband signal, its magnitude, and its phase were all then calculated. Though each of the spectrum analyzers was externally triggered and locked to a 10 MHz reference signal, there was still a phase offset between the two. This phase offset did not affect the correlation coefficient of the complex baseband or magnitude. However, it could affect the correlation coefficient of the phase. In order to account for this, the magnitude of the correlation coefficient of the phase of the two signals was maximized for several different phase offsets. The result was that the magnitude of the coefficient of all three was equal to 0.01. This is expected since the two measurements were of the internal noise of each spectrum analyzer, which should be uncorrelated. Next, each spectrum analyzer was 24

Table 2.4: Verifying the multi-antenna setup

Name Frequency Complex Magnitude Phase wo1 wo2 Day Time 50 ohm Load 201 MHz 0.01 0.01 0.01 -100.1 dBm -100.4 dBm 6/13 7:24 AM Function Gen 86 MHz 1.00 0.99 0.96 -68.6 dBm -68.0 dBm 6/9 3:30 PM Single Antenna 208 MHz 1.00 0.99 0.96 -62.9 dBm -62.6 dBm 6/9 3:55 PM Double Antenna 208 MHz 1.00 0.99 0.93 -57.5 dBm -68.7 dBm 6/9 4:17 PM

connected to a single function generator using a T-connector, and a measurement of AM modulated white noise was taken. The correlation coefficient of the complex baseband signal was calculated to be 1.00, the magnitude 0.99, and the phase was 0.96. This shows that the spectrum analyzers are able to find correlation in highly correlated signals. The next step was to connect each of the spectrum analyzers to a single antenna and take measurements again of AM modulated white noise. The correlation coefficient was calculated to be 1.00 for the complex baseband signal, 0.99 for the magnitude, and 0.96 for the phase. Finally, the entire system was tested by having each spectrum analyzer connected to separate antennas, and a measurement was taken in the presence of a strong AM modulated white noise signal. The correlation coefficient of the complex baseband was 0.98, the magnitude was 0.92, and the phase was 0.93. A summary of the results can be found in Table 2.4. The table has 9 columns. The first two columns have the description of the measurement as well as the center frequency in MHz. The next three columns have the maximum value of the magnitude of the correlation coefficient for the complex baseband, magnitude, and phase respectively.

The columns marked wo1 and wo2 are the average power of each measurement in dBm. The last two columns are the day and time that the measurement was taken. It is clear from the test that the measurement setup is able to distinguish between correlated and uncorrelated signals. 25

2.2.4 Results

As stated previously the two parameters measured are the correlation coefficient and the average power. The correlation coefficient was calculated for the complex baseband, its magnitude, and its phase. Similarly, for constant power signals the correlation coefficient of the magnitude can be quite low even when the two antenna are receiving the same level of power. This is because the mean is subtracted from the series before the correlation coefficient is calculated. Therefore small uncorrelated deviations from the mean make it appear as if the magnitudes are uncorrelated when, in fact there results are actually quite similar.

Tables 2.5- 2.7 contain the data from the measurements. The table is setup just as in the table for the system verification measurements with the addition of a column for the distance of separation of the two antennas. Some of these measurements were taken on different days, and it appears that the level of correlation is different at different times. Additionally, the measurement at the center frequency of 63 MHz is highly correlated because there was a strong constant signal during all of the measurement in this band. Conversely, because the signal in the 63 MHz band could be constant for long periods of time, the correlation of the magnitude could sometimes be much lower than that recorded in other bands. As mentioned previously, this is because the mean is subtracted out before calculating the correlation. It can also be seen that the correlation of the phase in the 63 MHz channel is much higher than the rest of the measurements. This is likely because of the presence of the strong signal during measurements.

While more data would certainly be helpful, a few observations can be made. First, there was a wide variety in the the amount of correlation in the measurements ranging from 0.09 to 0.83. The correlation of the magnitude ranged from 0.11-0.97 and was usually higher than the correlation of the complex signal. Furthermore, the amount of correlation can change signiﬁcantly at diﬀerent times of day. For example, the measurements around center frequency of 57 MHz had a correlation of 0.35 on one day and 0.72 two days later. Figures 2.7 26

Correlation Coefficient vs. Distance/Wavelength 1 69 0.9 79 0.8 85

0.7

0.6

0.5

0.4

Correlation Coefficient 0.3

0.2

0.1

0 0 0.5 1 1.5 2 Distance/Wavelength

Figure 2.7: Correlation coefficient for Channels 3-5 vs. fractional wavelength and 2.8 show the correlation coefficient as a fraction of the wavelength of the center frequency for each of the bands measured. The correlation of measurements at center frequencies 57 and 63 MHz did not seem to be affected by the distances used. However, for the three higher channels, there seems to be a slight drift downward in correlation as the antennas are moved farther apart. However, more data would need to be taken to be able to draw more conclusions. There are many possible sources of correlation between the measurements such as antenna coupling and interfering signals. It is not possible to distinguish between these with the measurement setup used.

2.3 Conclusions

In this chapter, we described the measurement campaign taken on Virginia Tech’s campus to measure the presence and power of impulsive noise as well as its correlation between two antennas. It is important to make sure that the measurements are actually of external noise, and not internal to the receiver. It has been shown in [37] that the median value of external 27

Table 2.5: Multi-antenna measurements

Name Frequency Distance Complex Magnitude Phase wo1 wo2 Day Time 00140 57 MHz 1 m 0.40 0.56 0.05 -74.2 dBm -74.6 dBm 6/3 9:33 AM 00141 57 MHz 1 m 0.44 0.66 0.06 -74.0 dBm -74.9 dBm 6/3 9:34 AM 00142 57 MHz 1 m 0.35 0.51 0.08 -73.8 dBm -74.6 dBm 6/3 9:35 AM 00170 57 MHz 1 m 0.64 0.79 0.17 -73.2 dBm -75.1 dBm 6/5 10:22 AM 00171 57 MHz 1 m 0.72 0.82 0.18 -71.0 dBm -72.4 dBm 6/5 10:23 AM 00172 57 MHz 1 m 0.71 0.82 0.20 -68.3 dBm -67.4 dBm 6/5 10:27 AM 00185 57 MHz 3 m 0.83 0.97 0.23 -69.7 dBm -70.2 dBm 6/5 10:55 AM 00186 57 MHz 3 m 0.55 0.86 0.19 -70.4 dBm -71.2 dBm 6/5 10:56 AM 00187 57 MHz 3 m 0.50 0.77 0.09 -75.8 dBm -75.4 dBm 6/5 10:59 AM 00200 57 MHz 4 m 0.53 0.84 0.06 -72.7 dBm -76.3 dBm 6/5 11:51 AM 00201 57 MHz 4 m 0.47 0.75 0.06 -73.0 dBm -77.2 dBm 6/5 11:52 AM 00202 57 MHz 4 m 0.40 0.67 0.05 -73.0 dBm -77.5 dBm 6/5 11:55 AM 00215 57 MHz 6 m 0.44 0.72 0.10 -71.9 dBm -72.0 dBm 6/5 12:23 PM 00216 57 MHz 6 m 0.53 0.69 0.07 -71.7 dBm -74.6 dBm 6/5 12:25 PM 00217 57 MHz 6 m 0.61 0.77 0.08 -72.0 dBm -75.9 dBm 6/5 12:26 PM 00143 63 MHz 1 m 0.80 0.58 0.78 -67.8 dBm -65.8 dBm 6/3 9:37 AM 00144 63 MHz 1 m 0.89 0.29 0.76 -68. dBm7 -65.6 dBm 6/3 9:38 AM 00145 63 MHz 1 m 0.88 0.28 0.74 -68.5 dBm -65.9 dBm 6/3 9:39 AM 00173 63 MHz 1 m 0.53 0.58 0.52 -61.6 dBm -63.3 dBm 6/5 10:31 AM 00174 63 MHz 1 m 0.78 0.27 0.62 -63.6 dBm -65.5 dBm 6/5 10:32 AM 00175 63 MHz 1 m 0.71 0.40 0.58 -62.9 dBm -65.3 dBm 6/5 10:33 AM 00158 63 MHz 3 m 0.80 0.14 0.65 -60.1 dBm -64.9 dBm 6/3 10:13 AM 00159 63 MHz 3 m 0.36 0.11 0.64 -60.1 dBm -64.8 dBm 6/3 10:14 AM 00160 63 MHz 3 m 0.83 0.15 0.68 -60.1 dBm -64.7 dBm 6/3 10:15 AM 00188 63 MHz 3 m 0.81 0.30 0.58 -65.6 dBm -66.6 dBm 6/5 11:01 AM 00189 63 MHz 3 m 0.82 0.90 0.52 -61.9 dBm -61.6 dBm 6/5 11:03 AM 00190 63 MHz 3 m 0.74 0.63 0.56 -64.8 dBm -65.2 dBm 6/5 11:04 AM 00203 63 MHz 4 m 0.49 0.27 0.37 -75.2 dBm -63.4 dBm 6/5 12:00 PM 00204 63 MHz 4 m 0.50 0.23 0.42 -74.6 dBm -63.6 dBm 6/5 12:03 PM 00205 63 MHz 4 m 0.41 0.21 0.31 -76.3 dBm -62.5 dBm 6/5 12:04 PM 00218 63 MHz 6 m 0.82 0.36 0.69 -58.2 dBm -62.1 dBm 6/5 12:40 PM 00219 63 MHz 6 m 0.88 0.39 0.76 -58.9 dBm -62.6 dBm 6/5 12:37 PM 00220 63 MHz 6 m 0.89 0.42 0.76 -59.1 dBm -63.0 dBm 6/5 12:39 PM 28

Table 2.6: Multi-antenna measurements

Name Frequency Distance Complex Magnitude Phase wo1 wo2 Day Time 00146 69 MHz 1 m 0.66 0.75 0.26 -70.2 dBm -73.1 dBm 6/3 9:41 AM 00147 69 MHz 1 m 0.65 0.79 0.27 -68.9 dBm -72.9 dBm 6/3 9:43 AM 00148 69 MHz 1 m 0.67 0.73 0.27 -70.1 dBm -73.3 dBm 6/3 9:44 AM 00176 69 MHz 1 m 0.42 0.65 0.26 -76.6 dBm -73.4 dBm 6/5 10:37 AM 00177 69 MHz 1 m 0.49 0.62 0.24 -77.6 dBm -75.6 dBm 6/5 10:38 AM 00178 69 MHz 1 m 0.57 0.67 0.26 -76.9 dBm -74.8 dBm 6/5 10:39 AM 00161 69 MHz 3 m 0.44 0.66 0.09 -72.1 dBm -75.0 dBm 6/3 10:18 AM 00162 69 MHz 3 m 0.43 0.74 0.11 -72.3 dBm -75.3 dBm 6/3 10:19 AM 00163 69 MHz 3 m 0.38 0.79 0.10 -71.6 dBm -74.9 dBm 6/3 10:19 AM 00191 69 MHz 3 m 0.23 0.68 0.14 -76.6 dBm -76.7 dBm 6/5 11:07 AM 00192 69 MHz 3 m 0.49 0.69 0.11 -75.4 dBm -73.6 dBm 6/5 11:09 AM 00193 69 MHz 3 m 0.42 0.70 0.11 -75.9 dBm -75.5 dBm 6/5 11:10 AM 00206 69 MHz 4 m 0.18 0.63 0.10 -73.2 dBm -75.3 dBm 6/5 12:06 PM 00207 69 MHz 4 m 0.29 0.77 0.09 -72.5 dBm -72.8 dBm 6/5 12:07 PM 00208 69 MHz 4 m 0.28 0.74 0.08 -72.8 dBm -73.5 dBm 6/5 12:08 PM 00221 69 MHz 6 m 0.54 0.85 0.07 -71.6 dBm -72.1 dBm 6/5 12:43 PM 00222 69 MHz 6 m 0.33 0.58 0.10 -73.7 dBm -73.9 dBm 6/5 12:44 PM 00223 69 MHz 6 m 0.33 0.55 0.09 -73.7 dBm -74.5 dBm 6/5 12:47 PM 00149 79 MHz 1 m 0.38 0.72 0.15 -70.4 dBm -71.3 dBm 6/3 9:45 AM 00150 79 MHz 1 m 0.44 0.83 0.19 -69.9 dBm -70.8 dBm 6/3 9:47 AM 00151 79 MHz 1 m 0.37 0.78 0.18 -70.6 dBm -71.2 dBm 6/3 9:48 AM 00179 79 MHz 1 m 0.81 0.87 0.43 -74.9 dBm -76.8 dBm 6/5 10:40 AM 00180 79 MHz 1 m 0.56 0.67 0.27 -74.4 dBm -75.9 dBm 6/5 10:41 AM 00181 79 MHz 1 m 0.70 0.77 0.36 -74.8 dBm -76.6 dBm 6/5 10:42 AM 00164 79 MHz 3 m 0.37 0.68 0.13 -71.0 dBm -71.2 dBm 6/3 10:22 AM 00165 79 MHz 3 m 0.40 0.65 0.13 -70.6 dBm -71.6 dBm 6/3 10:23 AM 00166 79 MHz 3 m 0.38 0.64 0.11 -70.6 dBm -71.7 dBm 6/3 10:24 AM 00194 79 MHz 3 m 0.41 0.66 0.08 -74.1 dBm -74.3 dBm 6/5 11:14 AM 00195 79 MHz 3 m 0.52 0.67 0.11 -74.4 dBm -74.9 dBm 6/5 11:15 AM 00196 79 MHz 3 m 0.48 0.73 0.10 -73.7 dBm -73.6 dBm 6/5 11:16 AM 00209 79 MHz 4 m 0.37 0.54 0.10 -74.0 dBm -73.0 dBm 6/5 12:10 PM 00210 79 MHz 4 m 0.43 0.62 0.13 -74.0 dBm -73.3 dBm 6/5 12:12 PM 00211 79 MHz 4 m 0.37 0.59 0.12 -73.8 dBm -73.2 dBm 6/5 12:13 PM 00224 79 MHz 6 m 0.05 0.26 0.11 -81.0 dBm -75.3 dBm 6/5 12:49 PM 00225 79 MHz 6 m 0.03 0.31 0.10 -81.2 dBm -75.4 dBm 6/5 12:50 PM 00226 79 MHz 6 m 0.07 0.35 0.10 -80.1 dBm -75.0 dBm 6/5 12:52 PM 29

Correlation Coefficient vs. Distance/Wavelength 1 57 0.9 63 0.8

0.7

0.6

0.5

0.4

Correlation Coefficient 0.3

0.2

0.1

0 0 0.5 1 1.5 2 Distance/Wavelength

Figure 2.8: Correlation coeﬃcient for Channels 1-2 vs. fractional wavelength

Table 2.7: Multi-antenna measurements

Name Frequency Distance Complex Magnitude Phase wo1 wo2 Day Time 00152 85 MHz 1 m 0.42 0.62 0.15 -72.8 dBm -72.3 dBm 6/3 9:49 AM 00153 85 MHz 1 m 0.51 0.67 0.20 -69.9 dBm -64.7 dBm 6/3 9:51 AM 00154 85 MHz 1 m 0.60 0.76 0.20 -69.4 dBm -64.8 dBm 6/3 9:52 AM 00182 85 MHz 1 m 0.33 0.54 0.17 -74.5 dBm -74.9 dBm 6/5 10:44 AM 00183 85 MHz 1 m 0.31 0.60 0.15 -74.2 dBm -74.7 dBm 6/5 10:45 AM 00184 85 MHz 1 m 0.40 0.67 0.11 -74.6 dBm -74.4 dBm 6/5 10:45 AM 00167 85 MHz 3 m 0.46 0.61 0.16 -70.9 dBm -68.4 dBm 6/3 10:26 AM 00168 85 MHz 3 m 0.41 0.57 0.12 -70.7 dBm -67.9 dBm 6/3 10:27 AM 00169 85 MHz 3 m 0.28 0.44 0.13 -70.0 dBm -68.4 dBm 6/3 10:28 AM 00197 85 MHz 3 m 0.25 0.62 0.13 -72.7 dBm -75.0 dBm 6/5 11:20 AM 00198 85 MHz 3 m 0.22 0.62 0.13 -73.4 dBm -74.7 dBm 6/5 11:21 AM 00199 85 MHz 3 m 0.28 0.50 0.16 -69.3 dBm -73.4 dBm 6/5 11:48 AM 00212 85 MHz 4 m 0.38 0.66 0.10 -70.8 dBm -72.6 dBm 6/5 12:16 PM 00213 85 MHz 4 m 0.36 0.68 0.11 -71.1 dBm -72.6 dBm 6/5 12:18 PM 00214 85 MHz 4 m 0.36 0.62 0.09 -70.8 dBm -72.1 dBm 6/5 12:19 PM 00227 85 MHz 6 m 0.15 0.35 0.05 -74.9 dBm -73.8 dBm 6/5 12:53 PM 00228 85 MHz 6 m 0.25 0.47 0.08 -75.4 dBm -73.3 dBm 6/5 1:02 PM 00229 85 MHz 6 m 0.22 0.47 0.07 -74.1 dBm -72.4 dBm 6/5 1:03 PM 30 man-made noise decreases linearly with the logarithm of frequency according to

Fm = c − d log f, (2.11)

where Fm is the median external noise figure, f is the frequency in MHz, and c and d are parameters that are specific to the location. The external noise figure is defined as the ratio of the antenna temperature to the reference temperature, To, in dB. This is a reflection of how much man-made noise power there is in excess of thermal noise. An example of this can be found in Figure 2.9 which shows the median external noise figure for a city location in the frequencies 57 to 213 MHz using (2.11) and location data from [37]. In order to confirm that our measurements are indeed of external noise, we would like to see a drop in noise power of about 16 dB from the lowest measured frequencies to the highest measured frequencies. Examining the resulting APD graphs from AppendixC shows that the measured noise power is not clearly decreasing with frequency. Without this it is difficult to be certain that we are in fact measuring external rather than internal noise. However, there are also convincing reasons that we are measuring external noise. First, the nose power measured varied greatly according to location and time. This is most pronounced when comparing the outdoor and indoor measurements. The outdoor measurements had an average power as much as 30 dB below the indoor measurements. Second, the presence of impulsive noise in the measurements indicates that at least the impulses are external to the measurement system since no impulses were recorded when the antenna was replaced with a 50 Ohm load. Finally, the strongest evidence that we were in fact measuring external noise is that during the multi- antenna measurements varying amounts of correlation were recorded. If the measurement system were only measuring internal noise then the multi-antenna measurements would not be correlated.

One reason that we may not have seen the decreasing noise power with frequency is that we had a small number of measurements taken at different times and on different days. It would be useful to take more measurements to see if we can find such a decrease in noise 31

18 External Noise Figure 16

12 60 80 90 100 120 140 160 180 200 Frequency [MHz]

Figure 2.9: External noise ﬁgure over the measured frequencies for a city environment using (2.11). The parameters c = 76.8 and d = 27.7 are for a city environment taken from [37] power.

In the case of the single antenna measurements, it was found that impulsive noise was common in almost all of the measured bands both indoors and out. The measurements taken indoors had a much higher average noise power than the outdoor measurements. It was noticed that the impulsive noise measurements changed from measurement to measurement. This was most apparent in the indoor measurements, which were taken over the course of several days. The outdoor measurements, which were taken all at one time, showed much less variation. It was also observed that a few of the measurements of impulsive noise had signiﬁcant periodicities while most did not. The dual antenna measurements, which were all taken indoors, showed a wide variety in the level of correlation of the noise between the two antennas though most often it was between 0.3 and 0.8. It was also noticed that the level of correlation could change for diﬀerent times of day. Chapter 3

Impulsive Noise Model

In this chapter, two models for non-Gaussian impulsive noise from the literature will be introduced and compared to measured results. In this thesis, simplified versions of Mid- dleton’s canonical statistical-physical model are used [22]. This chapter begins with a brief introduction of Middleton’s model, followed by two simplified versions. The analytical expressions for the APD graph of the components of this model are described. The APD graph was explained in further detail in Section 2.1.3. This chapter concludes with an explanation of estimating the parameters for these simplified models and a few representative examples comparing computer simulations using these models to actual measurements.

3.1 Introduction

Many modern communication systems have been designed to work in a channel corrupted by additive white Gaussian noise. In the presence of non-Gaussian impulsive noise, the performance may be signiﬁcantly degraded. In order to analyze the performance of such systems in impulsive noise, an appropriate noise model is needed. The simpliﬁed models in this chapter are based on the estimated amplitude probability distribution of noise measurements taken

32 33 in Chapter2. These models do have some limitations. The simplified models ignore the underlying physics of the impulsive noise and instead attempt to match measured data to known probability distributions. Since the models are matched to the output of the measurement system, the statistics of the impulsive noise prior to the antenna are not known. Transferring these results to receivers of different bandwidths is further complicated by the fact that the power of Gaussian noise is proportional to the received bandwidth while the power of impulsive noise is proportional to the square of the received bandwidth. Further- more, most measurements are necessarily done over a finite period of time, and the statistics of the impulsive noise may change for different times of day. Finally, basing the model on the amplitude distribution alone ignores the time correlation of the noise.

3.2 Middleton’s Model

The noise model developed by Middleton in [22] is a detailed statistical-physical model of man-made noise. In particular, it models the first order statistics of the amplitude of man- made noise after the final IF filter of the measurement system. Therefore, the physical nature of the underlying waveforms of man-made noise are not described. The parameters for the model would need to be adjusted for different receivers and scenarios. Noise is defined as any undesired signal at the receiver. It is assumed that there are an infinite number of possible noise emission sources. The amplitudes, frequencies, and durations of the emission pulses are considered to be random. The locations of the sources as well as the emission times are assumed to be Poisson-distributed. There are three basic classes of models used based on the effect of man-made noise on the receiver: Class A, B, and C. Class A models the amplitude of the noise when the response of the receiver is dominated by the amplitude of the pulse itself. This means that the duration of a given pulse must be much longer than the inverse of the bandwidth of the receiver used. Put another way, the bandwidth of a given pulse must be much smaller than the receiver bandwidth so that it passes through the receiver largely unchanged. Class B models noise pulses when the response of the receiver is dominated by 34 the transients, or ”ringing” in the receiver [22]. This means that the duration of the pulse must be much shorter than the inverse of the bandwidth of the receiver. Therefore, the bandwidth of a given pulse will be much larger than the receivers filter. The Class C model is a combination of both Class A and B models. These are summarized in Table 3.1.

Since noise modeled by Class A has a bandwidth narrower than the receiver, it will pass largely unchanged through the ﬁnal IF ﬁlter, and its transient response will be negligible compared to the pulse amplitude. This causes the APD graph to have a sharp increase at low probabilities. This sharp increase usually appears as a step function. Example sources of noise, which have been modeled using Middleton’s Class A, include an ore crushing machine in a mine using a receive bandwidth of 1-1.2 kHz and power lines [22], [23]. It is important to note that the class of the model is determined by the ratio of the receiver bandwidth and the bandwidth of the noise impulses. Noise modeled with Class A for one receiver may be better modeled with Class B for another narrower bandwidth receiver. Middleton models the CCDF of the Class A model for impulsive noise, which is the probability that the normalized instantaneous amplitude will exceed a given threshold with a Gaussian mixture model

∞ m −A X AA − 2/2ˆσ2 P [ > ] = e A e 0 mA , (3.1) 1 0 m! m=0 where is the normalized instantaneous amplitude and 0 is the threshold. AA is the impulsive index, the average number of pulses arriving at the receiver per second multiplied by

2 the average duration of the pulse. The quantity 2ˆσmA is given by

2 0 0 2ˆσmA = (m/AA + ΓA )/(1 + ΓA ), (3.2)

0 2 2 where ΓA = σG/Ω2A is the ratio of the power of the Gaussian component of the noise, σG, to the power of the impulsive component Ω2A. The instantaneous amplitude, , is normalized by the power in the impulsive component and the ratio between the Gaussian and impulsive components 35

q 0 = E/ 2Ω2A(1 + ΓA ), (3.3)

where E is the absolute instantaneous amplitude of the noise. So then, the Class A model 0 is described by three parameters (AA, ΓA , Ω2A).

In contrast to Class A, the Class B model represents the situation where the amplitude is dominated by the transient response of the receiver. Therefore, instead of a step function, there will be an increasing slope in the APD graph at low probabilities. Figure 3.1 shows the diﬀerence between Class A and B using the simpliﬁed models to be discussed in Section 3.3. The slope of the Class A model has a sharp increase, almost like a step function at low probabilities, while the Class B model has a more gentle change in slope. The APD for Class

B noise is broken into two parts: one for small and intermediate values (0 < 0 < B), and

another for large amplitudes (B < 0 < ∞). The APD for larger values is of the same form as Class A. The APD for small and intermediate values is more complicated, adding three parameters given by

∞ X (−1)nAˆn αn αn P [ > ] = 1 − ˆ 2 Γ 1 + F 1 + ; 2; −ˆ 2 , (3.4) 1 0 0 n! 2 1 1 2 0 n=0

ˆ α α where A = Aα/2 GB, ˆ0 = (0NI )/2GB, and α is a spatial density-propagation parameter 2 −2 0 −1 4−α 0 on the interval 0 < α < 2. GB = 2 (1 + ΓB) 2−α + ΓB , and 1F1 is a hypergeometric

function. In addition to α, two new parameters are added. Aα is the eﬀective impulsive index which is proportional to AB, and NI is a scaling factor. In all, the Class B model uses 0 six diﬀerent parameters. Three of the parameters, AB,ΓB , and ΩmB are similar to the ones from the Class A model.

Class B noise requires six parameters as well as an infinite series of hypergeometric functions. It can be quite difficult to estimate these parameters. A simpler model based on Middleton’s work is developed in [15]. Further, it was found in [14] that this simplified Class B model was sufficient to match all the data in their measurement campaign. In the following sections, 36

Table 3.1: Summary of models for man-made noise

Class Description Class A Noise with bandwidth less than the receiver bandwidth Class B Noise with bandwidth greater than the receiver bandwidth Class C A combination of both Class A and Class B noise

simpliﬁed versions of Class A and B models will be examined.

Class B Class A 40

35 B o

30 dB above kT 25

0.01 percent exceeding ordinate

Figure 3.1: Comparison of APD graph of Class A and Class B models at low probabilities with bandwidth B = 5 MHz

3.3 Simpliﬁcation of Middleton’s Models

While Middleton’s model for impulsive noise has been shown to have excellent results [24], [25] it can be too complicated to easily use in practice. Middleton’s Class A model uses a Gaussian mixture with an inﬁnite number of terms. It has been shown that the Class A 37

model can be approximated with only two mixture terms [26]. The authors of [14], [15] have introduced simplified versions of Middleton’s Class B model, which uses a train of impulses and a filter to mimic the properties of noise impulses with a bandwidth greater than the receiver. This section will begin with the simplification of the Class B model and analytical descriptions of the APD curves. Following this there will be a discussion of the simplification of the Class A model.

3.3.1 Simpliﬁed Class B Model

For the model developed by [15], it was desired to not only simplify Middleton’s Class B model but also to model the noise prior to the IF filter instead of after it. It was desired to have a model for the noise prior to the IF filter so that results would be transferable to receivers of differing bandwidths. The noise was modeled as the sum of two components, one Gaussian and the other non-Gaussian. The non-Gaussian component was represented with a Weibull distribution. The model for the complex baseband representation of the noise prior to the final IF filter is

jθ nˆk = (xekbk + gk)e , (3.5) wheren ˆk is the complex baseband representation of the noise, θ is a uniformly-distributed random variable on the interval (0 ≤ u < 2π), gk is the Rayleigh-distributed amplitude of the Gaussian noise component, bk is the Weibull-distributed amplitude of the impulsive noise component, and xek is a binary random variable determining whether a pulse is present. The author of [15] used a common phase parameter shared by the nominal and impulsive components of the noises while the author of [14] used independent phase values for each component. In practice, there is no real diﬀerence when the amplitudes for the impulsive component of the noise are much larger than the nominal component. The value of xek is given by 38

 1, with probability γ∆t xek = (3.6) 0, with probability 1 − γ∆t.

Pulses were assumed to arrive randomly with a Poisson distribution in time. The parameter γ is the mean pulse arrival rate, and ∆t is the sampling period. Therefore, the quantity γ∆t is the probability that a pulse will be present in a given sample and can be related to

Middleton’s impulsive index AA. It is assumed that prior to the IF filter the bandwidth of the pulse is much larger than the bandwidth of the IF filter. Therefore, the pulses are represented as impulses with a duration much shorter than the sampling period. In order to get the complex baseband representation of the noise after the final IF filter,n ˆk must be filtered using a filter that represents the receiver to be modeled. In the rest of this section, the statistics of Gaussian noise and Weibull-distributed impulsive noise as well as some observations of their APD will be examined.

3.3.2 Statistics for Gaussian Noise

The Gaussian component of the noise, gk, is described by a single parameter, the average power wog. The APD graph is based on the CCDF of the instantaneous power. The probability density function for the instantaneous power of Gaussian noise is given in [15]

1 − w p(w) = e wog (3.7) wog where p(w) is the probability density function of the instantaneous power, w, and wog is the average power of the signal. The CCDF of the instantaneous power can be derived from the probability density function

Z ∞ 0 −w/wog FW (w) = P [W > w] = p(u)du = e (3.8) w 39

Solving (3.8) for the instantaneous power, w, is given by [14]

1 w = wog ln 0 . (3.9) FW (w)

0 The instantaneous power w is plotted in dB. The probability of exceeding that power, FW (w), 1 is scaled as 1−log ln 0 . Making these algebraic manipulations, the equation describing FX (x) the APD graph is given by

1 10 log w = 10(log wog + 1) − 10 1 − log ln 0 (3.10) FW (w)

0 1 There are two important observations to be made about this equation. When FW (w) = e ≈

0.368 then 10 log (w) = 10 log (wog). Therefore, the average power for complex Gaussian- distributed noise can be read from approximately the 37% mark on the APD graph. The second important observation is that the slope of complex Gaussian noise on a Rayleigh scaled APD graph will always be -10 dB per unit distance. Examples of a unit distance would be between 0.368 and 0.905, or 0.905 and 0.999. This slope is independent of the average power in the noise.

3.3.3 Statistics for Impulsive Noise Using Weibull Distribution

The amplitude of the non-Gaussian component of the noise, bk, has a Weibull distribution. The CCDF of a Weibull distribution is given by [14]

w 1/α 0 −( w ) FW (w) = e ow (3.11)

Rearranging the terms and Rayleigh scaling for the APD graph gives the expression

1 10 log(w) = 10[α + 10 log(wow)] − 10α 1 − log ln 0 . (3.12) FW (w) 40

There are two parameters that describe the Weibull distribution, wow and α. The parameter 0 wo shifts the plot up and down on the y-axis. When FW (w) = 0.368 then 10 log(w) =

10 log(wow) so that just as in the case of Gaussian noise, wo can be read from the intersection of the Weibull distribution and the 37% mark on the APD graph. Equation(3.12) shows that the slope of a Weibull distribution is 10α.

An example comparing the APD graph of Weibull and Gaussian noise can be found in Figure

3.2. The Gaussian distributed noise has an average power, wog = 0 dB. Two diﬀerent distributions of Weibull noise are plotted. All three have average power, wow = 0 dB. The parameter for α is set to 2 and 3. The average power of all three distributions can be read directly from the 37% mark. It can be seen from the ﬁgure that the Gaussian noise has a slope of -10 dB for a unity distance on the APD graph. Similarly, the slopes for the two Weibull noise distributions are -20 dB and -30 dB.

40 Gaussian 30 Weibull α = 2 20 Weibull α = 3

−10

−20

Amplitude [dB] Average Power −30

−40

−50

−60 37 90.5 percent exceeding ordinate

Figure 3.2: Comparison of the APD graph for Weibull and Gaussian distributed noise 41

3.3.4 Simpliﬁed Class A Model

Middleton’s Class A model can be simplified by using a fixed number of terms in the Gaussian mixture model instead of an infinite series. In this case, two terms will be used. The complex baseband simplified Class A model is given by

jθ1 jθ2 nˆk = xekg2ke + (1 − xek)g1ke , (3.13) where xek is a binary random variable that indicates the presence of impulsive noise and is calculated much the same way as in (3.6), except the probability that xek = 1 is given by the term . For the simpliﬁed Class A model, both the nominal and impulsive components of the noise have a complex Gaussian distribution. Therefore both g1k and g2k are Rayleigh distributed random variables. The variance of g1k is given by w1 and the variance of g2k is

given by κw1. The variables θ1 and θ2 are the phase of the respective Gaussian terms. All the variables g1k, g2k, θ1, and θ2 are independent.

3.4 Estimating the Parameters for the Two Models

There are a total of four parameters that describe the simplified Class B model for impulsive noise: wog, wow, α and γ. As noted previously, wog can be estimated from where the measured data crosses the 37% mark. The average power for the impulsive component, wow, can be estimated by fitting a line to the impulsive component and measuring where it would cross the 37% mark. The parameter α can be estimated from the slope of the low probability impulsive noise on the APD graph. The parameter γ is the mean pulse arrival rate and is chosen based on where the noise deviates from a Rayleigh distribution on the APD graph. This is one area of weakness of this model because the value of γ cannot be read directly from the APD graph. Rather the product γ∆t is what can be estimated. Therefore, the choice of sampling period for the noise prior to the final IF filter will have an effect on 42

what value of γ is used. One challenge to choosing the correct parameters is that the APD graph will change after filtering. The power in both components will be reduced based on the fractional bandwidth of the digital filter used. Furthermore, for the non-Gaussian component the slope will tend to flatten out after filtering. Therefore, some guess work is involved for all the parameters execpt for wog which is easier to predict after the filter.

The simpliﬁed Class A model for noise only requires three parameters: w1, , and κ. Pa- rameter w1 can be estimated from where the noise measurement crosses the 37% mark, and κ can be estimated from the low probability high amplitude values. The parameter is estimated from where the measured results depart from a Gaussian distribution.

Below are the results of using simplified Class A and B models to fit some representative measurements taken in Chapter2. The simplified Class B model includes a filter, while the simplified Class A model does not. For all of the Class B model simulations, the sampling frequency before the filter was 21 MHz, which is three times the sampling frequency used by the measurement system. The Class B model was then filtered with a 6th order Chebyshev filter with a passband ripple of 0.1 dB and cutoff frequency of 2.8 MHz. This filter was chosen to mimic the actual filter used in the measurements. After filtering the both the

Class A and Class B models have a bandwidth of 5 MHz. The reference temperature, To, is 290 K. Table 3.2 contains the model parameters used for the simulations in this chapter.

In order to compare the results from the two models the Kullback-Leibler distance was estimated between the model and the measured data. The Kullback-Leibler distance between two probability densities p and q is given by [27]

Z ∞ p(x) D(p||q) = p(x) log2 dx, (3.14) −∞ q(x)

0 where 0 log 0 is set to 0. In order to estimate the Kullback-Leibler distance, p(x) was set to the empirical pdf of the measured data, and q(x) was set to the empirical pdf calculated from the model used. Since a base 2 logarithm is used, the units of the Kullback-Leibler distance 43

are in bits. In order to get a sense of what minimum distance could be using empirical pdfs, the Kullback-Leibler distance between two Gaussian random variables generated with Matlab was measured to be 0.0004 bits. To compare the impulsive noise model with AWGN, the Kullback-Leibler distance was measured between the data and a complex Gaussian random variable with the same average power. All results can be found in Table 3.3.

The APD graph of Class A, Class B, and AWGN models for noise were all compared to three representative measurements: two from an office cubicle and one from near the construction site. Figure 3.3 shows the first measurement taken near a cubicle. The non-Gaussian component of the noise has a steep slope at approximately 0.1%, which then begins to flatten out at approximately 0.01%. The α parameter for the Class B model is estimated from the more gradual slope at the lower probabilities. The steeper slope is achieved by the filtering process. For this measurement the pre-amplifier was turned off and the maximum amplitude in this measurement is about 73 dB above kToB. The 1 dB compression point with the pre-amplifier turned on and 15 dB of attenuation is approximately 76.3 dB above kToB. With the preamplifier turned off the 1 dB compression point would be much higher, and therefore the spectrum analyzer should not be in compression.

Figure 3.4 shows the construction site measurements. In this case, the slope of the low probability noise does not ﬂatten out as in the case of Figure 3.3. This indicates that not enough measurement samples have been taken to show the limit of the impulsive component of the noise. The α parameter is estimated from this slope. For this measurement the pre-ampliﬁer was turned on and the attenuation was set to 5 dB. From Table 2.1 the 1 dB compression point is 66.5 dB above kToB and the maximum measured amplitude is 40 dB

above kToB. Therefore, the spectrum analyzer should not be in compression.

For both of these sites, it can be seen by visual inspection and from Table 3.3 that there is good agreement between the models and the measurements. In both cases, the Class B model has the smallest Kullback-Leibler distance. In the case of Figure 3.4 the Class A model is only marginally superior to the AWGN model. This is likely because the high amplitude 44 noise is so rare, and the Class A model is not able to match the slope very closely.

The noise measurement in Figure 3.5 is an example of a noise measurement that is more difficult to model. Two parts of this measurement cause the difficulty. The first problem is the gradual increase in the slope of the APD graph at about the 37% mark. The second problem is that the slope of the low probability high amplitude noise is too low. The Class A model has difficulty with both of these. As stated previously, the Class A model is good for a sharp increase in the slope of APD graph and cannot match the gradual increase in slope. Furthermore, since the Class A model is only made up of Gaussian components, it cannot match the slope of the low probability high amplitude noise samples. This difficulty can be seen in that the Kullback-Leibler distance is 0.0638 bits for the Class A model. Since the Class B model makes use of a Weibull distribution which allows for a variable slope, it can more closely match this measurement. Therefore, as can be seen from the figure, the Class B model matches the slope of the low probability high amplitude noise more closely. Since the Class B model that has been used only has two terms, it also has a difficult time matching the gradual slope change at about the 37% mark. This could be improved by adding a third term. The Class B model has a distance of 0.0384 bits, and the AWGN model has a distance of 0.188 bits. For this measurement, both the Class A and Class B models have smaller Kullback-Leibler distances than the AWGN model as would be expected. For this measurement the pre-amplifer was turned on and the attenuation was set to 0 dB with a maximum amplitude of 47 dB above kToB. From Table 2.1 at this attenuation setting is the

1 dB compression point is 51.4 db above kToB which is 4.4 dB greater than the maximum amplitude. The spectrum analyzer should not be in compression.

By closely examining Figure 3.5, it can be seen that at the lowest probabilities the slope of the Class B model steps up a few dB right at the end. This is caused when two impulses occur closer together than the time resolution of the ﬁlter and are smeared. For very low probability impulsive noise as seen in Figures 3.3 and 3.4, this is unlikely to happen. However, as the impulse rate, γ, is increased and impulses occur with greater frequency, this problem becomes more likely. 45

Table 3.2: Parameters used for the simulation

Figure wog [dBm] wow [dBm] α[dB] γ κ[dB] Figure 3.3 -71.5 -35 0.5 3.5e3 0.002 26 Figure 3.4 -101.0 -88 3.2 5.25e3 0.0008 18 Figure 3.5 -88.0 -61 0.1 40e3 0.049 14 Figure 3.6 -89.5 -69 0.2 525e3 - -

Table 3.3: Kullback-Leibler divergence test

Figure Model A [bits] Model B [bits] AWGN [bits] Figure 3.3 0.0081 0.0022 0.1537 Figure 3.4 0.0015 0.0011 0.0017 Figure 3.5 0.0638 0.0384 0.1888

The problem of pulses occurring close together for the Class B model can be more clearly seen in Figure 3.6. On the left hand plot, at low probabilities the model continues to rise such that it is about 3 dB larger than the actual measurement for the largest amplitudes. One approach to solve this problem is to spread out the locations of the pulses in time. Any pulses that were within two sampling periods were spread out to random locations. The right plot in Figure 3.6 shows the model when spreading is used to prevent impulses from occurring close together. As can be seen this approach while ad hoc improves the match between the model and data. An inherent problem with this method is that it introduces correlation into the model. This is because pulses are never located within two sampling periods of each other. Therefore, there arrival time is inﬂuenced by recent pulses and is correlated in time. 46

80 Measured Data Class B 70 Class A AWGN

60 B o

dB above kT 40

20 0.01 1 5 20 37 60 70 80 90 95 98 99 percent exceeding ordinate

Figure 3.3: APD for measurements taken in an oﬃce outside of a cubicle along with Class A and B noise models with measurement bandwidth B = 5 MHz

50 Measured Data Class B 40 Class A AWGN

30 B o

dB above kT 10

−10 0.01 1 5 20 37 60 70 80 90 95 98 99 percent exceeding ordinate

Figure 3.4: APD for measurements taken outdoors near a construction site along with Class A and B noise models with measurement bandwidth B = 5 MHz 47

55 Measured Data 50 Class B 45 Class A AWGN 40 B o 35

25 dB above kT 20

5 0.01 1 5 20 37 60 70 80 90 95 98 99 percent exceeding ordinate

Figure 3.5: APD for measurements taken in an oﬃce outside of a cubicle along with Class A and B noise models with measurement bandwidth B = 5 MHz

Model Without Spreading Model With Spreading 50 50 Measured Data Measured Data 45 45 Class B Class B 40 40

35 35 B B o 30 o 30

25 25

20 20 dB above kT dB above kT 15 15

10 10

5 5

0 0 0.01 1 5 20 37 60 70 80 90 95 98 99 0.01 1 5 20 37 60 70 80 90 95 98 99 percent exceeding ordinate percent exceeding ordinate

Figure 3.6: APD for measurements taken near a refrigerator along with the Class B noise model with measurement bandwidth B = 5 MHz 48

3.5 Conclusion

In this chapter we reviewed popular models for impulsive noise. It is important to remember philosophically what it is that we are modeling. We are attempting to model the effect of impulsive noise on a receiver operating with a bandwidth of 5 MHz in the broadcast television bands. This is different than modeling the actual underlying waveforms of impulsive noise. It would be preferable to have a model for the actual impulse noise waveforms independent of the measurement equipment used. This would allow the results to be applied directly to different receivers. However, such a model would require more time and equipment than we had. These models and our measurements are still useful in that they can be applied to receivers with the same bandwidth operating in these frequencies. It would be difficult to apply these results to receivers of differing bandwidths, since the power of impulsive noise does not vary linearly with receiver bandwidth.

Simplified versions of Class A and B models for impulsive noise were compared to some representative measurements as well as the AWGN model. The Class B model clearly has a closer fit. However, the Class A model still has a good fit according to the Kullback-Leibler distance and is closer than AWGN for many types of impulsive noise. It is difficult to use the Class B model because of the need for a filter and the need to spread out the pulses to prevent overly large impulsive noise. Both the filtering process and the spreading serve to introduce temporal correlation into the simulated noise. This makes it more difficult to use for simulations since the algorithms we use in later chapters make the assumption that noise is not correlated in time. Furthermore, the introduction of a filter complicates the analysis. Therefore, since the Class A model is able to achieve reasonable results for at least some types of noise, we choose to use this model for the analysis in the remainder of this thesis. Chapter 4

Eﬀects of Impulsive Noise on the Energy Detector

In this chapter the effects of impulsive man-made noise on an energy detector will be discussed. The performance of an energy detector for deterministic signals in additive Gaussian noise is well understood [7]. These results have been extended to the detection of random signals in fading channels [28], [29]. The performance of an Lp-norm detector, of which an energy detector is a special case in the presence of non-Gaussian noise was analyzed in [16] where only the moments of the noise were assumed to be known. Our interest in this chapter is on the limits of signal detection in low SNR for additive non-Gaussian impulsive noise. We will extend the work of [16] by analyzing the SNR Wall for energy detectors in the presence of impulsive noise. The first section will have a description of the system model as well as the detector used for the analysis. While other models have been discussed in Chapter3, a Gaussian mixture model was selected in order to keep the analysis tractable. The following sections will discuss three observations about the effects of impulsive noise on the energy detector. Section 4.2 analyzes the effect of impulsive noise on the SNR Wall. Specifically, that impulsive noise requires more parameters to describe it, introducing more uncertainty and thus increasing the SNR Wall. Section 4.3 discusses the increased sensing time required

49 50 by the energy detector when noise corresponds to a Gaussian mixture to achieve the same performance as AWGN. Finally, Section 4.4 shows that the decision variable of the energy detector corrupted by impulsive noise will more slowly approach a normal distribution than that corrupted by AWGN, causing the receiver operating curve to worsen.

4.1 System Model and Detector Statistics

4.1.1 System Model

The detection of a weak signal in additive noise is a binary hypothesis problem [30]. Under hypothesis H1, there is a signal plus noise. Under hypothesis H0, there is only noise. The system model for the two hypotheses is given by

H0 : y[n] = w[n] (4.1)

H1 : y[n] = x[n] + w[n], (4.2) where y[n] are the received samples, w[n] are the samples of additive noise, and x[n] is the sampled unknown signal to be detected. All three of these signals are modeled in complex baseband, assumed to be bandlimited to some bandwidth B, and sampled at the Nyquist rate. All the signals are assumed to be both ergodic and wide-sense stationary over the interval of interest. Further, it is assumed that the signal x[n] is independent of the additive noise and that both x[n] and w[n] are independent identically distributed (i.i.d.) random variables. It is further assumed that x[n] and x[n+1] are i.i.d, and that w[n] and w[n+1] are 2 σX i.i.d. For this analysis, the eﬀects of fading will be ignored. The SNR is deﬁned as snr = 2 , σW 2 h 2i 2 h 2i where σX = E |x[n]| and σW = E |w[n]| . 51

4.1.2 Detector Statistics

In order to examine the eﬀects of impulsive noise on the energy detector, the Lp-norm

detector will be used of which the energy detector is a special case. The Lp-norm detector as introduced by [16] is used to investigate the eﬀects of non-Gaussian noise on signal detection.

The Lp-norm detector is useful because the mean and variance of the test statistic have been derived based only on the moments of the noise, w[n], and the variance of the signal,

2 2 2 σX . These derivations are based on the assumption of low SNR, i.e., σX << σW . This is a reasonable assumption since the interest of this thesis is the limits of low SNR signal

detection. For this system the test statistic of the Lp-norm detector is deﬁned as

N 1 X δ = |y[n]|p, (4.3) N n=1 where δ is the decision statistic, N is the number of samples used for the sensing interval,

and p is the order of the norm. In the case where p = 2, the Lp-norm detector is equal to the energy detector. Since the decision statistic is a sum of i.i.d. random variables, as N becomes large the distribution of δ will approach a normal distribution. Therefore, the probabilities of false alarm and missed detection are approximated by

τ − µ0 PFA = P [δ > τ|H0] = Q (4.4) σ0 µ1 − τ PMD = P [δ < τ|H1] = Q , (4.5) σ1

where PFA is the probability of false alarm, PMD is the probability of missed detection, τ 2 is a chosen threshold, µ0 and σ0 are the mean and variance of the decision statistic under 2 hypothesis H0, and µ1 and σ1 are the mean and variance under hypothesis H1. The authors 2 of [16] developed expressions for µ0 and σ0 that only depend on the moments of w[n] for a 2 multi-antenna conﬁguration. Furthermore, they developed expressions for µ1 and σ1 that rely only on the moments of w[n] and the variance of x[n] based on a Taylor series expansion 52 which is valid for low values of SNR. The simpliﬁed expressions for the means for a single antenna system without fading for an energy detector (p = 2) are given by

µ0 = MW (p) = MW (2) (4.6) p2 µ = M (p) + σ2 M (p − 2) = M (2) + σ2 , (4.7) 1 W X 4 W W X

th where MW (p) refers to the p moment about zero of the noise and is deﬁned as

h pi MW (p) = E |w[n]| , (4.8) where E[·] is the statistical expectation. The expressions for the variance of the decision statistic under the two hypotheses are given by

1 1 σ2 = M (2p) − M 2 (p) = M (4) − M 2 (2) (4.9) 0 N W W N W W 1 p2 σ2 = M (2p) − M 2 (p) − σ2 (M (p)M (p − 2) − 2M (2p − 2)) 1 N W W X 2 W W W 1 = M (4) − M 2 (2) + 2σ2 M (2) , (4.10) N W W X W where these expressions have been simpliﬁed by letting p = 2. Using these expressions for the means and variances of the test statistic, the probabilities of false alarm and missed detection can be calculated using (4.4) and (4.5). 53

4.2 SNR Walls for an Energy Detector

4.2.1 Reviewing the SNR Wall for Gaussian Noise of Uncertain Power

The author of [10] described the required SNR to detect spread spectrum signals in additive white Gaussian noise of uncertain power. The author of [11] coined the term SNR Wall and extended this analysis to distributions of noise that are nearly Gaussian with a certain degree of uncertainty for its moments. The SNR Wall is deﬁned as the minimum SNR such that robust detection below that SNR is impossible regardless of how many sample are used. Before examining the SNR Wall for impulsive noise, it is helpful to review the SNR Wall for Gaussian noise. In order to see the location of the SNR Wall, it is helpful to examine the sample complexity, which is the number of samples needed to achieve a certain detection performance, of an energy detector corrupted by AWGN of known power. By combining (4.4) and (4.5), the sample complexity of the energy detector is given by

Q−1(P )σ − Q−1(1 − P )σ 2 N = FA 0 MD 1 , (4.11) µ1 − µ0

2 where σ0 is given by

2 2 2 σ0 = Nσ0 = MW (4) − MW (2) , (4.12)

2 and σ1 is given by

2 2 2 2 σ1 = Nσ1 = MW (4) − MW (2) + 2σX MW (2) . (4.13)

Equation (4.11) demonstrates that as µ1 −µ0 → 0 the number of samples required to achieve a given PFA and PMD increase to inﬁnity. By calculating the values for µ0, µ1, σ0, and σ1 the sample complexity can be related directly to the SNR. In order to calculate these values 54

the moments about zero for a complex Gaussian random variable are needed and are given by [16]

p M (p) = Γ + 1 σp , (4.14) W 2 W

where Γ() is the Gamma function. Using these moments and (4.6), (4.7), (4.9) and (4.10)

2 1 4 1 the test statistic under hypothesis H0 is distributed according to ∼ N (σW , N σW ) . Under 2 2 1 4 2 2 hypothesis H1 it is distributed according to ∼ N (σW + σX , N (σW + 2σX σW )). Substituting these values into (4.11) and simplifying gives the sample complexity in terms of the SNR

√ 2 −1 −1 Q (Pfa) − Q (1 − Pmd) 1 + 2snr N = . (4.15) snr2

It is clear that if the noise power is perfectly known, then reliable detection for a given PFA

and PMD can be achieved for an arbitrarily low snr by simply increasing the the number of samples used for detection. However, as [10] and [11] have shown, if the noise power is not known perfectly, there is an snr below which detection becomes impossible. If noise has an uncertain power, then the means under both hypothesis H0 and H1 will have a range of possible values depending on the degree of uncertainty. It has been shown that if the means of a test statistic under the two hypotheses overlap, then reliable detection becomes impossible [11], [31]. The author of [11] has shown that if the actual noise power, deﬁned as

2 2 1 2 2 2 σA, is allowed to diﬀer from the nominal noise power, σW , according to α σW ≤ σA ≤ ασW , then when the snr is low enough, the means of δ under the two hypotheses will overlap and and the value of N will go to inﬁnity. Hence, robust detection will be impossible. Note that the valid range for α is greater than or equal to one (α ≥ 1).

2 Robust detection requires that for any actual noise power, σA, in the given uncertainty set the

desired PFA and PMD must be met. Therefore, the number of samples required, N, should

be chosen for the worst case scenario. From (4.4) it can be seen that PFA will be maximized

1Where ∼ N (µ, σ2) is a normally distributed random variable with mean µ and variance σ2. 55

2 when µ0 is maximized or when the actual noise power, σA, is at its highest possible value.

Similarly, from (4.5) it can be seen that PMD will be maximized when µ1 is at its minimum or when the actual noise power is at its minimum. Therefore, in the worst case scenario under hypothesis H0, the noise will be at its maximum possible power and the test statistic will 2 1 2 4 be distributed as ∼ N(ασW , N α σW ). Similarly, in the worst case scenario for hypothesis

H1, the noise power will be smaller than expected and the test statistic will be distributed 2 σW 2 1 4 2 2 as ∼ N α + σX , α2N (σW + 2σX σW α) . As previously mentioned, when the means of the test statistic under both hypotheses intersect, reliable detection becomes impossible. This is conﬁrmed by (4.11), which shows that the number of required samples goes to inﬁnity as

µ1 − µ0 → 0. Setting these two means equal and solving for the SNR gives

µ0,max = µ1,min 1 ασ2 = (σ2 + σ2 ) W α W X 2 2 σX α − 1 2 = snrwall = . (4.16) σW α

A simulation demonstrating the SNR Wall for additive white Gaussian noise can be seen in Figure 4.1. The uncertainty parameter α, was 0.35 dB and the probability of detection was plotted against the SNR for diﬀerent values of N. The SNR Wall is calculated using (4.16). It can be seen that the probability of detection goes to zero for all SNR values below the SNR Wall, which for an uncertainty of α = 0.35 is equal to approximately -8.0 dB according to (4.16). Furthermore, as the sensing time, N, is increased, the performance improves in that the probability of detection curve gets closer to the SNR Wall but is never able to pass it. 56

1 Theory: N = 1e3 0.9 Theory: N = 1e4 0.8 Theory: N = 1e5 Sim: N = 1e3 0.7 Sim: N = 1e4 Sim: N = 1e5 0.6 SNR Wall

0.5

0.4

0.3 Probability of Detection

0.2

0.1

0 −9 −8 −7 −6 −5 −4 Signal to Noise Ratio [dB]

Figure 4.1: SNR Wall for additive white Gaussian noise (AWGN) noise. The location of the SNR Wall is calculated using (4.16)

4.2.2 SNR Wall for Impulsive Noise

This section begins the contribution of this work, namely the location of the SNR wall for non-Gaussian impulsive noise. In order to examine the SNR wall for impulsive man-made noise, a two term Gaussian mixture model will be used. Unlike a Gaussian distribution which is fully described by a single parameter, a two term Gaussian mixture, known as -mixture noise, requires three parameters to be described. The pdf of -mixture noise is given by

|w|2 1 − |w|2 p(w) = 2 exp − 2 + 2 exp − 2 (4.17) πσI σI πσN σN

2 where is the probability of an impulse occurring, σI is the variance of the impulsive com- 2 ponent of the noise, and σN is the variance of the nominal component of the noise. The total 2 2 2 2 σI variance, σW is given by σW = (1 − + κ)σN . Where κ = 2 and is the ratio of the variance σN of the two components of the noise. -mixture noise is completely described by these three parameters (, κ, σN ). For this work, it is assumed that is much less than one and that κ is much greater than one. Essentially, -mixture noise is made of the combination of a high 57

2 probability low power Gaussian with variance σN and a low probability high power Gaussian 2 with variance, κσN . Since this Gaussian mixture requires three parameters to describe it, three uncertainty parameters, (β,γ,α), will be used to describe the uncertainty with which

h 1 i the process is known. Therefore, the actual value of is on the interval β ≤ A ≤ β . h 1 i The actual value of κ lies on the interval γ κ ≤ κA ≤ γκ . Similarly, the actual value of 2 1 2 2 2 σN lies on the interval α σN ≤ σA ≤ ασN . In order to calculate the mean and variance of the decision statistic, the moments about zero for a two term Gaussian mixture model are needed. They are given by [16]

p M (p) = Γ + 1 (1 − + κp/2)σp , (4.18) W 2 N

where p is the order of the moment about zero. The SNR wall can be found for the -mixture noise by calculating where the means of the decision statistic, δ, intersect. Again, since the noise is of uncertain power there is now a distribution of possible means for each hypothesis

H0 and H1. Just as in the case of Gaussian noise, a value for the number of samples, N, must be chosen that meets the desired PFA and PMD for any possible noise power in the

uncertainty set. PFA will be maximized when µ0 is maximized, and PMD will be maximized

when µ1 is minimized. µ0 will be maximized when all three parameters (κ, , σN ) are the

maximum possible. Similarly, µ1 will be minimized when all three parameters are minimized.

Setting µ0,max = µ1,min we can ﬁnd the SNR at which robust detection becomes impossible.

µ0max = µ1min (4.19) κ 1 (1 − β + κβγ)ασ2 = (1 − + ) σ2 + σ2 N β βγ α N X (1 − β + κγβ)α2 − (1 − + κ ) snr = β βγ , (4.20) W all α(1 − + κ)

Here the SNR is deﬁned as the ratio of signal power to the assumed noise power, snr = 2 2 σX σX 2 2 = 2 , not the actual noise power σWA since the actual noise power is unknown. σW (1−+κ)σN 58

This relationship shows that the SNR wall for -mixture noise is not only dependent on the uncertainty parameters β, γ,and α but also on the structure of the noise as described by the parameters and κ. Furthermore, if and κ are perfectly known, then β = γ = 1 and the expression will simplify to

(1 − + κ)α2 − (1 − + κ) α2 − 1 snr = = . (4.21) W all α(1 − + κ) α

Thus, in the case where and κ are perfectly known and only the level of noise is uncertain, then the SNR wall for -mixture noise will be equal to that for the Gaussian distribution. This makes sense since we are using an energy detector. Figure 4.2 shows the SNR Wall for -mixture noise with = 0.01 and κ = 100. It can be seen that when κ and are perfectly known, the SNR wall is the same as that for additive white Gaussian noise. Figure 4.3 shows the eﬀect that the value of and κ have on the SNR wall for -mixture noise. The SNR Wall is plotted against κ for diﬀerent values of when α = β = γ = 0.1dB. It can be seen that the SNR Wall is larger for increasing values of and κ. The maximum value for the SNR Wall as κ becomes large is given by,

(1 − + κ)α2 − (1 − + κ) snrW all = lim κ→∞ α(1 − + κ) κγβα2 − κ = lim βγ κ→∞ ακ α2β2γ2 − 1 = (4.22) αβγ

As κ → 1 and → 0 the SNR Wall is bounded by

(1 − + κ)α2 − (1 − + κ) snrW all = lim →0 α(1 − + κ) κ→1 α2 − 1 = lim (4.23) →0 α κ→1 59

Figure 4.3 conﬁrms these two bounds on the SNR Wall for -mixture noise. The lower bound makes sense because as κ → 1 and → 0 the -mixture noise becomes AWGN and would have the same noise uncertainty. If κ → ∞, the -mixture noise becomes totally dominated by the impulsive component. In this case, the SNR Wall is determined by the uncertainty in impulsive noise power and uncertainty in how often an impulse occurs, which are all described by the uncertainty terms (α, β, γ). Since the SNR Wall for -mixture noise is determined not only by the uncertainty but also by the parameters of the mixture (, κ), this expression shows that when κ → ∞ the impulse rate, as described by does not have an eﬀect, only the uncertainty in that rate.

−2

−4 AWGN γ, β = 0 dB −6 γ, β = 0.5 dB γ, β = 1 dB

SNR Wall [dB] −8

−10

−12

−14 0 0.5 1 1.5 2 2.5 3 α [dB]

Figure 4.2: SNR Wall for -mixture noise with = .002 and κ = 26dB

4.3 Eﬀects of Impulsive Noise on the Sample Complex- ity of an Energy Detector

As can be seen from (4.10), the variance of the decision statistic will be determined by the fourth moment of the noise. For κ >> 1 and for very small , there is the potential for the 60

−8.5

−9

−9.5

−10

−10.5 ε = 0.01 ε = 0.001 −11 ε = 0.0001 −11.5 Bounds SNR Wall [dB] −12

−12.5

−13

−13.5 0 10 20 30 40 50 60 Ratio of σ /σ ( κ [dB] ) I N

Figure 4.3: Eﬀect of the values of κ and on the SNR Wall

fourth moment of -mixture noise to be much greater than that of a Gaussian distribution. This will cause the variance of the decision statistic to increase. Thus the sample complexity will need to be increased to achieve the same performance. In the case of Gaussian noise, the variance of the decision statistic under the two hypotheses is given by

1 σ2 = σ4 (4.24) 0 N W 1 σ2 = σ4 (1 + 2snr) , (4.25) 1 N W

where σW is the standard deviation of the noise. The decision statistic for -mixture under both hypotheses, when p = 2, can be calculated from (4.9) and (4.10) and is given by

1 2(1 − + κ2) σ2 = σ4 − 1 (4.26) 0 N W (1 − + κ)2 1 2(1 − + κ2) σ2 = σ4 − 1 + 2snr (4.27) 1 N W (1 − + κ)2 61

Equation (4.27) shows that the variance of δ will be larger for the -mixture model depending on the values of and κ. Figure 4.4 shows the ratio of the variance of the test statistic, δ, for the two noise models for diﬀerent values of κ and . This is important because as (4.11) shows, the number of samples required to achieve a given performance is proportional to the

2 variance of the test statistic. As κ becomes large, the ratio will approach . Therefore, the largest ratio will occur for very large κ and very small . This requirement for more samples results in the worsening of the overall receiver operating characteristics as can be seen in Figure 4.5. The receiver operating curve is shown for snr = −15dB, = .01, N = 10000 and diﬀerent values of κ. As κ is increased, the performance of the detector is degraded.

35 ε = 0.001 30 ε = 0.01 ε = 0.05 ε = 0.1 25 ) [dB] δ 20

Increase in VAR( 10

0 0 5 10 15 20 25 30 35 40 κ [dB]

Figure 4.4: Ratio of variance of δ for -mixture and Gaussian noise. The increased variance of the test statistic requires a proportionally longer sensing time to get the same performance as in the AWGN case.

4.4 Eﬀect on the Central Limit Theorem Convergence

All of the analysis in this chapter hinges on the assumption that as N becomes large, the decision statistic will approach a normal distribution. This is valid in the case of Gaussian 62

0.9

0.8 D 0.7

0.6 Gauss κ = 10 0.5 κ = 30 0.4 κ = 100

0.3 Probability of Detection P 0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 Probability of False Alarm P FA

Figure 4.5: Receiver operating curve for N = 10, 000, = 0.01, and snr = −15 dB. Increasing values of κ negatively impact the ROC curve for GMM noise mixture noise because it can be assumed to be i.i.d. and has both a finite mean and variance. However, since impulsive noise has rare high amplitude components, it will converge to a normal distribution more slowly than would an energy detector corrupted by Gaussian noise. Figure 4.6 compares the normalized cumulative distribution function of the decision statistic to a normal distribution for different values of N. On the left, -mixture noise with = .005 and κ = 100 is used. It can be seen that for values of N as high as 5,000, there is still visible difference between the distribution of δ and a normal one. On the right hand side, the normalized distribution of δ with AWGN is plotted. In this case it can seen that there is good agreement with the normal distribution for values of N as low as 100. Figure 4.7 plots the receiver operating curve when = .01, κ = 100, and snr = −10dB. It can be seen that this departure from the normal distribution will cause the actual probability of detection to be lower than expected for a given probability of false alarm. 63

Normalized CDF for Gaussian Mixture Noise Normalized CDF for Gaussian Noise 1 1 N = 100 N = 100 0.9 N = 500 0.9 N = 500 0.8 N = 1000 0.8 N = 1000 N = 5000 N = 5000 0.7 Normal PDF 0.7 Normal PDF

0.6 0.6

0.5 0.5 F(x) F(x) 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x x

Figure 4.6: Comparing the convergence of the Central Limit Theorem (CLT) for -mixture noise and AWGN

Receiver Operating Curve 0 10 Actual ROC Expected ROC

−1 10 Probability of Detection

−2 10 −3 −2 −1 10 10 10 Probability of False Alarm

Figure 4.7: Eﬀect of slow convergence of the CLT on the receiver operating curve 64

4.5 Conclusions

In this chapter, we have investigated the eﬀect that impulsive noise has on the SNR Wall, sample complexity, and rate of convergence to the Central Limit Theorem for energy de-

2 tectors. Unlike AWGN noise, which is described by one parameter for the noise, σW , and another parameter for the uncertainty of the noise power, α, a two term Gaussian mixture is described by three terms. Since it is not possible that these additional terms will be known perfectly, they require the addition of two more terms, β and γ, to describe their uncertainty. This additional uncertainty causes the SNR Wall to increase. An example of the SNR Wall for Gaussian mixture noise can be seen in Figure 4.8 where the probability of detection is plotted against the SNR for a Gaussian mixture noise with = 0.002, κ = 26 dB, α = β = 0.2 dB and increasing values of N. The location of the SNR Wall is calculated according to (4.20). It can be seen from the plot that as the sensing time, N, is increased, the performance improves. However, as the SNR falls below the SNR Wall, the probability of detection falls to zero for any chosen value of N.

In the previous example a threshold is chosen such that a desired PMD and PFA will be met for any actual noise in a range of uncertainty. This assumes the worst case in that the threshold is set assuming the maximum possible power, and the actual power is the minimum. In the following example we demonstrate the SNR Wall by estimating the parameters of the

noise and then calculating PFA and PMD as SNR decreases. If the noise power is lower than estimated, PMD will be higher than expected. If the noise power is higher than estimated, then PFA will be higher than expected. If the signal power is lower than the diﬀerence between the actual noise power and the estimated noise power then it will be impossible to diﬀerentiate the signal from signal plus noise. Solving for the SNR when the signal power is less than the error in the estimate gives the expression

(1 − + κ) − (1 − β + κγβ)α snr = , (4.28) W all (1 − β + κγβ)α 65

Table 4.1: Single-antenna measurements

Name κ[dB] woN [dBm] Actual 26.0 0.00190 -2.4 Estimate Figure 4.10 27.1 0.00203 -2.4 Estimate Figure 4.9 24.7 0.00188 -2.5

where both the actual and estimated noise are modeled with -mixture noise. The parameters of the actual noise are κ and . The estimate of the ratio of impulsive to nominal components isκ ˆ = γκ, the estimate of the probability of an impulse occurring is ˆ = β, and the estimate

of the average power of the nominal component of the noise isw ˆoN = αwoN . Unlike (4.20) where γ, β, and α were the degree of uncertainty, in (4.28) γ, β, and α represent the actual error in the measurement. The diﬀerence between (4.28) and (4.20) is that in (4.20) the worst case is assumed and in the above equation the actual measurements are used.

In order to demonstrate how detection fails at lower values of SNR we implemented the Expectation-Maximization algorithm to estimate the parameters of a Class A noise model. The actual noise parameters were the same as above which were taken from actual noise measurements near a cubicle. The results can be found in Figures 4.9 and 4.10. In Figure 4.9 the noise power was underestimated and so the detection fails when the SNR falls below the

SNR Wall because PFA goes to 1. Similarly, in Figure 4.10 the noise power was overestimated and so signal detection fails because PMD goes to 1 as the SNR falls below the SNR Wall. Table 4.1 contains the actual parameters for the noise as well as the estimates for Figures 4.9 and 4.10. The ﬁrst column is whether the parameters are the actual numbers or estimates, the second column is κ the ratio of impulsive and nominal noise power, the third column is the probability of an impulse, and the ﬁnal column is the average power of the nominal

component of the noise woN .

In addition to increasing the SNR Wall, the presence of impulsive noise increases the variance 66 of the test statistic and thus requires more samples to achieve the same performance as in the AWGN case. The required extra sensing time is based on both the size of the impulses and their probability of occurrence. The worst case scenario being for very large and very rare impulses. Finally, it was shown that an energy detector in the presence of impulsive noise converges to the central limit theorem more slowly, which can cause a decrease in the expected performance.

0.9

0.8

0.7

0.6

0.5

0.4 Probability of Detection

0.3

Theory: N = 1e5 0.2 Theory: N = 1e6 Theory: N = 1e7 Sim: N = 1e5 0.1 Sim: N = 1e6 Sim: N = 1e7 SNR Wall 0 −9 −8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 Signal to Noise Ratio [dB]

Figure 4.8: Simulation to demonstrate the SNR Wall. The noise power uncertainty was set to 0.35 dB, PFA = 0.1. The SNR Wall was calculated using (4.20) 67

1 Theory: N = 1e5 0.9 Theory: N = 1e6 0.8 Theory: N = 1e7 Sim: N = 1e5 0.7 Sim: N = 1e6 Sim: N = 1e7 0.6 SNR Wall

0.5

0.4

0.3 Probability of False Alarm 0.2

0.1

0 −9 −8 −7 −6 −5 −4 Signal to Noise Ratio [dB]

Figure 4.9: Failure of signal detection when the noise power is underestimated. The decision threshold was set using the estimated noise so that PMD = 0.1. The SNR Wall was calculated using (4.28)

1 Theory: N = 1e5 0.9 Theory: N = 1e6 0.8 Theory: N = 1e7 Sim: N = 1e5 0.7 Sim: N = 1e6 Sim: N = 1e7 0.6 SNR Wall

0.5

0.4

0.3

Probability of Missed Detection 0.2

0.1

0 −9 −8 −7 −6 −5 −4 Signal to Noise Ratio [dB]

Figure 4.10: Failure of signal detection when the noise power is overestimated. The decision threshold was set using the estimated noise so that PFA = 0.1. The SNR Wall was calculated using (4.28) Chapter 5

Eﬀects of Spatially Correlated Impulsive Noise on the Energy Detector

In this chapter, we examine the eﬀect of spatially correlated impulsive noise on the energy detector in a multi-antenna system in the presence of Rayleigh fading. The performance of the Lp-norm detector, of which an energy detector is a special case, was described in [16] for several types of non-Gaussian noise. This analysis assumed the noise was independent in both time and space. In this chapter, we would like to extend this analysis, for the speciﬁc case of an energy detector, to correlated impulsive noise. In Section 5.1, we will describe the system model. In Section 5.2, we will introduce a model for correlated noise that is based on the measurement campaign described in Chapter2. In Section 5.3, we will review the performance of the energy detector in the presence of uncorrelated non-Gaussian noise. In Section 5.4, we will extend this analysis to the case of correlated impulsive noise based on the model suggested in Section 5.2. In Section 5.5, we will conclude with some numerical simulations to demonstrate the analysis.

68 69

5.1 System Model

Just as in Chapter4, the detection of the presence of an unknown signal is a binary hypothesis

testing problem. The two hypotheses, H0, the sampled received signal is only noise, and H1, the sampled received signal consists of a combination of unknown signal and noise are given by

H0 : y[n] = w[n] (5.1)

H1 : y[n] = h[n]x[n] + w[n], (5.2) where y[n] is a L×1 vector of samples taken from the received ﬁltered signal, w[n] is a L×1 vector of samples taken from the noise, x[n] are samples taken from the unknown signal to be detected, and h[n] is the L × 1 vector representing the channel gain which is assumed to be a complex Gaussian random variable. It is assumed that y[n], w[n], and x[n] are all samples taken from band-limited complex baseband signals and are assumed to be ergodic and widesense stationary over the interval of observation. The channel gain is independent

2 and identically distributed on each antenna with covariance matrix Rh = σhIL, where IL is an L × L identity matrix. A block fading model is assumed such that h[n] is constant over a given sensing interval N. The unknown signal to be detected, x[n], are samples taken from a random process which results in independent, identically distributed random variables with

2 variance σx. The signal , x[n], is the same for each antenna. The noise, w[n], is identically distributed across all antennas and has covariance matrix given by,

  1 ρw1,2 ··· ρw1,L     2 ρw2,1 1 ··· ρw2,L Rw = σ   (5.3) w  . . .. .   . . . .    ρwL,1 ρwL,2 ··· 1

th th 0 Samples of the noise taken from the i and j antennas, wi[n] and wj[n ], are uncorrelated 70 for all i and j when n 6= n0. The energy detector for this system calculates the test statistic

L N 1 X X 2 δ = |y [n]| , (5.4) LN l l=1 n=1

th th where yl[n] is the n sample on the l antenna of the received signal. The decision statistic is compared to a given threshold, τ. If δ > τ, then hypothesis H1 is decided. If δ ≤ τ, then a decision is made for hypothesis H0. In this chapter, the eﬀect of spatially correlated impulsive noise will be examined. In order to clarify the discussion, a list of terms to be used can be found below as a reference.

5.1.1 Deﬁnition of Terms gI [n] : Impulsive component of the Gaussian mixture model gN [n] : Nominal component of the Gaussian mixture model th wI,l[n] : Samples of the noise from the l antenna after setting all values below a certain threshold to 0

th wN,l[n] : Samples of the noise from the l antenna after setting all values above a certain threshold to 0 σ2 : Variance of the impulsive component of the Gaussian mixture model, gI h i σ2 = E |g [n]|2 gI I σ2 : Variance of the nominal component of the Gaussian mixture model, gN h i σ2 = E |g [n]|2 gN N 2 σ|w| : Variance of the instantaneous power of the noise, 2 2 h 4i h 2i σ|w| = E |w[n]| − E |w[n]| σ2 : Variance of the instantaneous power of the impulsive component of the |gI | h i h i2 Gaussian mixture model, σ2 = E |g [n]|4 − E |g [n]|2 |gI | I I σ2 : Variance of the noise after setting all values below a certain threshold to 0 wI,l on the lth antenna 71

σ2 : Variance of the noise after setting all values above a certain threshold to 0 on wN,l the lth antenna

th th ρwi,j : Correlation coeﬃcient of the noise on the i and j antennas h i h i h i E wi[n]wj [n] −E wi[n] E wj [n]

ρwi,j = σwi σwj

ρgI i,j : Correlation coeﬃcient of the impulsive component of the Gaussian mixture h i h i2 E gIi [n]gIj [n] −E gI [n] model on the ith and jth antennas, ρ = gI i,j σ2 gI

ρgN i,j : Correlation coeﬃcient of the nominal component of the Gaussian mixture h i h i2 E gNi [n]gNj [n] −E gN [n] model on the ith and jth antennas, ρ = gN i,j σ2 gN th th ρ|w|i,j : Correlation coeﬃcient of the noise power on the i and j antennas, h i h i h i 2 2 2 2 E |wi[n]| |wj [n]| −E |wi[n]| E |wj [n]| ρ|w|i,j = σ σ |wi| |wj |

ρ|gI |i,j : Correlation coeﬃcient of the power of the impulsive component of the Gaussian mixture model on the ith and jth antennas, h i h i h i 2 2 2 2 E |gIi [n]| |gIj [n]| −E |gIi [n]| E |gIj [n]| ρ|g | = I i,j σ|g |σ|g | Ii Ij

5.2 Correlated Noise Measurements

For the studies in this chapter, the model used for impulsive noise is the two-term Gaussian mixture model described in Section 4.2.2. From the multi-antenna measurements made in Chapter2, it is clear that there is a non-zero amount of correlated noise on the two antennas for the frequencies and distances measured. The magnitude of the correlation coeﬃcient of the complex noise usually ranged from 0.3 to 0.8. It was suspected that most of the correlation was due to the large amplitude impulses present on both antennas, which were likely coming from external sources. These sources would be correlated across the antennas. In order to test this hypothesis, it was desired to separately measure the correlation coeﬃcient of the higher and lower amplitude noise. To do this, a threshold was chosen to separate the 72

impulses from the nominal noise. All amplitudes above the threshold would be considered impulsive, and those below would be considered nominal. To simplify the process of choosing the value of the threshold, each measurement was normalized by dividing it by the square root of its average power. This served to simplify the choice of a threshold so that diﬀerent measurements could be more easily compared. The nth sample of the impulsive component

th of the noise on the l antenna, wI,l[n], was given by

  wˆl[n] :w ˆl[n] ≥ τ wI,l[n] = (5.5)  0 :w ˆl[n] < τ,

th th wherew ˆl[n] is the n sample from the normalized received signal on the l antenna, and τ is the chosen threshold. Similarly, the nth sample on the lth antenna of the nominal component of the noise is given by

  0 :w ˆl[n] ≥ τ wN,l[n] = (5.6)  wˆl[n] :w ˆl[n] < τ.

The value of the threshold, τ, was chosen to be the amplitude where the slope of the APD graph increased, thus, deviating from the Gaussian distribution. This choice of threshold is somewhat subjective. An example of choosing this threshold from a measurement can be found in Figure 5.1. It can be seen that the threshold is placed at the 5 dB point where the slope of the APD graph begins to increase. A Gaussian distribution was added to the plot to show the diﬀerence between nominal and impulsive noise. The increase in slope between 0 and 5 dB indicates the presence of impulsive noise.

Once the noise samples have been separated according to equations (5.5) and (5.6), the crosscorrelation coeﬃcient between the two antennas for the impulsive and nominal components of the noise was calculated. The crosscorrelation of the impulsive component is calculated by 73

h ∗ i h i h ∗ i E wI,1[n]wI,2[n] − E wI,1[n] E wI,2[n] ρ1,2 = (5.7) σwI,1 σwI,2

Where ρ1,2 is the crosscorrelation coeﬃcient of the noise between antennas one and two, σwI,l is the standard deviation of the impulsive component of the noise on the lth antenna, and

∗ th wI,2[n] is the complex conjugate of the n sample on the second antenna of the impulsive component of the noise. Equation (5.7) was used to calculate the correlation coefficient of the nominal component of the noise in the same manner. When using (5.7), the correlation coefficient was calculated for a few different delays and the maximum was chosen to account for the possibility that the two receivers may have had a slight time offset.

The placement of the threshold has an effect on the makeup of the two components and thus on the value of the two coefficients. An example of this can be found in Figure 5.2. There are three plots on the graph: the total cross correlation which is independent of the chosen threshold, the crosscorrelation for the impulsive component, and the crosscorrelation for the nominal component. From the figure, it can be seen that as the threshold increases, the crosscorrelation for both of the components increases. The correlation of the impulsive component reaches its maximum when the threshold is placed between 5 and 10 dB. As the threshold increases, more and more of the impulsive noise is present in the low amplitude component, and it can be seen that the correlation increases for this component as the threshold is increased. The correlation of the high amplitude values begins to sharply decrease after the threshold is raised above 15 dB. From the APD graph in Figure 5.1, it can be seen that amplitudes above 15 dB are quite rare. Since they are such rare events, the two signals would not be likely to achieve such a high amplitude at the same time. So one of the signals would be reduced to zero, causing the correlation to decrease. The figure implies that the angle of arrival of the nominal component of the noise is uniformly distributed, which results in a much lower spatial correlation. On the other hand the high degree of spatial correlation implies that the angle of arrival of the impulsive component of the noise is limited to specific angles. Therefore, it is likely coming from specific external sources. 74

The correlation coefficients of the impulsive and nominal components were calculated separately for several of the measurements taken at 79 MHz. Each measurement had a threshold chosen manually to be approximately where the distribution deviated from the Gaussian. The 79 MHz measurements were chosen because they showed a good diversity of correlations at different antenna distances. A summary of the results can be found in Table 5.1. The name of each measurement in the left hand column corresponds to the names given in Ap- pendixC. The measurements were taken at different distances ranging from 1 to 4 meters. The fourth column contains the chosen threshold used to separate the impulsive and nominal components. It can be seen from the table that in every case the higher amplitude values have the higher values for the correlation coefficient. With few exceptions, the correlation of the lower amplitude component is at or below 0.2 while the correlation of the higher amplitude component ranges from 0.4 to 0.8. Based on these observations, the suggested model for correlated impulsive noise will emphasize the correlation of the impulsive component over the nominal component.

5.2.1 Correlated Noise Model

Based on the results from the previous section, a model for correlated impulsive noise was developed. The model is based on two key assumptions. First, only the impulsive component of the noise is correlated. Second, if an impulse is present, it will be present on all antennas. With this model, the noise is given by

w[n] = xe[n]gI[n] + (1 − xe[n])gN[n], (5.8) where xe[n] is a binary random variable which determines the presence of an impulse, gN[n] is a L × 1 vector of zero mean complex Gaussian random variables with covariance matrix R = σ2 I , and g [n] is a L × 1 vector of zero mean complex Gaussian random variables gN gN L I with covariance matrix given by 75

20 First Antenna 15 Second Antenna Threshold 10 Impulsive Gaussian 5

−5 Gaussian −10

Normalized Amplitude [dB] −15

−20

−25 0.01 1 5 20 37 60 70 80 90 95 98 99 percent exceeding ordinate

Figure 5.1: Example of using a threshold to compare the correlation of the impulsive and nominal components of the noise. The threshold, τ, is set to 5 dB where the APD deviates from the Gaussian distribution. The measurement was taken at 79 MHz with an antenna separation of 3 m. 76

Table 5.1: Correlation for impulsive and Gaussian components of noise

Name Center Freq [MHz] Dist [m] Threshold [dB] Impulsive Nominal 00151 79 1 5 0.41 0.17 00179 79 1 7 0.94 0.66 00180 79 1 7 0.67 0.39 00181 79 1 7 0.81 0.56 00164 79 3 0 0.38 0.17 00165 79 3 -2 0.40 0.17 00166 79 3 -2 0.39 0.12 00194 79 3 6 0.59 0.10 00195 79 3 5 0.72 0.17 00210 79 4 2 0.54 0.24 00211 79 4 2 0.48 0.21 77

1 Total 0.9 Impulsive 0.8 Nominal

0.7

0.6

0.5

0.4

0.3

0.2

Empirical Cross−correlation Coefficient 0.1

0 −10 −5 0 5 10 15 20 Threshold [dB]

Figure 5.2: Comparison of the crosscorrelation coeﬃcients of the impulsive and nominal components of the noise plotted against a chosen threshold. Measurements were taken at a center frequency of 79 MHz with an antenna separation of 3 m.

  1 ρgI ··· ρgI  1,2 1,L  ρ 1 ··· ρ  2  gI 2,1 gI 2,L  Rg = σ   (5.9) I gI  . . .. .   . . . .   

ρgI L,1 ρgI L,2 ··· 1

where ρgI i,j is the correlation coeﬃcient of the impulsive component of the noise between the th th i and j antennas. Both gI [n] and gN [n] are independent for diﬀerent values of n. The variable xe[n] is the same for all L antennas. Therefore, if an impulse is present, it is present on all the antennas.

This is a two term Gaussian mixture model as described in Sections 3.3.4 and 4.2.2. The three parameters that describe a two term Gaussian mixture model are (, κ, σN ). The parameter

, the probability that an impulse is present, is equal to the probability that xe = 1. The parameter σN = σgN is equal to the standard deviation of the nominal component of the noise, and κ is equal to the ratio of the variance of the impulsive noise to the variance of the 78 nominal noise

σ2 κ = gI . (5.10) σ2 gN

The average power of the noise for a single antenna is given by

σ2 = (1 − + κ)σ2 . (5.11) w gN

5.3 Performance of an Energy Detector in Uncorre- lated Impulsive Noise

Before analyzing the performance of an energy detector in the presence of correlated impulsive noise, it will be helpful to review the probabilities of false alarm and missed detection of an energy detector for uncorrelated impulsive noise. Probabilities of false alarm and missed detection were calculated in [16] for multiple antennas with uncorrelated impulsive noise and Rayleigh fading. First, the probabilities of false alarm and missed detection for a single channel realization will be described. This will be followed by examining the average probabilities for false alarm and missed detection averaged over the Rayleigh fading channel gains.

5.3.1 Probabilities of False Alarm and Missed Detection for a Sin- gle Channel Realization

Assuming that the number of samples, N, is suﬃciently large so that the test statistic, δ, can be approximated as a normal random variable and that the noise on each antenna is uncorrelated, the probabilities of missed detection and false alarm are given by [16] 79

τ − µ0 PFA = P [δ > τ|H0] ≈ Q (5.12) σ0 µ1 − τ PMD = P [δ < τ|H1] ≈ Q . (5.13) σ1

where just as in (4.4) and (4.5) µ0, µ1, σ0, and σ1 are the means and standard deviations

of the test statistic under the hypotheses H0 and H1 respectively. Therefore, all that is required to calculate the probabilities of false alarm and missed detection are the means and variances of the test statistic. Assuming that the noise samples on each diﬀerent antenna are identically distributed, these values can be calculated using (4.6) and (4.7):

L 1 X µ = σ2 = σ2 (5.14) 0 L w w l=1 L L 1 X 1 X µ = σ2 + σ2|h |2 = σ2 + σ2|h |2, (5.15) 1 L w x l w L x l l=1 l=1

th where hl is the Rayleigh fading channel gain on the l antenna. The variance of the test statistic for a given channel realization can be approximated by using (4.9) and (4.10):

L 1 X (1 − + κ2) σ4 2(1 − + κ2) σ2 = 2σ4 − σ4 = w − 1 (5.16) 0 NL2 w,l (1 − + κ)2 w,l NL (1 − + κ)2 l=1 L σ4 X 2(1 − + κ2) σ2 = w − 1 + 2snr (5.17) 1 NL2 (1 − + κ)2 l=1

2 2 2 where snr = σx|hl| /σw. These expressions give the probabilities of false alarm and missed detection for a single channel realization. In order to get the average probabilities, they must

be averaged over the possible values of hl. 80

5.3.2 Average Probabilities of False Alarm and Missed Detection

Using (5.13) along with the given values for the means and variances for the test statistic under both hypotheses, the probabilities of false alarm and missed detection can be calculated

for a single channel realization. Under hypothesis, H0, obviously fading has no impact,

and the probability of false alarm remains unchanged. In the case of hypothesis, H1, the average probability of missed detection can be calculated as the expected value of (5.13)

for the possible channel realizations, hl. To make this calculation easier, [16] assumes that 2 2 the detection takes place in the low SNR region, i.e. σx << σw, and therefore σ0 ≈ σ1. Furthermore, after rearranging the variables, the probability of missed detection is given by

µ1 − τ Pmd ≈ E Q = E [Q (γ − ξ)] , (5.18) σ0

2 2 where γ = PL σx|hl| and ξ = τ−µ0 . In order to calculate this expectation, the pdf of γ is l=1 σ0L σ0 needed and is given by [16], [32]

γL−1 γ f(γ) = exp − , (5.19) L 2L 2 (L − 1)!α σh ασh

2 where α = σx . Using this pdf the probability of missed detection is given by σ0L

Z ∞ γL−1 γ P ≈ E {Q (γ − ξ)} = Q (γ − ξ) exp − dγ (5.20) md L 2L 2 0 (L − 1)!α σh ασh

5.4 Performance of an Energy Detector in Spatially Correlated Impulsive Noise

The analysis of PFA and PMD for a multi-antenna system in the presence of correlated impulsive noise will proceed in two steps. First, since the model assumes that only the 81

impulsive component of the noise is correlated, an expression will be developed to calculate the correlation coefficient between the noise power of two antennas in terms of the correlation coefficient of the impulsive component of the noise. Second, the effect of this correlation coefficient on the performance characteristics of the energy detector will be analyzed.

5.4.1 Calculating the Correlation in Noise Power for Impulsive Noise

The correlation coeﬃcient of the power of the noise on two antennas is given by

h 2 2i h 2i h 2i E |wi[n]| |wj[n]| − E |wi[n]| E |wj[n]|

ρ|w|i,j = (5.21) σ|wi|σ|wj |

2 2 th th where |wi[n]| and |wj[n]| are the instantaneous received power on the i and j antennas, and σ2 and σ2 are the variance of the instantaneous power on the ith and jth antennas. |wi| |wj | Since the noise on each antenna is assumed to be identically distributed, this expression can be rewritten as

2 h 2 2i h 2i E |wi[n]| |wj[n]| − E |w[n]| ρ|w|i,j = 2 . (5.22) σ|w|

h 2 2i h 2i 2 Therefore, if the quantities E |wi[n]| |wj[n]| , E |w[n]| , and σ|w| are known, then the correlation of the received power for the entire mixture model can be calculated. The mean of the received power on each antenna is already known from (5.14) and is given by

h 2i 2 E |w[n]| = σw. (5.23)

The variance of the instantaneous noise power on each antenna can be estimated from (4.9) 82

2(1 − + κ2) σ2 = σ4 − 1 (5.24) |w| w (1 − + κ)2

Since it is assumed that an impulse occurs on all antennas at the same time,

h 2 2i E |wi[n]| |wj[n]| can be broken up into its impulsive and nominal components and is given by

h 2 2i h 2 2i h 2 2i E |wi[n]| |wj[n]| = E |gI,i[n]| |gI,j[n]| + (1 − )E |gN,i[n]| |gN,j[n]| (5.25)

2 2 where |gI,i[n]| , and |gI,j[n]| are the instantaneous noise power for the impulsive component th th 2 2 of the noise on the i and j antennas, and |gN,i[n]| and gN,j[n]| are the instantaneous noise th power of the nominal component of the noise on the ith and j antennas. Since the model of correlation being used assumes that the nominal component of the noise is uncorrelated, h i h i h i h i E |g [n]|2|g [n]|2 = E |g [n]|2 E |g [n]|2 = σ4 . The value for E |g [n]|2|g [n]|2 N,i N,j N N gN I,i I,j can be found directly from the deﬁnition of the correlation coeﬃcient and is given by

ρ|gI |i,j = (5.26) σ|gI,i|σ|gI,j |

where ρ|gI |i,j is the correlation coeﬃcient of the power of the impulsive component of the noise between the ith and jth antennas. Since it is assumed that the impulsive noise is identically h i h i distributed across the L antennas E |g [n]|2 = E |g [n]|2 = σ2 and σ = σ . Since I,i I,j gI |gI,i| |gI,j | 2 |gI [n]| is the square of a complex Gaussian, it is distributed as a chi-squared random variable with 2 degrees of freedom [32]. The variance of the power of the impulsive component is given by σ2 = σ4 . Noting that σ2 = κσ2 and substituting these values into (5.26) and |gI | gI gI gN rearranging the terms gives

h i E |g [n]|2|g [n]|2 = ρ σ4 + σ4 I,i I,j |gI |i,j gI gI = κ2σ4 (ρ + 1). (5.27) gN |gI |i,j 83

h 2 2i 2 4 4 E |wi[n]| |wj[n]| = κ σ (ρ|g | + 1) + (1 − )σ gN I i,j gN (5.28) = (1 − + κ2(ρ + 1))σ4 . |gI |i,j gN

It has been shown that the correlation coefficient of the power of a complex Gaussian random process is equal to the square of its correlation coefficient [33]. Therefore, the correlation coefficient of the power of the impulsive component of the noise is related to the correlation coefficient of the impulsive component of the noise by

2 ρ|gI |i,j = |ρgI i,j | , (5.29)

Where ρgI i,j is the correlation coefficient of the impulsive component of the noise. Substi- h 2 2i tuting this result along with the values for E |wi[n]| |wj[n]| into (5.22) gives the value for the correlation coefficient of the received power between two antennas for -mixture noise, based on the correlation coefficient of the impulsive component of the noise

(1−+κ2(|ρ |2+1))−(1−+κ)2 gI i,j (5.30) ρ|w|i,j = 2(1−+κ2)−(1−+κ)2 .

In order to demonstrate the relationship between the correlation coeﬃcient of the impulsive

component of the noise, ρgI i,j , and the correlation coeﬃcient of the power of the total noise,

ρ|w|i,j , a numerical simulation was conducted. One million samples were taken from an -mixture model with varying amounts of correlation in the impulsive component. The parameters of the Gaussian mixture model were = 0.01 and κ = 100. Figure 5.3 shows the results of this test plotted alongside the theoretical curve. It can be seen that even if the impulsive component is uncorrelated, as long as the impulses arrive at both antennas at the same time and κ is sufficiently large, the power of the total noise will still have a correlation coefficient of nearly 0.5. It should be noted that the correlation of the nominal component 84 is always assumed to be zero. This means that if impulses occur across the antennas at the same time, even if the impulsive component of the model is uncorrelated, the total received power will still have a significant amount of correlation.

1 Simulation 0.9 Theory 0.8

0.7

0.6 i,j

|w| 0.5 ρ 0.4

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 ρ g Ii,j

Figure 5.3: Simulating the correlation of the instantaneous power of a Gaussian mixture model based on the correlation of the impulsive component with = 0.01, κ = 100, and 1e6 samples.

5.4.2 Probabilities of False Alarm and Missed Detection for Spa- tially Correlated Noise

Using the value for the correlation coeﬃcient of the received power in Equation (5.30), the eﬀect of that correlation on the probabilities of missed detection and false alarm can now be calculated. Equations (5.12) and (5.20) depend on the values of µ0 and σ0. In the previous analysis in [16] it was assumed that the average energy on the individual antennas was uncorrelated. For this analysis, it will still be assumed that the detection takes place in the

2 2 low SNR environment, σx << σw and that σ0 ≈ σ1. Since the noise is independent in time, the average energy on each antenna will be the same as the uncorrelated case. Therefore, 85

the mean and variance of the statistic for the lth antenna are given by

2 µ0,l = σw (5.31) σ4 (1 − + κ2) σ2 = W 2 − 1 (5.32) 0,l N (1 − + κ)2

Due to the spatial correlation of the impulsive noise, the energy on the ith and jth antennas will be correlated with each other. Since δ is the average of the energy of the L antennas, its mean and variance can be found by ﬁnding the mean and variance of a sum of correlated random variables. The mean of a sum of random variables is equal to the sum of the means of the random variables regardless of the statistical dependence of those values [20]. Therefore, the mean of δ does not change in the case of spatially correlated impulsive noise and is given by

L 1 X µ = µ = σ2 (5.33) 0 L 0,l w l=1 (5.34)

The variance of a sum of L correlated random variables is given by [20]

L ! L L X X X VAR Xl = VAR(Xl) + COV(Xi,Xj) (5.35) l=1 l=1 i,j: i6=j

Where V AR(·) refers to the variance and COV (·) refers to the covariance. Using the def- inition of the correlation coeﬃcient and the assumption that the energy on each of the L

antennas is identically distributed, the variance of the test statistic, δ, under H0 is given by

L ! σ2 2(1 − + κ2) X σ2 = w − 1 L + ρ , (5.36) 0 NL2 (1 − + κ)2 |w|i,j i,j: i6=j 86

The probabilities of false alarm and missed detection for spatially correlated impulsive noise can be calculated by substituting the correlation coeﬃcient of the received powers from (5.30) into the equation for the variance of the test statistic under spatially correlated impulsive noise in (5.36). This result can then be used with (5.12) and (5.18) to calculate the probabilities of false alarm and missed detection.

5.5 Numerical Simulations

Some numerical simulations were done to show the performance degradation for multi- antenna systems in the presence of correlated impulsive noise. In both of these simulations, the uncorrelated impulsive noise means that impulses arrive at the diﬀerent antennas at diﬀerent times. In the correlated case, the impulses arrive at each antenna at the same

time with diﬀering amounts of correlation. In Figure 5.4 PMD is plotted against the number of samples used to make the detection decision for diﬀerent numbers of antennas. For the

correlated case, the correlation coeﬃcient used was ρgI i,j = 0.1 for all the antennas. For the simulation PFA = 0.1, = 0.01 and κ = 100. It is clear that even for small amounts of correlation in the impulsive noise, the performance quickly degrades. This is because from (5.30) the correlation coeﬃcient of the total noise power will be approximately 0.5 even when the impulsive component is uncorrelated. This is because the impulses are arriving at the same time on each of the antennas.

In Figure 5.5 PMD is plotted against the number of antennas used for uncorrelated impulsive noise as well as noise with ρgI i,j = 0.1 and ρgI i,j = 1. The simulation was performed with N = 20000 samples for the sensing period. The Gaussian mixture noise had parameters = 0.01 and κ = 100. It can be seen that when the impulsive noise is uncorrelated, and pulses do not arrive at the same time, the performance gets better as more antennas are added to the system. However, when correlation is added to the impulsive noise and pulses arrive at the same time, performance degrades and additional antennas do not greatly improve the 87 performance. From this ﬁgure, it can be seen that the assumption of impulse arrival at the same time across the antennas is what drives the performance degradation, as there is not

much diﬀerence between ρgI i,j = 0.1 and ρgI i,j = 1.0.

0 10

L = 1 L = 2 L=4 L = 3 ] MD

−1 10 Probability of Missed Detection [P

Uncorrelated − Theory ρ = 0.1 − Theory g I ρ = 0.1 − Simulation g I Uncorrelated − Simulation −2 10 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of Samples 4 x 10

Figure 5.4: Comparing the eﬀect of sensing time on correlated impulsive noise that arrives at all antennas simultaneously vs. impulsive noise that arrives randomly. For this simulation,

PFA = 0.1, = 0.01, κ = 100 88

0 10 ] MD

−1 10

Probability of Missed Detection [P Uncorrelated − Theory ρ = 0.1 − Theory g N ρ = 1 − Theory g N Uncorrelated − Simulation ρ = 0.1 − Simulation g N ρ = 1 − Simulation g N −2 10 1 2 3 4 5 6 7 8 9 10 Number of Antennas

Figure 5.5: Comparing the eﬀect of additional antennas for correlated impulsive noise that arrives at all antennas simultaneously vs. arriving at diﬀerent antennas randomly. For this simulation = 0.01, κ = 100, N = 20e3, PFA = 0.1. 89

5.6 Conclusions

In this chapter we analyzed the performance of a multi-antenna energy detector in the presence of correlated impulsive noise and Rayleigh fading. We made use of the Lp-norm statistics from [16], where the performance was analyzed for uncorrelated non-Gaussian noise. Based on our measurements in Chapter2, we suggested a model for correlated impulsive noise in which impulses arrive on all antennas simultaneously, and only the impulsive component of the noise is correlated between the antennas. This is intuitive because impulsive noise is generally from an external source and would therefore be highly correlated across the antennas. We showed, that using our model, even if the amplitudes of the impulsive component of the noise are uncorrelated, the overall received power will be highly correlated for large enough values of κ. For example, we showed that for large values of κ, the correlation coef- ﬁcient of the total received power between two antennas would be nearly 0.5 even when the impulses themselves are uncorrelated. This generally agrees with our measurements which found that the correlation coeﬃcient between the two antennas of the received power was usually above 0.5. Using this model expressions were developed for PFA and PMD in the presence of correlated impulsive noise. A simulation was used to show agreement with the theory and demonstrated that if impulses occur on all antennas at the same time, adding antennas to a multi-antenna energy detector does not greatly improve the performance. Chapter 6

Receiver Design

6.1 Overview of Project

In this chapter a receiver designed to measure background RF noise in the low VHF band will be presented. This receiver was designed in conjunction with the class ECE 4984 RF Spectrum Sensing. The purpose was to design a RF spectrum sensing receiver operating in two bands: 25-80 MHz and 116-174 MHz. Conventional spectrum analyzers are not well suited for spectrum monitoring. This is because spectrum analyzers sweep across the band of interest with a narrow band filter and risk missing intermittent signals that are present when the filter is at another frequency. The RF receiver presented here measures each band instantaneously without sweeping, making it much easier to see bursting or short term signals. The entire system consisted of a monopole antenna, RF board for gain and filtering of the signal, an FPGA for signal processing, and a user laptop for the display of data. A picture of the system can be found in Figure 6.1. The contribution of this thesis was the design, construction, and testing of the RF board.

This chapter will examine the design process and decisions made concerning the RF board, as well as some results to demonstrate how it works. The techniques that were used are

90 91

Figure 6.1: Full working RF spectrum sensing system 92 based on the methods described in [34,35]. A circuit diagram, bill of materials, and system interconnects can be found in AppendixB.

6.2 Design Methodology

When designing a RF receiver the three main trade-offs are: sensitivity, linearity, and selectivity. Sensitivity can be defined as the smallest signal power required to produce a detection. This is called the minimum detectable signal and defines the minimum signal power required for the receiver to register a detection. MDS is given by:

MDS = δkT0BF (6.1) where δ is the ratio of signal to noise power needed to declare a detection, k is Boltzmann’s

−23 constant 1.38 × 10 J/K, T0 is the reference noise temperature set to be 290K by the IEEE [36], B is the detection bandwidth measured in Hz, and F is the noise factor of the receiver, a unit less quantity. The noise factor of the receiver is deﬁned as the ratio of signal to noise (SNR) at the output to the SNR at the input:

SNR F = in (6.2) SNRout

The noise factor is also known as the noise figure when measured in decibels. The lower the minimum detectable signal, the more sensitive the receiver. If the receiver contains an ADC then it is desirable that the signal power be greater than the quantization noise. Gain in the receiver is used to lift the received signal power above the quantization noise. Typically, the noise figure of a receiver will also increase with gain. There is an inherent limitation as too how low the MDS can realistically be lowered. While gain can be increased to raise the signal power above the quantization noise floor, and thus improve sensitivity, if it results in too much of a raise in noise figure then according to equation 6.1 MDS will suffer. 93

Linearity is the property of a system which states that the output of a system should be proportional to its input so that it will satisfy the superposition principle:

f(a1x1 + a2x2) = a1f(x1) + a2f(x2) (6.3)

Where a1 and a2 are scalars, x1 and x2 are the inputs to the system, and f(x) is the output of the system driven by the input x. All real systems are at best approximately linear. The linearity of RF receivers is often described in terms of the 1 dB compression point (P1), and the input-referred second and third order intercept points (IIP2, IIP3). An ampliﬁer will maintain approximately linear operation for low levels of input power. As the input power increases the gain of the ampliﬁer will begin to drop. The 1 dB compression point is the level of input power required to cause the gain to drop by 1 dB. Intermodulation products are caused by the non-linear response of a system. When two tones are input at frequencies

ω1 and ω2, second order intermodulation products will be generated at ω1 ± ω2. Third order intermodulation products will be generated at 2w1 ± w2 and w1 ± 2w2. As the input power increases, the level of intermodulation products will increase as well. For every 1 dB increase in input power, second order intermodulation products increase by 2 dB, and third order intermodulation products increase by 3 dB. The second order intercept point, (IP2), is the level of power required such that the second order intermodulation product is equal in power to the linear response of the system. Similarly, the third order intercept point (IP3) is the level of power required such that the third order intermodulation product is equal in power to the linear response of the system. Both intercept points can either be referred to the input power required (IIP2, IIP3) or to the output power required (OIP2,OIP3). For this thesis the intercept points will be input-referred. Together, the 1 dB compression point and second and third order intercept points help to describe the linearity of a system and give an idea of what the allowable input power is to maintain linear operation. Sensitivity and linearity are trade-oﬀs. Generally, as the gain is increased, the values for P1, IIP2, and IIP3 will decrease. When designing an RF receiver that is made up of multiple stages, it is helpful 94 to keep track of the gain, noise ﬁgure, and IIP3 of the system as a whole. Together these three features help to describe the sensitivity and linearity of the system. This can be done using stage cascade analysis. Equations to calculate these terms are given in AppendixA.

Selectivity refers to passing some frequencies, known as the passband, and rejecting others, known as the stopband. Selectivity can help to balance sensitivity and linearity. If selectivity is reduced, and thus the passband is made smaller, this will decrease the undesired input power to the system and help increase linearity. For example, using a narrow passband the receiver can reject nearby strong signals which may cause intermodulation products. However, if a wide passband is desired as in the case of this thesis, then more power will be allowed into the system and it will be more diﬃcult to achieve acceptable linearity.

The ﬁrst step to designing the RF board is to decide the required gain. In order to do that it is important to understand the three types of noise that are dealt with in a receiver. The three types of noise are internal, external and quantization noise. Internal noise is the noise present in the electrical components of the receiver itself. Examples of internal noise would be thermal and shot noise. External noise originates from the environment and is both natural in the case of Galactic noise and man made such as unwanted electrical emissions as well as undesirable transmissions. The third component, quantization noise, is caused by the error made when converting an analog signal to a digital one. It is desired that the receiver is externally noise limited such that the noise from the environment, which is irreducible, dominates over the internal and quantization noise. The primary purpose of gain in a receiver is to elevate the signal to a level that is suﬃciently higher than the quantization noise. Quantization noise can be modeled as a white stochastic process, and its power is given by Equation 6.4.

−6Nb /10 PQ = Pclip10 (6.4)

Where PQ is the quantization noise power, Pclip is the clipping point of the ADC, and Nb is the number of bits used in the ADC. If the gain is too high then it will cause the ADC to 95

clip. Therefore, the bounds for the useful amount of gain, in linear units, can be given by:

P δ P γ clip r ≥ G ≥ Q q (6.5) Pt Pext

Where Pt = Pext + Ps is the total input power to the receiver, Pext is the external noise

power, Ps is the power of the signal of interest, δr is chosen to prevent the received signal from temporarily going beyond the clipping point of the ADC, and γq is the chosen ratio of external noise power to quantization noise power. For both δr and γq a value of 10 has been chosen for this design. This means that the nominal gain will raise the total input power to about 10 dB below the ADC clipping point, and the external noise power will be ampliﬁed

so that it is at least 10 dB above the quantization noise. The value for Pext can be estimated from the measurements taken in [37]. A helpful table putting the results of [37] into antenna temperate is given in [35]. In order to design the RF board to detect signals that are about

the same power level as Pext it was assumed that the weakest signal would be Ps = Pext, and therefore a value for Pt was used that is 3dB above the expected value of Pext. The RF board was designed to be used with an ADC that had 11.3 eﬀective bits. Eﬀective number of bits describes the quality of an ADC and refers to actual number of bits of precision that can be realistically used. A more useful description of ADC quality may be SINAD, which stands for signal to noise and distortion ratio. In the case of an ADC with full-scale power of +10dBm, this would equate to a SINAD of about 67.8 dBFS, where dB FS means with reference to full scale power. The ADC actually used with this design had a full scale SINAD of 69.8 dBFS. It can be substituted for Nb in equation 6.4. Table 6.1 contains a summary of these values for the board. Table 6.2 shows what the useful minimum and maximum gain would be based on these assumptions.

The value of Pext is taken from the measurements done in [37] and the table put together in [35]. Using these values for the desired gain, the next section presents the architecture used for the RF board. 96

Table 6.1: Design considerations for the RF board.

Parameter Value Deﬁnition

Pclip +10.0 dBm Used FMC150 (2 Vp-p into 50 Ohms)

PQ -57.8 dBm ADC Quantization noise power based on ENOB 11.3

γq +10.0 dBm Desired ratio Pext to PQ

δr -10.0 dBm Maximum acceptable input power relative to Pclip

Table 6.2: Design implications for the RF board.

Frequency Pt Pext Nb Gmin Gr 25 − 80MHz -78.7 dBm -81.7 dBm 11.3 33.9 dB 78.7 dB 116 − 174MHz -93.1 dBm -96.1 dBm 11.3 48.3 dB 93.1 dB

6.3 System Architecture

6.3.1 Signal Flow and Design Decisions

The architecture used for the RF board can be seen in Figure 6.2. The RF board uses a direct sampling receiver architecture. In a direct sampling architecture, the incoming RF signal is directly sampled without any kind of downconversion to either baseband or an intermediate frequency. Any mixing of the signal is done in the digital domain. Generally, the sampling rate must be at least twice as high as the highest frequency present. Therefore, it is important to filter the signal received from the antenna prior to sampling. However, in this design undersampling, or bandpass sampling, is used. Both bands are sampled at 200 Msps. Each band is separately filtered to diminish aliasing. For the lower band this sampling rate is higher than twice the highest frequency which is 80 MHz after filtering. On the other 97

Figure 6.2: System architecture for the RF board hand the highest frequency present in the higher band is 174 MHz. In this case the sampling rate is much lower than twice the highest frequency present. The sampling process will cause the signals in the higher band to be reproduced periodically in the frequency domain with a period equal to the sampling rate. Therefore, the 116-174 MHz signals will be reﬂected into 26-84 MHz. This is allowable because the RF board will separate the incoming band into two signal paths. There are two main purposes of the RF board. First, it ampliﬁes the incoming RF signal so that it is above the quantization noise of the ADC, as previously discussed. Second, it separates the incoming signal into two bands so that they are not aliased into each other.

The RF board handles the two bands simultaneously. In order to split the incoming signal into two bands with a minimal amount of signal power loss, a diplexer is used. The diplexer is two ﬁlters in parallel, one low pass and one high pass, which allow each of the bands of interest to pass through separately. The design of the diplexer can be found in Appendix 98

B. In addition to separating the two bands, it is also important to reject any strong signals that may cause intermodulation products. In the case of receivers operating in the low VHF bands, FM broadcast signals are likely to be the strongest signals present. Therefore, a secondary goal of the diplexer is to reduce the power of incoming FM signals as much as possible before the ﬁrst ampliﬁer. Broadcast FM signals are located in the frequencies between 87.5 and 108 MHz.

The attenuators in both signal chains are voltage controlled and distributed throughout the signal chain. This is so that the attenuation can be spread throughout the chain to trade- off linearity and sensitivity. Attenuators placed at the end of the signal chain will have a much smaller impact on the noise figure, and thus sensitivity, than attenuators placed at the beginning. However, attenuators at the end of the signal chain will not help to improve the linearity of the system. By placing them throughout the chain and making them voltage controlled, it is possible to both vary the gain for different levels of signal power as well as trade-off linearity and sensitivity.

6.3.2 GNI Analysis

The RF board is made up of several stages of amplification, filtering, and attenuation. Each of these stages has its own gain, noise figure, and IIP3. It is helpful to know these values for the system as a whole. Using the methods described in AppendixA, the cascaded values can be calculated. It is also helpful to examine how the values change for differing amounts of gain. The results for the 25-80 MHz band can be found in Tables 6.3 and 6.4. The results for the 116-174 MHz band can be found in Tables 6.5 and 6.6. 99

Table 6.3: GNI analysis for the 25-80 MHz band maximum gain

Part G [dB] F [dB] IIP3 [dB] RF Switch -1 1 48.5 Diplexer -1 1 200 Var. Atten. -2.2 2.2 30 GALI-74 25.1 2.7 12.9 BPF -1 1 200 Var. Atten. -2.2 2.2 30 GALI-74 25.1 2.7 12.9 Var. Atten. -2.2 2.2 30 BPF -1 1 200 GALI-74 25.1 2.7 12.9 Gmax 64.7 6.9 -26.9 100

Table 6.4: GNI analysis for the 25-80 MHz band minimum gain

Part G [dB] F [dB] IIP3 [dB] RF Switch -1 1.0 48.5 Diplexer -1 1.0 200.0 Var. Atten. -2.2 2.2 30.0 GALI-74 25.1 2.7 12.9 BPF -1 1.0 200.0 Var. Atten. -22.2 22.2 30.0 GALI-74 25.1 2.7 12.9 Var. Atten. -14.2 14.2 30.0 BPF -1 1.0 200.0 GALI-74 25.1 2.7 12.9 Gmin 32.7 9.2 2.0 101

Table 6.5: GNI analysis for the 116-174 MHz band maximum gain

Part G [dB] F [dB] IIP3 [dB] RF Switch -1 1.0 48.5 Diplexer -1 1.0 200.0 GALI-74 25.1 2.7 12.9 Var. Atten. -2.2 2.2 30.0 BPF -1 1.0 200.0 ERA-6sm 12.8 4.5 23.8 Var. Atten. -2.2 2.2 30.0 GALI-74 25.1 2.7 12.9 BPF -1 1.0 200.0 GALI-74 25.1 2.7 12.9 Gmax 79.7 4.5 -41.7 102

Table 6.6: GNI analysis for the 116-174 MHz band minimum gain

Part G [dB] F [dB] IIP3 [dB] RF Switch -1 1.0 48.5 Diplexer -1 1.0 200.0 GALI-74 25.1 2.7 12.9 Var. Atten. -22.2 22.2 30.0 BPF -1 1.0 200.0 ERA-6sm 12.8 4.5 23.8 Var. Atten. -14.2 14.2 30.0 GALI-74 25.1 2.7 12.9 BPF -1 1.0 200.0 GALI-74 25.1 2.7 12.9 Gmin 47.7 8.2 -9.9 103

6.4 Results

After constructing the board, a series of tests were performed to verify proper functionality. A picture of the RF board can be seen in Figure 6.3. First, the system transfer function was measured for each of the two bands. In order to measure the transfer function, a white noise source with known power spectral density was placed at the input of the RF board, and the output was measured. Assuming linear operation of the system, the transfer function can be calculated using Equation 6.6.

Y (ω) = X(ω)H(ω) (6.6)

Where Y (ω) refers to the output of the system, X(ω) is the white noise source at the input, and H(ω) is the transfer function. The estimated system transfer function for the 25-80 MHz band can be found in Figure 6.4. The estimated system transfer function for the 116-174 MHz band can be found in 6.5. A spike can be seen in Figure 6.4 at approximately 105 MHz. This is not part of the system transfer function, instead this is RFI picked up in the cable between the spectrum analyzer and the RF board from a local radio station. The transfer function for the low band shows that there is a rejection of out-of-band frequencies of approximately 45 dB. There is a hump between 80 and 120 MHz in the transfer function. This is a result of the elliptical filters that were used in the diplexer, which have sharp cutoff frequencies but large ripples in the stop-band. The transfer function in the high-band also shows a rejection in the stop-band of at least 40 dB for out-of-band frequencies. From the picture it can be seen that there is still a gain of 50 dB at 200 MHz. This means that frequencies above 200 MHz will be aliasing into the 0-100 MHz band when the upper band is in use. However, this is not a problem because only frequencies above 225 MHz will alias into our band of interest (25-80 MHz). The transfer function in the high band has a steep enough cutoff that frequencies above this limit will be low enough. 104 Figure 6.3: Final design for the RF board 105

Measuring the Transfer Function with a Noise Generator 70

30 Transfer Function H(f) [dB]

10 0 20 40 60 80 100 120 140 160 180 200 Frequency [MHz]

Figure 6.4: System transfer function of the RF board for the 25-80 MHz band

The RF board was then used to take some live RF measurements outdoors near the Virginia Tech campus. For the measurements, a monopole antenna was attached to the input of the RF board, and the output of the RF board was connected directly to a spectrum analyzer. The tests were performed in the presence of strong RF signals. Therefore, attenuation was required between the antenna and the RF board input. This raised the noise figure of the board for these measurements. However, expected RF signals can be seen clearly and this indicates that the RF board is operating correctly. The measurement taken in the 25-80 MHz band can be found in Figure 6.6. Between 60 and 66 MHz broadcast television station 3 can be clearly seen. The spike at 46 and 73 MHz should be land mobile radio, and the spike at 28 MHz is in either CB radio or else in the 10m Ham radio band. The results for the the 116-174 MHz band can be seen in Figure 6.7. As can be seen, the noise floor is much higher than that predicted in the GNI analysis. This is due to the attenuation between the antenna and the RF board. The first spike at approximately 133 MHz is aeronautical mobile radio and the second at 146 MHz is also amateur radio. Most of the rest of the spikes are 106

Measuring the Transfer Function with a Noise Generator 80

Transfer Function H(f) [dB] 30

10 0 20 40 60 80 100 120 140 160 180 200 Frequency [MHz]

Figure 6.5: System transfer function of the RF Board for the 116-174 MHz Band land mobile radio, though in the approximately the 162 MHz range is NOAA weather radio.

6.5 Conclusions

In this chapter we designed and built an RF receiver to take wideband measurements in the lower VHF band. The wide bandwidth makes it appropriate for impulsive noise measurements as well as spectrum sensing. The RF board was designed to simultaneously receive two bands without sweeping: 25-80 MHz and 116-174 MHz. This wide bandwidth is excellent for taking impulsive noise measurements so that very narrow pulses can be resolved. For spectrum sensing purposes it allows you to sense a large amout of spectrum continuously so that short term or bursty signals can still be detected. With such large bandwidths interfering signals can be a problem. The strongest signals near the bands of interest are likely to be broadcast FM. Therefore, the ﬁrst ﬁlter in the RF board not only divides the incoming 107

RF Board Calibrated to Antenna Terminals − Field Test 20 dB Attenuation −60

−70

−80

−90

−100

Power Spectral Density [dBm / 100 kHz] −110 Calibrated RF Board MDS−F=25.4 MDS−F=5.4 −120 25 30 35 40 45 50 55 60 65 70 75 80 Frequency [MHz]

Figure 6.6: RF measurements taken in the 25-80 MHz band near the campus of Virginia Tech signal notches out the FM band as well. The board uses variable attenuators in order to maintain linearity while in the presence of strong signals and high sensitivity when strong signals are absent. 108

RF Board Calibrated to Antenna Terminals − Field Test 20 dB Attenuation −60

−70

−80

−90

−100

Power Spectral Density [dBm / 100 kHz] −110 Calibrated RF Board MDS−F=24.4 MDS−F=4.4 −120 120 125 130 135 140 145 150 155 160 165 170 Frequency [MHz]

Figure 6.7: RF measurements taken in the 116-174 MHz band near the campus of Virginia Tech Chapter 7

Conclusions

The problem for this thesis has been to measure impulsive noise in the VHF broadcast digital TV bands and to analyze its eﬀect on the energy detector. This is relevant because of the increasing demand for RF spectrum and the possibility of cognitive radio using the broadcast TV bands on a secondary basis. When using spectrum on a secondary basis, robust signal detection is necessary in order to prevent the primary users from suﬀering interference. Since energy detectors are often the preferred method of spectrum sensing and radios operating in these frequencies are likely to encounter impulsive noise, it is important to analyze their performance in such a scenario.

The contributions of this thesis are taking impulsive noise measurements on the Virginia Tech campus, analyzing the eﬀect of impulsive noise on an energy detector particularly the SNR Wall, examining the eﬀect of spatially correlated impulsive noise on an energy detector, and building an RF board that can be used for spectrum sensing and impulsive noise measurements.

109 110

7.1 Summary

Measurement Campaign

A measurement campaign was conducted on Virginia Tech’s campus to measure impulsive noise in the broadcast digital TV bands. Dual-antenna measurements were taken at various antenna separations to measure the amount of spatial correlation in the noise. It was found that most measurements in the broadcast TV bands contained impulsive noise, and the nature of that noise varied some from measurement to measurement. It was found that for most frequencies and antenna separations below two wavelengths, the correlation of the noise on the two antennas was between 0.3 and 0.8.

Impulsive Noise Model

The measurements from Chapter2 were compared to established models. It was found that the simpliﬁed models of impulsive noise were able to reasonably match the amplitude probability distributions of our measurements.

Eﬀects of Impulsive Noise on the Energy Detector

We extended the work of [11] to apply the SNR Wall concept to energy detectors in the presence of impulsive noise. We found that impulsive noise causes the SNR Wall and the required sensing time to meet a desired performance to increase. In an AWGN channel it is often assumed that the test statistic of an energy detector will follow a normal distribution. We showed that while this is a valid assumption for a channel corrupted by impulsive noise, it takes a much longer sensing interval for the test statistic to approach a normal distribution. This results in performance that is lower than expected when the test statistic is assumed to be normally distributed. 111

Eﬀects of Spatially Correlated Impulsive Noise on the Energy Detector

In this chapter, we analyzed the performance of a multi-antenna energy detector in the presence of correlated impulsive noise and Rayleigh fading. We made use of the Lp-norm statistics from [16], where the performance was analyzed for uncorrelated non-Gaussian noise. Based on our measurements in Chapter2, we suggested a model for correlated impulsive noise in which impulses arrive on all antennas simultaneously and only the impulsive component of the noise is correlated between the antennas. Using this model for correlated impulsive noise, we found that even if the impulsive component of the noise is uncorrelated, if the impulses arrive at the same time on all antennas the total received power will be highly correlated. Since the energy detector only averages the power, adding antennas does not improve the performance much.

Receiver Design

We designed and built an RF receiver that operated simultaneously in two bands: 25-80 MHz and 116- 174 MHz. Such a wide bandwidth makes this a good choice for taking future impulsive noise measurements. Sensing the two bands simultaneously without sweeping makes the RF receiver well suited to spectrum sensing so that short-term signals can still be detected.

7.2 Future Work

A key assumption for analyzing the performance of an energy detector in the presence of impulsive noise was that noise impulses would arrive at all antennas at the same time. An interesting avenue of future work would be to analyze the performance if pulse arrival times were merely correlated instead of exactly the same for diﬀerent antennas.

We were not able to use our RF board to take new impulsive noise measurements due to 112 time constraints. It would be interesting to take very wideband measurements to get a better understanding of the actual make up of the noise impulses for diﬀerent sources as in [13,19]. Appendix A

GNI Analysis

Stage-cascade analysis is a way of combining all the components in receiver chain to describe the linearity and sensitivity of the entire system. It is necessary for these equations that all quantities are in linear units and not decibels. The gain of a receiver chain made up of N components is given by

G = GG1G2 ··· G3, (A.1)

where G is the gain of the entire system and G, G1,...,GN are the gains of the N components. The overall noise ﬁgure of the system can be calculated by

F2 − 1 F3 − 1 FN − 1 F = F1 + + + , (A.2) G1 G1G2 G1G2 ··· GN

where F is the noise ﬁgure, and F1,F2,...FN are the noise ﬁgures of the N components. The input referred third order intercept point is given by

1 G1 G1G2 G1G2 ··· GN−1 IIP3 = + + + ··· + , (A.3) IIP3,1 IIP3,1 IIP3,2 IIP3,N where IIP3 is the input referred third order intercept point of the entire system and

113 114

IIP3,1,IIP3,2, ··· ,IIP3,N are the IIP3 points of the individual components. If the output referred intercept point is desired it is given by

OIP3 = IIP3G (A.4) Appendix B

RF Board Details

The design of the RF board was discussed in Chapter6. The RF board has 7 interconnects between it and other subsystems. There are two RF input ports, labeled RFIN1 and RFIN2, which connect to the antenna and noise source respectively. Both of these ports are Female SMA jacks. The noise source is used for calibration, there is a optical switch which controls which of the two inputs is connected to the rest of the board. In addition to the two RF inputs, there are also two RF outputs. These are labeled RFOUT 1 and 2 respectively, and are MMCX female jacks. These are the outputs for the high and low band signal chains which are input to an ADC. There are two interconnects for power. The first is to input 12V at approximately 1 A and uses a molex connector that is listed in the Bill of Materials. This is connected to the power supply, and is labeled X2. The second power connector outputs 12 V of power at approximately 40-70 mA to power the noise source, is labeled X6 and is a header pin. This output power is controlled by an optical switch which so that the noise source can be turned off when not in use. This final interconnect is 40 shrouded header pins which inputs low voltage CMOS control signals which control the attenuators, and optical switches. This connector is labeled X1. These seven interconnects are summarized in Table B.1.

115 116

Table B.1: Interconnects for the RF board

Function Name Description Antenna Input RFIN 1 SMA Input for the Antenna Noise Source Input RFIN 2 SMA Input for the Noise Source Low Band Output RFOUT 1 MMCX Output for the Low Band High Band Outut RFOUT 2 MMCX Output for the High Band Digital Control X1 40 Shrouded Header Pins for Digital Control Power In X2 Molex Power Connector 12V Power Out X6 Header Pins 12V DC Out

B.1 Diplexer Design

The purpose of the diplexer is to split the incoming RF signal into two seperate paths for further gain and filtering. The diplexer is designed so that the maximum possible power will transfer into each of the two signal paths. The most likely strong out-of-band signals will be broadcast FM which is located in 88-105 MHz. Since these frequencies lie between the two bands of interest, 25-80 and 116-174 MHz, it is also desired that the diplexer should suppress these frequencies to the greatest extent possible. With these two goals in mind, splitting the RF signal into two bands and suppressing the broadcast FM frequencies, the diplexer was designed as a high-pass and low-pass filter connected in parallel. An elliptical filter structure was used since this will give a steep slope to the filter after the cutoff frequency. This was necessary because the pass band ends at 80 MHz and the broadcast FM begins at 88 MHz which only gives 8 MHz for the filters transition region. A diagram of the components can be seen in Figure B.1 as well as the component values in Table B.2. Ideal values were calculated using a software package, and then standard values were chosen from available components. A plot of the expected transfer function of the diplexers can be found in Figure B.2. 117

Table B.2: Values for the diplexer

Name Value L1 120 nH L2 150 nH L3 68 nH L4 56 nH L5 220 nH L6 82 nH L7 220 nH C1 33 pF C2 15 pF C3 22 pF C4 18 pF C5 36 pF C6 47 pF C7 12 pF 118

Figure B.1: Diplexer design for the RF board

B.2 Schematic, Layout and Bill of Materials

The schematics for the RF board are separated into three diﬀerent sheets. The interconnects, power supply, voltage translators, and digital control can all be found in Figure B.3. The RF signal chains for the low and high band can be found in Figures B.4 and B.5 respectively. A four layer PCB was used for the layout. The top layer contained all of the components and can be found in Figure B.6. The bottom layer contained the control lines and can be found in Figure B.7. The middle two layers served as the power and ground.The Bill of Materials can be found in Figures B.8, B.9, and B.10. 119

0 LP − Std Values HP − Std Ideal −10

−20 H(f) −30

−40

−50 0 50 100 150 200 Frequency [MHz]

Figure B.2: Measured transfer function for diplexer ﬁlters connected in parallel 120 Figure B.3: RF board schematic 121 Figure B.4: RF board schematic 122 Figure B.5: RF board schematic 123 Figure B.6: RF board layout 124 Figure B.7: RF board layout 125 DESCRIPTION Inductor 0805 Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Resistor 1/8W 1% 0805 SMD Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Inductor 0805 Ceramic Capacitor 0805 SMD Inductor 0805 Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Tantalum Capacitor 2917 SMD 08052U330JAT2A-ND 478-3352-1-ND 478-1411-1-ND 08052U5R6BAT2A-ND RMCF0805FT10K0CT-ND 08052U120JAT2A-ND 08052U150JAT2A-ND 08052U180JAT2A-ND 495-2037-1-ND 08052U220JAT2A-ND 495-1852-1-ND 08052U390JAT2A-ND 08052U470JAT2A-ND 478-3325-1-ND Digikey DigikeyDigikey 495-1845-1-ND Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey 08055C104JAT2A B82498F1222J 0805YD105KAT2A 08052U5R6BAT2A RMCF0805FT10K0 08052U120JAT2A 08052U150JAT2A 08052U180JAT2A B82498F3180J 08052U220JAT2A B82498F3330J 08052U330JAT2A 08052U390JAT2A 08052U470JAT2A TPSD476K016R0080 Figure B.8: Bill of Materials $0.23 AVX Corporation $0.06 AVX Corportation $0.12 $0.06 8 0.1u $0.10 $0.80 AVX Corporation 7 2.2u $0.93 $6.51 EPCOS Inc. 5 18p $0.06 $0.31 AVX Corporation 2 1u $0.26 $0.52 AVX Corporation 2 0.47u $0.592 5.6p $1.18 AVX Corporation 1 12p 1 15p4 $0.06 18n1 22p $0.06 AVX Corporation 4 33n $0.771 33p $0.06 $3.08 EPCOS Inc. 1 $0.77 36p $0.06 AVX Corporation 1 $0.06 47p $3.08 EPCOS Inc. 2 $0.06 47u $0.06 AVX Corporation $0.06 $0.06 AVX Corporation $3.28 $0.06 AVX Corporation $6.56 AVX Corporation 40 10k $0.02 $0.65 Stackpole Electronics QTY VALUE PRICE sub-total MANUFACTURER MAN# DIST DIST# C3, C6, C9, C17, C23, C27, C33, C40 REFERENCE L1, L31, L32, L33, L34, L35, L36 C10, C38, C46, C50, C51 CIN1, CIN4 CIN2, CIN3 C48, C53 R7, R9, R10, R11, R12, R13, R14, R15, R16, R17, R18, R19, R20, R21, R22, R23, R24, R25, R26, R28, R29, R30, R31, R32, R33, R34, R35, R36, R37, R38, R39, R40, R41, R42, R43, R44, R46, R47, R48, R49 C45 C36 L22, L24, L27, L29 C37 L9, L15, L37, L38 C35 C39 C44 C_TANT2, C_TANT3 126 Optical Relay 9V LDO Voltage Regulator 1W Resistor 2512 SMD 1W Resistor 2512 SMD Inductor 0805 Ceramic Capacitor 0805 SMD Inductor 0805 5% Ceramic Capacitor 0805 SMD Inductor 0805 Thin Film Resistor 0805 Ceramic Capacitor 0805 SMD Inductor 0805 Inductor 0805 Inductor 0805 Inductor 0805 Inductor 0805 Ceramic Capacitor 0805 SMD Ceramic Capacitor 0805 SMD Schottky Diode 20V 1A D041 Header Pins .100' Pitch 80 ct 2.5 V LDO Voltage Regulator TO-92 5V LDO Voltage Regulator MMIC Amplifier 541-53.6AFCT-ND 541-562AFCT-ND 495-2041-1-ND 08052U560JAT2A-ND 495-1856-1-ND 08052U680JAT2A-ND 495-1858-1-ND RMCF0805JT100RCT-ND 08052U101JAT2A-ND 495-3433-1-ND 495-3434-1-ND AISC-0805-R18G-TCT-ND 495-1851-1-ND 495-1853-1-ND 478-6045-1-ND 478-1323-1-ND 1N5817FSCT-ND A34344-40-ND MCP1702-2502E/TO-ND LM2940IMP-5.0CT-ND Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey Digikey DigikeyDigikey CLA230CT-ND Digikey Digikey Digikey Digikey LM2940IMP-9.0CT-ND CRCW251253R6FKEG CRCW2512562RFKEG B82498F3560J 08052U560JAT2A B82498F3680J 08052U680JAT2A B82498F3820J RMCF0805JT100R 08052U101JAT2A B82498F3121J B82498F3151J AISC-0805-R18G-T B82498F3221J B82498F3331J 08055A331FAT2A 08055A391JAT2A CPC1002NTR 1N5817 9-146258-0 MCP1702-2502E/TO LM2940IMP-5.0 LM2940IMP-9.0 Figure B.9: Bill of Materials 6 53.65 56n $0.40 $2.39 Vishay Dale 5 $0.56 68n6 68p $2.80 EPCOS Inc. $0.77 $0.06 $3.85 EPCOS Inc. $0.37 AVX Corporation 1 -- $0.50 $0.50 Fairchild Semiconductor 1 56.2 $0.404 56p $0.40 Vishay Dale $0.061 82n2 100 $0.24 AVX Corporation 4 100p $0.771 120n $0.031 150n $0.77 $0.06 EPCOS Inc. 2 180n $0.06 $0.774 Stackpole Electronics 220n $0.77 $0.24 AVX Corporation 4 330n $0.25 $0.77 EPCOS Inc. $0.77 $0.774 EPCOS Inc. 330p $0.50 Abracon Corporation $0.77 $3.08 EPCOS Inc. $0.49 $3.08 EPCOS Inc. 1 -- $1.96 AVX Corporation 4 $3.90 --1 -- $3.90 Minicircuits1 -- $0.001 -- $3.47 $0.00 Use Breakaways from x4 1 -- $0.52 ERA-6SM $3.47 TE Connectivity $2.42 $0.52 Microchip Technology $2.42 $2.42 National Semiconductor Minicircuits $2.42 National Semiconductor ERA-6SM 2 -- $1.71 $3.42 Clare 13 390p $0.17 $2.21 AVX Corporation R1, R2, R3, R4, R6, R8 L5, L21, L25, L26, L30 L4, L10, L14, L16, L20 C2, C5, C19, C21, C30, C32 D1 R5 C47, C49, C52, C54 L7 R27, R45 C16, C20, C26, C31 L2 L3 L23, L28 L6, L8, L12, L18 L11, L13, L17, L19 C18, C22, C29, C34 C1, C4, C7, C8, C13, C14, C15, C24, C25, C41, C42, C55, C56 U6 X3, X6, X7, X8 X4, X5 LVR1 LVR3 LVR2 SSR1, SSR2 127 MMIC Amplifier 6CKT Header IC RF Switch SMA Connector Logic Level Translator Digital Step Attenuator Shrouded Connector MMCX Connector Gali-74 WM4260-ND 568-1206-5-ND WM5544-ND 296-23280-1-ND TOAT-4815 OR950-ND Minicircuits Digikey Digikey Digikey Gali-74 39310060 SA630D/01,112 SN74LVC8T245DWR Figure B.10: Bill of Materials $429.22 6 -- $2.35 $14.10 Minicircuits 2 --1 --2 -- $3.602 --5 $6.02 -- $7.20 Molex Inc.1 $3.78 -- $1.26 $6.02 NXP Semiconductors $7.56 $64.95 Molex Inc. $2.52 Texas Instruments $2.03 73415-1471 $324.75 Minicircuits $2.03 Omron Electronics 73391-0070 Digikey TOAT-4815 XG4C-4031 WM5557-ND Digikey Minicircuits Digikey 1 -- $1.61 $1.61 Molex Inc. Q1, Q2, Q3, Q4, Q5, Q6 RFOUT1, RFOUT2 SW1 RFIN1, RFIN2 LV1, LV2 U1, U2, U3, U4, U5 X1 X2 Appendix C

Measurement Results

In this Appendix, there are APD plots of all the data from the measurement campaign from all four measurement locations. Each of the measurements was given a numerical index number. An example of the measurement that was done to ﬁnd the 1 dB compression point can be found in Figure C.1. The 1 dB compression point is located where the measured power is 1 dB lower than the input power. Tables C.1-C.7 include all of the measurements with the index number, average power in dBm, time of day, date, and amount of attenuation used for each of the measurements. The APD plots can be found in Figures C.2-C.49. At the end of the appendix in Figures C.50 and C.51 there are also plots of the data for the average power vs. frequency at diﬀerent times of day.

128 129

−29 Actual Measurements Linear Performance −30

−31

−32

−33 Measured Power [dBm]

−34

−35

−36 −36 −35 −34 −33 −32 −31 −30 −29 Input Power [dBm]

Figure C.1: Measurements for the 1 dB compression point using 15 dB of attenuation

Table C.1: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00024 57 MHz 38.1 dBm 5/25 12:04 PM 10 dB Cubicle 00030 57 MHz 38.2 dBm 5/25 12:15 PM 10 dB Cubicle 00054 57 MHz 37.8 dBm 5/31 2:39 PM 10 dB Cubicle 00102 57 MHz 34.0 dBm 6/1 8:53 AM 10 dB Cubicle 00025 63 MHz 38.6 dBm 5/25 12:06 PM 10 dB Cubicle 00034 63 MHz 38.0 dBm 5/25 12:34 PM 10 dB Cubicle 00055 63 MHz 50.6 dBm 5/31 2:40 PM 10 dB Cubicle 00103 63 MHz 31.9 dBm 6/1 8:54 AM 10 dB Cubicle 130

Table C.2: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00026 69 MHz 33.9 dBm 5/25 12:08 PM 5 dB Cubicle 00031 69 MHz 34.6 dBm 5/25 12:37 PM 5 dB Cubicle 00056 69 MHz 31.3 dBm 5/31 2:41 PM 10 dB Cubicle 00104 69 MHz 21.9 dBm 6/1 8:56 AM 5 dB Cubicle 00023 79 MHz 28.8 dBm 5/25 12:00 PM 5 dB Cubicle 00027 79 MHz 28.2 dBm 5/25 12:10 PM 5 dB Cubicle 00029 79 MHz 28.3 dBm 5/25 12:13 PM 5 dB Cubicle 00032 79 MHz 28.3 dBm 5/25 12:40 PM 5 dB Cubicle 00057 79 MHz 34.2 dBm 5/31 2:42 PM 10 dB Cubicle 00105 79 MHz 31.2 dBm 6/1 8:57 AM 5 dB Cubicle 00028 85 MHz 27.0 dBm 5/25 12:11 PM 5 dB Cubicle 00033 85 MHz 26.9 dBm 5/25 12:43 PM 5 dB Cubicle 00058 85 MHz 32.4 dBm 5/31 2:43 PM 10 dB Cubicle 00106 85 MHz 33.1 dBm 6/1 8:57 AM 5 dB Cubicle 00035 177 MHz 25.0 dBm 5/25 12:55 PM 0 dB Cubicle 00059 177 MHz 39.8 dBm 5/31 2:54 PM 20 dB Cubicle 00107 177 MHz 25.8 dBm 6/1 8:58 AM 5 dB Cubicle 00036 183 MHz 23.1 dBm 5/25 12:56 PM 0 dB Cubicle 00060 183 MHz 40.1 dBm 5/31 2:55 PM 20 dB Cubicle 00061 183 MHz 42.3 dBm 5/31 2:56 PM 20 dB Cubicle 00108 183 MHz 26.4 dBm 6/1 9:02 AM 5 dB Cubicle 00037 189 MHz 31.4 dBm 5/25 12:57 PM 0 dB Cubicle 00062 189 MHz 41.0 dBm 5/31 2:57 PM 20 dB Cubicle 00109 189 MHz 37.7 dBm 6/1 9:07 AM 5 dB Cubicle 131

Table C.3: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00038 195 MHz 32.0 dBm 5/25 12:58 PM 0 dB Cubicle 00063 195 MHz 45.1 dBm 5/31 2:58 PM 20 dB Cubicle 00110 195 MHz 37.7 dBm 6/1 9:08 AM 5 dB Cubicle 00039 201 MHz 27.5 dBm 5/25 12:59 PM 0 dB Cubicle 00064 201 MHz 41.2 dBm 5/31 2:58 PM 20 dB Cubicle 00111 201 MHz 27.6 dBm 6/1 9:10 AM 5 dB Cubicle 00040 207 MHz 26.0 dBm 5/25 1:00 PM 0 dB Cubicle 00065 207 MHz 39.9 dBm 5/31 3:00 PM 20 dB Cubicle 00112 207 MHz 27.9 dBm 6/1 9:11 AM 5 dB Cubicle 00041 213 MHz 27.1 dBm 5/25 1:01 PM 0 dB Cubicle 00066 213 MHz 40.1 dBm 5/31 3:01 PM 20 dB Cubicle 00113 213 MHz 28.1 dBm 6/1 9:13 AM 5 dB Cubicle 00043 57 MHz 29.3 dBm 5/25 1:22 PM 10 dB Elevator 00127 57 MHz 37.8 dBm 6/1 10:03 AM 10 dB Elevator 00044 63 MHz 23.7 dBm 5/25 1:23 PM 10 dB Elevator 00128 63 MHz 29.6 dBm 6/1 10:04 AM 10 dB Elevator 00045 69 MHz 23.1 dBm 5/25 1:24 PM 10 dB Elevator 00129 69 MHz 27.6 dBm 6/1 10:06 AM 10 dB Elevator 00042 79 MHz 33.5 dBm 5/25 1:21 PM 10 dB Elevator 00046 79 MHz 31.2 dBm 5/25 1:25 PM 10 dB Elevator 00130 79 MHz 37.0 dBm 6/1 10:07 AM 10 dB Elevator 00047 85 MHz 38.9 dBm 5/25 1:26 PM 10 dB Elevator 00131 85 MHz 42.4 dBm 6/1 10:08 AM 10 dB Elevator 132

Table C.4: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00048 177 MHz 25.3 dBm 5/25 1:28 PM 10 dB Elevator 00132 177 MHz 28.6 dBm 6/1 10:09 AM 10 dB Elevator 00049 183 MHz 25.4 dBm 5/25 1:29 PM 10 dB Elevator 00133 183 MHz 27.6 dBm 6/1 10:11 AM 10 dB Elevator 00050 189 MHz 25.0 dBm 5/25 1:33 PM 10 dB Elevator 00134 189 MHz 26.8 dBm 6/1 10:14 AM 10 dB Elevator 00051 195 MHz 23.6 dBm 5/25 1:36 PM 5 dB Elevator 00135 195 MHz 26.2 dBm 6/1 10:16 AM 10 dB Elevator 00052 201 MHz 21.3 dBm 5/25 1:38 PM 5 dB Elevator 00136 201 MHz 25.3 dBm 6/1 10:18 AM 10 dB Elevator 00053 207 MHz 20.6 dBm 5/25 1:41 PM 5 dB Elevator 00137 207 MHz 24.9 dBm 6/1 10:19 AM 10 dB Elevator 00054 213 MHz 21.4 dBm 5/25 1:43 PM 5 dB Elevator 00138 213 MHz 25.4 dBm 6/1 10:20 AM 10 dB Elevator 00114 57 MHz 29.2 dBm 6/1 9:22 AM 5 dB Refridgerator 00290 57 MHz 36.0 dBm 6/21 2:58 AM 10 dB Refridgerator 00115 63 MHz 19.5 dBm 6/1 9:24 AM 5 dB Refridgerator 00291 63 MHz 26.1 dBm 6/21 3:03 AM 10 dB Refridgerator 00116 69 MHz 25.8 dBm 6/1 9:25 AM 5 dB Refridgerator 00292 69 MHz 29.5 dBm 6/21 3:10 AM 5 dB Refridgerator 00003 79 MHz 26.7 dBm 5/23 10:43 AM 5 dB Refridgerator 00117 79 MHz 32.8 dBm 6/1 9:26 AM 5 dB Refridgerator 00293 79 MHz 30.2 dBm 6/21 3:14 AM 5 dB Refridgerator 133

Table C.5: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00004 85 MHz 30.0 dBm 5/23 11:00 AM 5 dB Refridgerator 00118 85 MHz 34.6 dBm 6/1 9:28 AM 5 dB Refridgerator 00294 85 MHz 24.9 dBm 6/21 3:18 AM 5 dB Refridgerator 00005 177 MHz 23.3 dBm 5/23 1:29 PM 5 dB Refridgerator 00119 177 MHz 26.9 dBm 6/1 9:30 AM 10 dB Refridgerator 00295 177 MHz 27.6 dBm 6/21 3:23 AM 10 dB Refridgerator 00006 183 MHz 25.0 dBm 5/23 1:37 PM 5 dB Refridgerator 00120 183 MHz 24.5 dBm 6/1 9:32 AM 5 dB Refridgerator 00296 183 MHz 30.4 dBm 6/21 3:30 AM 0 dB Refridgerator 00007 189 MHz 24.5 dBm 5/23 1:45 PM 5 dB Refridgerator 00121 189 MHz 27.0 dBm 6/1 9:34 AM 10 dB Refridgerator 00297 189 MHz 28.5 dBm 6/21 3:34 AM 5 dB Refridgerator 00008 195 MHz 25.7 dBm 5/23 1:52 PM 5 dB Refridgerator 00122 195 MHz 27.7 dBm 6/1 9:39 AM 5 dB Refridgerator 00123 195 MHz 27.6 dBm 6/1 9:40 AM 5 dB Refridgerator 00298 195 MHz 28.9 dBm 6/21 3:38 AM 0 dB Refridgerator 00009 201 MHz 29.0 dBm 5/23 2:10 PM 5 dB Refridgerator 00010 201 MHz 28.9 dBm 5/23 2:32 PM 5 dB Refridgerator 00124 201 MHz 24.5 dBm 6/1 9:42 AM 5 dB Refridgerator 00299 201 MHz 25.9 dBm 6/21 3:42 AM 0 dB Refridgerator 00011 207 MHz 26.7 dBm 5/23 2:39 PM 5 dB Refridgerator 00125 207 MHz 24.4 dBm 6/1 9:43 AM 5 dB Refridgerator 00300 207 MHz 28.4 dBm 6/21 3:46 AM 0 dB Refridgerator 134

Table C.6: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00012 213 MHz 24.2 dBm 5/23 2:58 PM 15 dB Refridgerator 00126 213 MHz 22.4 dBm 6/1 9:45 AM 5 dB Refridgerator 00301 213 MHz 26.0 dBm 6/21 3:50 AM 5 dB Refridgerator 00067 57 MHz 23.1 dBm 5/31 3:18 PM 0 dB Construction Site 00079 57 MHz 19.4 dBm 5/31 3:40 PM 0 dB Construction Site 00091 57 MHz 20.0 dBm 5/31 3:57 PM 0 dB Construction Site 00068 63 MHz 18.4 dBm 5/31 3:18 PM 0 dB Construction Site 00080 63 MHz 16.4 dBm 5/31 3:41 PM 0 dB Construction Site 00092 63 MHz 16.8 dBm 5/31 3:58 PM 0 dB Construction Site 00069 69 MHz 17.7 dBm 5/31 3:19 PM 5 dB Construction Site 00081 69 MHz 10.6 dBm 5/31 3:42 PM 5 dB Construction Site 00093 69 MHz 10.9 dBm 5/31 3:59 PM 0 dB Construction Site 00070 79 MHz 24.1 dBm 5/31 3:20 PM 5 dB Construction Site 00082 79 MHz 15.6 dBm 5/31 3:43 PM 5 dB Construction Site 00094 79 MHz 15.3 dBm 5/31 4:00 PM 5 dB Construction Site 00071 85 MHz 56.7 dBm 5/31 3:24 PM 5 dB Construction Site 00083 85 MHz 11.0 dBm 5/31 3:47 PM 5 dB Construction Site 00095 85 MHz 10.9 dBm 5/31 4:03 PM 5 dB Construction Site 00072 177 MHz 9.7 dBm 5/31 3:27 PM 0 dB Construction Site 00084 177 MHz 9.5 dBm 5/31 3:48 PM 0 dB Construction Site 00096 177 MHz 9.4 dBm 5/31 4:03 PM 0 dB Construction Site 00073 183 MHz 10.2 dBm 5/31 3:28 PM 0 dB Construction Site 00085 183 MHz 10.3 dBm 5/31 3:49 PM 0 dB Construction Site 00097 183 MHz 10.8 dBm 5/31 4:04 PM 0 dB Construction Site 135

Table C.7: Single-antenna measurements

Name Frequency wo Day Time Att. Location 00074 189 MHz 11.1 dBm 5/31 3:28 PM 0 dB Construction Site 00086 189 MHz 10.5 dBm 5/31 3:50 PM 0 dB Construction Site 00098 189 MHz 10.5 dBm 5/31 4:05 PM 0 dB Construction Site 00075 195 MHz 9.9 dBm 5/31 3:37 PM 5 dB Construction Site 00087 195 MHz 11.6 dBm 5/31 3:51 PM 5 dB Construction Site 00099 195 MHz 12.6 dBm 5/31 4:09 PM 5 dB Construction Site 00076 201 MHz 14.7 dBm 5/31 3:38 PM 0 dB Construction Site 00088 201 MHz 15.3 dBm 5/31 3:52 PM 0 dB Construction Site 00100 201 MHz 14.5 dBm 5/31 4:10 PM 0 dB Construction Site 00077 207 MHz 13.9 dBm 5/31 3:38 PM 0 dB Construction Site 00089 207 MHz 11.7 dBm 5/31 3:53 PM 0 dB Construction Site 00101 207 MHz 12.5 dBm 5/31 4:10 PM 0 dB Construction Site 00078 213 MHz 12.6 dBm 5/31 3:39 PM 5 dB Construction Site 00090 213 MHz 10.5 dBm 5/31 3:54 PM 5 dB Construction Site 00102 213 MHz 12.3 dBm 5/31 4:11 PM 0 dB Construction Site 136

C.1 Near a Cubicle

In this section all of the APD plots for the measurements near a cubicle can be found. Likely sources of impulsive noise at this location would include flurorescent lights and computer equipment. The measurements taken in the 63 MHz band are unique in that during most of the measurements there was a strong carrier in the band. This carrier casued the APD plots to flatten out. It can also be seen that the average power in the band of interest varies from measurement to measurement. This is likely because of the changing nature of the RF enviornment as measurements were taken at different times on different days.

60 00024 50 00030 00054 40 00102 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.2: Measurements near a cubicle with a center frequency of 57 MHz and measurement bandwidth B = 5 MHz 137

60 00025 50 00034 00055 40 00103 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.3: Measurements near a cubicle with a center frequency of 63 MHz and measurement bandwidth B = 5 MHz

60 00026 50 00031 00056 40 00104 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.4: Measurements near a cubicle with a center frequency of 69 MHz and measurement bandwidth B = 5 MHz 138

60 00023 50 00027 00029 40 00032 00057 00105 B

o 30 50 Ohm Load

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.5: Measurements near a cubicle with a center frequency of 79 MHz and measurement bandwidth B = 5 MHz

50 00028 40 00033 00058 00106 30 50 Ohm Load B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.6: Measurements near a cubicle with a center frequency of 85 MHz and measurement bandwidth B = 5 MHz 139

60 00035 50 00059 00107 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.7: Measurements near a cubicle with a center frequency of 177 MHz and measurement bandwidth B = 5 MHz

80 00036 70 00060 60 00061 00108 50 50 Ohm Load B o 40

20 dB above kT 10

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.8: Measurements near a cubicle with a center frequency of 183 MHz and measurement bandwidth B = 5 MHz 140

60 00037 50 00062 00109 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.9: Measurements near a cubicle with a center frequency of 189 MHz and measurement bandwidth B = 5 MHz

80 00038 70 00063 60 00110 50 Ohm Load 50 B o 40

20 dB above kT 10

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.10: Measurements near a cubicle with a center frequency of 195 MHz and measurement bandwidth B = 5 MHz 141

60 00039 50 00064 00111 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.11: Measurements near a cubicle with a center frequency of 201 MHz and measurement bandwidth B = 5 MHz

60 00040 50 00065 00112 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.12: Measurements near a cubicle with a center frequency of 207 MHz and measurement bandwidth B = 5 MHz 142

60 00041 50 00066 00113 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.13: Measurements near a cubicle with a center frequency of 213 Mhz and measurement bandwidth B = 5 MHz 143

C.2 Near an Elevator

In this sectin APD plots of all the measurements taken near an elevator can be found. Elevators have been shown to be sources of impulsive noise. Most of our measurements in this area also had varying amounts of impulsive noise. It was found that the average power of the noise measurements in the hallway where the elevator was located was much stronger than that inside the oﬃce where the cubicle measurements were taken. While it is not clear why the noise would be stronger in the hallway, there was also a greater prevalance of strong signals in this area when the measurements were taken.

50 00043 40 00127 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.14: Measurements near and elevator with a center frequency of 57 MH and measurement bandwidth B = 5 MHz 144

40 00044 00128 30 50 Ohm Load

20 B o

dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.15: Measurements near and elevator with a center frequency of 63 MH and measurement bandwidth B = 5 MHz

50 00045 40 00129 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.16: Measurements near and elevator with a center frequency of 69 MH and measurement bandwidth B = 5 MHz 145

60 00042 50 00046 00130 40 50 Ohm Load B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.17: Measurements near and elevator with a center frequency of 79 MH and measurement bandwidth B = 5 MHz

60 00047 50 00131 50 Ohm Load 40 B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.18: Measurements near and elevator with a center frequency of 85 MH and measurement bandwidth B = 5 MHz 146

50 00048 40 00132 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.19: Measurements near and elevator with a center frequency of 177 MH and measurement bandwidth B = 5 MHz

50 00049 40 00133 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.20: Measurements near and elevator with a center frequency of 183 MH and measurement bandwidth B = 5 MHz 147

50 00050 40 00134 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.21: Measurements near and elevator with a center frequency of 189 MH and measurement bandwidth B = 5 MHz

50 00051 40 00135 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.22: Measurements near and elevator with a center frequency of 195 MH and measurement bandwidth B = 5 MHz 148

40 00052 00136 30 50 Ohm Load

20 B o

dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.23: Measurements near and elevator with a center frequency of 201 MH and measurement bandwidth B = 5 MHz

40 00053 00137 30 50 Ohm Load

20 B o

dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.24: Measurements near and elevator with a center frequency of 207 MH and measurement bandwidth B = 5 MHz 149

40 00054 00138 30 50 Ohm Load

20 B o

dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.25: Measurements near and elevator with a center frequency of 213 MH and measurement bandwidth B = 5 MHz 150

C.3 Near a Refrigerator

This section has all of the APD plots for the measurements taken near a microwave and refrigerator. We made certain that the microwave was on while the measrurement were being taken. Many of the samples of noise here appear to include coherent signals in our band of interest. The coherent signals in our band of interest can be identiﬁed by the ﬂattening out of the slope in the higher probability amplitudes such as measurement 00297 taken in the 189 MHz band. It should be noted that this measurement site was also near a lab with electrical test equipment such as function generators, spectrum analyzers, oscilliscopes etc. which frequently have electromagnetic emissions.

60 00114 50 00290 50 Ohm Load 40 B

o 30

10 dB above kT

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.26: Measurements near a Microwave with a center frequency of 57 MH and measurement bandwidth B = 5 MHz 151

40 00115 00291 30 50 Ohm Load

20 B o

dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.27: Measurements near a Microwave with a center frequency of 63 MH and measurement bandwidth B = 5 MHz

50 00116 40 00292 50 Ohm Load

30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.28: Measurements near a Microwave with a center frequency of 69 MH and measurement bandwidth B = 5 MHz 152

50 00003 40 00117 00293 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.29: Measurements near a Microwave with a center frequency of 79 MH and measurement bandwidth B = 5 MHz

50 00004 40 00118 00294 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.30: Measurements near a Microwave with a center frequency of 85 MH and measurement bandwidth B = 5 MHz 153

50 00005 40 00119 00295 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.31: Measurements near a Microwave with a center frequency of 177 MH and measurement bandwidth B = 5 MHz

50 00006 40 00120 00296 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.32: Measurements near a Microwave with a center frequency of 183 MH and measurement bandwidth B = 5 MHz 154

50 00007 40 00121 00297 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.33: Measurements near a Microwave with a center frequency of 189 MH and measurement bandwidth B = 5 MHz

50 00008 40 00122 00123 00298 30 50 Ohm Load B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.34: Measurements near a Microwave with a center frequency of 195 MH and measurement bandwidth B = 5 MHz 155

50 00009 40 00010 00124 00299 30 50 Ohm Load B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.35: Measurements near a Microwave with a center frequency of 201 MH and measurement bandwidth B = 5 MHz

50 00011 40 00125 00300 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.36: Measurements near a Microwave with a center frequency of 207 MH and measurement bandwidth B = 5 MHz 156

50 00012 40 00126 00301 50 Ohm Load 30 B o 20

10 dB above kT 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.37: Measurements near a Microwave with a center frequency of 213 MH and measurement bandwidth B = 5 MHz 157

C.4 Outdoors Near Construction

This section includes all of the outdoor impulsive noise measurements. A few observations can be made. First, the average noise power is much lower in the outdoor measurements than indoors. Second, in the measurements in the 79 MHz band a coherent signal appears to be interfering with the measurements 00082 and 00094. Also, there is impulsive noise in virtually every mesurement made. The outdoor impulsisve noise appears to occur much more rarely than in the indoor case, in that the break from the complex-Gaussian distribution happens at about the 0.1% mark. Also, there is less deviation from measurement to measurement in noise power for the outdoor measurements. This may be because all of the outdoor measurements were taken in a single setting, unlike the indoor ones which were spread out over multiple days.

CFreq = 57 MHz 50 00067 40 00079 00091 50 Ohm Load 30

10 dB above kTB

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.38: Outdoor measurements taken near a construction site with a center frequency of 57 MHz 158

CFreq = 63 MHz 35 00068 30 00080 25 00092 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.39: Outdoor measurements taken near a construction site with a center frequency of 63 MHz

CFreq = 69 MHz 35 00069 30 00081 25 00093 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.40: Outdoor measurements taken near a construction site with a center frequency of 69 MHz 159

CFreq = 79 MHz 35 00070 30 00082 25 00094 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.41: Outdoor measurements taken near a construction site with a center frequency of 79 MHz

CFreq = 85 MHz 50 00071 40 00083 00095 50 Ohm Load 30

10 dB above kTB

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.42: Outdoor measurements taken near a construction site with a center frequency of 85 MHz 160

CFreq = 177 MHz 30 00072 25 00084 00096 20 50 Ohm Load

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.43: Outdoor measurements taken near a construction site with a center frequency of 177 MHz

CFreq = 183 MHz 35 00073 30 00085 25 00097 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.44: Outdoor measurements taken near a construction site with a center frequency of 183 MHz 161

CFreq = 189 MHz 30 00074 25 00086 00098 20 50 Ohm Load

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.45: Outdoor measurements taken near a construction site with a center frequency of 189 MHz

CFreq = 195 MHz 35 00075 30 00087 25 00099 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.46: Outdoor measurements taken near a construction site with a center frequency of 195 MHz 162

CFreq = 201 MHz 35 00076 30 00088 25 00100 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.47: Outdoor measurements taken near a construction site with a center frequency of 201 MHz

CFreq = 207 MHz 40 00077 00089 30 00101 50 Ohm Load

10 dB above kTB 0

−10

−20 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.48: Outdoor measurements taken near a construction site with a center frequency of 207 MHz 163

CFreq = 213 MHz 35 00078 30 00090 25 00102 50 Ohm Load 20

5 dB above kTB 0

−5

−10

−15 0.0010.1 1 5 10 20 37 50 70 80 90 95 99 percent exceeding ordinate

Figure C.49: Outdoor measurements taken near a construction site with a center frequency of 213 MHz 164

C.5 Average Power vs. Frequency for Diﬀerent Times of Day

In this section the average noise power in each of the broadcast TV channels in the low VHF band is plottted vs. time of day. The goal of these measurements was to get an idea of how the noise power would change throughout the day. Both the average and maximum power were measured for approximately 2 minutes at a time for each channel.

Average Power in TV Broadcast Channels −55 57 MHz 63 MHz −60 69 MHz 79 MHz 85 MHz −65

−70

−75

−80 Power in 5 MHz Channel [dBm/5MHz]

−85 8 10 12 2 PM 4 6 8 Time of Day

Figure C.50: Average Power in Broadcast TV Channels for VHF Low 165

Max Power in TV Broadcast Channels −55 57 63 −60 69 79 85 −65

−70

−75

−80 Power in 5 MHz Channel [dBm/5MHz]

−85 8 10 12 2 PM 4 6 8 Time of Day

Figure C.51: Max Power in Broadcast TV Channels for VHF Low Bibliography

[1] Federal Communications Commission, “Spectrum policy task force,” Rep. ET Docket no. 02-135, Nov. 2002.

[2] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201 – 220, Feb. 2005.

[3] R. Chiang, G. Rowe, and K. Sowerby, “A quantitative analysis of spectral occupancy measurements for cognitive radio,” in Vehicular Technology Conference, 2007. VTC2007-Spring. IEEE 65th, Apr. 2007, pp. 3016 –3020.

[4] Shared Spectrum Company Ltd. Spectrum Occupancy Measurements: Chicago, Illinois, November 16-18. http://www.sharedspectrum.com/wp-content/uploads/NSF Chicago 2005-11 measurements v12.pdf. Available [ONLINE].

[5] S. Ellingson, “Spectral occupancy at vhf: implications for frequency-agile cognitive radios,” in Vehicular Technology Conference, 2005. VTC-2005-Fall. 2005 IEEE 62nd, vol. 2, Sep. 2005, pp. 1379 – 1382.

[6] T. Taher, R. Bacchus, K. Zdunek, and D. Roberson, “Long-term spectral occupancy ﬁndings in chicago,” in New Frontiers in Dynamic Spectrum Access Networks (DyS- PAN), 2011 IEEE Symposium on, May 2011, pp. 100 –107.

[7] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proceedings of the IEEE, vol. 55, no. 4, pp. 523–531, Apr. 1967.

166 167

[8] T. Y¨ucekand H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” Commun. Surveys Tuts., vol. 11, no. 1, pp. 116 –130, 2009.

[9] D. Cabric, S. Mishra, and R. Brodersen, “Implementation issues in spectrum sensing for cognitive radios,” in Signals, Systems and Computers, 2004. Conference Record of the Thirty-Eighth Asilomar Conference on, vol. 1, Nov. 2004, pp. 772 – 776 Vol.1.

[10] A. Sonnenschein and P. Fishman, “Radiometric detection of spread-spectrum signals in noise of uncertain power,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 3, pp. 654–660, Jul. 1992.

[11] R. Tandra and A. Sahai, “SNR Walls for Signal Detection,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008.

[12] K. Blackard, “Measurements and models of radio frequency impulsive noise inside build- ings,” Master’s thesis, Va. Poly. Inst. and State Univ., Blacksburg, VA, 1991.

[13] M. Sanchez, L. de Haro, M. Ramon, A. Mansilla, C. Ortega, and D. Oliver, “Impulsive noise measurements and characterization in a uhf digital tv channel,” IEEE Trans. Electromagn. Compat., vol. 41, no. 2, pp. 124–136, May 1999.

[14] Mass Consultants Limited, “Man-Made Noise Measurement Programme Final Report,” Issue 2, page 58, September 2003.

[15] Achatz R.J.,Lo Y., Papazian P.B., Dalke R.A., Huﬀord G.A., “Man-Made Noise in the 136 to 138-MHz VHF Meteorological Satellite Band,” NTIA Report 98-355, 1998.

[16] F. Moghimi, A. Nasri, and R. Schober, “Adaptive Lp-Norm Spectrum Sensing for Cog- nitive Radio Networks,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1934–1945, Jul. 2011.

[17] S. Shellhammer, A. Sadek, and W. Zhang, “Technical challenges for cognitive radio in the tv white space spectrum,” in Information Theory and Applications Workshop, 2009, Feb. 2009, pp. 323 – 333. 168

[18] Federal Communications Commission, “Second Report and Order and Memorandum Opinion and Order the Matter of Unlicensed Operation in the TV Broadcast Bands, Additional Spectrum for Unlicensed Devices Below 900 MHz and in the 3 GHz Band,” Document 08-260, Nov. 14 2008.

[19] K. Blackard, T. Rappaport, and C. Bostian, “Measurements and models of radio frequency impulsive noise for indoor wireless communications,” IEEE J. Sel. Areas Com- mun., vol. 11, no. 7, pp. 991–1001, Sep. 1993.

[20] A. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineer- ing, 3rd ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2008.

[21] S. M. Kay, Intuitive probability and random processes using MATLAB. New York, NY: Springer, 2006.

[22] D. Middleton, “Statistical-Physical Models of Electromagnetic Interference,” IEEE Trans. Electromagn. Compat., vol. 19, pp. 106–127, Aug. 1977.

[23] J.W. Adams, W.D. Bensema, and M. Kanda, “Electromagnetic noise, Grace mine,” National Bureau of Standards, Boulder, CO, Report NBSIR 74-388, 1974.

[24] A.D. Spaulding and L.R Espeland, “Man-made noise characteristics on and in the vicin- ity of military and other government installations,” U.S. Dep. Commerce, Boulder, CO, Oﬃce Telecommun. Tech. Memo OT-TM-48, 1971.

[25] L. R. Espeland and A. D. Spaulding, “Amplitude and time statistics for atmospheric radio noise,” U.S. Dep. Commerce, Boulder, CO, ESSA Tech. Memo. ERL-TM-ITS 250, 1970.

[26] P. Delaney, “Signal detection in multivariate class-a interference,” IEEE Trans. Com- mun., vol. 43, no. 234, pp. 365 –373, Feb 1995.

[27] T. Cover and J. Thomas, Elements of Information Theory, 2nd ed. Hoboken, New Jersey: John Wiles and Sons, Inc., 2006. 169

[28] F. F. Digham, M.-S. Alouini, and M. K. Simon, “On the Energy Detection of Unknown Signals Over Fading Channels,” IEEE Trans. Commun., vol. 55, no. 1, pp. 21–24, Jan. 2007.

[29] V. Kostylev, “Energy detection of a signal with random amplitude,” in Communications, 2002. ICC 2002. IEEE International Conference on, vol. 3, 2002, pp. 1606–1610.

[30] S. M. Kay, ”Fundamentals of Statistical Signal Processing: Detection Theory”. Upper Saddle River, New Jersey: Prentice Hall, 1998, vol. 2.

[31] P. Huber, “A robust version of the probability ratio test,” Ann. Math. Statist., vol. 36, pp. 1753–1758, 1965.

[32] J. Proakis, Digital Communications, 4th ed. New York, NY: McGraw-Hill, 2001.

[33] R. LaMaire and M. Zorzi, “Eﬀect of correlation in diversity systems with rayleigh fading, shadowing, and power capture,” ”IEEE J. Sel. Areas Commun.”, vol. 14, no. 3, pp. 449 –460, Apr. 1996.

[34] D. Taylor, “Design of ultrawideband digitizing receivers for the vhf low band,” Master’s thesis, Va. Poly. Inst. and State Univ., Blacksburg, VA, 2006.

[35] M. Hasan, “New concepts in front end design for receivers with large, multiband tuning ranges,” Ph.D. dissertation, Va. Poly. Inst. and State Univ., Blacksburg, VA, 2009.

[36] “Description of the noise performance of ampliﬁers and receiving systems,” Proc. IEEE, vol. 51, no. 3, pp. 436 – 442, Mar. 1963.

[37] International Telecommunication Union, “Radio noise,” Recommendation ITU-R P.372- 8, 2003.