A Cognitive Phy-Mac Cooperative Protocol for Low-Power Short-Range Wireless Ad-Hoc Networks Using Uwb Ppm Radios

A COGNITIVE PHY-MAC COOPERATIVE PROTOCOL FOR LOW-POWER SHORT-RANGE WIRELESS AD-HOC NETWORKS USING UWB PPM RADIOS

By JOSE M. ALMODOVAR-FARIA

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 c 2014 Jose M. Almodovar-Faria

2 To my parents, Mabel and Joe, for their support, encouragement, and inspiration.

3 ACKNOWLEDGMENTS Throughout my years as a graduate student I have received help, support, and encouragement from many individuals. They all have contributed in different ways to this dissertation and for that I will be forever thankful. First, I want to thank my advisor, Dr. Janise McNair, for all her help and guidance through the entire doctorate program. She was always available and very helpful. I could not have had a better advisor and mentor. The support and encouragement from my family kept me always motivated and has been one of the main reasons I have come this far. A special feeling of gratitude goes to all of them, in particular to my parents, Mabel and Joe. Without their effort and inspiration, I would not be where I am today. I thank my sister, Amarilys, and brothers, Arturo and Jose Angel, for their unconditional support and for always being there for me. They have been and will always be a great motivation for me. Likewise I thank all my friends for being supportive and believing in me uncondition- ally. A special acknowledgement goes to Pablo Rivera, my housemate and an old-time dear friend who followed my progress as a graduate student and was always very en- couraging, and to Edward Latorre, my ECE partner and a very good friend throughout all these years at UF. I want to thank also all the students in the WAMS Laboratory. They were cooperative and supportive throughout the entire doctorate program. That will be always appreciated. For agreeing to serve in my PhD supervisory committee, I want to acknowledge Dr. Xiaolin "Andy" Li, Dr. Richard Newman, and Dr. Haniph Latchman. I really appreciate their availability and time. Finally, I express my gratitude to Dr. David Wentzloff who was my ﬁrst advisor as a graduate student and awakened my interest in UWB communications. For that I am deeply thankful.

4 Contents page ACKNOWLEDGMENTS...... 4 LIST OF TABLES...... 9 LIST OF FIGURES...... 10 LIST OF ABBREVIATIONS AND VARIABLES...... 13 ABSTRACT...... 18

CHAPTER 1 INTRODUCTION...... 20 2 BACKGROUND AND MOTIVATION...... 23 2.1 Chapter Contributions...... 23 2.2 History of the Development of UWB Communications...... 23 2.3 Key Concepts in Wireless Communications...... 24 2.3.1 Multipath and Small-Scale Fading...... 24 2.3.2 Path Loss and Large-Scale Fading...... 25 2.3.3 Noise...... 26 2.3.4 Interference...... 28 2.4 UWB Deﬁnitions...... 30 2.5 UWB Beneﬁts...... 32 2.5.1 HDR and Low SNR Operation...... 32 2.5.2 Low Interference...... 33 2.5.3 Multipath Robustness...... 33 2.5.4 High Interference Rejection...... 34 2.5.5 Low-Cost, Low-Complexity, and Low-Energy Architectures..... 35 2.6 Common Digital Modulation Schemes For UWB Radios...... 36 2.6.1 Coherent Modulation Schemes...... 37 2.6.2 Non-Coherent Modulation Schemes...... 38 2.6.3 Other Coherent and Non-Coherent Modulation Schemes...... 39 2.7 UWB Applications...... 41 2.8 UWB Channel Modeling...... 43 2.8.1 Large-Scale Path Loss for Indoor UWB Channels...... 43 2.8.2 Small-Scale Fading Model for Indoor UWB Multipath Channels.. 44 3 OPTIMIZATION OF ENERGY-DETECTION PPM RECEIVERS...... 49 3.1 Chapter Contributions...... 50 3.2 Previous work...... 50 3.3 Energy-Detection Demodulation for PPM...... 51 3.3.1 Probability of Bit-Error...... 53

5 3.4 Optimal Receiver Bandwidth...... 53 3.4.1 Effect of Receiver Bandwidth Reduction...... 54 3.4.2 Modified Probability of Bit-Error and Optimal Receiver Bandwidth. 55 3.4.2.1 Probability of Bit-Error and Receiver Bandwidth...... 56 3.4.2.2 Optimal Receiver Bandwidth...... 57 3.4.3 Adjacent-Channel Interference...... 58 3.4.3.1 Effect of ACIon the Receiver Performance...... 59 3.4.3.2 An Approximation for the Optimal Receiver Bandwidth in the Presence of ACI...... 61 3.4.4 Simulation Setup and Validation...... 62 3.4.4.1 Setup...... 62 3.4.4.2 Validation...... 63 3.4.5 Analysis...... 64 3.4.5.1 Theory Corroboration...... 64 3.4.5.2 Numerical Results...... 65 3.5 Optimal Integration Time...... 66 3.5.1 Effect of Integration Time due to Multipath Fading...... 66 3.5.2 Modified Probability of Bit-Error and Optimal Integration Time... 68 3.5.2.1 Probability of Bit-Error and Integration Time...... 68 3.5.2.2 Optimal Integration Time...... 69 3.5.3 Inter-Symbol and Inter-Frame Interference...... 71 3.5.3.1 Effect of ISI and IFI on the Receiver Performance.... 71 3.5.3.2 Optimal Integration Time...... 73 3.5.4 Simulation Setup and Validation...... 73 3.5.4.1 Setup...... 73 3.5.4.2 Validation...... 75 3.5.5 Analysis...... 76 3.5.5.1 Theory Corroboration...... 76 3.5.5.2 Numerical Results...... 76 3.6 Summary...... 79 4 ENERGY-INTEGRATION DETECTION FOR PPM RECEIVERS...... 81 4.1 Chapter Contributions...... 81 4.2 Previous Work...... 82 4.3 Energy-Integration Detection...... 82 4.3.1 Motivation...... 82 4.3.2 Bit Decision...... 83 4.3.3 Example...... 86 4.4 Probability of Bit-Error for EID...... 88 4.4.1 Bit Decision...... 88 4.4.2 Probability of Bit-Error...... 89 4.4.3 Modified Probability of Bit-Error...... 91 4.4.4 Energy Scaling Factors...... 92 4.5 Simulation...... 94

6 4.6 Analysis...... 94 4.6.1 Theory Corroboration...... 94 4.6.2 Bit-Error Rate...... 95 4.6.3 Integration Time...... 95 4.6.4 Signal Bandwidth...... 97 4.7 Summary...... 98 5 COGNITIVE PHY-MAC COOPERATIVE PROTOCOL...... 100 5.1 Chapter Contributions...... 101 5.2 Previous Work...... 101 5.3 System Model...... 102 5.3.1 Network and Signal Model...... 102 5.3.2 Modulation and Demodulation Schemes...... 103 5.3.3 Optimal Integration Time...... 104 5.3.4 Carrier Sense Multiple Access with Collision Avoidance...... 105 5.4 Channel Estimation...... 107 5.4.1 Signal and Energy Model...... 107 5.4.2 Energy Difference...... 109 5.4.3 Estimation of the Energy Scaling Factor...... 109 5.4.4 Achieving an Optimal Transmission Data Rate...... 111 5.5 UWB Cooperative PHY-MAC Protocol...... 113 5.5.1 Receiver Architecture for Channel Estimation...... 113 5.5.2 Cooperative PHY-MAC Protocol...... 114 5.5.3 PHYand MACFrame Formats...... 116 5.6 Simulation Setup...... 117 5.6.1 Network Simulator...... 117 5.6.1.1 Node class...... 118 5.6.1.2 Channel class...... 119 5.6.1.3 Other important classes and functions...... 119 5.6.2 Simulation Parameters and Setup...... 120 5.7 Analysis...... 121 5.7.1 Message Delivery Ratio...... 122 5.7.2 Average Transmission Time...... 123 5.7.3 Throughput...... 125 6 CONCLUSIONS AND FUTURE WORK...... 126

APPENDIX A DERIVATION OF THE PROBABILITY OF BIT-ERROR FOR PPM-ED RE- CEIVERS...... 130 B DERIVATION OF THE PROBABILITY OF BIT-ERROR FOR PPM-EID RE- CEIVERS...... 132

P P 2 2 C MEAN AND VARIANCE OF i j Xj FOR Xj ∼ N µj , σ ...... 134

7 REFERENCES...... 137 BIOGRAPHICAL SKETCH...... 146

8 LIST OF TABLES Table page 2-1 EIRP Limits for Indoor and Outdoor UWB Systems...... 31 2-2 Comparison between coherent and non-coherent modulation schemes.... 37 2-3 Wireless applications and their potential beneﬁts from UWB technology.... 43 2-4 Path loss parameters for UWB channels in residential and commercial buildings...... 44 2-5 Model parameters for UWB multipath channels...... 48 3-1 Constant values for the exponential ﬁt given by Equation 3–15...... 58

3-2 Constant values for βopt ...... 58 3-3 Constant values for the exponential ﬁt given by Equation 3–35...... 70

3-4 Constant values for( Tw )opt ...... 71

3-5 Constant values for( Tw )opt when ISI is considered...... 74 3-6 Optimal integration times for different values of signal bandwidth and BER... 75

4-1 Constant values for γb (t) ...... 92

0 00 4-2 Constant values for γb (t) and γb (t) ...... 93 5-1 Simulation parameters...... 121

9 LIST OF FIGURES Figure page 1-1 Timeline of popular commercial short-range wireless systems...... 21 2-1 Example of multipath propagation...... 25 2-2 Example of small-scale and large-scale fading...... 26 2-3 Example of additive white Gaussian noise...... 27 2-4 Example of co-channel and adjacent-channel interference...... 29 2-5 Example of inter-symbol interference...... 29 2-6 FCC spectral mask for indoor and outdoor UWB systems...... 31 2-7 Comparison of the theoretical channel capacities between UWB and Wi-Fi systems...... 32 2-8 Example of UWB low interference with narrowband and wideband signals... 33 2-9 Example of the UWB robustness against multipath propagation...... 34 2-10 Examples of coherent modulation schemes for UWB communications..... 38 2-11 Examples of non-coherent modulation schemes for UWB communications.. 40 2-12 Some UWB wireless applications...... 42 2-13 Path loss for UWB channels in residential and commercial buildings...... 45 3-1 General architecture forED receivers...... 49 3-2 Signal processing for aED-PPM receiver...... 52 3-3 Power spectral densities of a square pulse, Gaussian pulse, and AWGN.... 54 3-4 Signal and noise energy proﬁle as a function of the receiver bandwidth..... 55 3-5 PSD of the transmitted signal and ACI signals...... 59 3-6 Simulator block diagram...... 62 3-7 Comparison between simulations and Equation 3–3 to validate the simulator. 64 3-8 Comparison between simulations, Equation 3–12, and Equation 3–22..... 65

−3 3-9 Required SNRbit to achieve a BER = 10 ...... 66 3-10 Normalized optimal receiver bandwidth versus the signal’s 10 dB-bandwidth −3 for Tw = 30 ns, α = 1, and BER = 10 ...... 67

10 3-11 Signal and noise energy proﬁle as a function of integration time...... 68

3-12 Energy scaling factor γ(Tw ) for each UWBCM reported in [28]...... 70 3-13 Illustration of ISI and IFI...... 72 3-14 Simulator block diagram...... 74 3-15 Comparison of simulations and Equation 3–3 to validate the simulator..... 76 3-16 Comparison between simulations and the modiﬁed BER equations with B = 2 GHz and Tw = 25, 30, 80, 100 ns forCM 1 through 4, respectively...... 77

−5 3-17 Required SNRbit to achieve a BER= 10 for B = 2 GHz...... 78

−5 3-18 Optimal integration time( Tw )opt to achieve BER= 10 ...... 78

−5 3-19 Required SNRbit to achieve BER = 10 ...... 79

4-1 Example of the actual and optimal probabilities of bit-error (Pe) for radios operating inCM 1 andCM2...... 84 4-2 General block diagram for an EID receiver...... 86 4-3 Example of a binary logic 1 demodulated usingED and EID...... 87 4-4 Energy scaling factors for UWB channels...... 93 4-5 Simulator Block Diagram...... 94 4-6 Comparison between simulation results and the derived BER equation for EID receivers...... 95 4-7 Probability of bit-error forED and EID forCM 1 through 4 and B = 2 GHz .. 96

−5 4-8 Required SNRbit forED and EID to achieve a BER = 10 for B = 2 GHz andCM 1 through 4...... 97

−3 4-9 Required SNRbit forED and EID to achieve BER = 10 ...... 98 5-1 Example of a wireless ad-hoc network...... 103 5-2 Illustration of the CSMA-CA protocol...... 106 5-3 Signal processing of the proposed channel estimation...... 111 5-4 Accuracy of the energy scaling factor estimation as more symbols are used.. 112 5-5 PPM ED receiver architecture with the proposed channel estimation...... 114 5-6 Cognitive PHY-MAC protocol summary...... 115 5-7 Frame format for each MPDU...... 117

11 5-8 Frame format of the PLDU...... 117 5-9 Diagrams of the two main classes used by the network simulator...... 118 5-10 Message delivery ratio as a function of (a) the message arrival rate and (b) the number of nodes in the network...... 122 5-11 Average transmission time as a function of (a) the message arrival rate and (b) the number of nodes in the network...... 124 5-12 Throughput as a function of (a) the message arrival rate and (b) the number of nodes in the network...... 125

12 LIST OF ABBREVIATIONS AND VARIABLES

Abbreviations ACI Adjacent-channel interference, page 28. ACK Acknowledgement packet, page 107. ADC Analog-to-Digital converter, page 54. AIC Akaike Information Criterion, page 45. ASK Amplitude shift keying, page 39. AWGN Additive white Gaussian noise, page 27. BCH Bose-Chaudhuri-Hocquenghem coding algorithm, page 118. BER Bit-error rate, page 36. BOK Bi-orthogonal keying, page 41. BPF Band-pass ﬁlter, page 29. BPSK Binary phase shift keying, page 37. CA-MAC Cognitive autonomous MAC protocol, page 102. CCI Co-channel interference, page 28. CCT Channel coding theorem, page 27. CIR Channel impulse response, page 46. CLT Central limit theorem, page 28. CM Channel model, page 47. CPLNC-MAC Cooperative PHY layer network coding MAC protocol, page 101. CRC cyclic redundancy check, page 116. CSMA Carrier sense multiple access, page 105. CSMA-CA CSMA with collision avoidance, page 105. CTS Clear-to-send packet, page 105. DATA Data packet, page 107. DCF Distributed coordination function, page 105. DCM Dual-carrier modulation, page 41.

13 DIFS DCF inter-frame spacing, page 106. DPSK Differential PSK, page 41. ED Energy detection, page 21. EID Energy-integration detection, page 81. EIRP Equivalent isotropically radiated power, page 30. ESD Energy spectral density, page 56. FCC Federal Communications Commission, page 21. FSK Frequency shift keying, page 39. HDR High data rate, page 32. HLDU Higher layer data unit, page 119. i.i.d Independent and identically distributed, page 47. IEEE Institute of Electrical and Electronics Engineers, page 20. IFI Inter-frame interference, page 51. IFS Inter-frame spacing, page 106. ISI Inter-symbol interference, page 28. JR Jam resistance, page 35. LNA Low-noise ampliﬁer, page 54. LOS Line of sight, page 25. MA Multiple access, page 35. MAC Medium access control sublayer, page 21. MPDU MAC protocol data unit, page 116. MUI Multi-user interference, page 102. NAV Network allocation vector, page 106. NLOS Non-LOS, page 33. OOK On-off keying, page 39. OPSM Orthogonal pulse-shape modulation, page 38. PG Processing gain, page 34.

14 PHY Physical layer, page 21. PLDU PHY layer data unit, page 116. PPM Pulse-position modulation, page 21. PSD Power spectral density, page 27. PSK Phase shift keying, page 37. PSM Pulse-shape modulation, page 38. QoS Quality of service, page 106. QPSK Quadrature phase shift keying, page 37. RF Radio frequency, page 30. RFID Radio-frequency identiﬁcation, page 20. RTS Request-to-send packet, page 105. Rx Receiver, page 25. SIFS Short inter-frame spacing, page 106. SINR Signal-to-interference-and-noise ratio, page 72. SIR Signal-to-interference ratio, page 35. SNR Signal-to-noise ratio, page 32. std. dev. Standard deviation, page 44. T-R Transmitter-receiver, page 25. TR Transmitted reference, page 41. Tx Transmitter, page 25. UWB Ultra-Wideband, page 21. WBAN Wireless body area networks, page 20. WLAN Wireless local area networks, page 20. WPAN Wireless personal area networks, page 20. WSN Wireless sensor networks, page 20.

Variables

α Channel spacing normalized to the B10dB of the Tx signal, page 60.

15 B Receiver bandwidth, page 67.

B10dB Signal 10dB-bandwidth, page 28.

Bf Fractional bandwidth, page 30.

β Receiver bandwidth normalized to the B10dB of the Tx signal, page 55.

βopt Optimal β , page 57. C Shannon’s channel capacity, page 32.

d0 Reference distance, page 43.

E0 Noise energy, page 67.

Eb Energy per bit, page 53.

Ei Interference energy, page 59.

fc Center frequency, page 28.

fch Channel frequency separation, page 28.

fs Sampling frequency, page 27.

γPL Path loss exponent, page 43.

I0 Interference PSD constant, page 59.

N0 Noise PSD constant, page 27.

P0 Noise power, page 27.

Pb Average power per bit, page 67.

PED Probability of bit-error forED receivers using PPM, page 53.

PEID Probability of bit-error for EID receivers using PPM, page 90. PL Path loss, page 44.

PL0 Path loss at the reference distance d0 , page 43. PL Average path loss, page 43.

SNRbit SNR per bit, page 57.

Tc Chip duration (same as pulse width), page 34.

Tp Pulse time width, page 56.

Ts Symbol period, page 34.

16 ts Sampling period, page 27.

Tw Integration window length or integration time, page 52.

(Tw )opt Optimal Tw , page 69.

XPL Shadowing parameter for the log-normal path loss model, page 44.

17 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A COGNITIVE PHY-MAC COOPERATIVE PROTOCOL FOR LOW-POWER SHORT-RANGE WIRELESS AD-HOC NETWORKS USING UWB PPM RADIOS By Jose M. Almodovar-Faria May 2014 Chair: Janise McNair Major: Electrical and Computer Engineering Nowadays low-power short-range wireless ad-hoc networks are becoming more popular as the demand for wireless applications such as sensor and personal area networks continue to grow. Recently, in particular since the Federal Communications Commission approval in 2002, ultra-wideband (UWB) communications have been proposed as a viable and efficient alternative to implement short-range wireless applications. For the past decade, numerous investigations and research works have been done in order to employ UWB technology in wireless applications that have been traditionally implemented with conventional narrowband technologies. The vast range of benefits offered by UWB makes it, in many cases, an ideal solution when implementing wireless radios and networks. Low-power operation, low-complexity and low-cost radio architectures, and high data rates are among the many advantages of UWB. In most short-range wireless ad-hoc networks, low-power operation as well as multiple access control (MAC) is crucial in the network design. Pulse-position modulation (PPM) is a well-known digital modulation scheme that when used in UWB radios can achieve simple low-cost architectures and more importantly a very low-power operation while offering relatively good data rates and bit-error rate (BER) performance. The DCF function described by the IEEE 802.11 WLAN standard is used quite often as the MAC protocol when implementing wireless networks in general and has proven to be efficient for many applications. This doctoral

18 dissertation presents a new cognitive and cooperative protocol between the physical (PHY) layer and the MAC sublayer for wireless ad-hoc networks using PPM UWB radios. By a cognitive estimation of the wireless channel and the cooperation between the MAC and PHY layers, the cognitive protocol can dynamically adjust the transmission data rate between two nodes optimizing their communication. Simulations show that the protocol improves the overall network performance in terms of message delivery ratio and average transmission delay.

19 CHAPTER 1 INTRODUCTION In the last three decades, as the wireless communications industry has advanced, we have seen a great increase in the development of not only long-range and medium- range wireless communications (e.g. radio and television broadcasts, satellite communications and cellular networks among others) but in short-range wireless systems as well. For the last 15 years, wireless systems involving radio-frequency identification (RFID), wireless sensor networks (WSN), wireless local area networks (WLAN), personal area networks (WPAN) and body area networks (WBAN) have been increasingly developed to meet the demands of our technology-hungry society. These systems are used in many of today’s wireless applications such as mobile devices, wireless routers, wireless audio and video systems, advanced remote controlling, and much more. Figure 1-1 shows a timeline of some popular short-range wireless applications that have been commercialized over the last six decades. Until early 1990s, there were few commercial applications for short-range systems (e.g. remote controls (RCs), cordless phones and few other applications). At the time, most of the wireless applications commercially available focused on medium and long-range communications. However, thanks to the research done during the 1970s and early 1980s [61], the first WLAN products started to appear at the end of the 1980s. By 1997, the original version of the IEEE1 802.11 standard for WLAN was finalized and with it came a rapid development of this technology for residential and commercial use (e.g. Wi-Fi routers). Other short- range systems such as WPAN and WBAN started around the idea of WLAN and have extended in the last decade to numerous other systems such as WSN and RFID. Today, these systems have a very wide range of applications such as bluetooth in mobile

1 IEEE stands for Institute of Electrical and Electronics Engineers

20 Figure 1-1: Timeline of popular commercial short-range wireless systems devices and remote controllers, high-speed wireless routers, RFID tags, and body implantable sensors to mention just a few. Recently, in particular since the Federal Communications Commission (FCC) approval in 2002, ultra-wideband (UWB) communications have been proposed as a viable and efficient alternative to implement short-range wireless applications. Thus, for the last decade, research and numerous investigations have been done in which the conventional narrowband communications are being substituted by UWB communications. This mainly due to the wide range of benefits offered by UWB communications including low-power operation, low-complexity and low-cost radio architectures, and high data rates among several others. In this doctoral dissertation, non-coherent UWB radios using pulse-position modulation (PPM) and energy-detection (ED) demodulation are studied and optimized. In addition, a modification to theED technique is proposed and discussed in detail. Based on the extensive study and the theory developed, a new cognitive and cooperative protocol involving the physical (PHY) layer and the medium access control (MAC) sublayer is introduced. The rest of this doctoral dissertation is divided in 5 additional chapters. Chapters 2,3, and4 correspond to the work done on UWB PPM radios. Chapter2, provides a literature overview of UWB communications along with the explanation of several

21 key concepts directly related to the discussion throughout this document. Chapter3 discusses in detailED and its optimal receiver bandwidth and optimal integration time in the presence noise and certain interference sources. The theory presented in this chapter is the base for the channel estimation that will be used for the cognitive and cooperative protocol. Chapter4 presents a modiﬁcation made to theED demodulation technique presented in Chapter3 that improves the receiver performance. This modiﬁ- cation along with the original demodulation method will be used in the implementation of the new protocol presented in Chapter5. In this chapter, the cognitive and cooperative PHY-MAC protocol is presented, discussed and analyzed. Finally, conclusions are presented in Chapter6. This chapter also provides a brief summary of the future work.

22 CHAPTER 2 BACKGROUND AND MOTIVATION The ﬁeld of UWB wireless communications has been rapidly growing in the past decade since its FCC approval in 2002. The wide variety of advantages that UWB offers has motivated a signiﬁcant interest toward its development and application to a vast range of wireless applications including medium and long range communications [30, 90]. However, at least 80% of UWB commercial applications are envisioned to be short-range wireless communications [38]. Although the concepts presented in this chapter focus on short-range wireless communications, they are still valid for potential applications in the medium-range and even in the long-range domain of UWB wireless communications.

2.1 Chapter Contributions

This chapter provides a literature overview of UWB communications. It presents the key concepts, beneﬁts, and challenges of UWB signaling as well as its applications and technologies. In addition, a review of UWB channel modeling is offered including a characterization of the channel models proposed by the IEEE P802.15.3a task group.

2.2 History of the Development of UWB Communications

UWB communications employ narrow pulses in order to achieve large bandwidths and, therefore, its early name was impulse communications –the term UWB became popular during the 1990s. The ﬁrst experiments with narrow pulses can be traced back to the late the 19th century when Heinrich Hertz experimented with spark discharges [42] to verify Maxwell’s equations on electromagnetic theory. The equipment Hertz used is probably the ﬁrst impulse radio in history. A few years later, Guglielmo Marconi’s experiments using spark-gap transmissions expanded Hertz’s work and demonstrated its practical application: wireless communications [8]. Ironically Marconi, known today as the inventor of radio, was using UWB communications for its radio applications by employing spark-gap transmissions. In fact, for about

23 20 years after Hertz’s first experiments, this was the dominant technology [37] for the early research in wireless communications. Later, mainly due to the lack of appropriate hardware for pulse-based modulation and demodulation as well as wideband interference mitigation techniques, sinusoidal waves became the leading form of wireless communications. It was not until the late 1960s and early 1970s that pulse-based communications resurfaced with the pioneering contributions of researchers like Henning Harmuth, Paul Van Etten, and Gerald Ross [7]. Harmuth publications presented the basic receiver and transmitter design for UWB while Van Etten’s experiments in UWB radar systems resulted in the development of the basic concepts for UWB antennas. In 1971, Ross filed a patent on the transmission and reception of pulse signals without distortion [68] and in 1973 it became the first US patent awarded for UWB communications. For the next two decades after Ross’ patent, UWB was mostly used by the military in communications, radar, sensing, and niche applications [18]. In the 1990s, a few startup companies –in particular, Time Domain Company (TDC) [80]– staged a movement toward the commercialization of UWB systems which, after years of much opposition, culminated with its approval by the FCC in April of 2002. A few months later, the PulsOn chipset from TDC became the first UWB communications product certified by the FCC. Since then, UWB has been a major research area in wireless communications as evidenced by the numerous articles and books published in the last 10 years.

2.3 Key Concepts in Wireless Communications

This section brieﬂy deﬁnes a few key concepts in wireless communications that are relevant to the discussion that will be developed in subsequent chapters. These key concepts are: small-scale fading, large-scale fading, noise, and interference.

2.3.1 Multipath and Small-Scale Fading

A radio signal that is transmitted through a wireless channel travels through multiple paths before reaching the receiver’s antenna. This phenomena is called multipath

24 (a) Multiple propagation paths (b) Transmitted and received signals Figure 2-1: Example of multipath propagation propagation, or simply multipath, and is the cause of the rapid amplitude ﬂuctuations that the radio signal undergoes over a short period of time, i.e. short distance. Figure 2-1(a) shows an example of multiple signal paths. The direct path from transmitter (Tx ) to receiver (Rx) is called the line-of-sight (LOS) path and it is usually the dominant multipath component of the received radio signal. Small-scale fading describes the effects caused by multipath propagation and other factors such as the signal bandwidth and receiver motion relative to the transmitter. The most important effects described by small-scale fading are: time dispersion due to multipath propagation delays, rapid changes in signal strength and polarity, and random frequency modulation due to Doppler shifts.1 Figure 2-1(b) shows an example of these effects on a transmitted pulse. As can be seen, the received pulse has been dispersed in time with a decreasing average signal strength and changes in frequency and polarity.

2.3.2 Path Loss and Large-Scale Fading

Small-scale fading describes the rapid changes in signal strength over short transmitter-receiver (T-R) separation distances. Large-scale fading, in contrast, describes the mean signal strength attenuation over longer T-R separation distances, i.e.

1 When a transmitter and a receiver are moving relative to each other, the frequency of the received signal changes based on their motion. This phenomenon is known as the Doppler effect (or Doppler shift).

25 (a) Linear Scale (b) Logarithmic Scale Figure 2-2: Example of small-scale and large-scale fading longer periods of times. Figure 2-2(a) shows an example of signal strength variations as a function of T-R distance illustrating the rapid small-scale fading and the slower large-scale fading. The ﬁgure also shows that the mean signal strength (i.e. large-scale fading) attenuates exponentially as the T-R distance increases. For this reason as well as to simplify related calculations, the signal strength attenuation is often described in logarithmic scales as shown in Figure 2-2(b). In wireless communications, the signal strength attenuation is commonly referred to as path loss. Since the signal attenuates more and more as it travels further along its wireless path, the path loss increases with larger T-R distances. Path loss is very useful when deﬁning that radio coverage area of a transmitter.2

2.3.3 Noise

There are several types of noise (e.g. shot noise, burst noise, Brownian noise) but the most common when it comes to wireless communications is thermal noise. This

2 The radio coverage area refers to how far a receiver can be from the transmitter so that the received signal strength is large enough to be detected.

26 (a) Time Domain (b) Frequency Domain Figure 2-3: Example of additive white Gaussian noise

unavoidable phenomenon is a random process caused by thermal motions of electrons in any conducting material and has three main properties:

1. It is an additive process because a received signal can be represented by the sum of the transmitted signal and the noise signal. 2. It has a constant power spectral density (PSD) for all frequencies (i.e. white noise).3 3. It follows a zero-mean Gaussian distribution with ﬁnite variance σ2 equal to the average noise power P0. Hence, thermal noise is often called additive white Gaussian noise (AWGN). Figure 2-

3(a) shows an example of AWGN when using a sampling period ts = 1/fs, where fs is the sampling frequency. Figure 2-3(b), on the other hand, shows AWGN in the frequency

4 domain, that is, a ﬂat spectrum with a PSD of N0/ 2. AWGN is fundamental in the understanding of wireless communications. A basic theorem of Information Theory is the Channel Coding Theorem (CCT) [14] from which

3 A signal with constant PSD for all frequencies is called white noise in analogy to white light which covers all wavelengths.

4 The PSD constant value for AWGN is N0 for all frequencies (positive and negative). However, real-world devices only use the positive half of the spectrum. Therefore, N0 / 2 is used instead.

27 can be concluded that the worst-case background noise in wireless channels is AWGN [50, 76] as it minimizes the channel information capacity. Furthermore, recently, it has been suggested that AWGN is also the worst-case additive noise in wireless networks in general [76, 77]. Intuitively, it makes sense the use of AWGN to model all noise since, in many cases, the combined noise sources should approach a Gaussian random distribution by the Central Limit Theorem (CLT).

2.3.4 Interference

Interference can be deﬁned as any unwanted signal from an external source that alters or disrupts the intended signal. In wireless communications there is a wide range of interference signals, or just interferers, that are taken into account when designing wireless systems. Broadly speaking, interference in wireless communications can be divided in:

1. Co-channel interference (CCI): the frequency bands (channels) of the interferer and the intended signal overlap. 2. Adjacent-channel interference (ACI): the interferer is at a neighboring channel and part of its energy is leaked into the frequency band of the intended signal 3. Inter-symbol interference (ISI): a previous intended signal interferes with the current intended signal due to the time dispersion caused by multipath propagation (see section 2.3.1).

Figure 2-4(a) shows an example of CCI for a signal with center frequency fc. In this example, the frequency bands of the interference signals overlap with the frequency

5 band of the intended signal, i.e. the channel frequency separation (fch) is less than

6 the 10dB-bandwidth of the signal (B10dB). Similarly, Figure 2-4(b) shows an example of ACI which, in contrast to CCI, fch is greater or equal than B10dB . In both ﬁgures, the

5 The channel frequency separation is the frequency spacing between the center frequencies of two channels. 6 In wireless communications, the frequency band of a signal is usually determined by its 10dB-bandwidth B10dB , i.e. the frequency band in which the signal’s PSD falls 10dB from its highest point.

28 (a) Co-channel interference (CCI) (b) Adjacent-channel interference (ACI) Figure 2-4: Example of co-channel and adjacent-channel interference dashed yellow lines represent a non-ideal band-pass filter (BPF). On the other hand, an example of ISI is illustrated in Figure 2-5. This example shows the wireless transmission of two symbols.7 As seen in the figure, the first received symbol interferes with the second since its time dispersion due to multipath propagation is larger than the symbol period.

(a) Transmitted Signal (b) Received Signal Figure 2-5: Example of inter-symbol interference

7 In digital communications, a symbol is a signal modulated to represent one or more logic bits.

29 2.4 UWB Deﬁnitions

The FCC final rule on UWB Transmission systems defines a UWB transmitter as “an intentional radiator 8 that at any point in time, has a fractional bandwidth equal to or greater than 0.20 or has a UWB bandwidth equal to or greater than 500 MHz, regardless of the fractional bandwidth”[26]. In this definition, the fractional bandwidth Bf is

Bf = (fH − fL) /fc (2–1) where fc = (fH + fL) /2 is the center frequency and fH and fL are, respectively, the upper and lower boundaries of the 3dB-bandwidth9 of the transmitted signal. In addition to the bandwidth requirement, the FCC also deﬁnes a maximum transmission power. The power limits for the equivalent isotropically radiated power (EIRP )10 of indoor and outdoor UWB systems when measured at a distance of 3 m with a bandwidth resolution of 1 MHz are tabulated in Table 2-1. The FCC also sets a power limit for unintentional radiators.11 For frequencies above 960 MHz, this type of radiators cannot exceed an electric ﬁeld strength of 500 µV /m measured at a T-R separation distance of 3 m over a 1 MHz frequency band [25]. To compare this limit to the UWB EIRP limit, the conversion equation given by [12]

8 Intentional radiators are devices that generate radio frequency (RF) energy on purpose such as wireless transmitters, imaging sensors, and ground penetrating radars among many others. 9 The 3dB-bandwidth of a signal is the frequency band in which its PSD falls 3dB from the highest point. 10 EIRP, as deﬁned by the FCC, refers to the highest signal power strength measured at 3 m from the source at any frequency and in any direction. 11 Unintentional radiators are devices not designed to emitRF energy on purpose such as digital electronics, electric chargers, and audio ampliﬁers among many others.

30 Table 2-1: EIRP Limits for Indoor and Outdoor UWB Systems

Frequency Range (MHz) Indoor EIRP( dBm/MHz) Outdoor EIRP( dBm/MHz) 960 − 1610 −75.3 −75.3 1610 − 1990 −53.3 −63.3 1990 − 3100 −51.3 −61.3 3100 − 10600 −41.3 −41.3 Above 10600 −51.3 −61.3

Figure 2-6: FCC spectral mask for indoor and outdoor UWB systems.

2 2 Prad = 4 · π · drad · Erad /Z0 (2–2) can be used. In this case, the distance from the measurement location to the radiator is drad = 3 m, the electric field strength from the radiator is Erad = 500 µV /m and the characteristic impedance of free space is Z0 = 120 · π Ω. With these values, the radiated power limit for unintentional radiators is Prad ≈ 75nW ≈ −41.3dBm per 1MHz. This limit is often called the noise floor and is shown in Figure 2-6 along with the FCC spectral mask for indoor and outdoor UWB transmitters. From the figure, it is evident that the best frequencies to operate UWB systems range from 3.1 GHz to 10.6 GHz where the EIRP limit is the highest.

31 Figure 2-7: Comparison of the theoretical channel capacities between UWB and Wi-Fi systems.

2.5 UWB Beneﬁts

UWB technology offers several advantages over the traditional narrowband technologies. Among these, the key beneﬁts can be summarized as: high data rate (HDR), low signal-to-noise ratio (SNR) operation, low interference, multipath robustness, high interference rejection, and low-cost, low-complexity, and low-energy architectures.

2.5.1 HDR and Low SNR Operation

The CCT states that information can be transmitted at any data rate R that does not exceed the channel capacity C [14], i.e. R ≤ C. This capacity is given by the well-known formula derived by Shannon in 1948 [75]

C = B · log2 (1 + SNR) (2–3) where B is the transmission bandwidth and SNR is the signal-to-noise ratio. From Equation 2–3, it is clear that UWB systems have the potential to achieve HDR due to their large transmission bandwidth. This is illustrated in Figure 2-7 which shows the theoretical channel capacity as a function of SNR for an UWB system with B = 500 MHz and current Wi-Fi systems. From the ﬁgure, it is easy to see that even at low values of SNR, UWB systems can still offer relatively large data rates as a result of the their large

32 Figure 2-8: Example of UWB low interference with narrowband and wideband signals. bandwidth. For instance, in theory, the UWB system (B = 500 MHz) with SNR= 0.5 dB has the same channel capacity than Wi-Fi 802.11ac (B = 80 MHz) at SNR= 20 dB.

2.5.2 Low Interference

As explained in section 2.4, the EIRP limit for UWB radios is −41.3 dBm/MHz which is the noise ﬂoor. Due to this power limit and large bandwidth, UWB signals appear as regular channel noise to traditional narrowband and wideband radios operating in the same frequency band, that is, UWB signals produce very low interference to in-band radios. This is illustrated in Figure 2-8.

2.5.3 Multipath Robustness

UWB pulses have a very short duration. This makes UWB systems less sensitive to multipath propagation than narrowband systems that use wider pulses. The reason is that the pulse propagating through a non-LOS(NLOS) path has a very small window of opportunity to collide with the pulse propagating through the LOS path which causes signal degradation [58]. To illustrate this concept, an example is shown in Figure 2-9. Let us assume that a transmitted pulse propagates only through a LOS path and a NLOS path as shown in Figure 2-9(a) with travel distances of 10 m and 11 m, respectively. Then, assuming a propagation speed equal to the speed of light c = 3 × 108 m/s, the signal propagating

33 (a) Propagation paths (b) Arrival time proﬁle Figure 2-9: Example of the UWB robustness against multipath propagation

through the LOS path arrives at the receiver after 33.3 ns while the NLOS signal arrives 3.3 ns later. The wide pulse representing a narrowband signal in Figure 2-9(b) has a time duration larger than the difference in the arrival times of the multipath signals which causes a collision. In contrast, the narrow pulse in the same ﬁgure has a smaller duration than 3.3 ns and thus no collision occurs.

2.5.4 High Interference Rejection

An approximate measure for the capability of a system to reject interference is the processing gain (PG. HigherPG results in greater ability to suppress in-band interference [66]. A common way to deﬁnePG is

R T PG = c = s (2–4) Rs Tc

where Rc = 1/Tc is the chip rate (Tc is the chip duration or pulse width) and Rs = 1/Ts is the symbol rate (Ts is the symbol period). In UWB systems, very narrow pulses are used in order to generate large signal bandwidths and, hence, Tc is very small in comparison to Ts. Consequently, the Ts/Tc ratio, sometimes called the spreading factor, is usually quite large.

34 In multiple access (MA) applications, in particular, highPG is desired since the dominant interference comes from in-band signals. The jam resistance (JR)12 margin offers a measure of how capable a system is when rejecting in-band interference and can be deﬁned as [95]

JR = PG − SIRmin [dB] (2–5) where SIRmin is the minimum signal-to-interference ratio (SIR) required to meet a desired system performance. Clearly, UWB is an attractive technology forMA applications since it can provide highPG resulting in a high resistance to narrowband interference signals.

2.5.5 Low-Cost, Low-Complexity, and Low-Energy Architectures

The low transmission power and very large signal bandwidth of UWB radios bring advantages in the hardware implementation such as small antennas and other passive elements, relatively simple architectures, and low-energy operation. The high frequency band (3.1 − 10.6 GHz) allocated for UWB results in signals with small wavelengths. This, in turn, helps reducing the size of antennas since it is typically proportional to the signal wavelength. In addition, passive elements such as inductors and capacitors used mainly for impedance matching and resonance are also reduced in size due to the UWB high frequency band.13 Having smaller antennas and passive elements signiﬁcantly reduce the size of UWB integrated circuits resulting in a considerable reduction in the cost of manufacturing.

12 Although the term “jamming” nowadays is usually used to refer to an intentional attempt of disrupting a communication, in the past, it was often used as a synonym to in-band interference. Thus,JR can be interpreted as interference resistance.

13 Circuits designed for higher frequencies need smaller values√ of inductance (L) and capacitance (C) as the resonance frequency is proportional to 1/ L·C.

35 The pulse-based transmission of UWB systems allows for low-complexity architectures. For instance, pulses can be generated directly in the UWB frequency band without requiring frequency translation [84]. This eliminates the need of an oscillator for frequency up-conversion reducing then the complexity of the transmitter and its energy consumption. Similarly, in the receiver end, oscillators for down-conversion can be omitted by employing non-coherent modulation schemes such as energy-detection (discussed in detail in Chapter3). In addition, the FCC limit of −41.3 dBm/MHz on the EIRP of UWB radios implies a very low power transmission which reduces the need of power ampliﬁers in the transmitter architectures. In general, the low transmission power and the potential low complexity of UWB radios result in low-energy and low-cost systems. These add to the list of beneﬁts that make UWB a very attractive technology for short-range wireless communications.

2.6 Common Digital Modulation Schemes For UWB Radios

This section brieﬂy discusses several basic modulation schemes used in UWB digital systems. These can be divided in two main groups: coherent and non-coherent modulation. Coherent modulation exploits the phase and shape of the carrier signal in order to transmit information. Non-coherent modulation, in contrast, uses only the instantaneous power of the signal eliminating the need for coherent carrier recovery[86]. Both coherent and non-coherent modulation techniques have advantages over each other (see Table 2-2) and choosing one over the other will strongly depend on the target application. For instance, in terms of data rate and bit-error rate (BER), coherent modulation schemes will typically provide a better system performance [67]. However, non-coherent modulation schemes require less energy to operate and can be realized with relatively simpler architectures mainly due to the fact that no coherent carrier recovery is needed.

36 Table 2-2: Comparison between coherent and non-coherent modulation schemes

Parameter Coherent Non-coherent Carrier Recovery Yes No Energy per bit Higher Lower Data Rate Higher Lower BER Lower Higher Complexity Higher Lower

2.6.1 Coherent Modulation Schemes

In digital radios, phase shift keying (PSK) is a commonly used coherent modulation technique. As its name suggests, the phase of the signal carries the digital information. A PSK signal can be modeled as

si (t) = α (t) · cos (2 · π · fc · t + φi ) (2–6) where α (t) is the envelope of the signal, fc is the center frequency, φi is the phase of the signal corresponding to the ith modulation state of a single symbol. For instance, the popular binary-PSK(BPSK) scheme uses two phases, i.e. φ1 = 0 and φ2 = π, to represent a binary bit. For these phase values, s1 (t) = α (t) · cos (2 · π · fc · t) and

14 s2 (t) = −s1 (t). This is shown in Figure 2-10 where s1 (t) and s2 (t) represent a binary logic 0 and 1, respectively. Quadrature-PSK(QPSK) is another common digital modulation scheme employed in UWB systems. It follows the same principle as BPSK but it uses four distinct phases, i.e. φ1 = 0, φ2 = π/2, φ3 = π, and φ4 = 3π/2, to represent four modulation states, that is, a 2-bit symbol.

14 For φ1 = 0 and φ2 = π, the signals representing the binary states (0 and 1) are oppo- site in sign and, hence, BPSK is sometimes interpreted as ASK or PSM.

37 Figure 2-10: Examples of coherent modulation schemes for UWB communications

Although less common than PSK, pulse-shape modulation (PSM) is a coherent scheme that has been proposed for UWB communications [24, 43, 52]. This modulation method, instead of signal phases as in PSK, uses different pulse shapes to represent each modulation state of the symbol to be transmitted. Orthogonal-PSM(OPSM) is a typical way of implementing this type of modulation scheme[40, 89]. It utilizes pulses that are orthogonal to each other. An example of OPSM is shown in Figure 2-10.

2.6.2 Non-Coherent Modulation Schemes

In many cases, coherent digital modulation schemes derived from conventional narrowband systems (e.g. BPSK and QPSK) are not feasible to implement low-power UWB radios [86]. This has led researchers to shift towards non-coherent schemes due to the potential of very low-power radio implementations. Pulse-position modulation (PPM), probably the most common modulation technique found in the UWB literature [37], uses the position in time of a pulse to represent the modulation states of the symbol to be transmitted. With PPM, a pulse is located in

38 one of two time slots to represent each modulation state (0 or 1) of a 1-bit symbol. An example of PPM is shown in Figure 2-11(a). Another common and very simple modulation scheme used for UWB communications is on-off keying (OOK). With this modulation technique, a pulse and its absence are used to represent each modulation state of a binary bit. An example of OOK is illustrated in Figure 2-11(a). Although less frequent, other non-coherent modulation schemes for UWB that can be found in the literature are frequency shift keying (FSK) [65, 78] and amplitude shift keying (ASK) [53]. FSK uses different center frequencies to represent two or more modulation states. Figure 2-11(b) shows an example for a 1-bit symbol in which the lower frequency represents a binary logic 0 while the higher frequency represents a logic 1. With ASK, the information is modulated in the amplitude of the signal, that is, each amplitude value represent a modulation state. Although ASK is not commonly in UWB applications, it is worth mentioning as OOK and even BPSK can be considered ASK. To

illustrate this, let the ASK signals s0 (t) and s1 (t) represent a binary logic 0 and logic 1, respectively, where   αi · cos (2 · π · fc · t) , 0 ≤ t ≤ tp si (t) = (2–7)  0, otherwise

th fc is the center frequency and αi is the signal amplitude corresponding to the i modulation state (in this case, 0 or 1). If α0 = 0 and α0 = 1, then ASK resembles OOK.

On the other hand, if α0 = −1 and α1 = 1, then ASK appears as BPSK. 2.6.3 Other Coherent and Non-Coherent Modulation Schemes

Over the last decade, several modulation schemes have been developed for UWB communications in order to achieve higher data rates and improve the BER performance. These go beyond the scope of the discussion in this work. Nevertheless,

39 (a) Most common schemes

(b) Other schemes

Figure 2-11: Examples of non-coherent modulation schemes for UWB communications

40 a few of them are mentioned next with references that the reader may look up for further information. In the non-coherent techniques domain, transmitted-reference (TR) signaling [45, 69] is an often used technique to achieve a non-coherent phase comparison of the carrier signal at the receiver. Similar to PSK schemes, it uses the phase of the carrier signal to modulate binary information. However, in contrast to PSK, a reference signal (sometimes called signal template) is transmitted along with the signal carrying the information. At the receiver, the signals can be correlated to perform the phase comparison eliminating the need for coherent carrier recovery. Another modulation scheme able to compare phases using a non-coherent demodulation is differential-PSK(DPSK) [11, 44]. 15 With this modulation technique, the information is modulated using the difference in phases of the carrier signal. At the receiver, similar toTR signaling, the current signal and the previous signal can be correlated to determine the change in phase and, hence, demodulate the signal. In the coherent techniques domain, some interesting modulation schemes are bi-orthogonal keying (BOK) [59] and the recently proposed dual-carrier modulation (DCM) [70]. Similar to OPSM, BOK utilizes different pulse shapes to modulate binary information. However, the set of pulses are bi-orthogonal to each other rather than orthogonal as in OPSM. On the other hand, DCM uses two QPSK symbols and a dual-frequency carrier to modulate the binary information.

2.7 UWB Applications

The vast beneﬁts offered by UWB communications qualify this technology as a promising alternative to existing and future short-range wireless applications. As shown in Figure 2-12, UWB can be used to implement a variety of today’s wireless technologies

15 For PSK, both coherent [19] and non-coherent [44] implementations can be found in literature.

41 Figure 2-12: Some UWB wireless applications. such as WSN, WBAN, WPAN, RFID, and radio localization systems. Table 2-3 shows some of these wireless technologies and how they can mainly beneﬁt from the UWB advantages discussed in section 2.5. For instance, WBAN and RFID require simple architectures that consume low energy and produce very small interference to other wireless systems. Therefore, they beneﬁt mainly from the low energy operation, low complexity, and low interference that UWB communications offer. UWB technology has been used in the last few years to realize wireless systems that were traditionally accomplished with narrowband (sometimes with wideband) communications. This demonstrate the feasibility of UWB as the alternative technology to current wireless systems. Take for example HDR wireless systems (e.g. Wi-Fi). In [97, 98], two UWB transceivers are reported to achieve data rates of up to 400 Mbps which is far more than the 150 Mbps offered by the popular 802.11n Wi-Fi standard and close to the 450 Mbps offered by the recent 802.11ac standard. Similarly, in [5, 17], [32, 83], [33, 62], and [81, 96] (see Table 2-3) wireless radios were implemented for WSN, WBAN, RFID, and positioning systems, respectively, demonstrating the ability of UWB communications to be an alternative to current technology in different wireless applications.

42 Table 2-3: Wireless applications and their potential beneﬁts from UWB technology

Examples Multipath Low Low Low Low High Application in HDR Robust- Comple- Interfe- Energy SNR PG Literature ness xity rence WSN[5, 17] WBAN[32, 83] RFID[33, 62] WPAN[97, 98] Localization [81, 96]

2.8 UWB Channel Modeling

2.8.1 Large-Scale Path Loss for Indoor UWB Channels

Wireless propagation models often use analytical expressions or ﬁtting curves to recreate empirical data measured in different environments. Indoor propagation models in particular have been extensively studied over the years and can be frequently found in the literature (e.g. [21, 35, 79, 93]). All of them agree that the average received power decreases exponentially with distance. Therefore, the average path loss PL can be approximated by the log-distance model given by [66]

PL (d) = PL0 + 10 · γPL · log10 (d/d0) , [dB] (2–8) and it is commonly used to estimate the average path loss as a function of distance d .

In Equation 2–8, d0 is the reference distance (usually chosen as 1 or 3 meters), PL0 is

the path loss at d0 , and γPL is the path loss exponent which indicates the slope of the average increase in path loss. One problem with the log-distance path loss model is that it does not take into account that the environmental clutter differs from one location to another resulting in different path losses even when the T-R separation distance is the same. This phenomenon is often called shadowing (or log-normal shadowing). Empirical observations

43 Table 2-4: Path loss parameters for UWB channels in residential and commercial buildings

Residential Commercial Parameter LOS NLOS LOS NLOS

d0 (m) 1 1 1 1 PL0 (dB) 45.9 50.3 43.7 47.3 γPL (dB) 2.01 3.12 2.07 2.95 XPL std. dev., σ (dB) 3.02 3.8 2.3 4.1

have shown that the path loss has a random component and follows a log-normal distribution [15]. Thus, the path loss PL for indoor environments is better represented by the log-normal shadowing model described by

PL (d) = PL0 + 10 · γPL · log10 (d/d0) + XPL, [dB] (2–9)

where XPL is a zero-mean Gaussian random variable with standard deviation (std. dev.) σ that models the shadowing effect. Using Equation 2–9, an UWB indoor path loss model for residential and commercial buildings was presented in [35]. Table 2-4

shows the values of the model parameters PL0 , γPL , d0 , and XPL for LOS and NLOS measurements in residential and commercial buildings. This log-normal shadowing model using the values in Table 2-4 is illustrated in Figure 2-13. Figure 2-13(a) shows PL as a function of T-R separation distance in residential buildings for both LOS and NLOS. Similarly, Figure 2-13(b) also shows PL but this time using the parameters values for commercial buildings. In both ﬁgures, the solid lines represent the average path loss PL and the dashed lines enclose the 98% conﬁdence-interval region (i.e. ±2.33 · σ).

2.8.2 Small-Scale Fading Model for Indoor UWB Multipath Channels

In contrast to narrowband systems, in UWB systems the sampling period is much smaller due its wideband nature and, hence, the number of resolvable multipath

44 (a) Residential (b) Commercial Figure 2-13: Path loss for UWB channels in residential and commercial buildings components within this period is too small to justify its approximation under the CLT. Therefore, it is often argued that Rayleigh and Rice fading16 are not good small-scale fading models for UWB wireless channels. Despite of this argument, there are empirical measurements that support the Rayleigh [60] and Rice [48] distributions to model UWB multipath fading scenarios. Furthermore, according to [74], the Akaike Information Criterion (AIC) [2] supports Rayleigh and Rice amplitude distributions to adequately model the UWB channels measured by the authors. Although there are measurements that support Rayleigh and Rice distributions to model small-scale fading for some UWB scenarios, extensive work can be found in the literature using other distributions to model UWB channels more accurately. Some of these distributions are Nakagami [13], Weibull [36], and log-normal [29]. Among these, the most commonly found in literature is the log-normal distribution probably because it is the one adopted by the IEEE P802.15.3a task group in its 2003 ﬁnal report [27].

16 For narrowband systems, Rayleigh and Rice are probably the most commonly used distributions to model small-scale fading.

45 Therefore, to model multipath fading, the IEEE P802.15 model is used in this work and it is presented next. The IEEE P802.15.3a model is derived from the Saleh-Valenzuela model [72] with minor modiﬁcations. It consists of the discrete-time channel impulse response (CIR) given by

L K X X i i i hi (t) = Xi · αk,l · δ t − Tl − τk,l (2–10) l=0 k=0 i i th where αk,l are the multipath gain coefﬁcients, Tl is the delay of the l cluster,

i th th τk,l is the delay of the k multipath component relative to the l cluster arrival time

i 17 th (Tl ), δ (·) is the delta function, i refers to the i realization, L is the total number of clusters, and K is the number of rays (multipath components) within the lth cluster. The

random variable Xi represents shadowing and follows a log-normal distribution such that

2 18 20 · log10 (|Xi |) ∼ N 0, σX or, equivalently,

(nX )/20 |Xi | = 10 (2–11)

2 where nX ∼ N 0, σX . The multipath gain coefﬁcients are deﬁned as

αk,l = pk,l · |ξl · βk,l | = pk,l · Yk,l (2–12) where pk,l is equiprobable ±1 to account for signal inversion due to reﬂections, ξl

th represents the fading associated with the l cluster, and βk,l models the fading associ-

th th ated with the k ray of the l cluster. In Equation 2–12, |ξl · βk,l | follows a log-normal

17 The Dirac delta function δ (t − t0) can be informally deﬁned as a function that is zero everywhere except at t = t0 with integral of one over all real numbers which implies that δ (t − t0) = ∞ at t = t0. However, in electrical engineering it is sometimes used as δ (t − t0) = 1 for t = t0 and zero otherwise. This is the case for Equation 2–10. 18 N µ, σ2 is the common notation for a normal (Gaussian) random variable with mean µ and variance σ2

46 2 2 distribution such that 20 · log10 (|ξl · βk,l |) ∼ N µk,l , σ1 + σ2 or

(µk,l +n1+n2)/20 Yk,l = |ξl · βk,l | = 10 (2–13)

2 2 where n1 ∼ N 0, σ1 and n2 ∼ N 0, σ2 are independent random variables corresponding to the fading on each cluster and ray, respectively, and

10 · [ ln (Ω ) − T /Γ − τ /γ ] ln (10) · σ2 + σ2 µ = 0 l k,l − 1 2 (2–14) k,l ln (10) 20

In Equation 2–14, Ω0 is the mean energy of the ﬁrst path of the ﬁrst cluster and Γ and γ

th are model parameters. The excess delay of the l cluster Tl with cluster arrival rate Λ can be modeled as

2 2 Tl = Tl−1 + (nT ,1) + (nT ,2) (2–15)

−1 for l = 1, 2, ... , L, where [nT ,1, nT ,2] ∼ N 0, (2 · Λ) are independent and identically

distributed (i.i.d) normal random variables and T0 = 0 for LOS channels or T0 =

2 2 th (nT ,1) + (nT ,2) for NLOS. Similarly, the delay of the k multipath component within the

th l cluster τk,l with ray arrival rate λ can be modeled as

2 2 τk,l = τk−1,l + (nτ,1) + (nτ,2) (2–16)

−1 for k = 1, 2, ... , K , where [nτ,1, nτ,2] ∼ N 0, (2 · λ) are i.i.d normal random variables

th and τ0,l = 0 by deﬁnition since it is relative to the cluster, i.e. the ﬁrst ray of the l cluster

arrives at time Tl . Based on the equations presented in this section, Table 2-5 shows all the necessary parameters to model CIRs for the four UWB channel scenarios presented in the IEEE P802.15.3a task group report [27]. The channel model (CM) 1 is based on LOS channel measurements with a T-R distance between 0 and 4 m.CM 2 andCM 3 correspond to NLOS measurements at T-R distances of 0 − 4 m and 4 − 10 m, respectively, and CM 4 was generated to represent an extreme NLOS multipath channel. These channel

47 Table 2-5: Model parameters for UWB multipath channels ParameterCM1CM2CM3CM4

σx (dB) 3.0000 3.0000 3.0000 3.0000 σ1 (dB) 3.3941 3.3941 3.3941 3.3941 σ2 (dB) 3.3941 3.3941 3.3941 3.3941 Γ 7.1000 5.5000 14.0000 24.0000 γ 4.3000 6.7000 7.9000 12.0000 Λ (nsec−1) 0.0233 0.4000 0.0667 0.0667 λ (nsec−1) 2.5000 0.5000 2.1000 2.1000

models are particularly important when optimizingED radios as will be discussed in Chapter3.

48 CHAPTER 3 OPTIMIZATION OF ENERGY-DETECTION PPM RECEIVERS For low complexity, low cost, and very low power wireless applications, non- coherent modulation schemes are in most cases the best solution as explained in section 2.6. Non-coherent architectures are often implemented using PPM to modulate the binary information to be transmitted. To demodulate a PPM signal, a well-known technique is energy detection (ED). In many practical applications found in the UWB literature (e.g. [16, 17, 34, 47, 51, 92, 97]),ED is frequently chosen as the demodulation technique mainly because the receiver architecture is relatively simple to implement and offers usually offers very low-power operation. With PPM (brieﬂy introduced in subsection 2.6.2), a pulse is transmitted in one of two time windows to represent a binary logic 1 or 0. Therefore, to demodulate the received signal it is necessary to determine in which window the pulse was transmitted in. To do this, withED, the energy in each window is calculated and compared. Then, the pulse is assumed to be transmitted in the window with the highest energy. Figure 3-1 shows the general architecture ofED receivers. As shown, after channel selection and ampliﬁcation, the transmitted signal is squared and integrated in order to determine its energy. A bit decision is then carried out by comparing the energies in the two time windows (also known as integration windows). Due to its basic principle of energy comparison,ED receivers are very sensitive to channel noise (i.e. an increase in noise energy increases the probability of bit-error).

Figure 3-1: General architecture forED receivers

49 Assuming an AWGN channel, the noise energy varies linearly with the length of the integration window and the ﬁlter bandwidth due to its constant PSD. Hence, the BPF bandwidth and the integration time of the receiver play a vital role in the receiver’s performance and they must be carefully chosen in order to minimize the BER (i.e. probability of bit-error).

3.1 Chapter Contributions

In this chapter, the effects of reducing the receiver’s bandwidth and integration time are discussed and general equations to determine the their optimal values are derived. As discussed in subsection 3.2, these equations are of special interest for both system design and potential cognitive radios. Furthermore, different interference sources, namely ACI, ISI, and IFI, are taken into consideration to accurately predict the optimal optimal bandwidth and integration time of PPM-ED receivers in realistic wireless scenarios.

3.2 Previous work

There has been previous work to show there are optimal bandwidths and integration times that minimize the bit-errors induced at the receiver by UWB. In [88] and [20], the well-known BER equation for PPMED receivers is used to graphically show that there exists optimal receiver bandwidths and optimal integration times, respectively. In [71], the authors proposed an adaptiveED receiver and, once again, graphically demonstrate the optimal integration interval. The previous work has been able to demonstrate the existence of optimal values for the receiver bandwidth and integration time. However, it fails to provide general closed- form equations that accurately predict these optimal values as a function of system parameters such as the bandwidth of the transmitted signal and the target BER. Such general equations would be a great advantage for system designers since, otherwise, ﬁnding the optimal values time could lead to spend valuable time running simulations or writing code to solve BER equations numerically.

50 General equations for optimal bandwidths and integration times not only facilitate the design process but they may also be beneficial in PPM cognitive radios. For instance, a PPM system that adjusts the receiver bandwidth and/or integration time to the optimal value based on parameters such as the signal bandwidth or the target BER would spend less processing power by having built-in equations instead of algorithms to numerically find the optimal value. In addition to the lack of closed-form equations for the optimal bandwidths and integration times, the previous work does not take into account interference. In particular, mainly due to the multipath fading, ISI and IFI1 cannot be ignored when determining the optimal integration time. On the other hand, ACI becomes more relevant when there are multiple transmission channels relatively close to each other. In this case, ACI cannot be ignored when finding the optimal receiver bandwidth.

3.3 Energy-Detection Demodulation for PPM

When using PPM, a binary bit is modulated by transmitting a pulse in one of two integration windows. For instance, in Figure 3-2a, a pulse is transmitted in the ﬁrst integration window to represent a binary logic 1. A pulse in the second window would

have represented a logic 0. Assume that p (t) is the transmitted pulse, h (t) is the CIR, and n (t) is abbrefAWGN. Then, the received signal (see Figure 3-2b) can be expressed as

s (t) = p (t) ∗ h (t) + n (t) (3–1) where (∗) represents the convolution of two functions. With energy detection (ED) demodulation, the received signal s (t) is squared and integrated to determine the total energy in each window and the bit decision is made by comparing these energies, i.e.

1 At this point, inter-frame interference (IFI) has not been introduced yet. It will be explained in section 3.5.

51 (a) Transmitted signal (binary logic 1) (b) Received signal (with AWGN)

(c) Received signal after squaring (d) Received signal after squaring and integration (i.e. signal energy) Figure 3-2: Signal processing for aED-PPM receiver

1 Tw 2·Tw 2 2 E1 = [s (t) + n (t)] dt ≷ E2 = [s (t) + n (t)] dt (3–2) ˆ0 ˆTw 0 where Tw is the integration time. Figure 3-2c shows the received signal after self-mixing, i.e. [s (t) + n (t)]2. Finally, Figure 3-2d shows the integration of the squared signals in each integration window, i.e. the energies in E1 and E2. Note that E1 < E2 which results in an error since the pulse was transmitted in the ﬁrst integration window. This is a bit-error caused by AWGN and an example of the previous argument thatED receivers are very sensitive to noise.

52 3.3.1 Probability of Bit-Error

In the previous example (Figure 3-2), a bit-error occurred because the energy in

the second integration window E2 was higher than the energy in the first window E1 and the pulse was transmitted in the first window. Consequently, if the same example is used, the probability of bit-error is PED = P (E1 ≤ E2) = P (E1 − E2 ≤ 0). The expression to calculate this probability is derived in [64] and summarized in AppendixA for the convenience of the reader. If the received signal is sampled at the Nyquist rate 2 and the number of samples in each integration window is sufficiently large (40 or more samples according to [20]) then, by employing the CLT, PED can be approximated by

  Eb/N0   PED ≈ Q p  (3–3) 2 · B · Tw + 2 · Eb/N0 where Eb is the energy per bit, Eb / N0 is often called the SNR-per-bit, B is the signal bandwidth, Tw is the integration time, and

√ ∞ Q (x) = 1/ 2 · π · exp −u2/2 du (3–4) ˆx

is the well-known Q-function.

3.4 Optimal Receiver Bandwidth

When designing wireless radios, the receiver bandwidth (i.e. BPF bandwidth) is typically chosen to be the 10 dB-bandwidth of the transmission signal [4]. However, in most cases, this bandwidth is not the optimal value. Typically, the optimal receiver bandwidth is smaller than the 10 dB-bandwidth of the signal as will be shown later in this chapter. Furthermore, a smaller receiver bandwidth relaxes the speciﬁcations on the

2 The Nyquist rate is the minimum sampling rate for which the sampled signal retains all the properties of the original signal.

53 receiver circuits (e.g. LNA bandwidth, ADC sample rate) while reducing the integration of noise energy into the system.

3.4.1 Effect of Receiver Bandwidth Reduction

In the introduction of this chapter, the importance of the BPF was motivated based on the fact thatED receivers are very sensitive to noise. Thus, the impact of noise on receiver’s BER performance can be reduced by maximizing the SNR.

This can be inferred from Equation 3–3 since Eb/N0 is proportional to SNR, i.e. SNR

−1 = (Eb/N0) · (B · Tw ) . By carefully choosing the BPF bandwidth, the SNR can be maximized. To illustrate this, let

  Ps − ∆Ps Ps 1 − ∆Ps/Ps SNR = = ·   (3–5) Pn − ∆Pn Pn 1 − ∆Pn/Pn where, as shown in Figure 3-3, ∆Ps and ∆Pn are the reduction in signal and noise power, respectively, and Ps and Pn are the nominal signal and noise power when the receiver bandwidth is equal to the 10 dB-bandwidth of the transmitted signal. As the BPF bandwidth is decreased from the 10 dB-bandwidth, the numerator of equation (3–5) decreases at a slower rate than the denominator, i.e. ∆Ps/Ps <

∆Pn/Pn. Hence, the SNR increases until ∆Ps/Ps is no longer smaller than ∆Pn/Pn. The

Figure 3-3: Power spectral densities of a square pulse, Gaussian pulse, and AWGN

54 Figure 3-4: Signal and noise energy proﬁle as a function of the receiver bandwidth

maximum value for SNR is achieved when the rates of change ∆Ps/Ps and ∆Pn/Pn are equal. This can be graphically seen in Figure 3-4. Both signal and noise power increase as the receiver bandwidth increases; however,∆Ps/Ps < ∆Pn/Pn remains true until the slope (i.e. rate of change) of the signal power is equal to the slope of the noise energy, i.e. ∆Ps/Ps = ∆Pn/Pn. At this point, the SNR is maximized. 3.4.2 Modiﬁed Probability of Bit-Error and Optimal Receiver Bandwidth

In this subsection, Equation 3–3 is modiﬁed to include the effect of reducing the BPF bandwidth. For convenience and comparison purposes, the receiver bandwidth is normalized to the commonly used 10 dB-bandwidth of the transmitted signal, i.e.

β= ∆f /B10dB where ∆f is the BPF bandwidth. Furthermore, square pulses are used as the transmitted signal since they are typically used in the implementation of UWB radios mainly due to the simplicity to generate them. Nevertheless, the results are similar for other pulse shapes such as Gaussian since the PSD of a Gaussian pulse is comparable to that of a square pulse inside the 10 dB-bandwidth (less than 3% of power difference) as shown in Figure 3-3.

55 3.4.2.1 Probability of Bit-Error and Receiver Bandwidth

The Fourier transform of a square pulse is given by the sinc function

sin π · Tp · f

VTp (f ) = Tp · (3–6) π · Tp · f 2 where Tp pulse time width. The ESD is given by SE (f ) = VTp (f ) . Then, integrating

SE (f ) over a range of frequencies gives the total energy of the signal over that range. Thus, the pulse energy (i.e. energy per bit) based on the BPF bandwidth ∆f is

2 ∆f /2 ∆f /2 sin π · Tp · f

Eb (∆f ) = SE (f ) · df = 2 · Tp · · df (3–7) ˆ−∆f /2 ˆ−∆f /2 π · Tp · f

cos π · Tp · ∆f + π · Tp · ∆f · Si π · Tp · ∆f − 1 E (∆f ) = 2 · (3–8) b π2 · ∆f x where Si (x) = 0 sin (t) /t · dt. The 10 dB-bandwidth of the transmitted signal can ´ be calculated using the Taylor series of a sinc function which yields the following approximation

B10dB ≈ 1.476/Tp (3–9)

With this approximation and the previously deﬁned β = ∆f /B10dB , Equation 3–8 becomes

cos (1.476 · π · β) + π · Tp · ∆f · Si (1.476 · π · β) − 1 β · Eb ( ) = 2 2 (3–10) π · β · B10dB To account for the reduction in the detected signal energy due to a receiver bandwidth reduction, an energy scaling factor can be deﬁned as

δ (β) = Eb (β) /Eb (β = 1) (3–11)

56 and incorporated into Equation 3–3. Then, the probability of bit-error can be rewritten as

  δ (β) · (Eb/N0)   PED (β) ≈ Q p  (3–12) 2 · Tw · β · B10dB + 2 · δ (β) · (Eb/N0)

and solving it for Eb/N0 gives the required SNR-per-bit SNRbit to achieve certain BER performance

v  u  2 u Eb K t 2 · Tw · β · B10dB β ·   SNRbit ( ) = = 1 + 1 + 2  (3–13) N0 δ (β)  K 

−1 −1 −1 where K = Q (PED) = Q (BER) and Q (·) is the inverse Q-function. Equation 3–13 can be used to obtain an optimal bandwidth as will be shown next.

3.4.2.2 Optimal Receiver Bandwidth

The optimal receiver bandwidth βopt is the value of β that minimizes Equation 3–13,

i.e. minimizes the required SNRbit . Hence,

  ∂ β = Solve  [SNR (β) = 0] , β (3–14) opt ∂β bit 

Equation 3–14 cannot be solved explicitly mainly due to the complexity of the scaling

factor δ (β) and the square root term. Thus, to obtain a solution for βopt , approximations

p 2 for 1 + 2 · Tw · β · B10dB/K and δ (β) are needed. For mathematical simplicity, a non-linear least-square regression is used with the exponential ﬁt

yi = ai + bi · exp (ci + di · x) (3–15) where x is the regression parameter (i.e. the independent variable), ai , bi , ci , di are √ constants, and i identiﬁes the desired approximation, i.e. i = 1 refers to 1 + X where

2 X = 2 · Tw · β · B10dB/K and i = 2 to δ (β)). The values for the constants are shown

57 Table 3-1: Constant values for the exponential ﬁt given by Equation 3–15

i Approximation Regression ai bi ci di Term Parameter 1 ≤ X ≤ 10 4.82 −0.66 1.73 −0.089 √ 1 1 + X 10 ≤ X ≤ 50 10.78 −1.35 1.89 −0.018 50 ≤ X ≤ 250 22.90 −11.77 0.49 −0.004 2 δ (β) 0.3 ≤ β < 1 1.138 −0.10 2.60 −2.35

Table 3-2: Constant values for βopt

Range Aβ Bβ Cβ Dβ 1 ≤ M ≤ 10 0.775 67.5 17.1 1.355 10 ≤ M ≤ 50 0.766 50.0 19.5 0.350 50 ≤ M ≤ 250 0.742 120.0 33.0 0.100

in Table 3-1. Now, using Equation 3–15 to estimate Equation 3–13 gives the following approximation

1 + a1 + b1 · exp (c1 + d1 · M · β) 2 SNRbit (β) ≈ K · (3–16) a2 + b2 · exp (c2 + d2 · β)

2 where M = 2 · Tw · B10dB/K . Equation 3–16 can be used to solve Equation 3–14. Taking its derivative and then solving for β gives the optimal normalized bandwidth βopt which can be approximated by

ln Bβ/M βopt = Aβ + (3–17) Cβ + Dβ · M The constants in Equation 3–17 are summarized in Table 3-2 for three different ranges of M and a maximum error of less than 1%.

3.4.3 Adjacent-Channel Interference

The analysis in this section assumes square pulses as the interference signal. The idea behind this analysis is to gain a better understanding on how the BPF bandwidth

58 changes the BER performance of a PPMED receiver in the presence of other transmitters with similar signals but operating in adjacent channels, therefore, producing ACI .

3.4.3.1 Effect of ACI on the Receiver Performance

Since the FCC approval for UWB wireless systems, several papers related to UWB -speciﬁc interference have been published and among them are [28, 31, 49, 57, 87]. Different approaches to model in-band UWB interference can be found in literature. However, many of them agree in a Gaussian approximation model (e.g. [28, 57, 87]). Consequently, here ACI is treated as AWGN and modeled with a ﬂat power spectrum. Figure 3-5 shows the spectrum of the transmitted signal, the BPF, the ACI signals, and the interference PSD I0. The shaded region represents the interference energy Ei . To calculate this energy, Equation 3–7 can be used with different integration limits. Then,

2 fch+∆f /2 sin π · Tp · f E (f , ∆f ) = T · · df (3–18) i ch ˆ p π · T · f fch−∆f /2 p

where fch is the frequency space between the transmitted signal and the interference signal (i.e. channel spacing) and ∆f is the BPF bandwidth (see Figure 3-5). Again, for convenience, fch and ∆f are normalized to the 10 dB-bandwidth of the signal, i.e.

Figure 3-5: PSD of the transmitted signal and ACI signals

59 α = fch/B10dB and β = ∆f /B10dB. Then, by solving (3–18), the ACI energy can be expressed as

cos (c · ρ1) cos (c · ρ2) 2 · β + + c · [Si (c · ρ ) + Si (c · ρ )] − ρ ρ 1 2 ρ · ρ α β 1 2 1 2 Ei ( , ) = 2 (3–19) π · B10dB

where c = 1.476 · π, ρ1 = β − 2 · α, and ρ1 = β + 2 · α. Now, recall the assumption of ﬂat

power spectrum for ACI. Then, the interference spectral density is given by I0 = Pi /∆f

where Pi is the interference average power and, for a single 1-sided adjacent interferer,

it can be approximated by Pi = Ei (α, β) /Ts where Ts = 2 · Tw . Therefore, the spectral density of multiple 2-sided adjacent interferers can be written as

m 2 X Ei (k · α, β) I0 (α, β) = · (3–20) Tw · β · B10dB εk k=1 where m is the total number of 2-sided interferers (e.g. m = 1 in Figure 3-5) and

th εk = Eb (β = 1) /Ei (0, 1) is the SIR of the k 2-sided interferer.

To account for ACI, an effective SNRbit , i.e. (Eb/N0)eff , can be used in Equation 3– 3. Since the interference is modeled by a Gaussian approximation as motivated earlier, the effective SNRbit can be expressed as (Eb/N0)eff = Eb/ (N0 + I0) which is often called the signal-to-interference-and-noise ratio (abbrSINRSignal-to-interference-and-noise ratio). Here, N0 = Eb/ (Eb/N0), I0 = I0 (α, β), and Eb ≈ Eb (β = 1). Thus,

   −1 Eb 1 I0 (α, β)   =  +  (3–21) N0 Eb/N0 Eb (β = 1) eff

By substituting Eb/N0 for (Eb/N0)eff to account for ACI, Equation 3–12 can be rewritten as

60   −1   1 I0 (α, β)   δ (β) ·  +     / β    Eb N0 Eb ( = 1)    PED (α, β) ≈ Q v  (3–22) u  −1 u   1 I0 (α, β)  u    u2 · Tw · β · B10dB + 2 · δ (β) · +  t Eb/N0 Eb (β = 1)

and solved for Eb/N0 which yields the required SNRbit as a function of the receiver bandwidth β and the channel spacing α, i.e.

 −1 δ (β) 1   SNRbit (α, β) = − (3–23)  2 p 2 SIRbit (α, β) K · 1 + 1 + 2 · Tw · β · B10dB/K

where SIRbit (α, β) = Eb (β = 1) /I0 (α, β). 3.4.3.2 An Approximation for the Optimal Receiver Bandwidth in the Presence of ACI

Equation 3–23 can only be solved numerically due to its complexity. However, by following the approach in subsection 3.4.2.2 and making some assumptions, an expression for βopt can be obtained. In this case, the assumptions are as follows: 1. Only the ﬁrst 2-sided ACI is signiﬁcant (i.e. m = 1) and it has a unitary SIR (i.e.

ε1 = 1). 2. Using the exponential ﬁt given by Equation 3–15,

1 E (α, β) ≈ · [exp (3.35 + 3.85 · ρ ) − exp (3.35 + 3.85 · ρ )] (3–24) i 2 1 2

3. The ACI signal has the same bandwidth as the transmitted signal. For these assumptions, the optimal bandwidth can be calculated by using the expression for SNRbit (α, β) given in Equation 3–23 and solving Equation 3–14 . This yields the following approximation

61 Figure 3-6: Simulator block diagram

ln Bβ/M − Eβ/α ln Fβ/α βopt ≈ Aβ + (3–25) Cβ + M · Dβ where Aβ, Bβ, Cβ, Dβ are given in Table 3-2, Eβ = 5.20, Fβ = 0.89, and 10 ≤ M < 50.

2 Recall that M = 2 · Tw · B10dB/K . 3.4.4 Simulation Setup and Validation

To support and corroborate the theory developed in this section, a simulator for a

PPM-ED receiver was built in MATLAB R . Its setup is discussed next followed by the simulations run to validate that it is properly working.

3.4.4.1 Setup

Figure 3-6 shows the simulator block diagram. It has three major parts: modulation, channel modeling, demodulation. In the modulation part, stream of binary bits are randomly generated. These are then modulated and up-converted using PPM and square pulses as the transmission signal. To simulate the wireless channel, the channel modeling part adds AWGN and ACI to the transmitted signal as follows:

62 1. AWGN: Since Eb/N0 is speciﬁed, then N0 can be calculated if the Eb is known. It can be calculated by squaring and integrating the modulated signal for one bit. Thus,

N Eb 1 X 2 N0 = = · si (3–26) (Eb/N0) (Eb/N0) i=1 th where si is the i sample of the received signal and N = 2 · Tw · β · B10dB is the total number of samples. The one-sided noise average power can be calculated as

N N0 fs X 2 P0 = · fs = · si (3–27) 2 2 · (Eb/N0) i=1 Finally, the noise signal is realized by generating N random values that are

2 normally-distributed with zero-mean and variance σ =P0 . 2. ACI: To generate interference, another stream of random bits is generated, modulated, and up-converted to an adjacent channel. The amplitude of the ACI

signals are scaled using the specified SIR, defined as εk in Equation3–20, such that the following equation is satisfied

N N X 2 X 2 si = ε1 · Ii (3–28) i=1 i=1 th where Ii is the i sample of the ACI signal. Finally, the demodulation part simulates the signal processing at theED receiver. The received signal is ﬁrst ﬁltered, then squared and integrated over periods of Tw . Each pair of integration windows is then compared and a bit decision is made based on the highest energy.

3.4.4.2 Validation

To validate the simulator, several simulations were run to compare the results with the well-known BER expression given by Equation 3–3. Figure 3-7 shows the simulated

63 Figure 3-7: Comparison between simulations and Equation 3–3 to validate the simulator

BER curves for two signal bandwidths (1 GHz and 2 GHz) and an integration time of 30 ns along with the ideal curves obtained using Equation 3–3. The simulated values closely agree with the ideal values. Hence, the simulator can predict with accuracy the BER performance for a PPM-ED receiver and it will be used next to corroborate the theory developed throughout this section.

3.4.5 Analysis

3.4.5.1 Theory Corroboration

The main equations derived in this section are Equation 3–12 and Equation 3– 22 which are expressions to calculate the BER of a PPM-ED receiver as a function of its bandwidth. Equation 3–12 does not take into account ACI while Equation 3–22 considers ACI under certain assumptions as stated in subsection 3.4.3.2. The other important equations are Equation 3–13, Equation 3–17, Equation 3–23, and Equation 3–25. These are just algebraic manipulations of Equation 3–12 and Equation 3–22. Thus, to corroborate the theory developed, it is sufﬁcient to verify that Equation 3–12 and Equation 3–22 hold. Figure 3-8(a) shows a plot of BER for the ideal values obtained using Equation 3–12 and the results from the simulations for β = 1. It can be seen that

64 (a) No ACI, Equation 3–12 (b) ACI with α = 0.8, Equation 3–22

Figure 3-8: Comparison between simulations, Equation 3–12, and Equation 3–22

Equation 3–12 yields values that are very close to those simulated and, therefore, it holds. Similarly, Figure 3-8(b) corroborates Equation 3–22.

3.4.5.2 Numerical Results

Designers in wireless communications use link budgets when implementing wireless radios. An important parameter for the link budget is the SNRbit (or Eb/N0) required to obtain a desired BER. This value can be calculated with Equation 3–13 and Equation 3–23 as a function of receiver bandwidth, desired BER and, in the case of

Equation 3–23, interference frequency spacing. Figure 3-9 shows the required Eb/N0 as a function of the normalized receiver bandwidth β to achieve a BER of 10−3. Note that the lowest value of Eb/N0 corresponds to the optimal receiver bandwidth. However, a smaller receiver bandwidth can provide hardware advantages (e.g. lower sampling rate, lower power consumption) at a minimal cost in the required Eb/N0. For instance, for a 1 GHz signal, the receiver bandwidth could be reduced by half (i.e. β = 0.5) with a loss in Eb/N0 of less than 1 dB. If the system can tolerate this degradation, the beneﬁts for the system may be signiﬁcant (e.g. half the sampling rate, lower power consumption, better input matching).

65 −3 Figure 3-9: Required SNRbit to achieve a BER = 10

From Figure 3-9, it is also clear that this optimal receiver bandwidth is a function of the signal’s 10 dB-bandwidth B10dB . Equation 3–17 is an accurate approximation for the normalized optimal receiver bandwidth βopt and it is plotted against B10dB in Figure 3-10. Note that having a larger signal bandwidth increases the savings in receiver bandwidth. In other words, the optimal receiver bandwidth becomes smaller with respect to the signal bandwidth (recall β = ∆f /B10dB ) as the latter increases. 3.5 Optimal Integration Time

AnED-PPM receiver essentially squares and integrates the received signal to determine its energy in two time windows. Then, by comparing both energies, a bit decision is made. As discussed before, this principle of energy detection makes the receiver more vulnerable to noise and, since both the signal and noise energies are proportional to the integration time, Tw must be carefully chosen so that the BER can be minimized.

3.5.1 Effect of Integration Time due to Multipath Fading

The energy of a transmitted pulse is spread in time due to the effect of multipath fading. Thus, to detect the pulse energy, the received signal must be integrated for a signiﬁcantly larger time than the pulse width. A larger integration time allows the

66 Figure 3-10: Normalized optimal receiver bandwidth versus the signal’s 10 dB- −3 bandwidth for Tw = 30 ns, α = 1, and BER = 10

detection of most of the signal energy (Eb ), however, it also integrates more noise energy (E0) into the system which degrades its BER performance. Hence, by carefully choosing the integration time, the SNR can be maximized just like choosing the optimal receiver bandwidth as discussed in section 3.4.

To show the concept of SNR maximization let SNR= Pb/P0, where Pb and

P0 are the average power of the transmitted signal and noise, respectively. Since

3 Eb =Pb ·Tw and E0 =P0 ·Tw = 2·N0 ·B ·Tw where B is the receiver bandwidth, then

SNR= Eb/E0. Figure 3-11 shows Eb and E0 as a function of Tw . As Tw increases

both Eb and E0 increase but at different rates. When the increase rate (slope) of Eb

becomes smaller than the constant slope of E0 , the SNR reaches its maximum.

3 The focus of this section is the integration time Tw and, hence, the receiver bandwidth is assumed to be B =B10dB .

67 Figure 3-11: Signal and noise energy proﬁle as a function of integration time

3.5.2 Modiﬁed Probability of Bit-Error and Optimal Integration Time

Following the approach used in section 3.4 for the receiver bandwidth, in this subsection a scaling factor for Eb is employed to modify Equation 3–3. The modiﬁed equation is then used to derive the optimal integration time.

3.5.2.1 Probability of Bit-Error and Integration Time

The received signal s(t) can be represented by the convolution of the CIR h (t) and the transmitted pulse p(t), i.e.

s(t) = h(t) ∗ p(t) (3–29)

Then, the energy of the received signal as a function of integration time can be calculated as

f ·T Tw Xs w E (T ) = s2(t) · dt = s2/f (3–30) b w ˆ i s 0 i th where si is the value of the i sample of the received signal s(t), fs is the sampling frequency and Tw is the integration time. Normalizing Equation 3–30 to the total energy of the transmitted signal, i.e. Eb(Tw ≈ ∞), gives the energy scaling factor

Eb(Tw ) γ(Tw ) = . (3–31) Eb(∞)

68 To obtain the modiﬁed BER expression, this factor can be applied to Equation 3–3. The new expression for the probability of bit-error that accounts for the energy spread of the signal is

! γ(Tw ) · (Eb/N0) PED (Tw ) ≈ Q p (3–32) 2 · Tw · B + 2 · γ(Tw ) · (Eb/N0)

Solving Equation 3–32 for Eb/N0 gives the SNRbit required to achieve a given BER, i.e.

r ! E K 2 2 · T · B b · w SNRbit (Tw ) = = 1 + 1 + 2 (3–33) N0 γ(Tw ) K where K = Q−1(BER).

3.5.2.2 Optimal Integration Time

For given values of BER and signal bandwidth B , Equation 3–33 predicts the

required SNRbit as a function of the integration time. Thus, by minimizing SNRbit (Tw ), the optimal integration window( Tw )opt can be obtained, i.e.

∂ (Tw )opt = Solve [SNRbit (Tw )] = 0 , Tw . (3–34) ∂Tw

Since the scaling factor γ(Tw ) is obtained numerically using the UWB channel modeling presented in subsection 2.8.2 and Tw appears in the square root term of Equation 3– 33, there is no explicit solution for Equation 3–34. Following the approach in subsection

p 2 3.4.2.2, approximations for 1 + 2 · Tw · B/K and γ(Tw ) are employed to get the solution of Equation 3–34. These approximations are obtained using a non-linear regression with an exponential ﬁt

Yj = aj + bj · exp cj + dj · Xj (3–35) √ where j identiﬁes the approximation parameter (i.e. Y1 ≈ γ(Tw ) and Y2 ≈ 1 + Z where

2 Z = 2 · Tw · B/K ), Xj is the independent variable (in this case Tw or Z), and aj , bj , cj ,

69 Table 3-3: Constant values for the exponential ﬁt given by Equation 3–35

j Approximation Regression aj bj cj dj Term Parameter 1 ≤ Z < 10 4.82 -0.66 1.73 -0.089 √ 1 1 + Z 10 ≤ Z < 50 10.78 -1.35 1.89 -0.018 50 ≤ Z < 300 22.90 -11.77 0.49 -0.004

2 ≤ Tw < 30 (CM 1) 0.993 -0.122 2.04 -0.178 8 ≤ Tw < 42 (CM 2) 1.025 -0.129 2.371 -0.094 2 γ(Tw ) 21 ≤ Tw < 102 (CM 3) 1.035 -0.112 2.517 -0.037 30 ≤ Tw < 147 (CM 4) 1.028 -0.081 2.847 -0.027

Figure 3-12: Energy scaling factor γ(Tw ) for each UWBCM reported in [28]

dj are constants summarized in Table 3-3 for each channel model (CM) presented in subsection 2.8.2.

Figure 3-12 plots the energy scaling factor γ(Tw ) with their corresponding exponential ﬁts. As shown, Equation 3–35 yields accurate approximations (< 1% of error). Thus, it is used to estimate Equation 3–33, i.e. SNRbit (Tw ) can be approximated by

2 1 + a2 + b2 · exp (c2 + d2 · M · Tw ) SNRbit ≈ K · (3–36) a1 + b1 · exp (c1 + d1 · Tw )

2 where M = (2 · B)/K . The optimal integration time( Tw )opt can be determined using this approximation in Equation 3–34 . The solution then yields

70 Table 3-4: Constant values for( Tw )opt

9 −9 RangeCM Aw Bw (×10 ) Cw Dw (×10 ) 0.1 ≤ M < 1 1 8.80 13.40 0.58 1.90 0.1 ≤ M < 1 2 21.90 23.50 0.46 2.30 0.1 ≤ M < 1 3 53.10 36.50 0.28 1.60 0.1 ≤ M < 1 4 73.10 36.40 0.23 1.50 1 ≤ M < 30 1 7.20 22.66 1.00 0.215 1 ≤ M < 30 2 20.40 21.35 0.95 0.250 1 ≤ M < 30 3 50.70 49.95 0.66 0.260 1 ≤ M < 30 4 69.90 55.00 0.60 0.190

ln (Bw /M) (Tw )opt ≈ Aw + (3–37) Cw + M · Dw

where( Tw )opt is given in nanoseconds and the values of constants Aw , Bw , Cw , Dw for eachCM can be found in Table 3-4.

3.5.3 Inter-Symbol and Inter-Frame Interference

ISI and IFI are inherit effects of multipath fading and, hence, should not be ignored particularly when optimizing aED-PPM wireless system. Thus, this subsection takes into account their effect.

3.5.3.1 Effect of ISI and IFI on the Receiver Performance

ISI and IFI can be approximated as Gaussian using the central limit theorem when the number of channel realizations and successive bits are sufﬁciently large [1]. Therefore, here ISI and IFI are treated as AWGN. Furthermore, they are taken as a whole so that the total interference energy is the energy of both ISI and IFI as it is explained next. Figure 3-13 shows two UWB pulses under multipath fading and transmitted in

different integration windows. In Figure 3-13(a), the signal energy Eb is that within the ﬁrst integration window, the IFI is the signal in the second integration window within the same symbol period, and theISI is the remaining signal that interferes with the next symbol. Figure 3-13(b) shows a pulse transmitted in the second integration window.

71 (a) Pulse in the ﬁrst window (b) Pulse in the second window Figure 3-13: Illustration of ISI and IFI

Note that, in this case, there is no IFI, i.e. the entire excess signal interferes with the next symbol. For a large number of transmitted bits that can randomly be in any of the two integration windows, ISI and IFI can be averaged together to determine the total interference (simulations corroborate this assumption). Consequently, from now on, both interferences are addressed only as ISI for simplicity and its total energy can be calculated as

∞ Tw 2 2 Ei (Tw ) = SRx (t) dt − SRx (t) dt (3–38) ˆ0 ˆ0 which, by using Equation 3–30, is the same as Ei (Tw ) = Eb(∞) − Eb(Tw ) where Eb(∞) is just the total bit energy. Recalling the assumption of the Gaussian approximation for ISI [1], the average interference spectral density can be estimated as

Eb(∞) − Eb(Tw ) Eb(∞) · [1 − γ(Tw )] I0(Tw ) = = (3–39) 2 · Tw · B 2 · Tw · B

Then, to account for ISI, an effective SNRbit , i.e. (Eb/N0)eff , can be applied to Equation

3–3. This effective SNRbit includes the energy scaling factor γ(Tw ) as well as I0(Tw ) and can be expressed as (Eb/N0)eff = γ(Tw ) · Eb/(N0 + I0). The ratio Eb/(N0 + I0) is often called the signal-to-interference-and-noise ratio per bit (SINR-per-bit). Thus, (Eb/N0)eff

72 is essentially the SINR-per-bit with the exception that the scaling factor γ(Tw ) has been

included. Now, let N0 = E(∞)/(Eb/N0) and I0(Tw ) = E(∞) · [1 − γ(Tw )]/(2 · Tw · B). Then, the effective SNRbit can be expressed as

−1 1 [1 − γ(Tw )] (Eb/N0)eff = γ (Tw ) · + . (3–40) (Eb/N0) 2 · Tw · B

By substituting (Eb/No)eff for (Eb/No) , Equation 3–3 and Equation 3–33 can be respectively rewritten as

! (Eb/N0)eff PED (Tw ) ≈ Q p (3–41) 2 · Tw · B + 2 · (Eb/N0)eff and

−1 " 2 # γ(Tw )/K 1 − γ(Tw ) SNRbit (Tw ) = − (3–42) p 2 1 + 1 + 2 · Tw · B/K 2 · Tw · B

3.5.3.2 Optimal Integration Time

The optimal integration time can be obtained by optimizing Equation 3–42 which has no explicit solution as explained in Section 3.5.2.2. Thus, once again, exponential

ﬁts are used resulting in the same approximation given by Equation 3–37, i.e.( Tw )opt

≈ Aw +ln(Bw /M)/(Cw +M ·Dw ), however, with different constant values. These can be found in table 3-5. For convenience, Table 3-6 shows( Tw )opt for several signal bandwidths and three different values of BER.

3.5.4 Simulation Setup and Validation

To corroborate and support the equations derived in this section, a simulator similar to the one presented in subsection 3.4.4 was built in MATLAB R . This section brieﬂy describes its implementation and validation.

3.5.4.1 Setup

The simulator’s block diagram is shown in Figure 3-14. Just as the simulator presented in subsection 3.4.4, it comprises three main parts: modulation, channel

73 Table 3-5: Constant values for( Tw )opt when ISI is considered

−9 RangeCM Aw Bw Cw Dw (×10 ) 0.1 ≤ M < 1 1 11.06 19.7 0.398 1.130 0.1 ≤ M < 1 2 24.20 23.9 0.305 0.988 0.1 ≤ M < 1 3 56.78 61.7 0.192 0.730 0.1 ≤ M < 1 4 77.67 50.1 0.150 0.628 1 ≤ M < 30 1 8.03 30.6 0.736 0.040 1 ≤ M < 30 2 20.95 60.9 0.629 0.115 1 ≤ M < 30 3 51.87 39.6 0.344 0.057 1 ≤ M < 30 4 71.28 50.0 0.300 0.054

Figure 3-14: Simulator block diagram modeling, and demodulation. The main difference is that here multipath fading is modeled and ACI and ﬁltering are not included as the main focus in this section is the integration time and not the receiver bandwidth. To simulate the wireless channel, the channel modeling part include AWGN and multipath fading as follows: 1. AWGN: Similar to the previous simulator, Equation 3–27 can be used to calculate

the noise power P0 . Then, the noise signal is is realized by generating random

2 values that are normally-distributed with zero-mean and variance σ =P0 .

74 Table 3-6: Optimal integration times for different values of signal bandwidth and BER

Bandwidth Optimal Integration Time( T ) (nsec) BER w opt (GHz) CM1 CM2 CM3 CM4 0.5 10−3 21.20 37.49 80.54 106.27 1.0 10−3 18.22 33.45 73.27 97.13 2.0 10−3 15.48 29.83 66.81 89.25 4.0 10−3 13.41 27.16 62.13 83.72 7.0 10−3 11.85 25.62 59.58 80.59

0.5 10−4 22.75 39.61 84.35 111.14 1.0 10−4 19.81 35.60 77.13 101.95 2.0 10−4 16.88 31.67 70.08 93.21 4.0 10−4 14.42 28.45 64.38 86.36 7.0 10−4 12.42 26.45 61.00 82.28

0.5 10−5 23.84 41.11 87.03 114.60 1.0 10−5 20.99 37.21 80.03 105.62 2.0 10−5 18.01 33.18 72.77 96.51 4.0 10−5 15.31 29.60 66.42 88.78 7.0 10−5 13.62 27.42 62.60 84.26

2. Multipath Fading: The CIR is generated using the multipath model presented in

subsection 2.8.2. Then, the convolution of the modulated signal and the CIR is carried out.

3.5.4.2 Validation

To validate the simulator, simulation results are compared to the well-known BER expression for aED-PPM receiver, Equation 3–3. Figure 3-15 shows BER curves obtained using both the simulator and Equation 3–3 for a 2 GHz signal and three different integration times. As can be seen, the simulator predicts with accuracy the BER performance of theED-PPM receiver. Thus, it is used next to corroborate the equations derived in this section.

75 Figure 3-15: Comparison of simulations and Equation 3–3 to validate the simulator

3.5.5 Analysis

3.5.5.1 Theory Corroboration

The main equations derived in this section are Equation 3–33, Equation 3–37, and Equation 3–42. These are obtained by algebraic manipulation from Equation 3–32 and Equation 3–41. Therefore, verifying that these two equations hold must be sufﬁcient to corroborate that the rest of the equations hold as well. In Figure 3-16(a), the ideal values obtained from Equation 3–32 and the simulation results are plotted for comparison. As illustrated, the BER curves closely agree which corroborate that Equation 3–32 holds. Likewise, Figure 3-16(b) corroborates Equation3–41.

3.5.5.2 Numerical Results

When designing wireless systems, an important parameter of the link budget is

the SNR-per-bit (Eb/N0) required by the system to yield certain probability of bit-error. Equation 3–33 and Equation 3–42 can be used to calculate this value as a function of the system parameters (i.e. signal bandwidth, integration time). In contrast to Equation 3–33, Equation3–42 takes into account ISI. An example is presented in Figure 3-17 which shows two plots. The ﬁrst, Figure 3-17(a), corresponds to the (Eb/N0) required to

76 (a) No ISI, Equation 3–32 (b) ISI , Equation 3–41

Figure 3-16: Comparison between simulations and the modiﬁed BER equations with B = 2 GHz and Tw = 25, 30, 80, 100 ns forCM 1 through 4, respectively achieve a BER = 10−5 when no ISI is considered while in the second, Figure 3-17(b),

ISI is taken into account. The lowest values of (Eb/N0) determine the optimal integration times which are marked with arrows in the figure. However, if the system can tolerate certain degradation in performance, the integration time can be decreased below its optimal value. For instance, forCM 3, the optimal integration time is around 72 nsec with (Eb/N0) = 20 dB. If the system can tolerate a 1 dB degradation in (Eb/N0), the integration time can be reduced to 41 nsec. This reduction could easily translate into valuable system benefits such as lower power consumption and higher data rate. From Figure 3-17, it is evident that the ISI increases the optimal integration time. This is because in order to reduce the ISI effect the integration windows must be larger compared to the case where ISI is ignored. This is also shown in Figure 3-18 where the optimal integration time for eachCM (with and without ISI) is plotted as a function of the signal bandwidth. In general, the integration time is significantly larger when considering ISI meaning that its effect on the system performance cannot be neglected.

77 (a) No ISI (b) ISI

−5 Figure 3-17: Required SNRbit to achieve a BER= 10 for B = 2 GHz.

−5 Figure 3-18: Optimal integration time( Tw )opt to achieve BER= 10

78 −5 Figure 3-19: Required SNRbit to achieve BER = 10

Figure 3-18 also shows that the optimal integration time decreases as the signal bandwidth increases. This is due to the fact that a larger bandwidth increases the noise power (recall P0 = 2·N0 ·B) and, consequently, the noise energy. Thus, to account for larger noise energy, a reduction in the integration window is necessary. However, the reduction has to be small enough to keep the effect of ISI at an acceptable level. Nevertheless, that reduction in integration time as the signal bandwidth increases comes at a cost. Figure 3-19 shows the SNR-per-bit required to achieve a BER of 10−5 when the integration time is chosen to be the optimal value. As illustrated, increasing the signal bandwidth degrades the overall system performance (i.e. a higher SNRbit is necessary to achieve the same BER). Again, here the SNRbit performance trade-off can been seen. For instance, in this case if the system could tolerate a 1 dB degradation in the required SNRbit , the signal bandwidth could be almost doubled resulting in a potential increase in data rate.

3.6 Summary

This chapter presented the derivation of analytical expressions to determine optimal values for the receiver bandwidth and integration time forED-PPM radios. The

79 discussion and analysis developed included the effect of three interference sources: ACI for the optimal bandwidth, and ISI and IFI for the optimal integration time. These equations are very convenient when designingED-PPM receivers for UWB

wireless channels. Furthermore, if the channel can be estimated speciﬁcally γ (Tw ) then the equations might be useful in cognitive radios to dynamically adjust the integration time.

80 CHAPTER 4 ENERGY-INTEGRATION DETECTION FOR PPM RECEIVERS As explained in Chapter3,ED receivers are very sensitive to noise, that is, they are more vulnerable to noise than coherent demodulation techniques (e.g. BPSK). Therefore, the size of the integration window (i.e. integration time) is a crucial design parameter since it is directly proportional to the signal and noise energy captured from the channel [3]. Previous works have investigated different approaches to improve the BER performance ofED receivers. In [3, 20, 71], the authors show that there is an optimal integration time to minimize the probability of bit-error (i.e. BER). In Chapter3, equations for this optimal integration time were derived. However, the usefulness of these equations relies on a good approximation of the multipath fading channel. Furthermore, even if the channel approximations are good, they are only valid for certain channel conditions. Therefore, a radio optimized for certain channel conditions will show a signif- icant degradation if it operates under different environments. To reduce this degradation, in this chapter, energy-integration detection (EID) is proposed and compared toED to show its advantages.

4.1 Chapter Contributions

In this chapter, a demodulation technique (EID) based on the integration of the received signal energy rather than the signal energy alone is proposed. Effectively, it is similar to a weightedED demodulation using linear decreasing weights in each sample of the received signal but does not increase signiﬁcantly the complexity of the receiver and yet reduces the BER in comparison toED. This is demonstrated by the BER equation derived later in the chapter and corroborated by simulations.

81 4.2 Previous Work

There has been previous work aiming to improveED receivers by applying weights to each sample of the received signal [56, 82]. Applying weights to the received signal requires a priori knowledge of the CIR in order to accurately determine the weight magnitude for each sample so that samples carrying higher energy have increased detectability. On the other hand, if the weights are dynamically determined by perform- ing a channel estimation [82], a considerable amount of signal processing is needed, increasing the complexity of the receiver and, hence, its power consumption. Thus, in applications where architecture simplicity and very low-power operation are required, this approach might not be a viable solution.

4.3 Energy-Integration Detection

4.3.1 Motivation

When transmitting a signal over a wireless channel, the multipath fading effectively spreads its energy over time (see the received signal in Figure 3-2). Thus, to avoid IFI and ISI (see subsection 3.5.3), the integration time of anED receiver must be large enough to capture most of the signal energy. This time is often determined based on the worst-case multipath scenario. For instance, in [54, 99, 85], integration times between 30 nsec and 50 nsec are used since they are good approximations forED receivers to capture at least 99% of the signal energy in worst-case multipath scenarios during short-range communications in LOS and NLOS channels, respectively. Typically, designing for worst-case multipath scenarios results in integration times that are signiﬁcantly larger than those required in the average-case multipath scenario.

From Equation 3–3, it is easy to see that the probability of bit-error PED increases

as the integration time Tw increases since the Q-function is a strictly decreasing function. Therefore, anED receiver designed for worst-case multipath scenarios (i.e.

larger Tw ) experiences a performance degradation when it operates in average-case multipath scenarios. In Chapter3, it was shown that there is an optimal integration

82 time that minimizes the SNRbit to achieve certain probability of error. This optimal integration time is obtained by assuming the receiver will operate under one of the UWB channel models described in the IEEE P802.15 working group report [28]. For instance, the optimal integration time for anED receiver to achieve a BER = 10−5 with a signal bandwidth of 2 GHz is around 18 nsec and 33 nsec forCM 1 andCM 2, respectively. If a receiver designed forCM 1 operates in a wireless channel similar to the channel approximated byCM 2 (quadrant I and II of Figure 4-1), then its probability of error increases to approximately 20 × 10−5, which represents a signiﬁcant degradation when compared to the original goal of BER= 10−5. On the other hand, if a receiver designed forCM 2 operates in a channel approximated byCM 1 (quadrant III and IV of Figure 4-1), then it has the same probability of bit-error but has the potential to be reduced to about 10−7 because the integration time could be decreased to 18 nsec. In general, a system designed to operate in a given wireless channel will show a considerable degradation in its actual or potential probability of bit-error when it operates in a different channel. Hence, for a system that requires proper functionality in different types of channels or a varying channel, it is desirable to at least reduce the inherent performance degradation caused by larger integration times. To that effect, EID is proposed and show in section 4.6 that it has a smaller performance degradation rate thanED.

4.3.2 Bit Decision

Assume that a binary logic 1 has been modulated using PPM (i.e. a pulse is transmitted in the ﬁrst integration window) and transmitted over a wireless channel. Then, the received signal in the ﬁrst and second window can be expressed by

Xi = µi + ni (4–1)

Yi = mi (4–2)

83 Figure 4-1: Example of the actual and optimal probabilities of bit-error (Pe) for radios operating inCM 1 andCM2.

th respectively, where µi is the i sample of the received pulse, ni and mi represent AWGN

in each integration window, i = 1, 2, ... , N and N =fs ·Tw is the number of samples per window. To demodulate the signal usingED, it is necessary to calculate the energy in

P 2 P 2 both windows, i.e. EX (N) = (1/fs) · i Xi and EY = (1/fs) · i Yi . The bit decision is then given by

EX (N) ≷ EY (N) (4–3) 0

Note that EX (N) = PX (N) · Tw = PX (N) · N/fs where PX (N) is the expected average

power in the ﬁrst integration window. Similarly, EY = PY (N) · N/fs. Hence, the condition given in Equation 4–3 is equivalent to PX (N) ≷ PY (N). It is known that the expected average power of a random process is given by the expectation of the sum of its square [41]. In this case, that is

" N # X 2 PX (N) = E (µi + ni ) (4–4) i=1

84 " N # X 2 PY (N) = E (mi ) (4–5) i=1 where E [·] represents the expected value. Since si and ni are independent processes and ni has mean zero for all i, then E [si ·ni ] = 0 and

" N # " N # X 2 X 2 PX (N) = E µi + E ni = Pµ (N) + Pn (N) (4–6) i=1 i=1 where Pµ (N) and Pn (N) are the expected average power of the received signal and noise in the ﬁrst integration window, respectively. Thus, the bit decision based onED is

Pµ (N) ≷ [Pm (N) − Pn (N)] (4–7) 0 where Pm (N) = PY (N) is the expected average power of noise in the second integration window. For the proposed EID, the bit decision is based on the comparison of the integration of the signal energy, i.e. N N X X EX (i) ≷ EY (i) (4–8) i i where EX (i) = PX (i) · i/fs and EY (i) = PY (i) · i/fs. Thus, the new bit-decision condition in P P terms of average power can be deﬁned as i i · PX (i) ≷ i i · PY (i) and is equivalent to

1 N N X X i · Pµ (i) ≷ i · [Pm (i) − Pn (i)] (4–9) i=1 i=1 0

If Pm−n (i) = Pm (i) − Pn (i), then the bit decisions given by the conditions in Equation 4–3 and Equation 4–9 simplify to

85 Figure 4-2: General block diagram for an EID receiver.

Pµ (N) − Pm−n (N) ≷ 0 (4–10) 0

1 N X i · Pµ (i) − Pm−n (i) ≷ 0 (4–11) i=1 0 respectively. Note that, since the channel thermal conditions are not likely to change signiﬁcantly in short periods of time (e.g. 2 · Tw ), the expected average power of noise in both windows is approximately the same, i.e. Pn (i) ≈ Pm (i), and Pm−n (i) ≈ 0. Then, in average, Equation 4–11 can be expected to be a stronger condition than Equation P 4–10 since i i · Pµ (i) > Pµ (N) and should result in a lower probability of error. Figure 4-2 shows a block diagram of the signal processing carried out by an EID receiver to make a bit decision. As illustrated, after squaring (self-mix), the signal is integrated twice (cumulative-sum and sum) in each integration window. Then, the highest value determines in which window the pulse was transmitted, that is a bit decision is made.

4.3.3 Example

In this part we show an example of the signal processing done by bothED and EID receivers. Once again, it is assumed that a pulse was transmitted in the ﬁrst integration window representing a binary logic 1. Figure 4-3(a) shows the received signal including

86 (a) Received signal (b) Signal energy (c) Energy integration

Figure 4-3: Example of a binary logic 1 demodulated usingED and EID

noise (AWGN) assuming a SNRbit at the receiver’s antenna of 3 dB, i.e. Eb/N0 = 3 dB. Figure 4-3(b) shows the integration (cumulative-sum) of the squared signal, i.e.

i i X 2 X 2 EX (i) = (1/fs) · Xj = (1/fs) · µj + nj (4–12) j=1 j=1

i i X 2 X 2 EY (i) = (1/fs) · Yj = (1/fs) · mj /fs (4–13) j=1 j=1

AnED receiver would have demodulated this signal as a binary logic 0 since EX (N) <

EY (N) or, in words, the energy in the second integration window (marked as Em in the ﬁgure) is higher than the energy in the ﬁrst window. Clearly, this is a bit-error since the transmitted bit was a logic 1 and not a logic 0. The bit decision based on EID involves the integration of energy in each window which is given by

i 2 X IX (i) = (1/fs) · EX (j) (4–14) j=1

i 2 X IY (i) = (1/fs) · EY (j) (4–15) j=1

87 for the ﬁrst and second integration windows, respectively. Then, the bit decision for EID

is IX (N) ≷ IY (N) which demodulates the signal to a binary logic 1 since IX (N) > IY (N)

as can be seen in Figure 4-3(d) where IX (N) is marked as Im. In this case, the previous argument of Equation 4–11 being a stronger condition than Equation 4–10 holds, i.e. the commonED produced a bit-error while the EID did not. At this point, it is probably worth mentioning again that Equation 4–11 is a stronger condition than Equation 4–10 in average. Therefore, there will be individual bit decisions in whichED outperforms EID. However, in general, EID performs better thanED. To compare their performances, we ﬁrst must calculate the probability of bit-error for EID which is done in the next section.

4.4 Probability of Bit-Error for EID

In this section, the equation for the probability of bit-error of EID is presented. First the condition required to make a bit decisions shown followed by the derivation of the probability of bit-error and its modiﬁed version to include the multipath fading effect.

4.4.1 Bit Decision

As before, let ni and mi be the noise in the ﬁrst and second integration windows, respectively, and µi be the transmitted pulse at the receiver’s antenna for i = 1, 2, ... , N

where N = 2·B ·Tw is the total number of samples in each integration window (assuming

1 the Nyquist sampling rate is used). Recall that mi and ni are assumed to be AWGN

and, hence, Xi and Yi are Gaussian random variables with means µi and 0, respectively.

2 Both Xi and Yi are assumed to have the same variance σ since the noise average power is not likely to change over small periods of time (e.g. 2 · Tw ). In short, we have

2 2 Xi ∼ N µi , σ and Yi ∼ N 0, σ .

1 The Nyquist rate is the minimum sampling rate for which the sampled signal retains all the properties of the original signal.

88 As discussed in the previous section and shown in the block diagram of Figure 4-2, the demodulation based on EID ﬁrst carries out the squaring (self-mix) and integration (cumulative-sum) of the received signal in each window just like the commonED

demodulation. Then, the resulting signals, EX (i) and EY (i), are integrated once again and compared to each other in order to make a bit decision, i.e.

1 N N X X EX (i) ≷ EY (i) (4–16) i=1 i=1 0 With simple algebraic manipulation, it can be shown that

N N i N X X X 2 X 2 EX (i) = Xi = (N + 1 − i) · Xi (4–17) i=1 i=1 j=1 i=1 and then Equation 4–16 can be expressed as

1 N N X 2 X 2 (N + 1 − i) · Xi ≷ (N + 1 − i) · Yi (4–18) i=1 i=1 0

2 Note that (N + 1 − i) · Xi is just the square of the signal weighted by (N + 1 − i) which decreases linearly as i increases. In other words, the signal samples will have lower weights as time passes which makes sense since the amplitude of the received signal in

P 2 average decreases with time due to multipath fading. Also, note that i (N + 1 − i) · Xi P 2 and i (N + 1 − i) · Yi follow a chi-square distribution since Xi and Yi are Gaussian random variables. Hence, the bit decision is just the comparison of two chi-square random variables similar to theED demodulation. The probability of bit-error of this comparison is discussed next.

4.4.2 Probability of Bit-Error

2 2 Recall that Xi ∼ N µi , σ and Yi ∼ N 0, σ and let

89 N i N X X 2 X 2 V = Xj = (N + 1 − i) · Xi (4–19) i=1 j=1 i=1

N i N X X 2 X 2 W = Yj = (N + 1 − i) · Yi (4–20) i=1 j=1 i=1 so that the condition for a bit decision given by Equation 4–18 can be expressed as

V ≷ W . Hence, a bit-error occurs when V ≤ W and, consequently, the probability of

bit-error is PEID = P (V ≤ W ). This probability is derived in AppendixB and for large N it can be approximated by

  s2/σ2 PEID ≈ Q   (4–21) 2 · pN 0 + s2/σ2

where

N i 2 X X 2 s = µj (4–22) i=1 j=1

N3 1 N i j 0 X X X N ≈ + · µ2 (4–23) 3 σ2 k i=1 j=1 k=1 and Q (·) is the Q-function as deﬁned by Equation 3–4. Note that the energy of the ﬁrst i

Pi 2 2 samples of the received signal is Eb (i) = (1/fs) · j=1 µj and σ = N0 · fs/2. Therefore, the

probability of bit-error PEID can be expressed as

 0  E /N0  b  PEID ≈ Q q (4–24)  3 00 0  (2 · B · Tw ) /3 + 2 · 2 · Eb − Eb /N0

0 PN 00 2 PN Pi where Eb = fs · i=1 Eb (i) and Eb = fs · i=1 j=1 Eb (i). The probability of error is

commonly expressed as a function of Eb/N0. Next we slightly modify Equation 4–24 in

order to express it in terms of Eb/N0 and take into account the multipath fading effect.

90 4.4.3 Modiﬁed Probability of Bit-Error

Let the received signal µi be the continuous function of time µ (t) such that (1/fs)·

Pi 2 t 2 j=1 µj = 0 µ (τ) dτ where i = fs · t = 2 · B · t (assuming again fs is the Nyquist sampling ´ rate). Then, the signal energy as a function of time is given by

t 2 Eb (t) = µ (τ) dτ (4–25) ˆ0

Furthermore, deﬁne the energy scaling factor γb (t) such that Eb (t) = γb (t) · Eb where

∞ 2 Eb = Eb (∞) = 0 µ (τ) dτ is the total energy of the received pulse, i.e. ´

Eb (t) Eb (t) γb (t) = = (4–26) Eb (∞) Eb

0 00 Then, the values Eb and Eb in Equation 4–24 can be expressed as

Tw Tw 0 Eb = fs · Eb (τ) dτ = 2 · B · Eb · γb (τ) dτ (4–27) ˆ0 ˆ0

Tw t Tw t 00 2 2 Eb = fs · Eb (τ) dτ dt = 4·B · Eb · γb (τ) dτ dt (4–28) ˆ0 ˆ0 ˆ0 ˆ0 0 0 00 00 If Eb = Eb · γb (Tw ) and Eb = Eb · γb (Tw ) where

Tw 0 γb (Tw ) = 2 · B · γb (τ) dτ (4–29) ˆ0

Tw t 00 2 γb (Tw ) = (2 · B) · γb (τ) dτ dt (4–30) ˆ0 ˆ0 then the probability of bit-error PEID given in Equation 4–24 can be rewritten as

 0  γ (Tw ) · (Eb/N0)  b  PEID ≈ Q q (4–31)  3 00 0  (2 · B · Tw ) /3 + 2 · 2 · γb (Tw ) − γb (Tw ) · (Eb/N0)

91 Table 4-1: Constant values for γb (t) Model Range* (nsec) a b ×109sec−1

CM1 Tw > 5 −0.9353 −0.1832 CM2 Tw > 13 −1.6221 −0.1207 CM3 Tw > 40 −2.4567 −0.0651 CM4 Tw > 50 −1.8616 −0.0389 * Constants are accurate (< 1% error) in the speciﬁed time range

and, hence, Equation 4–24 has been modiﬁed to be expressed in terms of Eb/N0. Next,

0 00 the scaling factors γb (Tw ) and γb (Tw ) are characterized based on the IEEE P802.15 UWB channel models.

4.4.4 Energy Scaling Factors

Since bothED and EID receivers only deal with the energy of the signal and not its phase, the only interest here is in how much of the total energy can be captured by the receiver as a function of time. Thus, the received signal µ(t) is normalized so that its total energy is 1 which is the convention used in [3, 20, 39], i.e.

s ∞ 0 µ(t) = µ(t)/ µ2(τ) dτ (4–32) ˆ0 where µ (t) = p (t)∗h (t), p (t) is the received pulse, h (t) is the CIR and (∗) represents the convolution of both functions. We can use µ0(t) to calculate the energy scaling factor 2 t 0 R γb (t) deﬁned by Equation 4–26, i.e. γb(t) = 0 µ(τ) dτ. Next, employing the MATLAB ´ code provided in the IEEE P802.15 working group report [28], CIR realizations are generated, convolved with p (t), and averaged in order to determine γb(t) which follows an exponential-like curve. Then a non-linear regression with an exponential ﬁt is carried out to approximate the energy scaling factor as

γb (t) ≈ 1 + a · exp (b · t) (4–33)

where a and b are constants with speciﬁc values for each channel model and are summarized in Table 4-1.

92 0 00 (a) γb (t) (b) γb (t) (c) γb (t) Figure 4-4: Energy scaling factors for UWB channels

0 00 Table 4-2: Constant values for γb (t) and γb (t) −9 −18 2 9 −1 Model Range* (nsec) C1 ×10 sec C2 ×10 sec C3 ×10 sec

CM1 Tw > 5 5.003 −27.792 −0.1832 CM2 Tw > 15 11.061 −98.971 −0.1207 CM3 Tw > 40 25.421 −467.307 −0.0651 CM4 Tw > 50 37.328 −1046.800 −0.0389 * Constants are accurate (< 2% error) in the speciﬁed time range

R Figure 4-4(a) shows γb (t) using Equation 4–33 and the IEEE P802.15 MATLAB model. As shown in the figure, the exponential fit yields accurate approximations with less than 1% of error within the time range specified in Table 4-1. Using Equation 4–33, Equation 4–29 and Equation 4–30 can be approximated as  

0 C1 γ (t) ≈ (2·B·t) · 1 + ·(exp (C · t) − 1)  (4–34) b  t 3 

 

00 1 C1 C2 γ (t) ≈ (2·B·t)2 ·  − + · (exp (C · t) − 1)  (4–35) b  2 t t2 3 

where the constants C1, C2, and C3 are summarized in Table 4-2. Figure 4-4(b) and

0 00 Figure 4-4(c) show γb (t) and γb (t), respectively, using their exponential approximations, i.e. Equation 4–34 and Equation 4–35, and the IEEE P802.15 model. Once again, from the figures, it can be seen that the exponential fits yield accurate approximations (less than 2% of error) within the time range specified in Table 4-2.

93 Figure 4-5: Simulator Block Diagram

4.5 Simulation

To corroborate and support the derived equation for the probability of bit-error of

an UWB receiver, a simulator was built using MATLAB R . Its block diagram is shown in Figure 4-5 and it is essentially the same as the one presented and validated in Section 3.5.4. The only difference is that when simulating EID a double integration is carried out.

4.6 Analysis

4.6.1 Theory Corroboration

To corroborate the theory presented in section 4.4.3, simulations for each one of the IEEE P802.15 UWB channel models were run and compared to Equation 4– 31 which is the derived equation for the probability of bit-error of a EID receiver. The simulation parameters are B = 2 GHz and Tw = 26, 43, 82, 130 nsec forCM 1 through 4, respectively. Figure 4-6 shows the probability of bit-error (BER) curves obtained with both the simulator and Equation 4–31. As can be seen from the ﬁgure, the theory and simulations closely agree which corroborates the derived equation.

94 Figure 4-6: Comparison between simulation results and the derived BER equation for EID receivers

4.6.2 Bit-Error Rate

In this section, the derived probability of bit-error forEID receivers, Equation 4–31 is used along with the well-known BER expression forED receivers, Equation 3–3, to compare both demodulation techniques and show the advantages of EID overED. Figure 4-7 compares the performances ofED and EID assuming a signal bandwidth

B = 2 GHz and integration times Tw = 26, 43, 82, 130 nsec for theCM 1 through 4, respectively, allowing at least 99% of the signal’s energy to be captured. It is clear from the ﬁgure that EID outperformsED. On average, it performs better by approximately 1.25 dB for the speciﬁed bandwidth and integration times. This translates to around 33% improvement in terms of the signal energy required to achieve the same performance. In other words, an EID receiver requires approximately 33% less signal energy than anED receiver to achieve the same BER performance.

4.6.3 Integration Time

Increasing the integration time also increases the total noise energy E0 detected by the receiver since in average E0 = Tw · fs · N0/2. This increase in noise energy degrades the receiver’s performance. This is shown in Figure 4-8 where the required

−3 SNRbit to achieve a BER = 10 is plotted as a function of Tw for B = 2 GHz. As the

95 (a)CM1( Tw =26 ns) andCM2( Tw =43 ns) (b)CM3( Tw =82 ns) andCM4( Tw =130 ns)

Figure 4-7: Probability of bit-error forED and EID forCM 1 through 4 and B = 2 GHz

integration time increases bothED and EID receivers require a higher SNRbit in order to achieve certain BER performance (in this case 10−3). The ﬁgure also shows the improvement of EID overED since EID always requires a lower SNRbit to achieve the same performance. As mentioned in subsection 4.3.1, when designing the receiver, the integration time is determined based on the worst-case multipath scenario to ensure the proper operation of the system. In subsection 4.3.1, it was argued that a receiver designed based on a speciﬁc wireless channel model will show a considerable degradation in its actual or potential BER performance when it operates in a different channel. This degradation is inherent to the need of larger integration times and, although it cannot be completely eliminated, it would be desirable to at least reduce it. Assume for instance that a receiver is designed so that it can operate in wireless channels modeled byCM 1 through 3. The worst-case multipath scenario is given by CM 3 and, hence, the integration time is chosen to be 82 nsec so that on average 99% of the signal energy can be captured by the receiver. In Figure 4-8(a), it can be seen that anED receiver will experience a degradation of about 1.2 dB when compared to the same receiver designed for a channel approximated byCM 2 which requires

96 (a)CM 1 and 2 (b)CM 3 and 4

−5 Figure 4-8: Required SNRbit forED and EID to achieve a BER = 10 for B = 2 GHz andCM 1 through 4

Tw = 42 nsec in order to capture 99% of the signal energy. However, when using EID this degradation reduces to approximately 0.8 dB. This can be inferred from Figure 4-8, since the degradation rate (i.e. the slope of the curves) of EID is smaller than that of ED. Hence, in general, EID reduces the performance degradation experienced by the receiver when it operates in better channels than the one it was originally designed for.

Furthermore, in the example above, an EID receiver designed forCM3( Tw =

82 nsec) still performs better than anED receiver designed forCM2( Tw = 43 nsec) as marked by the dashed horizontal line in Figure 4-8(a). When compared to theED

receiver designed forCM1( Tw = 26 nsec), the performance of theED receiver (designed forCM 3) is about the same even though its integration window is almost 3.2 times larger (i.e. Tw = 82 nsec). In the ﬁgure, this is marked by the solid horizontal line. 4.6.4 Signal Bandwidth

Recall that the average noise energy captured within an integration window is

E0 = Tw · fs · N0/2 where fs = 2 · B. Thus, E0 =Tw ·B·N0 meaning that the noise energy is not only higher when the integration time is increased but when the receiver bandwidth is increased as well. Consequently, the receiver’s performance degrades with increasing bandwidth just as with integration time as shown in the previous section. This

97 (a)CM1( Tw =26 ns) andCM2( Tw =43 ns) (b)CM3( Tw =82 ns) andCM4( Tw =130 ns)

−3 Figure 4-9: Required SNRbit forED and EID to achieve BER = 10

−3 is illustrated in Figure 4-9 where the required SNRbit to achieve a BER= 10 is plotted

as a function of B using Tw = 26, 43, 82, 130 nsec for theCM 1 through 4, respectively. The ﬁgure also shows once again the improvement of EID overED since EID always

requires a lower SNRbit to achieve the same performance. In contrast to what happens with higher integration times, increasing the signal bandwidth has approximately the same degradation rate (i.e. slope of the curves in Figure 4-9) for bothED and EID. This is due to the fact that changing the signal

bandwidth does not affect the ratio between the signal energy captured in Tw and the

total signal energy (i.e. the energy scaling factor γb (Tw ) does not depend on receiver bandwidth).

4.7 Summary

Energy-detection (ED) is often used as the demodulation technique for non- coherent PPM radios mainly due to its simplicity and potential low-power implementation. Nonetheless, one of its main disadvantages is its sensitivity to noise energy. In prior publications the approach to this problem was to optimize the integration time or dynamically apply weights to each sample of the received signal. In the case of the integration time optimization, anED receiver will signiﬁcantly degrade its performance if it operates in wireless channels different from the original channel it was designed for.

98 In the case of using dynamic weights, although it greatly improves its performance, it has been shown that the receiver complexity increases substantially and might not be a viable solution when architecture simplicity and low-power consumption is required. In this chapter, the theory for a demodulation technique based on the integration of energy rather than energy detection alone was presented and developed. The EID technique essentially adds an additional integrator to the receiver architecture, which does not represent a signiﬁcant increase in architecture complexity but gives a considerable improvement overED. It was also shown that EID not only performs better thanED in general, but that the degradation it experiences due to an increase of integration time is smaller than the degradation experienced by anED receiver with the same increase in integration time. This is particularly important since the receiver is usually designed for the worst-case multipath scenario (i.e. larger integration times) making the receiver miss its potential BER when operating under better multipath scenarios. With EID, this drawback has been shown to be reduced.

99 Chapter 5 COGNITIVE PHY-MAC COOPERATIVE PROTOCOL The wide range of beneﬁts that UWB technology offers, makes it an attractive solution for short-range wireless networks [91]. However, the intrinsic characteristics of UWB channels pose several challenges. One of the main challenges in UWB wireless networks is dealing with multipath fading [9, 10]. Multipath fading is the result of a wireless signal traveling through multiple paths. Due to the different signal paths, multiple copies of the same signal arrive at the receiver with different amplitudes, phases and delays. In terms of energy, multipath fading effectively spreads over time the energy of the transmitted signal. Therefore, to account for this energy spread, the window to receive an UWB pulse must be large enough to capture its energy [3]. A larger receiving window results in a smaller transmission data rate since the pulse repetition period, or inter-pulse spacing, is larger. Hence, worse channel conditions, i.e. larger energy spreading, require lower transmission data rates. In general, the transmission data rate of a UWB radio must be designed to be low enough so that the radio is operational under the worst channel conditions [55]. This, however, imposes the same low data rate even when the radio is operating under better channel conditions. In addition, if data rates are low, the transmission time is larger and so is the energy spent by the receiver as its circuitry must be kept running for a longer period of time [55]. Furthermore, the transmission data rate and, consequently, the receiving window are closely related to the bit-error rate (BER) of the UWB receiver. If the receiving window is too large or too small the BER performance is degraded. Previous work [3, 20, 71] has shown that, depending on the channel conditions, there is an optimal length for the receiving window that minimizes the BER. In this chapter, the UCP-MACUWB cooperative PHY-MAC protocol (short for UWB Cooperative PHY-MAC) protocol is proposed and discussed. UCP-MAC is a cognitive and cooperative protocol between the PHY and MAC layers that dynamically optimize

100 the transmission data rate based on the channel conditions in order to improve the communication between the transmitting and receiving nodes. The data rate optimization minimizes the BER of the receiver and increases the data rate when there are favorable channel conditions. In addition to the energy savings previously mentioned, an increased data rate combined with a lower BER improves considerably the network performance in terms of transmission time, message delivery ratio, and throughput as we discuss later. This performance improvement is proven by

simulations obtained from a network simulator built in MATLAB R

5.1 Chapter Contributions

The combination of the all the work presented in previous chapters led to the idea and design of the UCP-MAC protocol discussed in this chapter. The main contributions comprise the cognitive channel estimation procedure for UWB radios using PPM, the

UCP-MAC protocol itself, and the network simulator built in MATLAB R . To estimate the wireless channel, a cognitive estimation procedure was developed and it takes advantage of the signal processing carried out when receiving a packet usingED demodulation. With this channel estimation, the optimal data rate can be determined. The UCP-MAC protocol then allows both the transmitter and the receiver to synchronize their transmission data rates to the optimal value making a more efﬁcient communication. This is tested using the network simulator. It implements both PHY and MAC layers as well as UWB channel modeling and network topology in order to test and analyze the complete protocol in a more realistic scenario.

5.2 Previous Work

Often, when using rate adaptation techniques, the transmission data rate is ad- justed as a function of the SNR or the interference created by neighboring devices [22]. For instance, in [6], the author proposed the Cooperative PHY layer network coding MAC(CPLNC-MAC) protocol that adjusts the data rate to the Shannon’s maximum channel capacity based on the current SNR. However, for the impulse-based UWB, this

101 approach is not feasible since the channel capacity is constrained by the inter-pulse spacing required due to multipath fading [10]. On the other hand, in [63], the proposed Cognitive-Autonomous MAC(CA-MAC) protocol adjusts the transmission data rate to reduce the multi-user interference1 (MUI) caused by neighboring devices which should improve the receiver’s BER. However, this approach does not take into account that the BER performance is not only affected by interference but it also depends on the length of the receiving window and optimizing this length is crucial in reducing BER. The UCP-MAC protocol focuses on minimizing the BER by utilizing the optimal data rate. Rather than calculating the SNR to maximize the transmission data rate, UCP-MAC minimizes the BER by cognitively ﬁnding an optimal data rate. Furthermore, when channel conditions allow it, the optimization results in higher data rates which in turn reduces MUI since packet transmissions are faster and, therefore, devices in the network interfere with each other for a less amount of time.

5.3 System Model

5.3.1 Network and Signal Model

For the UCP-MAC protocol, a decentralized network structure is considered. In particular, it is focused for short-range wireless ad-hoc networks in which all nodes have equal status and, at any given time, may establish a communication link with any other node within transmission range. Figure 5-1 shows an example of a wireless ad-hoc network. In such a network, the signal received by a node comprises the transmitted signal (UWB pulses), AWGN and MUI. For UWB wireless networks, it has been shown that MUI can be approximated as MUI[94, 23]. Therefore, the signal received by a node can be expressed as

1 In a wireless network, multi-user interference refers to the interference signal seen by any receiving node due to transmissions from neighboring nodes.

102 Figure 5-1: Example of a wireless ad-hoc network

s (t) = p (t) ∗ h (t) + n (t) (5–1) where (∗) represents the convolution of two functions, p (t) is the transmitted UWB pulse, h (t) is the channel impulse response (CIR), and n (t) is AWGN (includes the channel’s thermal noise and MUI).

5.3.2 Modulation and Demodulation Schemes

At the PHY layer, the UCP-MAC protocol employs PPM (see Chapter3) for modulation. Recall that with PPM, a symbol (in this case a single binary bit) is modulated by transmitting a pulse in one of two integration windows. A binary logic 1 is modulated by transmitting a pulse in the ﬁrst integration window. A binary logic 0 is modulated by transmitting a pulse in the second window. To demodulate the PPM signal,ED or EID can be used. When usingED (discussed in Section 3.3), the receiver compares the signal energies in each of the integration windows. The integration window with the highest energy reveals the window in which the pulse signal was transmitted and a bit-decision is then made. Based on the signal deﬁned by Equation 5–1, the energy in an integration window is given by

t0+Tw E = s2 (t) dt (5–2) k ˆ t0

103 where k ∈ [1, 2] identiﬁes the integration window, Tw is the integration time (i.e. the

length of one integration window), t0 = 0 for k = 1 and t0 = Tw for k = 2. If E1 > E2, the received signal is demodulated as a binary logic 1. If E1 ≤ E2, the received signal is demodulated as a binary logic 0. On the other hand, if EID (see Section 4.3) is used, the receiver compares the integration of energy instead of the energy alone as done withED. Then, the value calculated in each integration window is

t0+Tw t 0 E = s2 (τ) dτ dt (5–3) k ˆ ˆ t0 t0 0 0 and, similar toED, if E1 > E2 then the received signal is demodulated as a binary logic 1

0 0 while if E1 ≤ E2 then the received signal is demodulated as a binary logic 0. 5.3.3 Optimal Integration Time

The BER equations for bothED and EID where previously discussed in Chapter3 and Chapter4, respectively. Equation 3–32 yields the BER forED while Equation 4–31 gives the BER for EID. Solving these equations for Eb/N0 gives the SNR-per-bit required to achieve the target BER as a function of integration time. ForED, this required SNR -per-bit can be expressed as

−1 " 2 # (ED) γ(Tw )/K 1−γ(Tw ) SNR (Tw ) = − (5–4) bit p 2 1+ 1+2 · Tw · B/K 2 · Tw · B

 −1 0 000 2 (EID)  γ (Tw ) · γ (Tw )/K 1−γ(Tw ) SNR (Tw ) =  −  (5–5) bit r h i 2 · T · B  000 2 3 2 w  1+ 1+[γ (Tw )] · (2 · Tw · B) /3 /K where K = Q−1 (BER) is the inverse Q-function (see Equation 3–4), B is the receiver’s bandwidth,

104 000 h 00 0 i−1 γ (Tw ) = 2 · γb (Tw ) /γb (Tw ) − 1 (5–6)

Tw t 00 2 γb (Tw ) = (2 · B) · γb (τ) dτ dt (5–7) ˆ0 ˆ0

Tw 0 γb (Tw ) = 2 · B · γb (τ) dτ (5–8) ˆ0 and γ(τ) is the energy scaling factor2 deﬁned by Equation 5–18. Minimizing Equations

5–4 and 5–5 gives the optimal integration time (Tw )opt forED and EID receivers, respectively. Since the energy scaling factor γ(Tw ) depends on the CIR, it must be estimated in real-time in order to accurately predict (Tw )opt as channel conditions vary over time. An estimation procedure for γ(Tw ) will be proposed in Section 5.4. 5.3.4 Carrier Sense Multiple Access with Collision Avoidance

At the MAC layer, the UCP-MAC protocol employs the carrier sense multiple access (CSMA) with collision avoidance (CSMA-CA) protocol described for the distributed coordination function (DCF) of the IEEE 802.11 MAC standard [46]. With CSMA-CA , the nodes in the wireless network contend for the channel using two control packets, namely request-to-send (RTS) and clear-to-send (CTS), to virtually sense the channel before transmitting. If a node senses the channel as idle then it is allowed to transmit. Otherwise, the node remains silent to avoid causing collisions in the network. The CSMA-CA protocol is illustrated in Figure 5-2. Assume a network comprises nodes A, B, C, and D where B and C are within transmission range of both A and D but A and D cannot hear each other. This is shown in Figure 5-2(a). Now suppose that node

2 Recall that the energy scaling factor γ(Tw ), as deﬁned in Section 3.5.2, is a ratio representing how much of the total signal energy is captured by the receiver as a function of the integration time Tw

105 (a) Network topology (b) Packet exchange

Figure 5-2: Illustration of the CSMA-CA protocol

A wants to transmit data to node B. Figure 5-2(b) shows the packet exchange under the CSMA-CA protocol. The protocol starts with node A sensing the channel for a period of time called DCF inter-frame spacing (DIFS).3 After an idle DIFS, node A transmits a RTS packet to node B. Both nodes B and C receive this packet as they are within transmission range of node A. Since the packet is intended for node B, node C sets its network allocation vector (NAV), an internal timer indicating how long the channel will be busy, and stays silent for that period. Node B now transmits a CTS packet after waiting a short inter- frame spacing (SIFS) period. This CTS packet is received by nodes A and D. Node D then sets its NAV and stays silent for the indicated period. Node A has now gained the

3 Inter-frame spacings (IFS) are periods of time that a node must wait before transmitting a packet. They can be used for channel sensing (e.g. DIFS) or to provide enough time for propagation delays and information processing (e.g. SIFS). In some cases, they might be also used to provide different priority levels to ensure quality of service (QoS) for certain transmissions.

106 channel to transmit its DATA packet since nodes C and D will remain silent. Node A now transmits the DATA packet. This is received by node B and, after waiting the SIFS period, it transmits an acknowledgement (ACK) packet to let node A know that it has successfully received the DATA packet. After the ACK packet is received by node A, their communication has successfully ﬁnished.

5.4 Channel Estimation

As explained in Section 5.3.2, to determine the optimal integration time (Tw )opt , the required SNRbit given in Equations 5–4 and 5–5 must be minimized. However, to do that, the energy scaling factor γ (Tw ), must be estimated ﬁrst. Recall that this factor is the ratio of how much of the total signal energy has been integrated by the receiver as a function of time. In this section, a channel estimation procedure for γ (Tw ) using theED technique is proposed.

5.4.1 Signal and Energy Model

Assume that a logic 1 has been modulated using PPM, i.e. the pulse signal is place in the ﬁrst integration window, and transmitted over a wireless channel. The samples of the received signal in the ﬁrst integration window (Xi ) and second integration window

(Yi ) can be expressed as

Xi = µi + ni (5–9)

Yi = mi (5–10)

th where µi is the i sample of the received pulse, ni and mi represent AWGN in the channel, and i = 1, 2, ... , N. N = fs · Tw = 2 · B · Tw is the number of samples per window assuming the Nyquist sampling rate (fs = 2 · B) is used. The energy samples in each integration window can then be expressed as

107 i i X 2 X 2 EX (i) = (1/fs) · Xj = (1/fs) · µj + nj (5–11) j=1 j=1

i i X 2 X 2 EY (i) = (1/fs) · Yj = (1/fs) · mj (5–12) j=1 j=1

Since the noise samples ni and mi are assumed to be AWGN, they can be represented as independent and identically distributed (i.i.d.) zero-mean Gaussian random variables

2 2 2 with variance σ = N0 · fs/2, i.e. ni ∼ N 0, σ and mi ∼ N 0, σ . Consequently,

2 2 Xi ∼ N µi , σ and Yi ∼ N 0, σ are i.i.d as well. At this point is worth mentioning that the variances of ni and mi can be assumed to be the same as they depend on the noise spectral density N0 which in turn depends on temperature. Since the time between samples is very small (a fraction of a nanosecond), the change in temperature between them can be neglected.

Now, note that EX (i) and EY (i) are the summation of the square of Gaussian random variables which are known to follow a chi-square random distribution with i

2 Pi 2 2 degrees of freedom and centrality parameters sX (i) = j=1 µi and sY (i) = 0, i.e.

2 2 2 EX (i) ∼ χi sX (i) and EY (i) ∼ χi (0), respectively.. In general, for a chi-square random variable with L degrees of freedom and centrality parameter s2, the mean is L · σ2 + s2 and the variance is 2 · N · σ4 + 4 · s2 · σ4 as shown in [64]. Therefore, the mean and variance for EX (i) are

2 2 µEX (i) = i · σ + sX (i) (5–13)

σ2 = 2 · i · σ4 + 4 · s2 (i) · σ2 (5–14) EX (i) X

2 while for EY (i), since sY (i) = 0, these are

2 µEY (i) = i · σ (5–15)

108 σ2 = 2 · i · σ4 (5–16) EY (i)

Now, that we have an energy model, we can develop the channel estimation procedure to approximate the energy scaling factor. This estimation procedure is based on the energy difference of the two integration windows.

5.4.2 Energy Difference

Following the energy model just presented, each energy sample in the ﬁrst integration window EX (i) contains energy from the transmitted pulse as well as from noise while EY (i) only contains noise energy. Intuitively, the difference EX (i) − EY (i) should then yield in average the transmitted pulse energy since the transmitted pulse and the noise in the channel can be assumed to be independent processes and the average noise power in both integration windows is approximately the same (assuming a negligible change in temperature).

To show this concept, let us quickly go through the math. First recall that Xi and Yi

are i.i.d. random variables. Hence, EX (i) and EY (i) are independent random variables

and the mean (i.e. expected value) of their difference EZ (i) = EX (i) − EY (i) is given by

2 µEZ (i) = µEX (i) − µEY (i) = sX (i) (5–17)

2 Pi 2 th where sX (i) = j=1 µi is the transmitted pulse energy accumulated up to the i signal

sample. Therefore, the expected value of the energy difference EZ (i) = EX (i) − EY (i) is the energy from the transmitted pulse.

5.4.3 Estimation of the Energy Scaling Factor

As explained in Section 5.3.2, the energy scaling factor γ represents how much of the total energy from the transmitted pulse is integrated by the receiver based on the length of the integration window. If the energy samples of the transmitted pulse are

109 2 Pi 2 given by sX (i) = j=1 µi as previously deﬁned, then the energy scaling factor up to the ith signal sample of a single symbol can be expressed as the ratio given by

2 sX (i) γ (i) = 2 (5–18) sX (N)

where, as before, N = fs · Tw = 2 · B · Tw is the number samples in each integration

2 2 window and sX (N) is the total energy of the transmitted pulse. Now, recall that sX (i) is

2 the mean of EX (i) − EY (i) as shown in Equation 5–17. Similarly, sX (N) is the mean of

EX (N) − EY (N). Therefore, an estimation for the energy scaling factor can be obtained using the difference of the energy samples, i.e. EX (i) − EY (i), of a single symbol. This is illustrated in Figure 5-3. Suppose again that a logic 1, i.e. a pulse is transmitted in the first integration window as shown in Figure 5-3(a). The received signal due to multipath fading only (assume no AWGN is present for now) may look like the one in Figure 5-3(b). The resulting signal after squaring and integrating is shown in Figure 5-3(c). As defined by Equation 5–18, normalizing this resulting signal to its maximum value yields the energy scaling factor. Now, let us assume that the same pulse is received but now in the presence of AWGN. Figure 5-3(d) shows the received noisy signal. After squaring and integrating, the resulting signal now looks like that of Figure 5-3(e). Finally, taking the energy difference between the first and second integration windows yields the signal shown in Figure 5-3(f) which is an approximation of the energy of the received pulse without AWGN shown in Figure 5-3(c). We note that this approximation is not very accurate. If more approximations are carried out with different symbols, then their average yields a more accurate estimation. This is illustrated in Figure 5-4 where the UWB channel model provided by the IEEE P802.15.3a report [27] was used to generate the CIRs. Increasing the number of symbols (5, 20, 50, 150) used for the estimation, increases its accuracy.

110 (a) Transmitted pulse (d) Received pulse with noise

(b) Received pulse (e) Energy of the received pulse with noise

Figure 5-3: Signal processing of the proposed channel estimation

5.4.4 Achieving an Optimal Transmission Data Rate

For PPM-ED radios, the transmission data rate is given by Rdata = 1/ (2 · Tw ) since it takes two integration windows to transmit a single bit. The nominal integration time Tw is determined by the worse-case multipath scenario [55]. Recall that the energy of the transmitted pulse is spread over time due to multipath fading. A larger energy spreading corresponds to a worse multipath scenario and requires a larger Tw so that the receiver can integrate most of the energy from the transmitted pulse. Therefore, designers usually choose the nominal Tw to be large enough so that the receiver works even at the worse-case multipath scenario. However, often the receiver will be operating in channel conditions where the nominal Tw is too large resulting in a performance degradation

111 (a) 5 bits (b) 20 bits

Figure 5-4: Accuracy of the energy scaling factor estimation as more symbols are used

(recall that noise energy is directly proportional to Tw ). Therefore, determining the

optimal integration time (Tw )opt improves the receiver performance in terms of BER as demonstrated in [3] and, at the same time, increases the transmission data rate since

(Tw )opt ≤ Tw , i.e. (Rdata)opt ≥ Rdata. In the worst-case scenario, the receiver exhibits its original design performance and operates at the nominal data rate. However, when the channel conditions improve so will the receiver’s BER performance and transmission data rate. For instance, in the example illustrated in Section 5.4.3, if the nominal Tw of an BER was 40 nsec and the received SNR-per-bit is Eb/N0 = 15 dB for a signal bandwidth of B = 2 GHz, the BER can be calculated using Equation 3–32 yielding 1.72 × 10−2. However, after estimating the energy scaling factor γ (Tw ) —see Figure 5-4(d)— and minimizing Equation 5–4, the

−2 optimal integration time (Tw )opt is approximately 31 nsec with a BER of 1.13 × 10 . This represents a 34% improvement in BER and a 29% increase in the transmission data rate.

112 5.5 UWB Cooperative PHY-MAC Protocol

As previously mentioned, the UCP-MAC protocol proposed in this section employs PPM modulation andED or EID demodulation at the PHY layer and the multiple access mechanism of the IEEE 802.11 CSMA-CA at the MAC layer. The modulated symbols (binary bits) of the control packets, RTS and CTS , are used at the PHY layer to determine the optimal integration time of the receiving and the transmitting nodes. Information on the optimal integration time is then exchanged using the MAC layer such that each node adapts its data rate accordingly.

5.5.1 Receiver Architecture for Channel Estimation

A possible receiver implementation for a PPM-ED radio employing the channel estimation technique proposed in Section 5.4.3 is depicted in Figure 5-5. The front-end of the receiver corresponds to the general architecture of a PPM-ED receiver in which the received signal is filtered, amplified, squared and integrated. After this analog signal processing, the resulting signal which represents the energy in each integration window is then passed through an analog-to-digital (A-to-D) converter. Now, in the digital domain, the bit-decision can be made by comparing the energy in each integration window. In addition, those energy signals are also used for the estimation of the energy scaling factor γ. The channel estimation block in Figure 5-5 starts with two delay components that account for the time it takes to make the bit-decision (i.e. the delay of the comparator). This is because the bit-decision must be made before the signal processing for the channel estimation in order to determine which integration window contains the transmitted pulse. Recall that a pulse in the first integration window was assumed for the energy model presented in Section 5.4.1 and thus the energy scaling factor was estimated with

the energy difference EX (i) − EY (i); however, if the pulse was transmitted in the second integration window, EY (i)−EX (i) must be used instead. The order of the integration windows is done by the second component called “window order” in the block diagram. This

113 Figure 5-5: PPM ED receiver architecture with the proposed channel estimation component is followed by a digital subtractor and a component to normalize the energy difference to its maximum (recall that the energy scaling factor must be between 0 to 1). After the normalization, the resulting signal is an approximation of the energy scaling factor. However, as previously explained, multiple approximations must be averaged to increase the accuracy of the estimation. Therefore, the approximation obtained from each received symbol is accumulated and the total is divided by the number of symbols L. The resulting signal is then an accurate estimation of the energy scaling factor for the current channel conditions.

5.5.2 Cooperative PHY-MAC Protocol

The general idea of the proposed UCP-MAC protocol can be explained with a simple example. Suppose node A wants to transmit data to node B. Figure 5-6 illustrates the general steps of their communication using the proposed protocol which

114 Figure 5-6: Cognitive PHY-MAC protocol summary is based on the RTS/RTS mechanism of the CSMA-CA protocol discussed in Section 5.3.4. Node A starts by sensing the channel during the DIFS period. If nothing is received during this period, node A assumes the channel is idle and transmits the RTS packet at the nominal data rate Rdata = 1/ (2 · Tw ) where Tw is the nominal integration time. While

receiving the RTS packet, node B is estimating the energy scaling factor γ (Tw ) which will be used to determine the optimal integration time (Tw )opt . Once it has ﬁnished receiving, node B calculates the (Tw )opt that minimizes its BER and sets its new receiving (Rx) data rate to (Rdata)opt = 1/ 2 · (Tw )opt . Then, node B prepares and transmits the

CTS packet which includes the (Tw )opt that it just calculated. Node A, while receiving this CTS packet, also estimates γ (Tw ) and calculates its own (Tw )opt and include it in the DATA packet. With the (Tw )opt received from node B in the CTS packet, node A sets its the new transmitting (Tx) data rate to (Rdata)opt = 1/ 2 · (Tw )opt . At this point, both nodes are synchronized with the same data rate for the exchange of the DATA packet. Node A now prepares and transmit the DATA packet that includes the data frame and its own (Tw )opt . When node B receives the DATA packet, it sets the new Tx data rate

115 based on the (Tw )opt received from node A. Then, it transmits the ACK packet at this new data rate which will be received by node A. After the exchange of the ACK packet, the communication has successfully finished and both nodes reset their Rx and Tx data rates to the nominal value Rdata = 1/ (2 · Tw ). In summary, the RTS and CTS packets are always transmitted at the nominal data rate and, in addition to virtually sensing the channel, they are also used to determine the optimal data rates at which the DATA and ACK packets will be transmitted. One of the benefits obtained with the proposed protocol is the reduced probability of bit-errors for the DATA and ACK packets since they are transmitted at the optimal data rate. Another benefit is the increase in data rate (recall that (Rdata)opt ≥ Rdata ) allowing the DATA and ACK packets to be transmitted faster and, as a consequence, the channel is busy for less amount of time. Next, we describe the frame formats for the MAC layer packets, or MAC protocol data units (MPDUs) and the PHY layer packet, or PHY layer data unit (PLDU).

5.5.3 PHYand MAC Frame Formats

Figure 5-7 shows the frame formats of the four MPDUs (RTS, CTS, DATA, ACK). These are similar to those used in IEEE 802.11 CSMA-CA protocol except for the ﬁeld

reserved for the optimal integration time (“Optimal Tw ”). The “ID” field identifies the type of MPDU(RTS, CTS, DATA, or ACK). The “Duration” field contains the time needed to finish the communication and is used by the receiving node to set its NAV. The “Source” and “Destination” fields are used to identify the node initiating the communication and the node for which the packet is intended, respectively. The “Data Frame” field contains the information to be transmitted while the “CRC” field is used for error detection using the well-known cyclic redundancy check (CRC). For the PHY layer, the frame format of its PLDU is shown in Figure 5-8. The PHY preamble comprises the “SYNC” and “SFD” fields. Both of them contain a specific sequence of logic 0s and 1s. The “SYNC” sequence is used for synchronization purposes

116 (a) RTS

(b) CTS

(d) ACK

Figure 5-7: Frame format for each MPDU

Figure 5-8: Frame format of the PLDU while the “SFD” is the start frame delimiter and is used to mark the beginning of the PHY header. This header contains the “Frame Length” field and a dedicated “CRC” field for error detection. The frame length refers to the length of the “Coded MPDU” field which contains the MPDU from the MAC layer and can be coded for different purposes such as error correction.

5.6 Simulation Setup

5.6.1 Network Simulator

In order to test the proposed UCP-MAC protocol, a complete network simulator was programmed in MATLAB R . The simulator contains over 30 classes that implement each one of its functionalities. A general diagram of the main classes, Node and Channel, is shown in Figure 5-9 where each block represents a parent class. The Node class implements all the processes carried out by a node in order to transmit or receive a message. A node comprises three communication layers: PHY, MAC, and a higher

117 (a) Node class

(b) Channel class

Figure 5-9: Diagrams of the two main classes used by the network simulator layer provides the messages (i.e. data bits) to be transmitted. On the other hand, the Channel class is used to model the wireless channel, i.e. the medium through which the transmitted signals travel.

5.6.1.1 Node class

To appropriately simulate the cognitive protocol, it is necessary to fully implement both PHY and MAC layers as the protocol utilizes the communication between them and its main feature, integration time optimization, is based on the signal processing of the received signal. The PHY layer takes care of error-correction and signal modulation while the MAC layer runs the RTS/CTS mechanism. In the PHY layer, the MPDU received from the MAC layer is coded for error- correction using the Bose-Chaudhuri-Hocquenghem (BCH) algorithm with a 7/4 coding ratio which has been shown to be more energy-efﬁcient than most block and convolutional codes[73]. The coded MPDU along with the PHY header and preamble make up the PLDU as previously shown in Figure 5-8. This PLDU is then modulated

118 using PPM and the resulting signal is transmitted. The inverse occurs when a signal is received. It is ﬁrst demodulated using theED or EID demodulation technique. At this

point the energy scaling factor γ (Tw ) is estimated and for the calculation of the optimal

integration time (Tw )opt . The demodulated bits are then checked. If the PHY preamble and header were received correctly, the remaining bits corresponding to the coded MPDU are decoded and error-corrected before being sent to the MAC layer. The MAC layer runs the UCP-MAC protocol with the support of the PHY layer. The protocol is executed for each message in the MAC buffer which in turn receives messages from a higher layer. This messages can be called higher-layer data units (HLDUs). For each HLDU in the MAC buffer, RTS, CTS, DATA, and ACK packets must be exchanged to achieve a successful transmission.

5.6.1.2 Channel class

This class models the changes that a signal undergoes while traveling through the wireless medium, i.e. multipath fading, path loss, interference, and additive noise. To model multipath fading, each signal is convoluted with the corresponding CIR

generated by the MATLAB R model provided by the IEEE P802.15.3a task group report [27]. Path loss, on the other hand, is modeled using a log-normal shadowing model and the parameters for UWB channels provided in [35]. Interference is modeled by adding up all convoluted signals with their respective path losses applied. Finally,

noise is modeled by adding AWGN with power spectral density N0 = kB · TK where

−23 kB = 1.3806488 × 10 Joules/Kelvin is the Boltzmann constant and TK is the environment temperature in Kelvins.

5.6.1.3 Other important classes and functions

In addition to the Node and Channel classes, there are two other major classes used in the simulator: Topology and Messages. The Topology class creates the network topology and models node mobility. The nodes are randomly distributed and moved within the network area. On the other hand, the Messages class generates random data

119 bits to simulate the HLDUs that are sent to the MAC layer for transmission. The HLDU arrival time is modeled as a Poisson process with arrival rate 1/λ sec−1. The simulator also uses several functions to properly work. Among these, the most most important are the RunSimulation and Results functions. The ﬁrst is mainly a loop that controls the network time but is also responsible for how everything is interconnected. The second goes through the simulator log where all actions were recorded during the simulation and calculates network performance metrics such as average transmission time and message delivery ratio. These metrics are used later to analyze and compare network performances.

5.6.2 Simulation Parameters and Setup

Table 5-1 shows the parameters used for the simulations run. The number of nodes and the message arrival rate were varied from 5 to 30 and from 4 to 32 sec−1, respectively. When the arrival rate was varied, the number of nodes was set to 25. When the number of nodes was varied, the arrival rate was set to 20 sec−1. The nodes were randomly placed within the network area which has dimensions 5 × 5 × 5 meters. During the simulation, nodes move randomly within the network area at a speed of 1 m/s. The backoff algorithm used by the MAC layer when a packet collision is detected is an exponential algorithm in which the backoff time is calculated as

tbackoff = tbackoff slot · Nslots (5–19) where tbackoff slot is the length of a single backoff time slot, Nslots is a number randomly chosen from the set [0, 1, ... , 2n − 1], and n is the retransmission number. For instance, assume a node is transmitting a new message and a packet collision is detected. Since the node will try to retransmit for the ﬁrst time, then n = 1 and a number from the set [0, 1] is randomly chosen. If the node is trying to retransmit for the second time then n = 2 and the set from which a number is randomly chosen is now [0, 1, 2, 3]. If

120 Table 5-1: Simulation parameters Parameter Value Number Of Nodes 5 − 30 Network Dimensions (m) 5 × 5 × 5 Node Speed (m/s) 1 Message Arrival Rate 1/λ (1/s) 4 − 32 Max. Number of Retransmissions 3 Backoff Time Slot (µs) 50 DIFS(µs) 40 SIFS(µs) 20 Temperature (K ) 300

Nominal Integration Time Tw (ns) 100 Pulse Bandwidth (GHz) 2 Transmission Power (dBm) −8.3 Length of MPDU Fields (bits) ID 4

Duration, Destination, Source, Optimal Tw 16 Data frame 4000 CRC 12 Length of PLDU ﬁelds (bits) SYNC, SFD 24, 8 Frame length, CRC 16, 12

n = 3, the set is [0, 1, 2, 3, ... , 7] and so on. The parameters for the exponential backoff algorithm used in the simulations are also shown in Table 5-1.

5.7 Analysis

Several simulations were run in order to compare the performance of the UCP-MAC protocol. Three parameters are plotted in this section as a function of the message arrival rate and the number of nodes in the network. Both the arrival rate and the number of nodes were chosen since these effectively increase the network trafﬁc4 giving a better idea of how the protocol behaves with increasing trafﬁc.

4 The trafﬁc in the network increases as more messages are sent either due to more messages being created or more nodes transmitting.

121 (a) (b)

Figure 5-10: Message delivery ratio as a function of (a) the message arrival rate and (b) the number of nodes in the network

5.7.1 Message Delivery Ratio

The message delivery ratio as a percentage is calculated as

nmsgs − ndrop DRmsgs = × 100 (5–20) nmsgs where nmsgs is the total number of messages to be transmitted in the network and ndrop is the number of messages dropped after unsuccessfully retransmitting for certain amount of times (in this case, 3 as shown in Table 5-1). In general, an unsuccessful transmission occurs due to uncorrectable bit-errors. These mainly happen due to transmission collisions and noise in the wireless channel. The message delivery ratio shows the percentage of messages that are successfully delivered. Figure 5-10 shows this percentage as a function of the message arrival rate and the number of nodes in the network. In comparison to the regular CSMA-CA protocol, the UCP-MAC protocol significantly improves the number of messages that are transmitted successfully in the network as the traffic increases. There are two main factors for this improvement. The first is the reduction in BER the optimal integration time. A smaller probability of bit-error reduces the probability of a packet error due to uncorrectable bit-errors. The second factor is the increase in data rate due to a smaller integration

122 time. A faster transmission of the DATA and ACK packets, reduces the time for which the channel is busy which in turn reduces the probability of collisions in the network. From the ﬁgure, it is also clear that the EID demodulation technique performs better than theED technique. This is mostly due to the improvement in bit-error rate of EID overED as discussed in Chapter4.

5.7.2 Average Transmission Time

The transmission time is defined here as the total time it takes to successfully finish the communication between two nodes. That time is measured from the moment the first RTS packet is transmitted until the ACK has been received. Therefore, the minimum transmission time for a message to be successfully delivered occurs when no retransmissions are required and can be calculated as

(tTx )min = tRTS + tCTS + tDATA + tACK + 3 · tSIFS (5–21) where tSIFS is the SIFS time and tRTS, tCTS, tDATA, and tACK are the times required to transmit the RTS, CTS, DAATA, and ACK packets, respectively. When the channel conditions are better than the worst-case scenario, (Rdata)opt > Rdata which reduces

tDATA + tACK resulting in a smaller transmission time. The total transmission time when retransmissions are required is calculated as the total time it took to successfully receive the ACK packet starting from the time the first RTS packet was transmitted. Therefore, the transmission time increases with the number of retransmissions. Figure 5-11 shows the average transmission time of the network as a function of message arrival rate and number of nodes. From the plots is easy to see that as the traffic in the network increases, the average transmission time increases as well. This is a consequence of an increase in the number of collisions yielding more retransmissions and, hence, increasing the average transmission time. The figure also shows the improvement of the UCP-MAC protocol in comparison to the regular CSMA-CA protocol. Once again, this improvement is mainly related to the reduced BER

123 (a) (b)

Figure 5-11: Average transmission time as a function of (a) the message arrival rate and (b) the number of nodes in the network and the increased data rate for the DATA and ACK packets. Smaller transmission times for these packets reduce the communication time between two nodes. In addition, as previously explained, these two factors reduce the probability of retransmissions. With lower number of retransmissions, the average transmission time is reduced as well (recall that transmission time is measured from the ﬁrst RTS transmitted). In contrast to the message delivery ratio discussed in Section 5.7.1, here the EID demodulation technique performs worse than theED technique when using the UCP- MAC. The main reason for this behavior is that the optimal integration time of EID is higher than that ofED as discussed in Chapter4. However, when using the CSMA-CA protocol, EID performs better thanED since it reduces the number of retransmissions caused by bit-errors. A possible soultion to decrease the average transmission time of EID is to use the optimal integration time calculated forED. Although, EID will not be operating at its optimal it should still have a similar or better BER performance thanED as can be concluded from the analysis done in Chapter4, in particular, from Figure 4-8. This possible solution will be studied and analyzed in the future.

124 (a) (b)

Figure 5-12: Throughput as a function of (a) the message arrival rate and (b) the number of nodes in the network

5.7.3 Throughput

Throughput, or effective data rate, can be deﬁned as

ldata Reff = (5–22) tTx

where ldata is the length of the data frame (in our case, 4000 bits) and tTx is the average transmission time. Figure 5-12 shows the calculated throughput and, as expected, it decays as the traffic in the network increases due higher collisions which translates into higher number of retransmissions. Since the UCP-MAC protocol, in average, reduces the number of retransmissions and increases the data rate of the DATA and ACK packets, it outperforms the regular CSMA-CA protocol as depicted in the figure. Figure 5-12 also shows that for the specific simulation parameters used (see Table 5-1), the effective data rate fluctuates from about 4 Mbps to less than 100 Kbps as the traffic in the network increases. This degradation is the result of higher number of retransmissions mainly due to an increase in the number of collisions in the network.

125 Chapter 6 CONCLUSIONS AND FUTURE WORK For the last three decades, as the wireless communications industry has advanced, there has been a great increase in the development of not only long-range and medium- range wireless communications but in short-range wireless systems as well. Short- range wireless networks such as WLAN, WPAN, and WSN are widely used in today’s technology applications. In many cases these applications have a limited power source and thus require very low-power communications to extend their time of operation. Recently, in particular since the FCC approval in 2002, UWB communications have attracted the interest of many researchers as an alternative technology to implement low-power wireless applications. UWB technology offers a wide range of benefits that makes it a viable solution for many short-range wireless systems. Among these benefits are low-power transmission, reduced interference, low cost and complexity in hardware, increased robustness against multipath fading and relatively high data rates. For the last decade, research and numerous investigations have proven UWB to be an efficient and feasible technology for digital communications. The work presented in this doctoral dissertation was focused on UWB radios using PPM which is a very popular digital modulation technique for very low-power wireless communications. Among the contributions of the work done, the key contribution is the application of non-coherent UWB radios using PPM to a new cognitive and cooperative protocol between the PHY and MAC layers, i.e. the UCP-MAC protocol. This protocol employs a cognitive channel estimation technique based onED in the PHY layer and a multiple access mechanism based on the CSMA-CA protocol in the MAC layer. These layers exchange information with each other in order to optimize the wireless communication. Previous to the development of the theory presented throughout this dissertation, extensive investigative work on UWB was performed and a general overview along with

126 key concepts were presented in Chapter2. This chapter included the FCC definitions and specifications, the main benefits, the commonly used modulation techniques, and a description of channel modeling for UWB wireless communications. From the discussion in this chapter, it was clear that non-coherent PPM radios are an excellent choice for applications requiring low-power and low-complexity architectures which is the general case of low-power short-range ad-hoc wireless networks. The most popular non-coherent demodulation technique for PPM is based on energy detection (ED) which is the base for the channel estimation technique used by the UCP-MAC. A more detailed discussion onED receivers was presented in Chapter3. It was shown that forED-PPM receivers there is an optimal bandwidth and an optimal inte-

gration time that minimizes the required SNRbit to achieve a desired BER. Analytical equations to approximate these optimal values were derived and presented taking into account non-idealities such adjacent-channel interference (ACI), inter-symbol interference (ISI), and inter-frame (IFI). These equations, corroborated by simulations, were used to analyze the effect of multipath fading onED-PPM receivers. Among the findings, it was shown that increasing the signal bandwidth reduces the optimal integration time but degrades the system performance in terms of BER. In addition, examples were briefly discussed in showing that a relatively small degradation in SNRbit can poten- tially offer valuable benefits such as lower power consumption and higher data rates. Moreover, the equations derived are not only useful when understanding the effects of multipath fading but they are convenient when designingED-PPM receivers operating in UWB channels. By using them, the designer can easily choose the appropriate integration time based on system parameters without the need of building and running simulators. In Chapter4, an improvement to the commonED technique was presented. The motivation was based on the fact that when approximating the best integration time, the optimized receiver will be optimal only if it operates under a wireless channel similar to

127 the channel model used for its optimization. Otherwise, the receiver will show a signif- icant performance degradation. To reduce this performance degradation experienced by an optimized receiver operating in different channel conditions, the EID technique was introduced and mathematically proved. As done for the optimization ofED-PPM receivers in Chapter3, analytical equations were derived in order to understand the effect of multipath fading on receivers using the EID demodulation technique and how it improves the receiver’s BER performance in comparison to theED technique. Simulations corroborated the equations derived which were used to numerically compare bothED and EID techniques. The results showed that, although EID is not an optimal solution1 , it reduces considerably the degradation experienced byED receivers while keeping a relatively simple architecture. Another interest ﬁnding was that EID also reduces the degradation in BER performance inherent to an increase of the integration time when compared toED. On the other hand, although EID improves in general the BER performance of the receiver, the main drawback is that its optimal integration time is usually higher than that of a receiver employing theED technique. This is a disadvantage since higher integration times translate in a slower bit transmission, i.e. smaller transmission data rates. Therefore, when deciding betweenED and EID techniques, the main tradeoff is between transmission speed and BER. In Chapter5, a channel estimation method for PPM receivers was presented and discussed along with the new UCP-MAC protocol which uses this channel estimation in the PHY layer. It also uses a MAC mechanism based on the IEEE 802.11 DCF function to optimally adjust the transmission data rate in both the transmitting and the receiving node. The channel estimation technique is based onED, however, it can be used for bothED and EID techniques since they have the same receiver front-end

1 EID is not an optimal solution in the sense that it does not completely eliminate the degradation of optimized receivers when operating in different channel conditions.

128 architecture up to the first integration stage. Using the channel estimation, a node can calculate its optimal integration time and, hence, its optimal data rate. This optimal data rate is then shared by the transmitting and the receiving nodes in order to optimize their communication in terms of transmission speed and a reduced BER. Both nodes exchange their optimal data rates through the MAC layer protocol. Then, the MAC layers of both nodes communicate with their respective PHY layers in order to adjust their transmission data rates allowing the nodes to synchronize their transmitting and receiving integration times. By simulating the new protocol, it was shown that the overall network performance improves in terms of the average transmission time and the message delivery ratio. Moreover, when using the EID demodulation technique, the message delivery ratio improves further. However, theED technique yields a better average transmission delay. Therefore, a tradeoff between delivery ratio and throughput exists and must be studied in a case-by-case basis depending on the type of application. Future work work will include additional simulations to help understand more the mentioned tradeoff and find other areas and scenarios in which the the new protocol may or may not improve the overall network performance. Improvements to the protocol will be considered as well. For instance, the channel estimation technique might be further improved or even substituted to yield more accurate results when calculating the optimal integration times. Better accuracy in this calculation should yield a lower BER which reduces the probability of a packet error. Also, additional improvements to the MAC layer protocol can be investigated in order to adapt the cognitive protocol to certain applications. An example would be, in very low-power or high traffic networks, having a scheme of idle periods could reduce the energy consumption and the collisions produced by high network traffic.

129 Appendix A DERIVATION OF THE PROBABILITY OF BIT-ERROR FOR PPM-ED RECEIVERS

Let Xi = µi + ni and Yi = mi be the signals in the ﬁrst and second integration

th windows, respectively, for i = 1, 2, ... , N. Here µi represents i sample of the received

signal (pulse) and mi and ni are independent zero-mean Gaussian random variables with the same variances σ2 representing AWGN. The condition for a bit decision based on the energy-detection (ED) is

1 N N X 2 X 2 Xi ≷ Yi (A–1) i=1 i=1 0 In this case, since the pulse is in the ﬁrst integration window, a bit-error occurs if

P 2 P 2 P 2 P 2 i Xi ≤ i Yi . Thus, we are interested in the probability P i Xi ≤ i Yi . In [64], PN 2 it has been shown that V = i=1 Xi follows the distribution of a non-central chi-square 2 P 2 random variable with a centrality parameter s = i µi and N degrees of freedom, i.e. 2 2 PN 2 V ∼ χN s , while W = i=1 Xi follows a central chi-square random variable with N

2 degrees of freedom, i.e. W ∼ χN (0). It was also shown that the probability of bit-error

PED = P (V ≤ W ) is given by

1 N−1 2 2 X n P = · es /σ · c · s2/σ2 (A–2) ED 22·N−1 n n=0 where

N−1−n 1 X c = · 2·N−1 (A–3) n n! k k=0 However, by the CLT, if N is sufﬁciently large , V and W can be approximated a Gaus-

2 2 2 2 sian random variables, i.e. V ∼ N µV , σV and W ∼ N µW , σW where µV = N · σ + s ,

2 2 4 2 2 2 4 µW = N · σ , σV = 2 · N · σ + 4 · s · σ , and σW = 2 · N · σ as shown in [64]. Hence, if we let Z = V − W , the probability of bit-error PED = P (Z ≤ 0) can be approximated by

130 q 2 PED ≈ Q µZ / σZ where Q (·) refers to the Q-function deﬁned by Equation 3–4 and the mean and variance of Z are

2 µZ = µV − µW = s (A–4)

2 2 4 2 2 µZ = σV + σW = 4 · N · σ + 4 · s · σ (A–5)

respectively. Now, assuming the signals in both integration windows are sampled at the

Nyquist frequency, the number of samples in each integration window is N = 2·B ·Tw

where B is the bandwidth of the received signal and Tw is the integration time in each

2 P 2 2 window. Also, note that s = i µi is the energy of the received signal Eb and σ = N0/2 q 2 where N0 is the two-sided noise spectral density. Therefore, PED ≈ Q µZ / σZ can

be expressed in terms of the received SNR-per-bit Eb/N0, bandwidth B, and integration time Tw as

  Eb/N0   PED ≈ Q p  (A–6) 2 · B · Tw + 2 · Eb/N0 which yields accurate results (< 5% error) for B·Tw > 20 [20].

131 Appendix B DERIVATION OF THE PROBABILITY OF BIT-ERROR FOR PPM-EID RECEIVERS

As in Appendix A, assume Xi and Yi are the received signals in the ﬁrst and second integration windows, respectively. Then, the energy integration in each window is

N i N X X 2 X 2 V = Xj = (N + 1 − i) · Xi (B–1) i=1 j=1 i=1

N i N X X 2 X 2 W = Yj = (N + 1 − i) · Yi (B–2) i=1 j=1 i=1

2 2 respectively, where Xi ∼ N µi , σ and Yi ∼ N 0, σ are statistically independent for all i = 1, 2 ... , N. Here we are interested in the P (V ≤ W ) = P (V − W ≤ 0) which is the probability of bit-error Pe for abbrefEID. To derive an expression for this probability we will need the means and variances of V and W . In Appendix C, the mean and variance of V are calculated to be

0 2 0 µV = N · σ + E (N) (B–3)

2 00 4 h 00 0 i 2 σV = 2 · N · σ + 4 · 2 · E (N) − E (N) · σ (B–4) where N 0 = N ·(N + 1) /2, N 00 = N ·(N + 1) · (2 · N + 1) /6, and E 0(N) and E 00(N) are given by EquationsC–16 and C–17, respectively. For W , the mean and variance simplify to

0 2 µW = N · σ (B–5)

2 00 4 σW = 2 · N · σ (B–6)

0 00 since the mean of Yi is zero, i.e. E (N) = E (N) = 0. If we let Z = V − W , then mean and

2 2 2 variance of Z are µZ = µV − µW and σZ = σV + σW , i.e.

132 0 µZ = E (N) (B–7)

2 00 4 h 00 0 i 2 σZ = 4 · N · σ + 4 · 2 · E (N) − E (N) · σ (B–8) which, with simple algebraic manipulation, can be expressed as

2 µZ = seff (B–9)

2 4 2 2 σZ = 4 · Neff · σ + 4 · σ · seff (B–10)

2 0 00 00 0 2 where seff = E (N) and Neff = N + 2· E (N) − E (N) /σ . Note that Equations B–9 and B–10 are in the same form as Equations A–4 and A–5 in Appendix A, respectively.

Hence, the probability of bit-error PEID = P (Z ≤ 0) can be calculated using Equation

00 3 2 A–2. However, if N is sufﬁciently large then N ≈ N /3, Z ∼ N µZ , σZ by the CLT, and q 2 PEID = P (Z ≤ 0) ≈ Q µZ / σZ , i.e.

  s2 /σ2  eff  PEID ≈ Q q (B–11)  2 2 2 · Neff + seff /σ or, equivalently,

  E 0/σ2 PEID ≈ Q   (B–12) pN3/3 + 2 · (2 · E 00 − E 0) /σ2 where Q (·) is the Q-function as deﬁned by Equation 3–4.

133 Appendix C P P 2 2 MEAN AND VARIANCE OF I J XJ FOR XJ ∼ N µJ , σ

2 Let us deﬁne the random variable V with mean µV and variance σV

N i N X X 2 X 2 V = Xj = (N + 1 − i) · Xi (C–1) i=1 j=1 i=1

where X1, X2, ... , XN are statistically independent Gaussian random variables with

2 2 2 means µi and variances σi = σ , i.e. Xi ∼ N µi , σ . Also, let us now deﬁne the follow- √ √ 2 ing Gaussian random variable Yi = N + 1 − i·Xi where Yi ∼ N N + 1 − i · µi , (N + 1 − i) · σ such that

N X 2 V = Yi (C–2) i=1 It is clear that V follows a non-central chi-square random variable with N degrees of freedom since Y1, Y2, ... , YN are independent Gaussian random variables with non-zero

2 means. Then, the mean and variance of Yi are given by [64]

2 2 µ 2 = σ + µ (C–3) Yi Yi Yi

2 4 2 2 σ 2 = 2 · σY + 4 · σY · µY (C–4) Yi i i i √ respectively, where µ = N + 1 − i · µ and σ2 = (N + 1 − i) · σ2. Since V = P Y 2 is the Yi i Yi i i

sum of independent random variables, then its mean µV can be calculated as

N N X X 2 2 µV = µ 2 = σ + µ (C–5) Yi Yi Yi i=1 i=1

N N 2 X X 2 µV = σ · (N + 1 − i) + (N + 1 − i) · µi (C–6) i=1 i=1 2 and the variance σV as

134 N N 2 X 2 X 4 2 2 σV = σ 2 = 2 · σY + 4 · σY · µY (C–7) Yi i i i i=1 i=1

N N 2 4 X 2 2 X 2 2 σV = 2 · σ · (N + 1 − i) + 4 · σ · (N + 1 − i) · µi (C–8) i=1 i=1 With some algebraic manipulation, it can be shown that

N X (N + 1 − i) = N · (N + 1) /2 (C–9) i=1

N X (N + 1 − i)2 = N · (N + 1) · (2 · N + 1) /6 (C–10) i=1

N N i X 2 X X 2 (N + 1 − i) · µi = µj (C–11) i=1 i=1 j=1

N N i j N i X 2 2 X X X 2 X X 2 (N + 1 − i) · µi = 2 · µk − µj (C–12) i=1 i=1 j=1 k=1 i=1 j=1 and, hence, the mean and variance of V can be expressed as

N i 0 2 X X 2 µV = N · σ + µj (C–13) i=1 j=1

 N i j N i  2 00 4 X X X 2 X X 2 2 σV = 2 · N · σ + 4 · 2 · µk − µj  · σ (C–14) i=1 j=1 k=1 i=1 j=1 respectively, where N 00 = N · (N + 1) · (2·N + 1) / 6 and N 0 = N ·(N + 1) /2. Furthermore, let us deﬁne

i X 2 E (i) = µj (C–15) j=1

i j i 0 X X 2 X E (i) = µk = E (j) (C–16) j=1 k=1 j=1

135 i j k i 00 X X X 2 X 0 E (i) = µl = E (j) (C–17) j=1 k=1 l=1 j=1

2 Then, µV and σV can be simpliﬁed to

0 2 0 µV = N · σ + E (N) (C–18)

2 00 4 h 00 0 i 2 σV = 2 · N · σ + 4 · 2 · E (N) − E (N) · σ (C–19)

136 REFERENCES [1] S. Ahmed and H. Arslan, “Iﬁ and isi in high data rate uwb coherent transceivers,” in IEEE Radio and Wireless Symposium, January 2008, pp. 883–886. [2] H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in International Symposium on Information Theory, 1973, pp. 267–281. [3] J. M. Almodovar-Faria and J. McNair, “Optimal integration time for energy-detection ppm uwb systems,” in IEEE Global Communications Conference, GLOBECOM, December 2012, pp. 4054–4059. [4] J. M. Almodovar-Faria, J. McNair, and D. Wentzloff, “Optimal receiver bandwidth for energy-detection ppm uwb systems,” in IEEE Wireless Communications and Networking Conference, WCNC, March 2011, pp. 1700–1705. [5] M. Anis, M. Ortmanns, and N. Wehn, “A 2.5mw 2mb/s fully integrated impulse-fm- uwb transceiver in 0.18 um cmos,” in IEEE International Microwave Symposium MTT-S, 2011, pp. 1–3. [6] A. Argyriou, “Mac protocol for wireless cooperative physical-layer network coding,” in IEEE Wireless Communications and Networking Conference, WCNC, 2012, pp. 1596–1601. [7] T. W. Barret, “History of ultra wideband (uwb) radar & communications pioneers and innovators,” in Progress in Electromagnetic Symposium, PIERS, 2000. [8] P. Bondyopadhyay, “Guglielmo marconi - the father of long distance radio communication - an engineer’s tribute,” in European Microwave Conference, vol. 2, 1995, pp. 879–885. [9] I. Broustis, S. Krishnamurthy, M. Faloutsos, M. Molle, and J. Foerster, “Multiband media access control in impulse-based uwb ad hoc networks,” in IEEE Transactions on Mobile Computing, vol. 6, no. 4, April 2007, pp. 351–366. [10] I. Broustis, A. Vlavianos, P. Krishnamurthy, and S. Krishnamurthy, “Mac layer throughput estimation in impulse-radio uwb networks,” in IEEE Transactions on Mobile Computing, vol. 10, no. 5, May 2011, pp. 700–715. [11] J. Bussgang and M. Leiter, “Error performance of differential phase-shift transmission over a telephone line,” in IEEE Transactions on Communication Technology, vol. 16, no. 3, 1968, pp. 411–419. [12] J. A. Camara, Electrical Engineering Reference Manual for Electrical and Elec- tronics, and Computer PE Exams, 8th ed. Professional Publications, Inc, 2009, ch. 59.

137 [13] D. Cassioli, M. Win, and A. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” in IEEE Journal on Selected Areas in Com- munications, vol. 20, no. 6, 2002, pp. 1247–1257. [14] T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, 2006. [15] D. Cox, “Universal digital portable radio communications,” Proceedings of the IEEE, vol. 75, no. 4, pp. 436–477, 1987. [16] M. Crepaldi, M. R. Casu, M. Graziano, and M. Zamboni, “A low-power cmos 2- ppm demodulator for energy detection ir-uwb receivers,” in IEEE International Conference on Ultra-Wideband, ICUWB, September 2007, pp. 461–466. [17] D. Daly, P. Mercier, M. Bhardwaj, A. Stone, Z. Aldworth, T. Daniel, J. Voldman, J. Hildebrand, and A. Chandrakasan, “A pulsed uwb receiver soc for insect motion control,” in IEEE Journal of Solid-State Circuits, JSSC, vol. 45, no. 1, 2010, pp. 153–166. [18] M. Di Benedetto, T. Kaiser, A. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview. Hindawi Publishing Corporation, 2006. [19] I. Dotlic and R. Kohno, “Performance analysis of impulse radio ultra-wideband differential detection schemes for body area networks,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications Workshops, PIMRC Workshops, 2010, pp. 72–77. [20] S. Dubouloz, B. Denis, S. de Rivaz, and L. Ouvry, “Performance analysis of ldr uwb non-coherent receivers in multipath environments,” in IEEE International Conference on Ultra-Wideband, ICUWB, September 2005, pp. 491–496. [21] A. Durantini and D. Cassioli, “A multi-wall path loss model for indoor uwb propagation,” in IEEE Vehicular Technology Conference, VTC, vol. 1, 2005, pp. 30–34 Vol. 1. [22] A. El Fawal, J.-Y. Le Boudec, R. Merz, B. Radunovic, J. Widmer, and G. Maggio, “Trade-off analysis of phy-aware mac in low-rate low-power uwb networks,” in IEEE Communications Magazine, vol. 43, no. 12, Dec 2005, pp. 147–155. [23] T. Erseghe and S. Tomasin, “Uwb wpan receiver optimization in the presence of multiuser interference,” in IEEE Transactions on Communications, vol. 57, no. 8, Aug 2009, pp. 2369–2379. [24] K. Fan, S. Zhang, and X. Liu, “A uwb pulse shapes modulation scheme based on modiﬁed hermite polynomials,” in International Conference on Wireless Communi- cations, Networking and Mobile Computing, WiCOM, 2008, pp. 1–4.

138 [25] F. Federal Communications Commision, “Unintentional radiators,” in Code of Federal Regulations (CFR) Title 47, Chapter 1, Part 15, Subpart B, October 2004. [26] F. Federal Communications Commission, “Ultra-wideband operation,” in Code of Federal Regulations (CFR) Title 47, Chapter 1, Part 15, Subpart F, May 2002, pp. 5–8. [27] J. Foerster, “Ieee p802.15 channel modeling sub-committee report,” IEEE P802.15 Working Group for Wireless Personal Area Networks, Tech. Rep., February 2003. [28] J. R. Foerster, “Interference modeling of pulse-based uwb waveforms on narrowband systems,” in IEEE Vehicular Technology Conference, VTC, vol. 4, 2002, pp. 1931–1935 vol.4. [29] J. Foerster, “The effects of multipath interference on the performance of uwb systems in an indoor wireless channel,” in IEEE Vehicular Technology Conference, VTC, vol. 2, 2001, pp. 1176–1180 vol.2. [30] R. Fontana, A. Ameti, E. Richley, L. Beard, and D. Guy, “Recent advances in ultra wideband communications systems,” in IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 129–133. [31] R. Fontana, “An insight into uwb interference from a shot noise perspective,” in IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 309–313. [32] Y. Gao, X. Liu, Y. Zheng, S. Diao, W. Toh, Y. Wang, B. Zhao, M. Je, and C.-H. Heng, “A low power interference robust ir-uwb transceiver soc for wban applications,” in IEEE International Symposium on Radio-Frequency Integration Technology, RFIT, 2012, pp. 153–155. [33] P. Gentner, M. Wiessflecker, H. Arthaber, A. Scholtz, and C. Mecklenbrauker, “Measured wideband near-field characteristics of an uwb rfid tag with on-chip antenna,” in IEEE International Conference on Ultra-Wideband, ICUWB, 2011, pp. 479–483. [34] A. Gerosa, M. D. Costa, A. Bevilacqua, D. Vogrig, and A. Neviani, “An energy- detector for non-coherent impulse-radio uwb receivers,” in IEEE International Symposium on Circuits and Systems, ISCAS, 2008. [35] S. Ghassemzadeh, L. Greenstein, A. Kavcic, T. Sveinsson, and V. Tarokh, “Uwb indoor path loss model for residential and commercial buildings,” in IEEE Vehicular Technology Conference, VTC, vol. 5, 2003, pp. 3115–3119. [36] S. Ghassemzadeh, R. Jana, C. Rice, W. Turin, and V. Tarokh, “Measurement and modeling of an ultra-wide bandwidth indoor channel,” in IEEE Transactions on Communications, vol. 52, no. 10, 2004, pp. 1786–1796.

139 [37] M. Ghavami, Ultra Wideband Signals and Systems in Communication Engineering. John Wiley & Sons, 2004. [38] G. B. Giannakis, “Ultra-wideband communications: an idea whose time has come,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP, vol. 1, April 2003. [39] S. Gishkori, G. Leus, and H. Deliç, “Energy detection of wideband and ultra- wideband ppm,” in IEEE Global Communications Conference, GLOBECOM, December 2010, pp. 1–5. [40] A. Gordillo, G. T. F. De Abreu, and R. Kohno, “Band-limited frequency efficient orthogonal pulse shape modulation for uwb communications,” in IEEE International Symposium on Spread Spectrum Techniques and Applications, 2004, pp. 498–502. [41] J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, 1st ed. Cambridge University Press, 2006. [42] H. Hertz, Electric Waves: Being Researches on the Propagation of Electric Action with Finite Velocity Through Space. Macmillan and Company, 1893. [43] R. Hidayat and Y. Miyanaga, “Ir-uwb pulse position modulation and pulse shape modulation through s-v channel model,” in International Conference on Communi- cation Software and Networks, ICCSN, 2010, pp. 214–217. [44] M. Ho, V. Somayazulu, J. Foerster, and S. Roy, “A differential detector for an ultra- wideband communications system,” in IEEE Vehicular Technology Conference, VTC, vol. 4, 2002, pp. 1896–1900 vol.4. [45] R. Hoctor and H. Tomlinson, “Delay-hopped transmitted-reference rf communications,” in IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 265–269. [46] Standard for Information technology–Telecommunications and information exchange between systems Local and metropolitan area networks–Specific requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standards Association Std., 2012. [47] O. Klymenko, G. Fischer, and D. Martynenko, “A high band non-coherent impulse radio uwb receiver,” in IEEE International Conference on Ultra-Wideband, ICUWB, vol. 3, September 2008, pp. 25–29. [48] J. Kunisch and J. Pamp, “Measurement results and modeling aspects for the uwb radio channel,” in IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 19–23. [49] D. Landi and C. Fischer, “The effects of uwb interference on gsm systems,” in International Zurich Seminar on Communications, 2004, pp. 86–89.

140 [50] A. Lapidoth, “Nearest neighbor decoding for additive non-gaussian noise channels,” in IEEE Transactions on Information Theory, vol. 42, no. 5, 1996, pp. 1520–1529. [51] F. S. Lee and A. P. Chandrakasan, “A 2.5nj/b 0.65v 3-to-5ghz subbanded uwb receiver in 90nm cmos,” in IEEE International Solid-State Circuits Conference, ISSCC, February 2007, pp. 116–590. [52] W. Li and T. Gulliver, “Optimal pulse shaping for pulse position modulation uwb systems with sparsity-driven signal detection,” in IEEE Vehicular Technology Conference, VTC, vol. 3, 2006, pp. 1455–1458. [53] J. Mao, D. Sarmiento M, Q. Zhou, J. Chen, P. Wang, Z. Zou, F. Jonsson, and L.-R. Zheng, “A 90nm cmos uhf/uwb asymmetric transceiver for rﬁd readers,” in Proceedings of the ESSCIRC, 2011, pp. 179–182. [54] P. Mercier, D. Daly, M. Bhardwaj, D. Wentzloff, F. Lee, and A. Chandrakasan, “Ultra- low-power uwb for sensor network applications,” in IEEE International Symposium on Circuits and Systems, ISCAS, 2008, pp. 2562–2565. [55] R. Merz, A. El Fawal, J.-Y. Le Boudec, B. Radunovic, and J. Widmer, “The optimal mac layer for low-power uwb is non-coordinated,” in IEEE International Symposium on Circuits and Systems, ISCAS, May 2006, pp. 2429–2432. [56] D. Mu and Z. Qiu, “Weighted non-coherent energy detection receiver for uwb ook systems,” in International Conference on Signal Processing, ICSP, 2008, pp. 1846–1849. [57] A. Nasri, R. Schober, and L. Lampe, “Performance of a bpsk nb receiver in mb- ofdm uwb interference,” in IEEE International Conference on Communications, ICC, vol. 10, 2006, pp. 4700–4705. [58] F. Nekoogar, Ultra-wideband Communications: Fundamentals and Applications, 1st ed. Upper Saddle River, NJ, USA: Prentice Hall Press, 2005. [59] K. Ohuchi and H. Habuchi, “Analysis of the bi-orthogonal modulation system using two different inner sequences,” in IEEE International Symposium on Spread Spectrum Techniques and Applications, vol. 2, 2000, pp. 743–746 vol.2. [60] P. Pagani and P. Pajusco, “Experimental assessment of the uwb channel variability in a dynamic indoor environment,” in IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC, vol. 4, 2004, pp. 2973–2977. [61] K. Pahlavan, “Preface of the workshop,” in 1st IEEE Workshop on Wireless Local Area Networks, May 1991. [62] M. Pelissier, J. Jantunen, B. Gomez, J. Arponen, G. Masson, S. Dia, J. Varteva, and M. Gary, “A 112 mb/s full duplex remotely-powered impulse-uwb rﬁd transceiver for wireless nv-memory applications,” in IEEE Journal of Solid-State Circuits, JSSC, vol. 46, no. 4, 2011, pp. 916–927.

141 [63] M. Perez-Guirao, T. Kaiser, K. Jobmann, and B. Kull, “Ca-mac: Autonomous, cognitive mac for ir-uwb networks,” in International Conference on Cognitive Radio Oriented Wireless Networks and Communications, CrownCom, Aug 2007, pp. 199–205. [64] J. Proakis, Digital Communications, 4th ed. McGraw-Hill Higher Education, 2000. [65] F. Ramirez-Mireles, T. Cooklev, and G. Paredes-Orozco, “Uwb-fsk: performance tradeoffs for high and low complexity receivers,” in IEEE Transactions on Consumer Electronics, vol. 56, no. 4, 2010, pp. 2123–2131. [66] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Prentice Hall, 2002. [67] L. Reggiani and G. M. Maggio, “Coherent vs. non-coherent detection for orthogonal convolutional modulation: A trade-off analysis,” in IEEE International Conference on Ultra-Wideband, ICUWB, September 2006, pp. 43–48. [68] G. F. Ross, “Transmission and reception system for generating and receiving base-band pulse duration pulse signals without distortion for short base-band communication system,” US Patent 3 728 632, 1973. [69] C. Rushforth, “Transmitted-reference techniques for random or unknown channels,” in IEEE Transactions on Information Theory, vol. 10, no. 1, 1964, pp. 39–42. [70] H.-S. Ryu, J.-S. Lee, and C. Kang, “Ber analysis of dual-carrier modulation (dcm) over rayleigh fading channel,” in International Congress on Ultra Modern Telecom- munications and Control Systems and Workshops, ICUMT, 2010, pp. 717–721. [71] M. E. Sahin, I. Guvenc, and H. Arslan, “Optimization of energy detector receivers for uwb systems,” in IEEE Vehicular Technology Conference, VTC, vol. 2, June 2005, pp. 1386–1390. [72] A. Saleh and R. Valenzuela, “A statistical model for indoor multipath propagation,” in IEEE Journal on Selected Areas in Communications, JSAC, vol. 5, no. 2, 1987, pp. 128–137. [73] Y. Sankarasubramaniam, I. Akyildiz, and S. McLaughlin, “Energy efﬁciency based packet size optimization in wireless sensor networks,” in IEEE International Work- shop on Sensor Network Protocols and Applications, SNPA, 2003, pp. 1–8. [74] U. Schuster and H. Bolcskei, “Ultrawideband channel modeling on the basis of information-theoretic criteria,” in IEEE Transactions on Wireless Communications, vol. 6, no. 7, 2007, pp. 2464–2475. [75] C. E. Shannon, “A mathematical theory of communications,” in Bell Systems Technical Journal, vol. 27, 1948.

142 [76] I. Shomorony and A. S. Avestimehr, “Is gaussian noise the worst-case additive noise in wireless networks?” in IEEE International Symposium on Information Theory Proceedings, ISIT, 2012, pp. 214–218. [77] I. Shomorony and A. Avestimehr, “Worst-case additive noise in wireless networks,” in IEEE Transactions on Information Theory, vol. 59, no. 6, 2013, pp. 3833–3847. [78] V. Sipal, B. Allen, and D. Edwards, “Multi-tone frequency shift keying for ultrawideband wireless communications,” in IET Communications, vol. 6, no. 10, 2012, pp. 1170–1178. [79] P. Supanakoon, S. Chaiyapong, S. Promwong, and J.-i. Takada, “Three-ray path loss based on peak power loss for ultra wideband impulse radio systems,” in International Symposium on Intelligent Signal Processing and Communications Systems, ISPACS, 2011, pp. 1–4. [80] Time Domain Company, TDC. [Online]. Available: http://www.timedomain.com/ profile.php [81] N. Van Helleputte, M. Verhelst, W. Dehaene, and G. Gielen, “A reconfigurable, 130 nm cmos 108 pj/pulse, fully integrated ir-uwb receiver for communication and precise ranging,” in IEEE Journal of Solid-State Circuits, JSSC, vol. 45, no. 1, 2010, pp. 69–83. [82] F. Wang, Z. Tian, and B. Sadler, “Weighted energy detection for noncoherent ultra- wideband receiver design,” in IEEE Transactions on Wireless Communications, vol. 10, no. 2, 2011, pp. 710–720. [83] X. Wang, A. Dinh, D. Teng, L. Chen, S. B. Ko, Y. Shi, J. Basran, and V. Bello-Hass, “Impulse based range-gated uwb wireless transceiver ic in 90nm cmos for medical sensing applications and communications,” in IEEE International Conference on Ultra-Wideband, ICUWB, 2009, pp. 194–199. [84] D. Wentzloff, “Pulse-based ultra-wideband transmitters for digital communication,” Ph.D. dissertation, Massachusetts Institute of Technology, June 2007. [85] D. Wentzloff, F. Lee, D. Daly, M. Bhardwaj, P. Mercier, and A. Chandrakasan, “Energy efficient pulsed-uwb cmos circuits and systems,” in IEEE International Conference on Ultra-Wideband, ICUWB, 2007, pp. 282–287. [86] K. Witrisal, G. Leus, G. J. M. Janssen, M. Pausini, F. Troesch, T. Zasowski, and J. Romme, “Noncoherent ultra-wideband systems,” in IEEE Signal Processing Magazine, vol. 26, no. 4, 2009, pp. 48–66. [87] S. Wong and F. Lau, “Impacts of uwb interference on selected radio systems used by the government,” in IEEE International Conference on Circuits and Systems for Communications, ICCSC, 2008, pp. 525–529.

143 [88] J. Wu, M. Dong, Q. Liang, and H. Xiang, “Consideration of pre-ﬁlter bandwidth optimization for non-coherent uwb receiver,” in International Conference on Wireless Communications, Networking and Mobile Computing, WiCOM, 2006, pp. 1–4. [89] L. Wu and A. Vosoughi, “Biorthogonal pulse shape modulation for ir-uwb systems over fading channels,” in IEEE Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2008, pp. 191–195. [90] J. Xia, C. Law, Y. Zhou, and K. S. Koh, “3-5 ghz uwb impulse radio transmitter and receiver mmic optimized for long range precision wireless sensor networks,” in IEEE Transactions on Microwave Theory and Techniques, vol. 58, no. 12, 2010, pp. 4040–4051. [91] L. Yang and G. Giannakis, “Ultra-wideband communications: an idea whose time has come,” in IEEE Signal Processing Magazine, vol. 21, no. 6, Nov 2004, pp. 26–54. [92] M. Yargholi and A. Nabavi, “Analog front end modules design in non-coherent uwb receivers for sensor networks,” in IEEE International Conference on Semiconductor Electronics, ICSE, November 2008, pp. 64–68. [93] T. Zasowski, F. Althaus, M. Stager, A. Wittneben, and G. Troster, “Uwb for noninvasive wireless body area networks: channel measurements and results,” in IEEE Conference on Ultra Wideband Systems and Technologies, 2003, pp. 285–289. [94] L. Zhao and A. Haimovich, “The capacity of an uwb multiple-access communications system,” in IEEE International Conference on Communications, ICC, vol. 3, 2002, pp. 1964–1968 vol.3. [95] ——, “Performance of ultra-wideband communications in the presence of interference,” in IEEE Journal on Selected Areas in Communications, vol. 20, no. 9, 2002, pp. 1684–1691. [96] Y. Zheng, S.-X. Diao, C.-W. Ang, Y. Gao, F.-C. Choong, Z. Chen, X. Liu, Y.-S. Wang, X.-J. Yuan, and C. Heng, “A 0.92/5.3nj/b uwb impulse radio soc for communication and localization,” in IEEE International Solid-State Circuits Conference, ISSCC, 2010, pp. 230–231. [97] Y. Zheng, Y. Tong, C. W. Ang, Y.-P. Xu, W. G. Yeoh, F. Lin, and R. Singh, “A cmos carrier-less uwb transceiver for wpan applications,” in IEEE International Solid-State Circuits Conference, ISSCC, 2006, pp. 378–387. [98] Y. Zheng, K.-W. Wong, M. Annamalai Asaru, D. Shen, W. H. Zhao, Y. J. The, P. Andrew, F. Lin, W.-G. Yeoh, and R. Singh, “A 0.18 mu;m cmos dual-band uwb transceiver,” in IEEE International Solid-State Circuits Conference, ISSCC, 2007, pp. 114–590.

144 [99] Z. Zou, Y. Ruan, L.-R. Zheng, and H. Tenhunen, “Impulse uwb energy detection receiver with energy offset synchronization scheme,” in IEEE International Conference on Ultra-Wideband, ICUWB, 2009, pp. 540–544.

145 BIOGRAPHICAL SKETCH Jose M. Almodovar-Faria received his B.S. degree in Electrical Engineering from the University of Puerto Rico - Mayaguez in 2006, and M.S. degrees in Electrical En- gineering from the University of Michigan - Ann Arbor in 2008 and from the University of Florida - Gainesville in 2011. In the spring of 2014, he received his Ph.D. degree in Electrical and Computer Engineering from the University of Florida. His research inter- ests include UWB wireless communications and cognitive radios and networks.

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