Reducing Effects of Multipath Propagation with a Blind Equalizer

Master of Science Thesis in Electrical Engineering Department of Electrical Engineering, Linköping University, 2018

Reducing Effects of Multipath Propagation With A Blind Equalizer

Emma Söderström Master of Science Thesis in Electrical Engineering Reducing Effects of Multipath Propagation With A Blind Equalizer Emma Söderström LiTH-ISY-EX--18/5171--SE

Supervisor: Kamil Senel isy, Linköpings universitet Mattias Avesten SAAB Aeronautics Rikard Bergsten SAAB Aeronautics Examiner: Mikael Olofsson isy, Linköpings universitet

Division of Communicationsystems Department of Electrical Engineering Linköping University SE-581 83 Linköping, Sweden

When transmitting data from an aircraft being prepared at the apron (the area in front of the hangar) telemetry data is transmitted to ground personnel. The transmitted data is subject to severe distortion due to multipath propagation created by the surroundings, resulting in erroneous detection. By equalizing the signal using the Constant Modulus Algorithm a significant increase in detection performance has been observed, both in simulations and tests on collected data. The most sufficient parameters were chosen after testing a set of different parameter combinations on simulations with single delays. These parameters were then used to equalize simulated multipath as well as collected data. The results show that short delays with low power can be resolved without any equalizer. Longer delays with relatively low power can be resolved using the proposed equalizer but long delays with high power cannot be resolved by the equalizer at all. The thesis shows that it is worth investigating implementation of the equalizer.

iii

Contents

List of Figures vii

List of Tables ix

Notation xi

1 Introduction 1 1.1 Motivation ...... 1 1.2 Purpose ...... 3 1.3 Problem statement ...... 3 1.4 Limitations ...... 4 1.5 Thesis outline ...... 4

2 Theory 5 2.1 Basic Concepts ...... 5 2.1.1 The Telemetry System ...... 5 2.1.2 Line Codes ...... 5 2.1.3 Randomizers ...... 7 2.1.4 Numerically Controlled Oscillators ...... 8 2.1.5 Analog Modulation ...... 9 2.1.6 Mixers ...... 12 2.2 Complex Matrix Calculus ...... 12 2.3 Multipath Propagation ...... 13 2.3.1 Fading Channel Model ...... 13 2.3.2 Multipath Eﬀects on Frequency Modulated Signals . . . . . 14

3 Channel Equalization 17 3.1 Equalizers ...... 17 3.1.1 Linear Adaptive Equalization Using Training Symbols . . . 19 3.1.2 Linear Adaptive Blind Equalization ...... 19 3.1.3 Equalizing PCM/FM ...... 19 3.1.4 Previous Research ...... 20 3.1.5 The Constant Modulus Algorithm ...... 20

v 4 Method 23 4.1 Test Bed ...... 23 4.1.1 Current Telemetry System ...... 23 4.1.2 Communication Model ...... 24 4.1.3 CMA Parameters ...... 25 4.2 Equalizer Analysis ...... 26 4.2.1 Simulation Setup ...... 26 4.2.2 Simulation correctness ...... 26 4.2.3 Test Cases ...... 27 4.2.4 Test on Real Data ...... 28

5 Results 31 5.1 Simulation Correctness ...... 31 5.2 Single Delays ...... 32 5.3 Multiple Delays ...... 34 5.4 Test on Real Data ...... 35

6 Discussion 39 6.1 Results ...... 39 6.1.1 Simulation Correctness ...... 39 6.1.2 Single Delays ...... 39 6.1.3 Multiple Delays ...... 40 6.1.4 Simulation Results in Relation to the Apron ...... 42 6.1.5 Real Data ...... 42 6.2 Method ...... 43 6.3 A wider perspective ...... 44

7 Conclusion 45 7.1 Future work ...... 45

A Communication Model Listings 49

B Simulation Results for Single delays 53

C Simulation Results for Multiple Delays 69

Bibliography 79

vi LIST OF FIGURES vii

List of Figures

1.1 An illustration of the surroundings of the apron and how the re- ﬂection causes multipath propagation...... 2 1.2 Example of ber as a function of snr. The increasing snr is retrieved by adding white Gaussian noise to a fm modulated signal before fm demodulation...... 3 1.3 ber as function of snr of a multipath signal. The receiver receives a strong reﬂection that has traveled 150m longer than the main signal, along with the main signal...... 4

2.1 Different line codes and their voltage responses to a binary sequence [8]...... 6 2.2 Randomized nrz encoder as defined in IRIG106 [17]...... 7 2.3 Randomized nrz decoder as defined in IRIG106 [17]...... 8 2.4 Phase wheel representing a period of a signal, the number of points is determined by 2n and M is the jump size...... 9 2.5 I/Q plot showing the envelope and phase of a passband signal . . 10 2.6 An example of fm modulation ...... 11 2.7 A Tapped Delay Line [2] ...... 14 2.8 The result of multipath on fm-modulated signals...... 15

3.1 Simple tapped delay line used as an equalizer for known delays. . 18

4.1 A general outline of the communication model...... 24 4.2 Transmitter part of generated signal for testing the simulation correctness ...... 27 4.3 RF receiver setup...... 27 4.4 Explanation of areas in the area plots ...... 28 4.5 Test-sender setup with two antennas to mimic the actual aircraft. . 29 4.6 Real setup of test-sender ...... 29

5.1 Figure showing the eye-diagrams of a generated, known, signal without multipath and the simulation of the same signal ...... 31 5.2 Figure showing the eye-diagrams of a generated, known, signal with 117m multipath and the simulation of the same signal . . . . 32 viii LIST OF FIGURES

5.3 Figure showing the eye-diagrams of a generated, known, signal with 150m multipath and the simulation of the same signal . . . . 32 13 5.4 Sub-plot group of one of the best results for cma with µ = 2− . . 33 14 5.5 Sub-plot group of one of the best results for cma with µ = 2− . . 33 5.6 Eye-diagram before and after cma when ber = 0 both before and after ...... 36 5.7 Eye-diagram before and after cma when ber is not zero before but is after equalizer ...... 36 5.8 Eye-diagram before and after cma when the cma is unable to resolve the multipath ...... 37

6.1 The three areas and their abbreviations ...... 41

10 B.1 Sub-plot group for µ = 2− and update every sample ...... 54 10 B.2 Sub-plot group for µ = 2− and update interval 12 ...... 54 10 B.3 Sub-plot group for µ = 2− and update interval 23 ...... 55 10 B.4 Sub-plot group for µ = 2− and update interval 46 ...... 55 10 B.5 Sub-plot group for µ = 2− and update interval 92 ...... 56 11 B.6 Sub-plot group for µ = 2− and update every sample ...... 56 11 B.7 Sub-plot group for µ = 2− and update interval 12 ...... 57 11 B.8 Sub-plot group for µ = 2− and update interval 23 ...... 57 11 B.9 Sub-plot group for µ = 2− and update interval 46 ...... 58 11 B.10 Sub-plot group for µ = 2− and update interval 92 ...... 58 12 B.11 Sub-plot group for µ = 2− and update every sample ...... 59 12 B.12 Sub-plot group for µ = 2− and update interval 12 ...... 59 12 B.13 Sub-plot group for µ = 2− and update interval 23 ...... 60 12 B.14 Sub-plot group for µ = 2− and update interval 46 ...... 60 12 B.15 Sub-plot group for µ = 2− and update interval 92 ...... 61 13 B.16 Sub-plot group for µ = 2− and update interval 12 ...... 61 13 B.17 Sub-plot group for µ = 2− and update interval 23 ...... 62 13 B.18 Sub-plot group for µ = 2− and update interval 46 ...... 62 13 B.19 Sub-plot group for µ = 2− and update interval 92 ...... 63 14 B.20 Sub-plot group for µ = 2− and update interval 12 ...... 63 14 B.21 Sub-plot group for µ = 2− and update interval 23 ...... 64 14 B.22 Sub-plot group for µ = 2− and update interval 46 ...... 64 14 B.23 Sub-plot group for µ = 2− and update interval 92 ...... 65 15 B.24 Sub-plot group for µ = 2− and update interval 12 ...... 65 15 B.25 Sub-plot group for µ = 2− and update interval 23 ...... 66 14 B.26 Sub-plot group for µ = 2− and update interval 46 ...... 66 14 B.27 Sub-plot group for µ = 2− and update interval 92 ...... 67

C.1 BER vs SNR of short resolvable multipath combinations without equalizer ...... 70 C.2 BER vs SNR of short resolvable multipath combinations with equalizer ...... 70 C.3 BER vs SNR of long resolvable multipath combinations without equalizer ...... 71 C.4 BER vs SNR of long resolvable multipath combinations with equalizer ...... 71 C.5 BER vs SNR of combination of Resolvable with equalizer multipath combinations without equalizer ...... 72 C.6 BER vs SNR of combination of Resolvable with equalizer multipath combinations with equalizer ...... 72 C.7 BER vs SNR of non-resolvable with equalizer multipath combinations without equalizer ...... 73 C.8 BER vs SNR of non-resolvable with equalizer multipath combinations with equalizer ...... 73 C.9 BER vs SNR of combination of resolvable with equalizer and resolvable multipath combinations without equalizer ...... 74 C.10 BER vs SNR of resolvable with equalizer and resolvable multipath combinations with equalizer ...... 74 C.11 BER vs SNR of non-resolvable with equalizer and resolvable multipath combinations without equalizer ...... 75 C.12 BER vs SNR of non-resolvable with equalizer and resolvable multipath combinations with equalizer ...... 75 C.13 BER vs SNR of non-resolvable with equalizer and resolvable with equalizer multipath combinations without equalizer ...... 76 C.14 BER vs SNR of non-resolvable with equalizer and resolvable with equalizer multipath combinations with equalizer ...... 76 C.15 BER vs SNR of resolvable, non-resolvable with equalizer and resolvable with equalizer multipath combinations without equalizer . . . 77 C.16 BER vs SNR of resolvable, non-resolvable with equalizer and resolvable with equalizer multipath combinations with equalizer . . . . . 77

List of Tables

4.1 Table of delay compositions for test of multiple multipath components. r = Resolvable, re = Resolvable with Equalizer, nr = Non- Resolvable ...... 28

5.1 Delays and powers used to create multipath signals ...... 34 5.2 ber before and after cma on test cases from the apron and taxiway 35

Notation

xi xii Notation

Abbreviations Abbreviation Deﬁnition agc Automatic Gain Control am Amplitude modulation awgn Additive White Gaussian Noise ber Bit Error Rate Bc Coherence Bandwidth BiΦ Bi-phase cma Constant Modulus Algorithm dc Direct Current dds Direct Digital Synthesis dfe Decision Feedback Equalizer fir Finite Impulse Response fm Frequency Modulation fpga Field Programmable Gate Array i In-phase if Intermediate Frequency isi Inter Symbol Interference lms Least Mean Square lo Local Oscillator nco Numerically Controlled Oscillator nr Neural Network nr Non-Resolvable nrz Non-Return to Zero pcm Pulse Code Modulation pcm/fm Pulse Code Modulation/Frequency Modulation pm Phase Modulation prbs Pseudo-Random Bit Sequence psk Phase Shift Keying q Quadrature-phase qpsk Quadrature Phase Shift Keying r Resolvable re Resolvable with Equalizer rf Radio Frequency rls Recursive Least Square rnrz Randomized Non-Return to Zero rz Return to Zero snr Signal to Noise Ratio soqpsk Shaped-Oﬀset Quadrature Phase Shift Keying Tc Coherence Time Td Multipath Delay Spread tdl Tapped Delay Line vfo Variable Frequency Oscillator Notation xiii

Terminology Word Explanation Apron Place in front of the hangar where the aircraft is being prepared. Runway The area where takeoﬀ and landing takes place. Taxiway The route taken by the aircraft from the apron to the runway. Telemetry A system that records system parameters and trans- System mits them. Multipath A phenomena where a transmitted signal is reﬂected Propagation of objects in its path, resulting in multiple versions of the signal being detected at the receiver.

1 Introduction

This chapter will introduce the problem to be studied, motivate the study and provide the questions that the thesis aims to answer. Furthermore, limitations and assumptions are stated and the chapter is concluded with an outline of the thesis.

1.1 Motivation

Flight test operations are utilized to verify aircraft systems and during the testing, it is important to convey the system parameters from the aircraft to a receiver on the ground. These parameters are transmitted from the aircraft using telemetry, which records the system parameters and transmits them.

Knowledge about the system parameters is not only important while the aircraft is in the air, it is also important to know that the system is ready for takeoff. The aircraft is prepared for takeoff at the apron where the necessary system parameters are sent to the test engineers. In most cases, the apron is surrounded by hangars, buildings, fences, fuel tanks and the apron itself is made of reinforced concrete. Electromagnetic waves at high frequencies, reflect of and are distorted by hard surfaces like the ones surrounding the apron, this is illustrated in Figure 1.1. At the receiver side a composite signal consisting of the transmitted signal and its delayed reflections, called a multipath signal, needs to be decoded.

In real systems, any channel between the transmitter and receiver will be subject to noise. The ratio between the power of the signal and the power of the noise is called Signal To Noise Ratio (snr). The bigger the diﬀerence between the power of the signal and the noise power, the higher the snr. It is thus desired to have as high snr as possible. The snr is directly connected to the Bit Error Rate (ber)

1 2 1 Introduction

100m

Fuel tanks Fences

other houses

200m 150m

Apron

Hangar 90m

Reflected signal Direct signal

Receiver

Transmitter

Figure 1.1: An illustration of the surroundings of the apron and how the reﬂection causes multipath propagation. 1.2 Purpose 3 at the receiver as shown in Figure 1.2. A higher snr results in less errors.

Figure 1.2: Example of ber as a function of snr. The increasing snr is retrieved by adding white Gaussian noise to a fm modulated signal before fm demodulation.

When introducing multipath propagation, the signal sometimes becomes so distorted that it is impossible for the receiver to detect the bits correctly even with a high snr, this is illustrated in Figure 1.3. It is clear from Figure 1.3 that increasing the snr does not decrease the amount of errors introduced by the multipath propagation. In order to correctly detect the bits, the multipath signal has to be altered to resemble the transmitted signal, this process is called Equalizing.

1.2 Purpose

The purpose of this thesis is to facilitate the correct detection of the telemetry data which in turn will help test engineers by improving the detection performance based on the received multipath data.

1.3 Problem statement

The thesis seeks to answer the following questions: 4 1 Introduction

Figure 1.3: ber as function of snr of a multipath signal. The receiver receives a strong reﬂection that has traveled 150m longer than the main signal, along with the main signal.

1. How well can a blind equalizer reduce the eﬀects of multipath on the received signal? 2. How should the equalizer be tuned to achieve suﬃcient results?

1.4 Limitations

The thesis will only focus on blind equalizers for pcm/fm modulation. Other modulation techniques or training-based equalization techniques are left for future work and will not be considered in this thesis.

1.5 Thesis outline

The theoretical background will be presented in Chapter 2, the diﬀerent equalization techniques will be described in greater detail in Chapter 3. Chapter 4 will describe the method used to test the equalizers and Chapter 5 will present the results. In Chapter 6, the results will be discussed providing the base for the conclusion in Chapter 7. 2 Theory

This chapter aims to give the reader a deeper understanding of the theory behind the multipath propagation problem and the algorithms used by the equalizers. It is expected that the reader has some previous knowledge about signals and systems. The ﬁrst part should be seen as introduction of basic concepts along with some technical terms. It is recommended that the sections are read in sequence since each section relies on theory presented in the previous one.

2.1 Basic Concepts

This section will brieﬂy review basic theory and concepts that are used through- out the thesis.

2.1.1 The Telemetry System A telemetry system collects data in a place where system monitoring is inconvenient and transmits the collected data to another location for evaluation [4, p.1- 5]. It is mostly used for testing moving vehicles such as aircrafts and cars, where on-board system overview is not always possible. The telemetry system, like any communication system, consists of three main parts [15], a transmitter, a channel and a receiver. The transmitter sends data to the receiver over a channel, which in this case is air.

2.1.2 Line Codes The analog data signal collected in the telemetry system needs to be quantized and converted into an n-bit digital word. The logic word needs to be converted into electrical voltage before transmission. The digital words are converted into

5 6 2 Theory voltage with the help of line codes [4, pp. 121-125]. Line codes, or pcm-codes, convert binary data into differentiable voltages. There are different ways to do this, such as Non-Return-to-Zero (nrz), Return-to-Zero (rz), and BiPhase (BiΦ), which are demonstrated in Figure 2.1. The extension L (level) means that the voltage level is set by the current bit, M (mark) changes voltage level when a binary 1 (a mark) appears and S (space) changes voltage level when a binary 0 (space) appears. nrz changes the voltage level as indicated by the extension and stays on that level for the rest of that bit period. rz represents 1 by high voltage for half a bit period and stays low for 0. BiΦ represents 1 with going high for the first half of the bit period and then goes low, 0 is represented by low voltage in the first half and then high for the last half. The extensions for BiΦ work in the same way as for nrz but with half bit periods. nrz is further divided into polar and non-polar. Polar nrz uses the voltage +V to indicate 1 and –V for 0 whereas non-polar uses +V as high voltage and zero voltage for binary 0. In telemetry systems polar nrz-L is most commonly used due to its low bandwidth requirement and ease of implementation.

Figure 2.1: Diﬀerent line codes and their voltage responses to a binary sequence [8]. 2.1 Basic Concepts 7

2.1.3 Randomizers It is not unusual to have long sequences of ones and zeros to be transmitted. The diﬀerence between the number of ones and zeros in a sequence is called disparity. Large disparity in a sequence may introduce a Direct Current (dc) component to the signal [5]. dc-components are in general diﬃcult to reliably transmit over long distances and can cause the receiver to loose track of bits. It is thus desirable to maximize the disparity of a code, or make it dc-free. This can be done by using a Randomizer.

Encoder The most common randomizer for randomized nrz-L (rnrz-L) uses a 15-stage shift register and 2 XOR (modulo-2 addition) gates to randomize the data [17, p.D-10]. Initially the 14th and 15th stages of the shift register are XOR:ed where the result is XOR:ed with the bit to be encoded. The resulting bit from the last XOR operation is both shifted into the register and the output of the encoder, see Figure 2.2. The rnrz-L randomizer has similar properties as an Pseudo Random Bit Sequence (prbs) generator, and will produce a maximum prbs of 215 1 bits if the input sequence is only zeros and there is a single high bit in the shift− register.

Figure 2.2: Randomized nrz encoder as deﬁned in IRIG106 [17].

Decoder The rnrz-L bit sequence is decoded by once again using the result of the XOR operation on the the 14th and 15th stages of the shift register and use as input to the second XOR gate together with the incoming bit. The result of the final XOR is, as in the encoder, the output of the decoder. The encoded incoming bit is then shifted into the shift register, see Figure 2.3. The decoder is self synchronizing 8 2 Theory but needs 15 initial bits to fill the shift register with valid data if the initial values of the shift registers of the encoder and decoder are not equal. Usually the registers are not equal since there are 215 different ways the bits can be arranged and encoder and decoder are aware of each other [17, pp.D-11:D-12].

Figure 2.3: Randomized nrz decoder as deﬁned in IRIG106 [17].

Furthermore, single bit errors will produce 2 extra errors 14 and 15 bits later [17, D-11]. This due to the fact that the erroneous bit is used in the 14th and 15th stage of the shift register to decode the incoming bits. Single bit errors therefore have an error multiplication factor of 3 in this decoding system.

2.1.4 Numerically Controlled Oscillators

A Numerically Controlled Oscillator (nco), also called Direct Digital Synthesis (dds), is a way to accurately generate analog waveforms at a speciﬁc frequency [6]. Con- sidering a time-continuous sinusoidal signal, its phase varies between 0 and 2π over a period at a speed determined by the frequency. In the digital case, a phase wheel from 0 to 2π represents the period of the signal and the number of incre- ments needed to overﬂow the wheel determines the frequency, see Figure 2.4. The wheel then acts as a modulo-M counter where M is the step-size. The number of points on the phase wheel is 2n where a larger n will increase the accuracy of the nco. With a sampling frequency of fs and desired nco output frequency f, the following relation can be derived

Mf f = s (2.1) 2n 2.1 Basic Concepts 9

Jump Size

Figure 2.4: Phase wheel representing a period of a signal, the number of points is determined by 2n and M is the jump size.

2.1.5 Analog Modulation In order to transmit signals to distant destinations, electromagnetic radiation is used, often referred to as Radio Frequency (rf) [9]. The electromagnetic wave is a high frequency sinusoid called carrier. The typical waveform of a carrier is given by

s(t) = A cos(2πfct + θ), (2.2) where A is the amplitude, fc is the carrier frequency and θ is the phase of the signal.The carrier waveform can be modiﬁed, or modulated, to carry messages. The most common modulation types are Amplitude Modulation (am), Phase Mod- ulation (pm) and Frequency Modulation (fm) where the amplitude, phase and frequency respectively are altered with a speciﬁc message. The non-modulated message is called a baseband signal and when the baseband signal is modulated onto a carrier it is called a passband signal.

Complex Baseband Representation A passband signal can be written in the form s (t) = s (t) cos(2πf t) s (t) sin(2πf t), (2.3) p I c − Q c where sI (t) and sQ(t) are baseband signals referred to as In-phase component (i) and Quadrature-phase component (q) [12]. i and q are orthogonal to each other with a phase shift of 90° and together they can represent any sinusoidal signal. The complex envelope, also called complex baseband representation [12], is deﬁned by sbb(t) = sI (t) + jsQ(t). (2.4) 10 2 Theory

e(t) θ(t) I

Figure 2.5: I/Q plot showing the envelope and phase of a passband signal

A relationship between the passband and baseband signals can be derived with Euler’s identity resulting in

j2πf t s = Re s (t)e− c . (2.5) p { bb } jθ(t) Furthermore, Equation 2.4 can be written in polar form as e(t)e− , where the envelope e(t) and phase θ(t) are deﬁned by q e(t) = s (t) = s2(t) + s2 (t), | bb | I Q ! (2.6) 1 sI (t) θ(t) = tan− . sQ(t) Figure 2.5 shows a plot of the envelope and phase where the i component corresponds to the real part and the q component is the imaginary part as deﬁned in Equation 2.4.

A signal that is represented by complex samples obtained from i and q is often referred to as a Quadrature signal, or equivalently an Analytic signal.

Frequency Modulation Recall that, a carrier waveform can be altered in diﬀerent ways to contain a message. In frequency modulation, the message determines the Instantaneous Fre- quency which is the derivative of phase with respect to time and is given by Z dθ = am(t) θ m(t) = a m(t)dt, (2.7) dt ←→ { } 2.1 Basic Concepts 11

Figure 2.6: An example of fm modulation where a is a scaling factor and m(t) is the message to be modulated. By combining Equation 2.2 and 2.7, the modulated signal can be deﬁned as Z s(t) = A cos(2πfct + a m(t)dt). (2.8)

The frequency deviation, fd, is given by a f (t) = m(t), (2.9) d 2π and attains its peak value when m(t) is at its maximum.

Figure 2.6 shows the result of modulating a baseband sinusoid (top Figure) with fm. Since all information about the message is encoded in the frequency, the envelope, e(t), will be constant while the phase varies. Note that, frequency modulation does not take exact phases into consideration, only the instantaneous frequency is considered. This means that the position on the circumference of the circle in Figure 2.5 is not important, only the angular velocity.

PCM/FM pcm/fm is a common modulation technique in telemetry. pcm means that digital line coding is used to convert the collected data bits into waveforms, then fm is used to modulate the baseband signal. It is commonly used with nrz-L as line code where the voltage level is between +1V and -1V. When sending data over long distances pcm/fm is preferred due to its constant envelope, also called constant modulus, a property which makes pcm/fm resistant to amplitude variations in the channel. 12 2 Theory

2.1.6 Mixers It is often inconvenient to operate at rf frequencies, as the electrical components might not work and the frequency deviations might be hard to distinguish [9]. Intermediate Frequency (if) has been introduced in wireless communications to be able to do signal operations at lower frequencies. The if frequency is between baseband and the carrier frequency. A mixer is a function that shifts the frequency up or down depending on if the conversion is from if to rf or from rf to if. The mixer uses a Variable Frequency Oscillator (vfo) to shift the frequencies according to equation f = f f , IF RF − VFO (2.10) fRF = fIF + fVFO. Converting to baseband directly can cause rf leaks which in turn may cause un- wanted dc-oﬀsets, for a detailed explanation see the discussion in [9].

2.2 Complex Matrix Calculus

This section will introduce the non-trivial calculus used in the thesis and provide simple examples. Deﬁnition 2.1 (Hermitian Transpose). The Hermitian transpose, also called conjugate transpose, of a matrix containing complex numbers is deﬁned as the transpose of the complex conjugate of the matrix. AH = A¯T

Example 2.2 shows a simple example of the Hermitian Transpose. Example 2.2 Suppose we want to take the Hermitian transpose of " # 2 3 j A = . 6 3j 8 +− j − − The resulting matrix would then be " # 2 6 + 3j AH = . 3 + j −8 j −

Deﬁnition 2.3 ( ¯ , "Gradient operator with respect to the elements of vector ∇W W¯ ). ¯ f is deﬁned as ∇W " #T ∂f ∂f ∂f w¯ f = , , ..., . ∇ ∂w1 ∂w2 ∂wn Where f is a scalar and W¯ is a vector. This is called scalar-by-vector operation in matrix calculus. 2.3 Multipath Propagation 13

2.3 Multipath Propagation

It is often assumed that the receiving antenna only receives the transmitted signal. However, this is not always the case. The transmitted signal can, like any wave, be reﬂected, refracted and scattered by hard objects. In addition to noise, the receiving antenna will receive a distorted version of the transmitted signal leading to Intersymbol Interference (isi) which in turn may result in erroneous demodulation. When a carrier-modulated signal experiences distortion due to multipath, it is often referred to as fading [16].

There are two important properties of a channel, coherence time, Tc and coherence bandwidth, Bc [10, p.200]. Tc deﬁnes the duration over which the system can be modeled as approximately time-invariant. If the duration of the transmitted signal, T, is smaller than Tc the channel can be regarded as time-invariant. If T

Td := max τi(t) τj (t) . (2.11) i,j | − |

The relation between Td and Bc is given by 1 Bc = . (2.12) Td

2.3.1 Fading Channel Model There are two types of fading [10, pp.204-205], frequency-flat and frequency- selective. If the bandwidth of the multipath signal, B, is much greater than Bc the channel is frequency-selective, otherwise it is frequency-flat. This can be summarized by B << B : Frequency-flat, c (2.13) B >> Bc : Frequency-selective. The typical model for the impulse response in wireless communication is defined as XM h(t) = A ejθk δ(t τ ). (2.14) k − k k=1 For the frequency-flat case, h(t) is approximately constant for the coherence time and all frequency components of the signal are scaled and delayed identically [10, p207].

For the frequency-selective case, Equation 2.14 can be modeled as a tapped delay line (tdl), also called Finite Impulse Response (fir) ﬁlter, see Figure 2.7. If the 14 2 Theory taps are spaced 1/W apart, the channel can be represented by

X∞ h(t) = α δ(t i/W ), (2.15) i − i=1 where the each {αi} can be seen as zero-mean complex Gaussian random variable [12, pp.380-384]. In this case, the signals at diﬀerent frequencies are not scaled and delayed identically. The channel considered in this thesis is frequency selec-

z–1 z–1 z–1

Figure 2.7: A Tapped Delay Line [2] tive. Assume one reﬂection taking a path 200 m longer than another, which will 6 5 result in a delay of approximatley 6.67 · 10− s. Using Equation 2.12, Bc is 1.5 · 10 Hz whereas the bandwidth of the transmitted signal is 734 kHz (7.34 · 106 Hz) due to the frequency deviation. It can be easily seen that Bc < B resulting in a frequency selective channel.

2.3.2 Multipath Effects on Frequency Modulated Signals The multipath environment simply sums up time shifted and amplitude deci- mated versions of the signal. In the case of fm, a delayed signal usually has a diﬀerent frequency than the one being transmitted. This case is shown in Figure 2.8, which demonstrates that the resulting signal is both frequency shifted and has diﬀerent amplitude than the transmitted signal. Thus, the property of constant envelope discussed in Section 2.1.5 is ruined. Further, and more severe, the phase variations interfere with the phase-modulated signal [18]. 2.3 Multipath Propagation 15

Figure 2.8: The result of multipath on fm-modulated signals.

3 Channel Equalization

Equalizers are designed to revert the changes made on the signal by the channel, for example removing the delayed version of the signal. If the channel is frequency-flat, the equalization task is easier since all signal frequencies are af- fected in the same way at the same time. To equalize frequency-flat signals different diversity techniques can be used, see [12, pp.387-397]. A frequency-selective channel on the other hand, affects the spectral components differently and varies with time and therefore requires more sophisticated equalization methods.

3.1 Equalizers

Equalizers are usually divided into two groups, adaptive and static [1]. The static equalizer does not update its taps and is therefore is not suited for unknown, time-varying channels. Adaptive equalizers on the other hand, automati- cally adapts its filter to adjust for the channel [13]. Adaptive equalizers are sub- divided into linear and non-linear and these can be either training-based or blind. One common type of training-based non-linear equalizer is called Decision Feed- back Equalizers (dfe) [19], which uses previously detected symbols to remove isi from the symbols currently being demodulated. Also, the use of Neural Networks (nn) has been subject to research showing promising results for both blind [22] and training-based [3] equalization. In order to undo the changes made by the channel, the linear adaptive equalizers are typically designed as fir filters due to the structure of the channel impulse response, see Equation 2.14 and Figure 2.7. The filter taps are updated in response to the error between the desired response and the output, y, from the equalizer [13]. The output from the equalizer can be written as y(k) = XT (k)W(k), (3.1)

17 18 3 Channel Equalization where X is the vector of signal values stored in the tapped delay line and W is the vector of tap coeﬃcients. The adaptive equalizer then uses the information about the error between the desired signal, d, and the output, y,

e(k) = d(k) y(k), (3.2) − to update the taps according to

W(k + 1) = W(k) + ∆W(k) (3.3) where the decision on the correction factor ∆W (k) is what diﬀers between the diﬀerent linear adaptive equalizer algorithms. Example 3.1 is intended to show the idea of an ideal linear adaptive equalizer.

Example 3.1 Imagine a multipath environment leading to one direct and two delayed paths, which delays the transmitted signal by one and two samples, respectively. Fur- thermore assume that there is no decimation of the amplitude for any signal. The resulting signal, r, will then be the sum of these three signals, represented as the vectors below. 1 0 0  1          2 1 0  3          3 + 2 + 1 =  6                  4 3 2  9  5 4 3 12 The resulting vector, Q, will then be the input to the equalizer. If we assume an ideal equalizer, it will in this case have two taps, one with delay 1, d1, and one with delay 2, d2, and a multiplication factor of -1, see Figure 3.1.

d2 d1

-1 -1

r Q

Figure 3.1: Simple tapped delay line used as an equalizer for known delays.

The steps can be seen as a loop where the following steps are taken:

1 : Q(i) = r(i) d1 d2 − − 2 : d1 = Q(i) 3 : d2 = Q(i 1) − 3.1 Equalizers 19

The index of Q in step 2 and 3 is in general i NumberOf samplesToDelay + 1. − In the very first loop, the first value of the vector reaches the equalizer. The initial values of the delays (d1 and d2) are 0 which gives Q(1) = r(1) = 1. Then d1 and d2 are updated resulting in d1=1 and d2 remains 0. The second time the loop is run, Q(2) = r(2) 1 = 2. Now d1=2 and d2=1. This process is repeated and after the final loop, the− equalized signal is equal to the transmitted signal.

3.1.1 Linear Adaptive Equalization Using Training Symbols

One of the most common ways to adapt the equalizer to match the channel is to periodically transmit known training symbols [19]. The knowledge about the transmitted signal, makes it possible to closely estimate the channel using the received signal and adapt the taps thereafter. Most training-based linear methods are based on the Least Mean Squares (lms) Algorithm [1] or the Recursive Least Square (rls) Algorithm [13].

3.1.2 Linear Adaptive Blind Equalization

Transmitting trainings symbols can sometimes be inconvenient, the system can have bandwidth limitations or the channel can vary rapidly with time. This requires training symbols to be re-transmitted frequently. Due to these limitations, blind equalizers have emerged. Blind equalizers do not use training symbols to estimate the channel, they estimate the channel using other techniques that make use of á priori knowledge about the transmitted signal [18]. Such á priori knowledge could be about the constant envelope of the signal or the shape of the frequency spectrum [14].

3.1.3 Equalizing PCM/FM

The current telemetry setup does not have any training symbols meaning that a blind equalizer has to be used. The transmitted signal has the constant modulus property, explained in Section 2.1.5. This means that algorithms using the knowledge of the constant modulus for equalization can be used. The shape of the frequency spectrum is known, but its mathematical representation is not exact and is only valid under certain circumstances [4]. This makes it more diﬃcult to use the information for equalization purposes. Linear equalizers does not usually perform as well when there are spectral nulls present [13], in those cases non- linear equalizers usually oﬀer better performance to the price of more complexity. Especially in the blind case, non-linear equalizers become harder to implement, for example nn is a possibility that will not be explored in this thesis. The best suited equalizer for test is thus one that make use of the constant modulus, for example the Constant Modulus Algoritm (cma) [18]. 20 3 Channel Equalization

3.1.4 Previous Research There exists vast amounts of newer equalization techniques for other modulation techniques. For example, there have been developments within the equalization for Phase Shift Keying (psk) and am, which, unfortunately, cannot be used to equalize pcm/fm modulated signals due to the fact that the angular frequency is lost in the process. Not much research has been done lately on new equalizing methods specifically on pcm/fm since new, more efficient modulation techniques for telemetry, such as Shaped-Offset Quadrature Phase Shift Keying (soqpsk), have emerged. The telemetry system in focus for this thesis uses pcm/fm and is not currently compatible with soqpsk or other modulation techniques. Numerous papers have researched the use of cma for pcm/fm-modulated telemetry signals, for example [7], [11], [21], with promising results. These papers, however, do not include any information about how parameters such as step size, number of taps or update interval (explained below) should be chosen or how they affect the performance.

3.1.5 The Constant Modulus Algorithm One of the most commonly used blind equalization algorithm is the cma, which is similar to the lms algoritm. It uses the á priori knowledge about the constant modulus to estimate the taps. In the approach by John R. Treichler and Brian G. Agee [18], a quadrature sampled multipath fm signal is transmitted through a fir filter with complex valued coefficients. The output of the filter is described by y(k) = XT (k)W(i). (3.4) Here X(k) is the signal values stored in the tapped delay line, written as

X(k) = [x(k) x(k 1) x(k N + 1)]T , (3.5) − ··· − and W(i) is the vector of adjustable coeﬃcients given by

T W(i) = [w0(i) wN 1(i)] . (3.6) ··· − The index i means that the coeﬃcients are adjustable with time, we will for sim- plicity, assume that the coeﬃcients are updated at each sampling instant and therefore replace the index i with k after this point.

The output, y(k), should be the same as the transmitted signal after ideal equalization. The property of constant modulus should therefore also be pre- served. The algorithm adjusts the taps in W to minimize a positive deﬁnite measure of the deviation from the constant modulus. This measure is called the cost function and is denoted by J. The general form of J is given by

J = d[F(y(k)),F(s(k))], (3.7) where the length metrics d and F should be defined for the specific algorithm. F=E{ · } where E{ · } denotes the statistical expectation function, for this specific 3.1 Equalizers 21 algorithm d and F are chosen such that J is given by 1 J = E y(k) 2 12 . (3.8) 4 {| | − } The choice of W that minimizes J is the coefficients used in the filter. Note that y(k) in Equation 3.8 is replaced with the definition given in Equation 3.4. There are many ways to find the best values for W and in the approach adopted from [18], a gradient search algorithm is employed to make hardware implementation easier.

A gradient search algorithm computes the derivative of a function, f ( · ), at a point a to see which way the function is increasing at point a, and takes a step, decided by the algoritm used, in the the negative direction [20]. This is due to the fact that a function f(x) decreases while going from a point a in the direction of the negative gradient of f at that point, - f (a), which allows an iterative search algoritm of the following form, ∇

a = a µ f (a ). (3.9) n+1 n − ∇ n

Hence, the newly reached point, an+1 is one step closer to the minimum, unless the minimum is passed, in which case the algorithm will oscillate between two points around the minimum.

Note that the step size µ has to be chosen with care, too large step size may result in the algorithm oscillating around the minimum. Utilizing a smaller step size requires more iterations and can, depending on the function, result in converging to local minimas.

The cma algorithm uses an n-dimensional version of Equation 3.9 where the tap-values are updated to minimize the cost function deﬁned as follows

W(k + 1) = W(k) µ J . (3.10) − ∇w k The gradient of J is given by [18]

2 J = E [ y(k) 1] · y(k)X∗(k) . (3.11) ∇w { | | − } The algorithm replaces the true gradient in Equation 3.10 with the instantaneous estimate given by 2 ˆ J = [ y(k) 1] · y(k)X(k)∗, (3.12) ∇w | | − which results in

2 W(k + 1) = W(k) µ[ y(k) 1] · y(k)X(k)∗. (3.13) − | | −

4 Method

The details of the method are described in this chapter. Since the most conve- nient á priori knowledge is the constant modulus, cma is the equalizer tested. The main approach is that the equalizer is ﬁrst tested through simulations in a testbed written in Matlab. The testbed is designed to mimic the current telemetry system while making it possible to test the system with and without the equalizer present, allowing comparison between the simulations. By running simulations with a set of diﬀerent cma parameters, described in Section 3.1.5, and comparing the results, the best parameters out of the set can then be chosen. These parameters are used when equalizing the reception of a known signal sent from a signal generator at the apron. The ber of the received signal with and without the equalizer will show how well the equalizer performs.

4.1 Test Bed

The test bed and its components will be described in this section, which should give the reader an idea of how the communication model is working.

4.1.1 Current Telemetry System The current telemetry system modulates rnrz-L data with pcm/fm. The carrier frequency is 2.2 GHz and the frequency deviation is 734 kHz. The system transmits 2163 kbps from two antennas and the receiver has a sampling frequency of 100 MHz, resulting in 46 or 47 samples per bit.

At the transmitter, the linear coded bits are low-pass filtered before modulation, called pre-mod filtering, to round off the sharp edges of the square wave. After the fm-modulation, the modulated signal is band-pass filtered to reduce spectral

23 24 4 Method

Transmitter

NCO/ Lowpass- FM- Bandpass- Bit Gen filter modulator filter Channel

Equalizer Receiver

Error rate Lowpass- FM- Bandpass- calculator filter demodulator filter

Figure 4.1: A general outline of the communication model. coverage. At the receiver, the signal is once again band-pass ﬁltered before fm- demodulation and low-pass ﬁltered before decision. Bit decisions are made from the integration of the middle 10 samples of the bit, if it is positive the bit is decoded as 1, otherwise it is decoded as 0.

4.1.2 Communication Model The ﬁrst step towards testing the equalizer is to have a communication system model with a transmitter, a channel and a receiver. The communication model used is shown in Figure 4.1. The basic implementation of the diﬀerent parts of the system is described in the following sections.

Numerically Controlled Oscillator

To obtain simulation data, a bit vector is created by generating a bit each time the nco overflows. The bit value is decided by a modulo-2 counter where 1 is added to the previous bit, resulting in 0 and 1 every other overflow. Each bit is first scrambled and then repeated for every step M, until the counter overflows, at which time the next bit is repeated in the same manner. This results in a vector of bits sampled at a sampling frequency of the output frequency of the nco. The bits are generated using the nco to resemble the real transmitter and sampled in order to be able to delay the signal with delays of higher precision than a whole bit. A simplified version of the nco code is provided in Listing A.1, where the 4.1 Test Bed 25 hexadecimal bit sequence FF00 is generated.

FM-Modulator

The transmitter fm modulates the input vector according to Equation 2.8 where the carrier frequency, sampling frequency, frequency deviation and time vector is given as input. Phase noise is added in order to make the simulations more like the real transmitter. Listing A.2 shows the code for the transmitter.

Channel

A function which creates the multipath version of an input signal with given delays and attenuations is shown in Listing A.3. Hence, the output of the function represents a signal transmitted through a multipath channel.

FM-Demodulator

The receiver demodulates the fm signal according to Listing A.4. The aim is to compare equalization techniques in terms of ber. A normal approach is to com- pute ber as a function of snr. The ber is calculated by comparing the generated bit sequence with the demodulated sequence. In the real receiver, the bit value is chosen based on an average of a few samples in the middle of a bit. In this model, the middle ten samples are used to make a decision. To plot the ber against the snr, the ber has to be calculated at diﬀerent snr values. To accomplish this, Additive White Gaussian Noise (awgn) with variance based on the given snr is generated and added to the modulated signal before demodulation.

4.1.3 CMA Parameters

In the cma algorithm, the number of taps, the µ value and the interval the taps are updated are variables that can be tweaked to get better results. The number of taps determines how long delays that can be detected, more taps allows cma to resolve longer delays. The µ value determines how fast we move in the direction of the zero, once the minimum is closer than the step size, the point reached by the cost function will oscillate around the minimum. Too large µ will result in oscillations around the minimum causing incorrect equalization whereas too small µ will lead to large convergence times. The update interval of the taps is also a variable to take into consideration. Updating on every sample will cause the ﬁlter taps to change 46-47 times per bit, updating too seldom will not resolve 10 15 the multipath. The cma parameter ranges tested are µ = 2− 2− , the number of taps tested are 50, 100, 200 and 300 and ﬁnally the update− intervals are every 12, 23, 46 and 92 samples as well as a few tests with update every sample. 26 4 Method

4.2 Equalizer Analysis

As previously described in Section 4.1.3, finding the optimal parameters for the simulations is tricky. This section will cover the method used for analyzing the equalizer and adjusting the algorithm parameters. The performance of the equalizer will first be simulated for different combinations of the parameters, the combination that results in the best equalization will then be tested on real data from the apron.

4.2.1 Simulation Setup All of the simulations have the following settings:

Carrier Frequency = 10 MHz, Sampling Frequency = 100 MHz, Frequency Deviation = 734 kHz, Simulation time = 0.01 s.

The carrier frequency is set to 10 MHz since 2.2 GHz would result in too much data for a short time period. Reducing the carrier frequency allows longer time periods and in result a more reliable results. Since the time vector is created with a step size equal to the inverse of the sampling frequency, the simulation time is limited to 0.01 s since longer simulation time results in such large vectors to be handled that Matlab runs out of memory or takes too long to ﬁnish. The alterna- tions from the real transmitter does not aﬀect the results of the multipath or the equalization.

4.2.2 Simulation correctness The correctness of the simulation setup is tested by comparing simulated multipath eye-diagrams with generated ones. If there is a signiﬁcant diﬀerence between the eye diagrams of the signals the simulated multipath is incorrect. If the eye diagrams are more or less the same, the simulation results are credible.

The generated bit sequence is generated from a bit generator and fm-modulated, from which i and q signals are created. These signals can be split where one part is delayed by a number of 100 MHz delays decided by the user, see Figure 4.2. The delayed i and q signals are then added to their non-delayed versions, converted to an analog signal and mixed to rf with a Local Oscillator (lo). The multipath signal is then sent to a Quasonix receiver where the i/q baseband components are fed to an oscillator, shown in Figure 4.3. The oscilloscope samples the signals at 10 Msps and saves the samples in binary form to a USB drive, allowing the results to be imported to Matlab. The capture is then up-converted to 100 Msps to mimic the preferred sampling rate. The idea is that this test setup can generate and record real multipath signals, as they appear in real life, and compare with the simulated multipath signals in order to evaluate the simulation credibility. 4.2 Equalizer Analysis 27

D D

DAC + Mixer

BIT GEN FM-MOD x RF + DAC

Q ~

D D

Figure 4.2: Transmitter part of generated signal for testing the simulation correctness

RF Quasonix Oscilloscope USB

Figure 4.3: RF receiver setup.

4.2.3 Test Cases Since the equalizers are supposed to resolve the multipath signals that the receiver is not able to resolve on its own, these cases are the primary targets for testing the equalizer. The equalizer is given 3000 bits for convergence in the single delay case and 5000 for the multiple delay case. The bits at the beginning will be erroneous before the cma algorithm has converged, these bits will be dis- regarded and not considered for performance assessment.

Single Delays Non resolvable single multipath components are found by running the simulations with increasing delay and power of the reflected signal component. If more 6 than 10− errors occurs for a specific power and delay, the position is marked. The area formed by the errors is filled with a solid color shown as an area in a grid, hereafter called an area plot. The area plots can be divided into 3 different areas, one area where the receiver can detect the bits without any errors despite the delayed signal, one where the equalizer is able to aid the receiver to detect bits without errors and one where the equalizer is unable to resolve signal, see Figure 4.4. By comparing the area before and after the equalizer it can be seen how well the equalizer works for different parameters.

Multiple Delays To test the equalizer on multiple delays, the parameters providing the smallest area of non-resolvable multipath in the single delay case are used. The multipath 28 4 Method

Error at receiver with equalizer Error at the receiver, resolved No errors at by equalizer the receiver, no errors with equalizer

Figure 4.4: Explanation of areas in the area plots signals are constructed by combining signals from the three areas, Resolvable (r), Resolvable with Equalizer (re) and Non-Resolvable (nr), as described in Table 4.1. The delayed signals are added together using the multipath generator described in Listing A.3. The main idea behind combining the diﬀerent areas is to see if minimizing the area of nr increases the chances of resolving the multipath signal. If that is the case, the focus can remain on minimizing the area of nr for the single delays to maximize the performance.

Type of multipath Combinations of Short r delays Combinations of Long r delays Combinations of re delays Combinations of nr delays r + re delays r + nr delays re + nr delays r + re + nr delays

Table 4.1: Table of delay compositions for test of multiple multipath components. r = Resolvable, re = Resolvable with Equalizer, nr = Non-Resolvable

4.2.4 Test on Real Data

To evaluate the equalizer in a realistic environment, a mobile test-sender was created to be able to retrieve real multipath data from the apron. The test-transmitter consists of the same transmitter as in Figure 4.2 but without the delays and with 4.2 Equalizer Analysis 29

DAC Mixer

BIT GEN FM MOD x 10dB 10dB PA 5W

DAC ~

Figure 4.5: Test-sender setup with two antennas to mimic the actual aircraft.

Figure 4.6: Real setup of test-sender added ampliﬁers as in Figure 4.5. The actual transmitter is shown in Figure 4.6. The receiver is the same as in Figure 4.3 where the rf comes from the antennas used for receiving the current telemetry communication. The whole reception works in the manner as in 4.2.2 where the Quasonix forwards the i/q components to the oscilloscope where the signal is sampled at 10 Msps and saved to an USB-drive. The transmitter transmits 8 ones and 8 zeros (FF00) using rnrz-L described in Section 2.1.3 which can be de-randomized and compared with the initial transmitted signal for an error rate calculation.

The data was manually collected (by pressing "save" on the oscilloscope) at in- stances where the receiver had trouble resolving the multipath and the error rate was high. The main focus is on the apron, but data sets were collected in front of the apron as well, following the route the aircraft will take whilst taxiing.

5 Results

This chapter will show the results of the tests described in Chapter 4.

5.1 Simulation Correctness

The result of the simulation correctness, explained in Section 4.2.2 is shown in Figures 5.1 to 5.3, where the eye diagrams of two diﬀerent delays are shown along with a signal without multipath for comparison.

Figure 5.1: Figure showing the eye-diagrams of a generated, known, signal without multipath and the simulation of the same signal

31 32 5 Results

Figure 5.2: Figure showing the eye-diagrams of a generated, known, signal with 117m multipath and the simulation of the same signal

Figure 5.3: Figure showing the eye-diagrams of a generated, known, signal with 150m multipath and the simulation of the same signal

5.2 Single Delays

The performance of the cma equalizer on single delays was tested as described 10 15 in Section 4.2.3. The tests included µ values from 2− to 2− , updates every 12, 23, 46 and 92 samples as well as 50, 100, 200 and 300 taps. The results are shown in Appendix B, but the two best results are presented below in Figure 5.4 and 5.5. 5.2 Single Delays 33

The plots show the results before (blue or darkest color) and after the cma equalizer, where the µ value and update interval is ﬁxed for each sub-plot group while the number of taps diﬀer within the sub-plots. The y-axis shows the strength of the delayed signal in comparison to the direct signal. The direct signal will always have power one, the delayed signal will then, in all real cases, be lower than that. The x-axis shows the delay in bits, where a delay value of one corresponds to 138m (1 sample is 3m delay, one bit has approximately 46 samples). The area then shows both at which delay and power the equalizer is needed along with how severe delay it can resolve.

13 Figure 5.4: Sub-plot group of one of the best results for cma with µ = 2−

14 Figure 5.5: Sub-plot group of one of the best results for cma with µ = 2− 34 5 Results

5.3 Multiple Delays

The best parameters from Section 5.2 are tested on diﬀerent multipath scenarios, as explained in 4.1. The results are shown in Appendix C. The powers and delays used for the simulations are shown in Table 5.1, where the results from Figure 5.4 is the basis for the choices.

Type of Multipath Delay (bits) Power 0.22 0.32 0.9 0.9 0.22 0.32 0.44 0.9 0.9 0.9 r (Shorter) 0.22 0.32 0.44 0.54 0.9 0.9 0.9 0.9 0.22 0.32 0.44 0.54 0.65 0.9 0.9 0.9 0.9 0.9 0.22 0.97 0.9 0.5 0.22 0.97 1.74 0.9 0.5 0.5 r (Longer) 0.22 0.32 0.97 1.74 0.9 0.9 0.5 0.5 0.22 0.32 0.97 1.74 1.96 0.9 0.9 0.5 0.5 0.5 0.54 1.00 0.9 0.9 0.54 0.76 1.00 0.9 0.6 0.8 re + r 0.54 0.76 1.00 1.74 0.9 0.6 0.8 0.7 0.54 0.76 1.00 1.74 2.17 0.9 0.6 0.5 0.7 0.7 1.00 1.08 0.9 0.8 1.00 1.08 1.30 0.9 0.8 0.8 re 1.00 1.08 1.30 1.52 0.9 0.8 0.8 0.8 1.00 1.08 1.30 1.52 1.74 0.9 0.8 0.8 0.8 0.8 1.00 1.08 1.0 1.0 1.00 1.08 1.30 1.0 1.0 1.0 nr 1.00 1.08 1.30 1.52 1.0 1.0 1.0 1.0 1.00 1.08 1.30 1.52 1.74 1.0 1.0 1.0 1.0 1.0 0.22 1.08 1.0 1.0 0.22 0.44 1.30 1.0 1.0 1.0 r + nr 0.22 0.44 1.30 1.52 1.0 1.0 1.0 1.0 0.22 0.44 1.30 1.52 1.74 1.0 1.0 1.0 1.0 1.0 1.08 1.30 0.8 1.0 1.08 1.30 1.52 0.8 1.0 1.0 re + nr 1.08 1.19 1.30 1.52 0.8 0.8 1.0 1.0 1.08 1.19 1.30 1.52 1.74 0.8 0.8 1.0 1.0 1.0 0.54 1.30 1.52 0.8 0.8 0.9 0.54 1.30 1.52 1.74 0.8 0.8 0.9 0.9 r + re + nr 0.54 1.08 1.30 1.52 1.74 0.8 0.8 0.8 0.9 0.9

Table 5.1: Delays and powers used to create multipath signals 5.4 Test on Real Data 35

5.4 Test on Real Data

Data was collected in two areas, at the apron and in front of the apron. The results are shown in Table 5.2, where data sets 1-15 are from the apron whilst 16-25 are from the area in front.

Data BER Be- BER After Data BER Be- BER After Set fore CMA CMA Set fore CMA CMA 1 0 0 14 0.000497 0 2 0 0 15 0.1221 1.42e-5 3 0.23742 0.00097 16 0 0 4 0 0 17 0.081 0 5 0 0 18 0 0 6 0 0 19 0.070 0 7 0.032 0 20 0.441 0.2478 8 0.021 0 21 0.097 0 9 0.391 1.45e-05 22 0.134 0 10 0 0 23 0.212 0.178 11 0.178 1.89e-5 24 0.052 0.382 12 0.316 0 25 0.67 0.089 13 0.000104 0

Table 5.2: ber before and after cma on test cases from the apron and taxiway

Figure 5.6 shows the eye-diagrams before and after cma for the case when the ber is 0 before and after the equalizer. Figure 5.7 shows the eye-diagrams when the ber is not zero before but after cma and Figure 5.8 when the cma is unable to resolve the multipath. 36 5 Results

Figure 5.6: Eye-diagram before and after cma when ber = 0 both before and after

Figure 5.7: Eye-diagram before and after cma when ber is not zero before but is after equalizer 5.4 Test on Real Data 37

Figure 5.8: Eye-diagram before and after cma when the cma is unable to resolve the multipath

6 Discussion

The simulation results have been presented in Chapter 5. This chapter aims to analyze and discuss the simulation results in order to answer the problem statement. Further more the method and the work in a wider perspective will be discussed.

6.1 Results

This section will analyze and discuss the results of the tests of the tests conducted.

6.1.1 Simulation Correctness Looking at the eye-diagrams in Figures 5.1, 5.2 and 5.3, it is clear that the simulations are very similar to the generated version. The transmitter generating the real signal does not have exactly the same filters as the simulated signal, making the eye of the generated signal slightly larger than the simulated one. There is also different phase jitter as well as different snr in the cables. The comparison is mainly focused on the similarity between the shapes of the eye-diagrams since the multipath determines the shape. Since there are no significant differences between the generated and simulated signals, the simulations should be sufficient enough to make a fair assessment of the cma equalizer.

6.1.2 Single Delays It is not intuitive from the results in Appendix B how the different parameters by themselves affect the performance of the equalizer. By looking at one sub-plot group, where µ and the update interval is fixed, it seems that longer taps tend to increase the performance and manage to resolve multipath signals with higher

39 40 6 Discussion powers. By focusing on only the tap length of 300 for a fix update interval, the µ value seems to increase the ability to equalize multipath signals with higher power in a similar manner as the tap-values. Too large µ however seems to decrease the power possible to equalize. Varying the update interval, and keeping µ and the tap-value fixed, results in decreased performance when having longer update intervals, whilst lower update intervals, apart from update every sample, results in higher performance. The results seem reasonable from a theoretical point of view. The discussion in Section 4.1.3 mention that too large µ value will result in oscillations and too small update interval or too few taps will not be able to resolve the multipath. This is also reflected in the results.

From the results in Appendix B, it is clear that the results shown in Figures 5.4 13 14 and 5.5, with µ = 2− and update interval of 23 or µ = 2− with update interval 12 while using 300 taps provides the best equalizer performance out of the parameter values tested.

The simulations loops through the power for each delay, where the power is increased with 0.1 for each loop and the delay is increased with 1 sample for each iteration. Since the power is increased with steps of 0.1, this results in an approximate area, which in theory could be larger. Looking at, for example, Figure 5.4, the bottom of the area could thus start at 0.81 instead of 0.9. Furthermore, in order to show the results in a way that allows comparison, the added noise pseudo random starting with the same seed for each simulation, meaning that there could be small deviations in the results depending on the noise. The simulations do, however, give an approximate area of equalization that makes comparison between parameter choices possible.

6.1.3 Multiple Delays

The resulting plots for the multiple delays are shown in Appendix C, using 300 13 taps and µ = 2− along with an update interval of 23.

It can be seen in Figures C.1 - C.8 that all r, re and nr multipath signals created from power-delay combinations conﬁned within their respective area (see ﬁgure 6.1) can be resolved in all cases. This can be expected for combinations of r and re since they are resolvable in the single delay case. It is, however, interesting that combining nr signals can be resolved, since the combined single delays are not resolvable. One possible explanation is that the delays chosen somehow cancel each other out to some extent.

Looking at the r +re and r +nr cases in Figures C.9 - C.12, it seems like it is possible to resolve multipath components to a certain threshold. The case with 5 combined delays in the r +nr case, Figure C.12 is not resolvable within the 5000 bits of convergence. In some cases, having multiple long delays with high power seems to cause a severe isi which cannot be resolved. 6.1 Results 41

NR RE

Figure 6.1: The three areas and their abbreviations

The combination of re + nr components seem to be more diﬃcult to equalize, see Figure C.13 and C.14. Furthermore, the combination of r + re + nr, shown in Figure C.15 and C.16 is not resolvable by the equalizer in any of the cases tested. The results show that all of the cases that are not resolvable include a combination of re and nr. It would thus seem reasonable to believe that minimizing the area where the equalizer is unable to resolve the single delay multipath signals should result in better performance of the equalizer.

It is important to note that not all combinations can be tested, these results are thus, once again, only basis for a general perception of how good the equalizer works. The case where the same delay was added multiple times was not tested, since it will only increase the power of the signal and not cause any more distortion. It is, however, important to have an Automatic Gain Controller (agc) before detection to prevent large power deviations. The equalizer will work better if the input power is closer to 1 since it calculates the deviation from one and tries to counteract that deviation. A power 4 times more or less than the original signal would imply a large deviation and cause the equalizers cost function to grow towards inﬁnity.

The problem with larger power was encountered when adding multiple multipath signals with each other. Since no agc is present in the simulation setup, this was counteracted by dividing signals with power larger than 2 by 2 before running the equalizer. This can not be seen as a real agc but it does have the same eﬀect in this case. 42 6 Discussion

6.1.4 Simulation Results in Relation to the Apron The simulation results have been very promising, the proposed parameters give reasonable results both in the single and the multiple delay case. Looking at the outline of the apron in Figure 1.1, it is possible to get a rough idea of the most likely delays that the receiver will encounter. The aircraft is most often placed as in Figure 1.1 and there are 4 parallel fuel tanks, spaced around 3 m apart. There is one transmitter on each side of the aircraft with a main power transmission sideways and straight forward, the power of the transmission behind the aircraft can be neglected due to the symmetry of the aircraft. When placed at the apron two antennas are transmitting directly into the fuel tanks which will cause shorter but stronger delays if the signal is reflected once. The signal can, however, bounce multiple times between the tanks causing longer delays. The receiver starts having a hard time resolving multipath signals at around one bit delay. Assuming close to complete reflectivity where the reflection has a power of 0.99 (input is 1) one bit delay would require roughly 46 bounces with a remaining power of around 0.6. Furthermore, surrounding buildings will cause around 250 m delay, or 1.8 bits, with relatively high power depending on the material.

The results show that both multiple short direct delays, such as direct reflection from one or more fuel tanks, and the long delays caused by multiple reflections, should be resolvable without any equalizer. The combination with longer reflections from buildings could result in erroneous detection even after equalizer. In the stationary case, reflections from the fence should be less likely than reflections from surrounding houses due to the positioning of the transmitters. The reflectivity of a normal brick house is much lower than the reflectivity of a metal fence, meaning that the long delays should have power low enough to be correctly equalized. Looking at the results and the surroundings on the apron, there should not be many multipath combinations that the equalizer is unable to resolve. When taxiing to the runway on the other hand, one of the transmitters will be transmitting directly into the fence which will create multipath signals that are severe enough that the equalizer is unable to resolve them.

6.1.5 Real Data The results in Table 5.2 show that at the apron (Data set 1-15), 9 out of 15 signals are not resolvable by the receiver, 5 out of the 9 signals can be resolved completely with the equalizer whilst 3 out of 9 produce 2 or 3 errors and only one is non-resolvable. These results follow the reasoning of the apron outline in Section 6.1.4 since only one out of the 15 datasets is far from reaching error free reception with the equalizer. These tests are, however, not exactly like the the real case when the aircraft is made ready at the apron. There will be both individuals and vehicles moving around causing other multipath signals than the ones recorded in the test. It is reasonable to assume that the reflections caused will be short and strong. However, it is not investigated how a time varying channel affects the equalization. Once the channel changes the equalizer has to converge again and that convergence time is unknown, but is assumed to be less or equal to the fixed 6.2 Method 43 convergence time of 5000 samples or roughly 50µs. Thus within the 50µs convergence time there will, most likely, be erroneous detection. It is also unknown how the channel will change when an object enters the signal path. The case could be that there is more or less complete fading, where the singal is unable to reach the receiver, a severe reflection might be faded making detection possible or it could cause no change at all. 50µs is, in this context, a very short convergence time, a person walking at normal speed would most likely only cause the channel to change slowly allowing the equalizer to converge and have a constant channel for a short while.

The data sets from the taxiway also reflects the reasoning in Section 6.1.4, where only 2 out of the 10 data sets are resolvable without any equalizer and the cma manages to resolve half of the remaining signals. It is therefore difficult to resolve any multipath signals from the taxiway. The data sets are furthermore taken when the test-sender is more or less stationary, due to the limitations of data col- lection, making the real channel vary with time on top of the strong delays. When taxiing there are usually no moving objects around, limiting the movement to the aircraft itself. It is impossible to say how the channel will change without testing, and it will probably be different each time an aircraft is taxiing. Equalizing the channel at the taxiway proves to be very difficult, and more or less impossible with the blind cma equalizer.

6.2 Method

The main problem with the method used is, as mentioned before, that it only analyzes a stationary channel and a stationary aircraft. This is a very simpliﬁed version of reality where there will be objects interfering with the channel and the aircraft is not always stationary. The only way to get a better understanding of how the equalizer reacts in the real environment is to implement and test the cma on a receiver. The tests do, however, show that cma can improve the signal reception and that implementation is worth testing.

The parameters tested are separated with relatively large step sizes, meaning that there could exist parameters that provide a better equalization. The number of taps could be increased and the increase could be ﬁner to ﬁnd the best settings. The same goes for the update interval, there are many update intervals that are not explored that could potentially provide a better equalization. The area plots could be generated with smaller power and delay steps, as well as with a larger data vector. There are millions of combinations of multiple delays that are, for obvious reasons, not tested. The way to test the equalizer in a more detailed manner is, once again, to implement the cma and test it in a real case scenario.

The sources used to develope the code for the cma are quite old, meaning that newer, more modern versions might exist. The main idea behind the algorithm is still the same but with some tweaks when it comes to the µ value and update in- 44 6 Discussion tervals. The main objective of the more recent cma versions focus on signals that are dependent on phase, for example Quadrature Phase Shift Keying (qpsk). The article chosen, [18], only focuses on the original cma algoritm and not on ﬁnding a correct phase. There are many articles that provide test results where cma has been used to equalize telemetry signals. Unfortunatley there is little information about the cma algoritm used and parameter settings. The articles that provide necessary information have not performed the tests in such severe environment as on the apron, making the settings diﬀerent.

6.3 A wider perspective

The fact that all SAAB branches focus on the development of military equipment is an ethical dilemma. SAAB is forced by the government to only sell equipment to countries currently not at war for defense purposes, but there is nothing keeping the countries from later on using the equipment in acts of war. Since the thesis aims to provide a solution for the testing of military equipment there is an indirect connection between my thesis and war. However, the results of this thesis can only be used to enable more accurate data acquisition and in turn a safer work-environment for the civil pilots during testing of the aircraft. Since it is impossible to use this work in acts of war, there should be no ethical problems directly associated with the work of this thesis. 7 Conclusion

In this thesis, an equalizer based on the constant modulus algorithm has been tested. Different combinations of parameters were compared to find a combination that would provide sufficient enough equalization. Initial tests on only single 13 multipath delays showed that using a µ value of 2− , 300 taps and an update interval of 23 samples provided good performance in these cases. The parameters were then used to test the equalizer on multiple delays generated from 3 different single delay combinations, where the receiver is either able to resolve the multipath with or without the equalizer as well as the case where the equalizer is unable to contribute to better reception. The equalizer can completely solve or significantly reduce the ber in 8 out of 9 cases at the apron or 4 out of 8 cases on the taxiway by using the proposed parameters. The proposed parameters seems to be able to provide a sufficient enough equalization when the aircraft is stationary at the apron.

The work shows that cma could potentially be implemented and used to equalize a distorted multipath signal from the apron. There are no guarantees that when implemented on hardware the equalization will work in a similar manner as the simulations, but the results shows that it is worth investigating implementation further.

7.1 Future work

To use the equalizer with a time variant channel caused by a moving aircraft needs extra analysis, all tests in the thesis have been focused on the time non- variant case when the aircraft is stationary at the apron. Furthermore, imple- menting the equalizer on a Field-Programmable Gate Array (fpga) with physi- cal limitations on both speed and available multiplicators would beneﬁt from a

45 46 7 Conclusion slower sampling frequency. Since the cost of lowering the sampling frequency is a decreased time resolution, a deeper study into how the sampling frequency aﬀects the equalization performance is needed. There are many areas that are interesting to analyze deeper, for example how increasing the bit rate would affect the results or how a diﬀerent modulation technique with constant modulus would have to be tuned. Appendix

A Communication Model Listings

Listing A.1: NCO Code function [bitStartVec , val , scrambled_sig] = NCO(Fs, t l e n ) %Fs= Sampling frequency %tlen = length of time vector f = 2163000; NrOfBitChanges = 0; M=2^32* f / Fs ; idx = 0 ; bitStartVec = zeros(1,tlen); nonscrambled_sig = zeros(1,tlen); scrambled_sig = zeros(1,tlen); scrambler = comm.Scrambler(2,[0 14 1 5 ] , . . . [11111111111111 1]);− − bitchange = ’false ’; currscr = 1; %assuming first bit is 1 while(idx < tlen) idx = idx +1; s = rememb; %Save the previous point rememb = mod(rememb+M, 2^32) ; %Count up one step if (rememb < s) %If overflow bitStart(idx) = mod(nonscr_sig(idx)+1,2);

49 50 A Communication Model Listings

bitchange = ’true ’; NrOfBitChanges = NrOfBitChanges + 1; end

% Create the FF00 sequence if (mod(NrOfBitChanges,16) < 8) nonscrambled_sig(idx) = 1; e l s e nonscrambled_sig(idx) = 1; end − %If bit change, start adding the samples of %that scrambled value until next bit %change/overflow if strcmp(bitchange ,’true ’) currscr = scrambler((nonscr_sig(idx)+1)/2); bitchange = ’false ’; end scrambled_sig(idx) = 2* currscr 1; end −

Listing A.2: Transmitter Code function y = modufm(x,Fc,Fs,freqdev ,t) int_x = cumsum(x)/Fs; % Integral of message y = cos (2* pi * Fc * t(1:length(x)) + 2* pi * freqdev * int_x ) ;

%Phase noise only operates on baseband signals [baseband, ~, ~] = IQdemod(y’, t, Fc); pnoise = comm.PhaseNoise(’Level ’ , 8 0 , . . . ’FrequencyOffset ’ ,500 ,... − ’SampleRate ’ ,Fs) ; y = IQmod(pnoise(baseband) ,t , Fc);

Listing A.3: Multipath Generator function [MultiPathSig] =... MultipathGen(Original ,DelayVec , AmplitudeVec)

MultiPathSig = Original; len = length(Original); 51

for i = 1:length(DelayVec) delay = DelayVec(i); ampl = AmplitudeVec(i) ; delayedSig = [zeros(1,delay) Original]; MultiPathSig = MultiPathSig+ampl* delayedSig(1:len) ; end

Listing A.4: Receiver Code function z = demodufm(y,Fc,Fs,freqdev ,t) yq = hilbert(y). * exp( 1 i *2* pi * Fc * t(1:length(y))); z = ( 1 / ( 2 * pi * freqdev )− ) * ... [zeros(1,size(yq,2)); diff(unwrap(angle(yq))) * Fs ] ; end

B Simulation Results for Single delays

53 54 B Simulation Results for Single delays

10 Figure B.1: Sub-plot group for µ = 2− and update every sample

10 Figure B.2: Sub-plot group for µ = 2− and update interval 12 55

10 Figure B.3: Sub-plot group for µ = 2− and update interval 23

10 Figure B.4: Sub-plot group for µ = 2− and update interval 46 56 B Simulation Results for Single delays

10 Figure B.5: Sub-plot group for µ = 2− and update interval 92

11 Figure B.6: Sub-plot group for µ = 2− and update every sample 57

11 Figure B.7: Sub-plot group for µ = 2− and update interval 12

11 Figure B.8: Sub-plot group for µ = 2− and update interval 23 58 B Simulation Results for Single delays

11 Figure B.9: Sub-plot group for µ = 2− and update interval 46

11 Figure B.10: Sub-plot group for µ = 2− and update interval 92 59

12 Figure B.11: Sub-plot group for µ = 2− and update every sample

12 Figure B.12: Sub-plot group for µ = 2− and update interval 12 60 B Simulation Results for Single delays

12 Figure B.13: Sub-plot group for µ = 2− and update interval 23

12 Figure B.14: Sub-plot group for µ = 2− and update interval 46 61

12 Figure B.15: Sub-plot group for µ = 2− and update interval 92

13 Figure B.16: Sub-plot group for µ = 2− and update interval 12 62 B Simulation Results for Single delays

13 Figure B.17: Sub-plot group for µ = 2− and update interval 23

13 Figure B.18: Sub-plot group for µ = 2− and update interval 46 63

13 Figure B.19: Sub-plot group for µ = 2− and update interval 92

14 Figure B.20: Sub-plot group for µ = 2− and update interval 12 64 B Simulation Results for Single delays

14 Figure B.21: Sub-plot group for µ = 2− and update interval 23

14 Figure B.22: Sub-plot group for µ = 2− and update interval 46 65

14 Figure B.23: Sub-plot group for µ = 2− and update interval 92

15 Figure B.24: Sub-plot group for µ = 2− and update interval 12 66 B Simulation Results for Single delays

15 Figure B.25: Sub-plot group for µ = 2− and update interval 23

14 Figure B.26: Sub-plot group for µ = 2− and update interval 46 67

14 Figure B.27: Sub-plot group for µ = 2− and update interval 92

C Simulation Results for Multiple Delays

69 70 C Simulation Results for Multiple Delays

Figure C.1: BER vs SNR of short resolvable multipath combinations without equalizer

Figure C.2: BER vs SNR of short resolvable multipath combinations with equalizer 71

Figure C.3: BER vs SNR of long resolvable multipath combinations without equalizer

Figure C.4: BER vs SNR of long resolvable multipath combinations with equalizer 72 C Simulation Results for Multiple Delays

Figure C.5: BER vs SNR of combination of Resolvable with equalizer multipath combinations without equalizer

Figure C.6: BER vs SNR of combination of Resolvable with equalizer multipath combinations with equalizer 73

Figure C.7: BER vs SNR of non-resolvable with equalizer multipath combinations without equalizer

Figure C.8: BER vs SNR of non-resolvable with equalizer multipath combinations with equalizer 74 C Simulation Results for Multiple Delays

Figure C.9: BER vs SNR of combination of resolvable with equalizer and resolvable multipath combinations without equalizer

Figure C.10: BER vs SNR of resolvable with equalizer and resolvable multipath combinations with equalizer 75

Figure C.11: BER vs SNR of non-resolvable with equalizer and resolvable multipath combinations without equalizer

Figure C.12: BER vs SNR of non-resolvable with equalizer and resolvable multipath combinations with equalizer 76 C Simulation Results for Multiple Delays

Figure C.13: BER vs SNR of non-resolvable with equalizer and resolvable with equalizer multipath combinations without equalizer

Figure C.14: BER vs SNR of non-resolvable with equalizer and resolvable with equalizer multipath combinations with equalizer 77

Figure C.15: BER vs SNR of resolvable, non-resolvable with equalizer and resolvable with equalizer multipath combinations without equalizer

Figure C.16: BER vs SNR of resolvable, non-resolvable with equalizer and resolvable with equalizer multipath combinations with equalizer

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