reducing effects of multipath propagation with a blind equalizer1252119/... · 2018. 9. 30. ·...

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

Reducing Effects ofMultipath Propagation WithA 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 Senelisy, Linköpings universitet

Mattias AvestenSAAB Aeronautics

Rikard BergstenSAAB Aeronautics

Examiner: Mikael Olofssonisy, Linköpings universitet

Division of CommunicationsystemsDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2018 Emma Söderström

Abstract

When transmitting data from an aircraft being prepared at the apron (the areain front of the hangar) telemetry data is transmitted to ground personnel. Thetransmitted data is subject to severe distortion due to multipath propagation cre-ated by the surroundings, resulting in erroneous detection. By equalizing thesignal using the Constant Modulus Algorithm a significant increase in detectionperformance has been observed, both in simulations and tests on collected data.The most sufficient parameters were chosen after testing a set of different param-eter combinations on simulations with single delays. These parameters were thenused to equalize simulated multipath as well as collected data. The results showthat short delays with low power can be resolved without any equalizer. Longerdelays with relatively low power can be resolved using the proposed equalizerbut long delays with high power cannot be resolved by the equalizer at all. Thethesis shows that it is worth investigating implementation of the equalizer.

iii

Contents

List of Figures vii

List of Tables ix

Notation xi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theory 52.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 The Telemetry System . . . . . . . . . . . . . . . . . . . . . 52.1.2 Line Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Randomizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Numerically Controlled Oscillators . . . . . . . . . . . . . . 82.1.5 Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . 92.1.6 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Complex Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Multipath Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Fading Channel Model . . . . . . . . . . . . . . . . . . . . . 132.3.2 Multipath Effects on Frequency Modulated Signals . . . . . 14

3 Channel Equalization 173.1 Equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Linear Adaptive Equalization Using Training Symbols . . . 193.1.2 Linear Adaptive Blind Equalization . . . . . . . . . . . . . . 193.1.3 Equalizing PCM/FM . . . . . . . . . . . . . . . . . . . . . . 193.1.4 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . 203.1.5 The Constant Modulus Algorithm . . . . . . . . . . . . . . . 20

v

4 Method 234.1 Test Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Current Telemetry System . . . . . . . . . . . . . . . . . . . 234.1.2 Communication Model . . . . . . . . . . . . . . . . . . . . . 244.1.3 CMA Parameters . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Equalizer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Simulation correctness . . . . . . . . . . . . . . . . . . . . . 264.2.3 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.4 Test on Real Data . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Results 315.1 Simulation Correctness . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Single Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Multiple Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4 Test on Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Discussion 396.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1.1 Simulation Correctness . . . . . . . . . . . . . . . . . . . . . 396.1.2 Single Delays . . . . . . . . . . . . . . . . . . . . . . . . . . 396.1.3 Multiple Delays . . . . . . . . . . . . . . . . . . . . . . . . . 406.1.4 Simulation Results in Relation to the Apron . . . . . . . . . 426.1.5 Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 A wider perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7 Conclusion 457.1 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 re-trieved by adding white Gaussian noise to a fm modulated signalbefore fm demodulation. . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 ber as function of snr of a multipath signal. The receiver receivesa strong reflection that has traveled 150m longer than the mainsignal, 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]. . . . . . . . 72.3 Randomized nrz decoder as defined in IRIG106 [17]. . . . . . . . 82.4 Phase wheel representing a period of a signal, the number of points

is determined by 2n and M is the jump size. . . . . . . . . . . . . . 92.5 I/Q plot showing the envelope and phase of a passband signal . . 102.6 An example of fmmodulation . . . . . . . . . . . . . . . . . . . . . 112.7 A Tapped Delay Line [2] . . . . . . . . . . . . . . . . . . . . . . . . 142.8 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. . . . . . . . . . . . 244.2 Transmitter part of generated signal for testing the simulation cor-

rectness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 RF receiver setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Explanation of areas in the area plots . . . . . . . . . . . . . . . . . 284.5 Test-sender setup with two antennas to mimic the actual aircraft. . 294.6 Real setup of test-sender . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Figure showing the eye-diagrams of a generated, known, signalwithout multipath and the simulation of the same signal . . . . . . 31

5.2 Figure showing the eye-diagrams of a generated, known, signalwith 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, signalwith 150m multipath and the simulation of the same signal . . . . 32

5.4 Sub-plot group of one of the best results for cma with µ = 2−13 . . 335.5 Sub-plot group of one of the best results for cma with µ = 2−14 . . 335.6 Eye-diagram before and after cma when ber = 0 both before and

after . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.7 Eye-diagram before and after cmawhen ber is not zero before but

is after equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.8 Eye-diagram before and after cma when the cma is unable to re-

solve the multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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

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

C.1 BER vs SNR of short resolvable multipath combinations withoutequalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.2 BER vs SNR of short resolvable multipath combinations with equal-izer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.3 BER vs SNR of long resolvable multipath combinations withoutequalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.4 BER vs SNR of long resolvable multipath combinations with equal-izer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.5 BER vs SNR of combination of Resolvable with equalizer multipathcombinations without equalizer . . . . . . . . . . . . . . . . . . . . 72

C.6 BER vs SNR of combination of Resolvable with equalizer multipathcombinations with equalizer . . . . . . . . . . . . . . . . . . . . . . 72

C.7 BER vs SNR of non-resolvable with equalizer multipath combina-tions without equalizer . . . . . . . . . . . . . . . . . . . . . . . . . 73

C.8 BER vs SNR of non-resolvable with equalizer multipath combina-tions with equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

C.9 BER vs SNR of combination of resolvable with equalizer and resolv-able multipath combinations without equalizer . . . . . . . . . . . 74

C.10 BER vs SNR of resolvable with equalizer and resolvable multipathcombinations with equalizer . . . . . . . . . . . . . . . . . . . . . . 74

C.11 BER vs SNR of non-resolvable with equalizer and resolvable multi-path combinations without equalizer . . . . . . . . . . . . . . . . . 75

C.12 BER vs SNR of non-resolvable with equalizer and resolvable multi-path combinations with equalizer . . . . . . . . . . . . . . . . . . . 75

C.13 BER vs SNR of non-resolvable with equalizer and resolvable with equal-izer multipath combinations without equalizer . . . . . . . . . . . 76

C.14 BER vs SNR of non-resolvable with equalizer and resolvable with equal-izer multipath combinations with equalizer . . . . . . . . . . . . . 76

C.15 BER vs SNR of resolvable, non-resolvable with equalizer and resolv-able with equalizer multipath combinations without equalizer . . . 77

C.16 BER vs SNR of resolvable, non-resolvable with equalizer and resolv-able with equalizer multipath combinations with equalizer . . . . . 77

List of Tables

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

5.1 Delays and powers used to create multipath signals . . . . . . . . . 345.2 ber before and after cma on test cases from the apron and taxiway 35

ix

Notation

xi

xii Notation

Abbreviations

Abbreviation Definition

agc Automatic Gain Controlam Amplitude modulationawgn Additive White Gaussian Noiseber Bit Error RateBc Coherence BandwidthBiΦ Bi-phasecma Constant Modulus Algorithmdc Direct Currentdds Direct Digital Synthesisdfe Decision Feedback Equalizerfir Finite Impulse Responsefm Frequency Modulationfpga Field Programmable Gate Arrayi In-phaseif Intermediate Frequencyisi Inter Symbol Interferencelms Least Mean Squarelo Local Oscillatornco Numerically Controlled Oscillatornr Neural Networknr Non-Resolvablenrz Non-Return to Zeropcm Pulse Code Modulationpcm/fm Pulse Code Modulation/Frequency Modulationpm Phase Modulationprbs Pseudo-Random Bit Sequencepsk Phase Shift Keyingq Quadrature-phaseqpsk Quadrature Phase Shift Keyingr Resolvablere Resolvable with Equalizerrf Radio Frequencyrls Recursive Least Squarernrz Randomized Non-Return to Zerorz Return to Zerosnr Signal to Noise Ratiosoqpsk Shaped-Offset Quadrature Phase Shift KeyingTc Coherence TimeTd Multipath Delay Spreadtdl Tapped Delay Linevfo Variable Frequency Oscillator

Notation xiii

Terminology

Word Explanation

Apron Place in front of the hangar where the aircraft is beingprepared.

Runway The area where takeoff and landing takes place.Taxiway The route taken by the aircraft from the apron to the

runway.Telemetry

System A system that records system parameters and trans-mits them.

MultipathPropagation A phenomena where a transmitted signal is reflected

of objects in its path, resulting in multiple versions ofthe signal being detected at the receiver.

1Introduction

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

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 onthe 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 aircraftis in the air, it is also important to know that the system is ready for takeoff. Theaircraft is prepared for takeoff at the apron where the necessary system param-eters are sent to the test engineers. In most cases, the apron is surrounded byhangars, buildings, fences, fuel tanks and the apron itself is made of reinforcedconcrete. Electromagnetic waves at high frequencies, reflect of and are distortedby hard surfaces like the ones surrounding the apron, this is illustrated in Figure1.1. At the receiver side a composite signal consisting of the transmitted signaland its delayed reflections, called a multipath signal, needs to be decoded.

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

1

2 1 Introduction

Hangar

Apron

Fences

other houses

150m

100m

200m

90m

Fuel tanks

Reflected signal

Direct signal

Receiver

Transmitter

Figure 1.1: An illustration of the surroundings of the apron and how thereflection 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 isretrieved by adding white Gaussian noise to a fm modulated signal beforefm demodulation.

When introducing multipath propagation, the signal sometimes becomes sodistorted that it is impossible for the receiver to detect the bits correctly evenwith a high snr, this is illustrated in Figure 1.3. It is clear from Figure 1.3 thatincreasing the snr does not decrease the amount of errors introduced by the mul-tipath propagation. In order to correctly detect the bits, the multipath signal hasto 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 telemetrydata which in turn will help test engineers by improving the detection perfor-mance 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 re-ceives 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 re-ceived 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. Othermodulation techniques or training-based equalization techniques are left for fu-ture work and will not be considered in this thesis.

1.5 Thesis outline

The theoretical background will be presented in Chapter 2, the different equal-ization techniques will be described in greater detail in Chapter 3. Chapter 4will describe the method used to test the equalizers and Chapter 5 will presentthe results. In Chapter 6, the results will be discussed providing the base for theconclusion in Chapter 7.

2Theory

This chapter aims to give the reader a deeper understanding of the theory behindthe multipath propagation problem and the algorithms used by the equalizers. Itis expected that the reader has some previous knowledge about signals and sys-tems. The first part should be seen as introduction of basic concepts along withsome technical terms. It is recommended that the sections are read in sequencesince 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 inconve-nient 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, whereon-board system overview is not always possible. The telemetry system, like anycommunication system, consists of three main parts [15], a transmitter, a channeland a receiver. The transmitter sends data to the receiver over a channel, whichin this case is air.

2.1.2 Line Codes

The analog data signal collected in the telemetry system needs to be quantizedand converted into an n-bit digital word. The logic word needs to be convertedinto 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 dothis, 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 thevoltage level is set by the current bit, M (mark) changes voltage level when abinary 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 levelfor the rest of that bit period. rz represents 1 by high voltage for half a bit periodand stays low for 0. BiΦ represents 1 with going high for the first half of the bitperiod and then goes low, 0 is represented by low voltage in the first half andthen high for the last half. The extensions for BiΦ work in the same way as fornrz 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 band-width requirement and ease of implementation.

Figure 2.1: Different line codes and their voltage responses to a binary se-quence [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. Thedifference between the number of ones and zeros in a sequence is called disparity.Large disparity in a sequence may introduce a Direct Current (dc) componentto the signal [5]. dc-components are in general difficult to reliably transmit overlong distances and can cause the receiver to loose track of bits. It is thus desirableto maximize the disparity of a code, or make it dc-free. This can be done by usinga Randomizer.

Encoder

The most common randomizer for randomized nrz-L (rnrz-L) uses a 15-stageshift 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 wherethe result is XOR:ed with the bit to be encoded. The resulting bit from the lastXOR operation is both shifted into the register and the output of the encoder, seeFigure 2.2. The rnrz-L randomizer has similar properties as an Pseudo RandomBit Sequence (prbs) generator, and will produce a maximum prbs of 215 −1 bits ifthe 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 XORoperation on the the 14th and 15th stages of the shift register and use as input tothe second XOR gate together with the incoming bit. The result of the final XORis, as in the encoder, the output of the decoder. The encoded incoming bit is thenshifted 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 val-ues of the shift registers of the encoder and decoder are not equal. Usually theregisters are not equal since there are 215 different ways the bits can be arrangedand 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 15thstage of the shift register to decode the incoming bits. Single bit errors thereforehave 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 2πover a period at a speed determined by the frequency. In the digital case, a phasewheel 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 num-ber of points on the phase wheel is 2n where a larger n will increase the accuracyof the nco. With a sampling frequency of fs and desired nco output frequencyf, the following relation can be derived

f =Mfs2n

(2.1)


Jump Size

Figure 2.4: Phase wheel representing a period of a signal, the number ofpoints 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 isused, often referred to as Radio Frequency (rf) [9]. The electromagnetic wave is ahigh frequency sinusoid called carrier. The typical waveform of a carrier is givenby

s(t) = A cos(2πfct + θ), (2.2)

where A is the amplitude, fc is the carrier frequency and θ is the phase of thesignal.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 fre-quency respectively are altered with a specific message. The non-modulated mes-sage is called a baseband signal and when the baseband signal is modulated ontoa carrier it is called a passband signal.

Complex Baseband Representation

A passband signal can be written in the form

sp(t) = sI (t) cos(2πfct) − sQ(t) sin(2πfct), (2.3)

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 otherwith a phase shift of 90° and together they can represent any sinusoidal signal.

The complex envelope, also called complex baseband representation [12], isdefined by

sbb(t) = sI (t) + jsQ(t). (2.4)

10 2 Theory

e(t)θ(t)

Q

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 withEuler’s identity resulting in

sp = Re{sbb(t)e−j2πfct}. (2.5)

Furthermore, Equation 2.4 can be written in polar form as e(t)e−jθ(t), where theenvelope e(t) and phase θ(t) are defined by

e(t) = |sbb(t)| =√s2I (t) + s2Q(t),

θ(t) = tan−1(sI (t)sQ(t)

).

(2.6)

Figure 2.5 shows a plot of the envelope and phase where the i component corre-sponds to the real part and the q component is the imaginary part as defined inEquation 2.4.

A signal that is represented by complex samples obtained from i and q is oftenreferred 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 mes-sage. In frequency modulation, the message determines the Instantaneous Fre-quency which is the derivative of phase with respect to time and is given by

dθdt

= am(t)←→ θ{m(t)} = a

∫m(t)dt, (2.7)


Figure 2.6: An example of fmmodulation

where a is a scaling factor andm(t) is the message to be modulated. By combiningEquation 2.2 and 2.7, the modulated signal can be defined as

s(t) = A cos(2πfct + a∫m(t)dt). (2.8)

The frequency deviation, fd , is given by

fd(t) =a

2πm(t), (2.9)

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) withfm. Since all information about the message is encoded in the frequency, theenvelope, e(t), will be constant while the phase varies. Note that, frequency mod-ulation does not take exact phases into consideration, only the instantaneous fre-quency is considered. This means that the position on the circumference of thecircle 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 dig-ital line coding is used to convert the collected data bits into waveforms, thenfm is used to modulate the baseband signal. It is commonly used with nrz-Las line code where the voltage level is between +1V and -1V. When sending dataover long distances pcm/fm is preferred due to its constant envelope, also calledconstant modulus, a property which makes pcm/fm resistant to amplitude vari-ations in the channel.

12 2 Theory

2.1.6 Mixers

It is often inconvenient to operate at rf frequencies, as the electrical componentsmight not work and the frequency deviations might be hard to distinguish [9].Intermediate Frequency (if) has been introduced in wireless communications to beable to do signal operations at lower frequencies. The if frequency is betweenbaseband and the carrier frequency. A mixer is a function that shifts the fre-quency up or down depending on if the conversion is from if to rf or from rfto if. The mixer uses a Variable Frequency Oscillator (vfo) to shift the frequenciesaccording to equation

fIF = fRF − fV FO,fRF = fIF + fV FO.

(2.10)

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 providesimple examples.

Definition 2.1 (Hermitian Transpose). The Hermitian transpose, also calledconjugate transpose, of a matrix containing complex numbers is defined as thetranspose of the complex conjugate of the matrix.

AH = AT

Example 2.2 shows a simple example of the Hermitian Transpose.

Example 2.2Suppose we want to take the Hermitian transpose of

A =[

2 3 − j−6 − 3j 8 + j

].

The resulting matrix would then be

AH =[

2 −6 + 3j3 + j 8 − j

].

Definition 2.3 (∇W , "Gradient operator with respect to the elements of vectorW ). ∇W f is defined as

∇wf =[∂f

∂w1,∂f

∂w2, ...,

∂f

∂wn

]T.

Where f is a scalar and W is a vector. This is called scalar-by-vector operation inmatrix calculus.

2.3 Multipath Propagation 13

2.3 Multipath Propagation

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

There are two important properties of a channel, coherence time, Tc and coherencebandwidth, Bc [10, p.200]. Tc defines the duration over which the system canbe modeled as approximately time-invariant. If the duration of the transmittedsignal, T, is smaller than Tc the channel can be regarded as time-invariant. IfT<Tc, the coherence bandwidth is defined as the frequency interval for which asystem H can be seen as essentially constant. Furthermore, the multipath delayspread, Td is defined as the maximum delay between the longest and the shortestpath from the transmitter to the receiver, defined as

Td := maxi,j|τi(t) − τj (t)|. (2.11)

The relation between Td and Bc is given by

Bc =1Td. (2.12)

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 Bcthe channel is frequency-selective, otherwise it is frequency-flat. This can besummarized by

B << Bc : Frequency-flat,

B >> Bc : Frequency-selective.(2.13)

The typical model for the impulse response in wireless communication is definedas

h(t) =M∑k=1

Akejθkδ(t − τk). (2.14)

For the frequency-flat case, h(t) is approximately constant for the coherence timeand 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 delayline (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

h(t) =∞∑i=1

αiδ(t − i/W ), (2.15)

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 scaledand 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 willresult in a delay of approximatley 6.67 · 10−6 s. Using Equation 2.12, Bc is 1.5 · 105

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 afrequency 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 adifferent frequency than the one being transmitted. This case is shown in Figure2.8, which demonstrates that the resulting signal is both frequency shifted andhas different amplitude than the transmitted signal. Thus, the property of con-stant envelope discussed in Section 2.1.5 is ruined. Further, and more severe, thephase variations interfere with the phase-modulated signal [18].

2.3 Multipath Propagation 15

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

3Channel Equalization

Equalizers are designed to revert the changes made on the signal by the chan-nel, for example removing the delayed version of the signal. If the channel isfrequency-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 differ-ent diversity techniques can be used, see [12, pp.387-397]. A frequency-selectivechannel on the other hand, affects the spectral components differently and varieswith time and therefore requires more sophisticated equalization methods.

3.1 Equalizers

Equalizers are usually divided into two groups, adaptive and static [1]. Thestatic equalizer does not update its taps and is therefore is not suited for un-known, 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 isifrom 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 thechannel, the linear adaptive equalizers are typically designed as fir filters due tothe 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 responseand the output, y, from the equalizer [13]. The output from the equalizer can bewritten 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 thevector of tap coefficients. The adaptive equalizer then uses the information aboutthe 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 thedifferent linear adaptive equalizer algorithms. Example 3.1 is intended to showthe idea of an ideal linear adaptive equalizer.

Example 3.1Imagine 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 asthe vectors below.

12345

+

01234

+

00123

=

1369

12

The resulting vector, Q, will then be the input to the equalizer. If we assume anideal equalizer, it will in this case have two taps, one with delay 1, d1, and onewith delay 2, d2, and a multiplication factor of -1, see Figure 3.1.

-1 -1

r Q

d1d2

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 samplesT oDelay + 1.

In the very first loop, the first value of the vector reaches the equalizer. The initialvalues of the delays (d1 and d2) are 0 which gives Q(1) = r(1) = 1. Then d1 andd2 are updated resulting in d1=1 and d2 remains 0. The second time the loop isrun, Q(2) = r(2) − 1 = 2. Now d1=2 and d2=1. This process is repeated and afterthe 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 isto periodically transmit known training symbols [19]. The knowledge about thetransmitted signal, makes it possible to closely estimate the channel using thereceived signal and adapt the taps thereafter. Most training-based linear methodsare based on the Least Mean Squares (lms) Algorithm [1] or the Recursive LeastSquare (rls) Algorithm [13].

3.1.2 Linear Adaptive Blind Equalization

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

3.1.3 Equalizing PCM/FM

The current telemetry setup does not have any training symbols meaning that ablind equalizer has to be used. The transmitted signal has the constant modu-lus property, explained in Section 2.1.5. This means that algorithms using theknowledge of the constant modulus for equalization can be used. The shape ofthe frequency spectrum is known, but its mathematical representation is not ex-act and is only valid under certain circumstances [4]. This makes it more difficultto use the information for equalization purposes. Linear equalizers does not usu-ally 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 bestsuited equalizer for test is thus one that make use of the constant modulus, forexample 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 modulationtechniques. For example, there have been developments within the equalizationfor Phase Shift Keying (psk) and am, which, unfortunately, cannot be used toequalize pcm/fmmodulated signals due to the fact that the angular frequency islost in the process. Not much research has been done lately on new equalizingmethods specifically on pcm/fm since new, more efficient modulation techniquesfor telemetry, such as Shaped-Offset Quadrature Phase Shift Keying (soqpsk), haveemerged. The telemetry system in focus for this thesis uses pcm/fm and is notcurrently compatible with soqpsk or other modulation techniques. Numerouspapers have researched the use of cma for pcm/fm-modulated telemetry signals,for example [7], [11], [21], with promising results. These papers, however, donot include any information about how parameters such as step size, number oftaps or update interval (explained below) should be chosen or how they affect theperformance.

3.1.5 The Constant Modulus Algorithm

One of the most commonly used blind equalization algorithm is the cma, whichis similar to the lms algoritm. It uses the á priori knowledge about the constantmodulus 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 afir filter with complex valued coefficients. The output of the filter is describedby

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

W(i) = [w0(i) · · ·wN−1(i)]T . (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 andtherefore replace the index i with k after this point.

The output, y(k), should be the same as the transmitted signal after idealequalization. The property of constant modulus should therefore also be pre-served. The algorithm adjusts the taps in W to minimize a positive definite mea-sure of the deviation from the constant modulus. This measure is called the costfunction 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

J =14E{|y(k)|2 − 12}. (3.8)

The choice of W that minimizes J is the coefficients used in the filter. Note thaty(k) in Equation 3.8 is replaced with the definition given in Equation 3.4. Thereare 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 implementationeasier.

A gradient search algorithm computes the derivative of a function, f ( · ), at apoint 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 thefact that a function f(x) decreases while going from a point a in the direction ofthe negative gradient of f at that point, -∇f (a), which allows an iterative searchalgoritm of the following form,

an+1 = an − µ∇f (an). (3.9)

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

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

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

W(k + 1) = W(k) − µ∇wJk . (3.10)

The gradient of J is given by [18]

∇wJ = E{[|y(k)|2 − 1] · y(k)X∗(k)}. (3.11)

The algorithm replaces the true gradient in Equation 3.10 with the instantaneousestimate given by

∇wJ = [|y(k)|2 − 1] · y(k)X(k)∗, (3.12)

which results in

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

4Method

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. Themain approach is that the equalizer is first tested through simulations in a testbedwritten in Matlab. The testbed is designed to mimic the current telemetry sys-tem while making it possible to test the system with and without the equalizerpresent, allowing comparison between the simulations. By running simulationswith a set of different cma parameters, described in Section 3.1.5, and compar-ing the results, the best parameters out of the set can then be chosen. Theseparameters are used when equalizing the reception of a known signal sent froma signal generator at the apron. The ber of the received signal with and withoutthe 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 shouldgive 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 carrierfrequency is 2.2 GHz and the frequency deviation is 734 kHz. The system trans-mits 2163 kbps from two antennas and the receiver has a sampling frequency of100 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. Afterthe fm-modulation, the modulated signal is band-pass filtered to reduce spectral

23

24 4 Method

NCO/Bit Gen

FM-modulator

Bandpass-filter

Transmitter

Error ratecalculator

FM-demodulator

Receiver

ChannelEqualizer

Lowpass-filter

Lowpass-filter

Bandpass-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 fromthe integration of the middle 10 samples of the bit, if it is positive the bit isdecoded 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 systemmodel with a transmitter, a channel and a receiver. The communication modelused is shown in Figure 4.1. The basic implementation of the different parts ofthe system is described in the following sections.

Numerically Controlled Oscillator

To obtain simulation data, a bit vector is created by generating a bit each timethe nco overflows. The bit value is decided by a modulo-2 counter where 1 isadded to the previous bit, resulting in 0 and 1 every other overflow. Each bit isfirst 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 vectorof bits sampled at a sampling frequency of the output frequency of the nco. Thebits are generated using the nco to resemble the real transmitter and sampled inorder to be able to delay the signal with delays of higher precision than a wholebit. 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 wherethe carrier frequency, sampling frequency, frequency deviation and time vectoris given as input. Phase noise is added in order to make the simulations more likethe real transmitter. Listing A.2 shows the code for the transmitter.

Channel

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

FM-Demodulator

The receiver demodulates the fm signal according to Listing A.4. The aim is tocompare 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 generatedbit sequence with the demodulated sequence. In the real receiver, the bit value ischosen 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 thesnr, 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 isgenerated 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 tapsare updated are variables that can be tweaked to get better results. The numberof taps determines how long delays that can be detected, more taps allows cma toresolve longer delays. The µ value determines how fast we move in the directionof the zero, once the minimum is closer than the step size, the point reachedby the cost function will oscillate around the minimum. Too large µ will resultin oscillations around the minimum causing incorrect equalization whereas toosmall µ will lead to large convergence times. The update interval of the taps isalso a variable to take into consideration. Updating on every sample will causethe filter taps to change 46-47 times per bit, updating too seldom will not resolvethe multipath. The cma parameter ranges tested are µ = 2−10 − 2−15, the numberof taps tested are 50, 100, 200 and 300 and finally the update intervals are every12, 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 thesimulations is tricky. This section will cover the method used for analyzing theequalizer and adjusting the algorithm parameters. The performance of the equal-izer will first be simulated for different combinations of the parameters, the com-bination that results in the best equalization will then be tested on real data fromthe 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 muchdata for a short time period. Reducing the carrier frequency allows longer timeperiods and in result a more reliable results. Since the time vector is created witha step size equal to the inverse of the sampling frequency, the simulation time islimited to 0.01 s since longer simulation time results in such large vectors to behandled 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 theequalization.

4.2.2 Simulation correctness

The correctness of the simulation setup is tested by comparing simulated mul-tipath eye-diagrams with generated ones. If there is a significant difference be-tween the eye diagrams of the signals the simulated multipath is incorrect. If theeye 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 partis delayed by a number of 100 MHz delays decided by the user, see Figure 4.2. Thedelayed i and q signals are then added to their non-delayed versions, convertedto an analog signal and mixed to rf with a Local Oscillator (lo). The multipathsignal is then sent to a Quasonix receiver where the i/q baseband componentsare fed to an oscillator, shown in Figure 4.3. The oscilloscope samples the signalsat 10 Msps and saves the samples in binary form to a USB drive, allowing theresults to be imported to Matlab. The capture is then up-converted to 100 Mspsto mimic the preferred sampling rate. The idea is that this test setup can generateand record real multipath signals, as they appear in real life, and compare withthe simulated multipath signals in order to evaluate the simulation credibility.

4.2 Equalizer Analysis 27

x

~

RF

DAC

DAC

Mixer

D D

+I

Q

BIT GEN FM-MOD

D D

+

Figure 4.2: Transmitter part of generated signal for testing the simulationcorrectness

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 re-ceiver is not able to resolve on its own, these cases are the primary targets fortesting the equalizer. The equalizer is given 3000 bits for convergence in the sin-gle delay case and 5000 for the multiple delay case. The bits at the beginningwill 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 simula-tions with increasing delay and power of the reflected signal component. If morethan 10−6 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 agrid, hereafter called an area plot. The area plots can be divided into 3 differentareas, one area where the receiver can detect the bits without any errors despitethe delayed signal, one where the equalizer is able to aid the receiver to detectbits without errors and one where the equalizer is unable to resolve signal, seeFigure 4.4. By comparing the area before and after the equalizer it can be seenhow well the equalizer works for different parameters.

Multiple Delays

To test the equalizer on multiple delays, the parameters providing the smallestarea of non-resolvable multipath in the single delay case are used. The multipath

28 4 Method

No errors atthe receiver,no errors withequalizer

Error at the receiver, resolved by equalizer

Error at receiver 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 describedin Listing A.3. The main idea behind combining the different areas is to see ifminimizing the area ofnr increases the chances of resolving the multipath signal.If that is the case, the focus can remain on minimizing the area ofnr for the singledelays to maximize the performance.

Type of multipathCombinations of Short r delaysCombinations of Long r delays

Combinations of re delaysCombinations of nr delays

r + re delaysr + nr delaysre + nr delaysr + re + nr delays

Table 4.1: Table of delay compositions for test of multiple multipath compo-nents. 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 cre-ated to be able to retrieve real multipath data from the apron. The test-transmitterconsists of the same transmitter as in Figure 4.2 but without the delays and with

4.2 Equalizer Analysis 29

x

~

DAC

DAC

Mixer

PA 5W10dB 10dBBIT GEN FM MOD

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 antennasused for receiving the current telemetry communication. The whole receptionworks in the manner as in 4.2.2 where the Quasonix forwards the i/q compo-nents to the oscilloscope where the signal is sampled at 10 Msps and saved to anUSB-drive. The transmitter transmits 8 ones and 8 zeros (FF00) using rnrz-Ldescribed in Section 2.1.3 which can be de-randomized and compared with theinitial 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 ratewas high. The main focus is on the apron, but data sets were collected in front ofthe apron as well, following the route the aircraft will take whilst taxiing.

5Results

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 inFigures 5.1 to 5.3, where the eye diagrams of two different delays are shown alongwith a signal without multipath for comparison.

Figure 5.1: Figure showing the eye-diagrams of a generated, known, signalwithout multipath and the simulation of the same signal

31

32 5 Results

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

Figure 5.3: Figure showing the eye-diagrams of a generated, known, signalwith 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 describedin Section 4.2.3. The tests included µ values from 2−10 to 2−15, updates every 12,23, 46 and 92 samples as well as 50, 100, 200 and 300 taps. The results are shownin 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 equal-izer, where the µ value and update interval is fixed for each sub-plot group whilethe number of taps differ within the sub-plots. The y-axis shows the strength ofthe delayed signal in comparison to the direct signal. The direct signal will al-ways have power one, the delayed signal will then, in all real cases, be lower thanthat. The x-axis shows the delay in bits, where a delay value of one correspondsto 138m (1 sample is 3m delay, one bit has approximately 46 samples). The areathen shows both at which delay and power the equalizer is needed along withhow severe delay it can resolve.

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

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

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 delaysused for the simulations are shown in Table 5.1, where the results from Figure5.4 is the basis for the choices.

Type of Multipath Delay (bits) Power

r (Shorter)

0.22 0.32 0.9 0.90.22 0.32 0.44 0.9 0.9 0.90.22 0.32 0.44 0.54 0.9 0.9 0.9 0.90.22 0.32 0.44 0.54 0.65 0.9 0.9 0.9 0.9 0.9

r (Longer)

0.22 0.97 0.9 0.50.22 0.97 1.74 0.9 0.5 0.50.22 0.32 0.97 1.74 0.9 0.9 0.5 0.50.22 0.32 0.97 1.74 1.96 0.9 0.9 0.5 0.5 0.5

re + r

0.54 1.00 0.9 0.90.54 0.76 1.00 0.9 0.6 0.80.54 0.76 1.00 1.74 0.9 0.6 0.8 0.70.54 0.76 1.00 1.74 2.17 0.9 0.6 0.5 0.7 0.7

re

1.00 1.08 0.9 0.81.00 1.08 1.30 0.9 0.8 0.81.00 1.08 1.30 1.52 0.9 0.8 0.8 0.81.00 1.08 1.30 1.52 1.74 0.9 0.8 0.8 0.8 0.8

nr

1.00 1.08 1.0 1.01.00 1.08 1.30 1.0 1.0 1.01.00 1.08 1.30 1.52 1.0 1.0 1.0 1.01.00 1.08 1.30 1.52 1.74 1.0 1.0 1.0 1.0 1.0

r + nr

0.22 1.08 1.0 1.00.22 0.44 1.30 1.0 1.0 1.00.22 0.44 1.30 1.52 1.0 1.0 1.0 1.00.22 0.44 1.30 1.52 1.74 1.0 1.0 1.0 1.0 1.0

re + nr

1.08 1.30 0.8 1.01.08 1.30 1.52 0.8 1.0 1.01.08 1.19 1.30 1.52 0.8 0.8 1.0 1.01.08 1.19 1.30 1.52 1.74 0.8 0.8 1.0 1.0 1.0

r + re + nr

0.54 1.30 1.52 0.8 0.8 0.90.54 1.30 1.52 1.74 0.8 0.8 0.9 0.90.54 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. Theresults are shown in Table 5.2, where data sets 1-15 are from the apron whilst16-25 are from the area in front.

DataSet

BER Be-fore CMA

BER AfterCMA

DataSet

BER Be-fore CMA

BER AfterCMA

1 0 0 14 0.000497 02 0 0 15 0.1221 1.42e-53 0.23742 0.00097 16 0 04 0 0 17 0.081 05 0 0 18 0 06 0 0 19 0.070 07 0.032 0 20 0.441 0.24788 0.021 0 21 0.097 09 0.391 1.45e-05 22 0.134 010 0 0 23 0.212 0.17811 0.178 1.89e-5 24 0.052 0.38212 0.316 0 25 0.67 0.08913 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 theber is 0 before and after the equalizer. Figure 5.7 shows the eye-diagrams whenthe ber is not zero before but after cma and Figure 5.8 when the cma is unableto resolve the multipath.

36 5 Results

Figure 5.6: Eye-diagram before and after cmawhen ber = 0 both before andafter

Figure 5.7: Eye-diagram before and after cma when ber is not zero beforebut is after equalizer

5.4 Test on Real Data 37

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

6Discussion

The simulation results have been presented in Chapter 5. This chapter aims toanalyze and discuss the simulation results in order to answer the problem state-ment. Further more the method and the work in a wider perspective will bediscussed.

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 simu-lations are very similar to the generated version. The transmitter generating thereal signal does not have exactly the same filters as the simulated signal, makingthe eye of the generated signal slightly larger than the simulated one. There isalso different phase jitter as well as different snr in the cables. The comparisonis mainly focused on the similarity between the shapes of the eye-diagrams sincethe multipath determines the shape. Since there are no significant differences be-tween the generated and simulated signals, the simulations should be sufficientenough 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 parametersby themselves affect the performance of the equalizer. By looking at one sub-plotgroup, where µ and the update interval is fixed, it seems that longer taps tend toincrease 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, theµ value seems to increase the ability to equalize multipath signals with higherpower in a similar manner as the tap-values. Too large µ however seems to de-crease the power possible to equalize. Varying the update interval, and keepingµ and the tap-value fixed, results in decreased performance when having longerupdate intervals, whilst lower update intervals, apart from update every sample,results in higher performance. The results seem reasonable from a theoreticalpoint of view. The discussion in Section 4.1.3 mention that too large µ value willresult in oscillations and too small update interval or too few taps will not be ableto 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 5.4and 5.5, with µ = 2−13 and update interval of 23 or µ = 2−14 with update interval12 while using 300 taps provides the best equalizer performance out of the pa-rameter values tested.

The simulations loops through the power for each delay, where the power is in-creased with 0.1 for each loop and the delay is increased with 1 sample for eachiteration. Since the power is increased with steps of 0.1, this results in an ap-proximate area, which in theory could be larger. Looking at, for example, Figure5.4, 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 noisepseudo random starting with the same seed for each simulation, meaning thatthere could be small deviations in the results depending on the noise. The sim-ulations do, however, give an approximate area of equalization that makes com-parison between parameter choices possible.

6.1.3 Multiple Delays

The resulting plots for the multiple delays are shown in Appendix C, using 300taps and µ = 2−13 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 cre-ated from power-delay combinations confined within their respective area (seefigure 6.1) can be resolved in all cases. This can be expected for combinations ofr and re since they are resolvable in the single delay case. It is, however, interest-ing that combining nr signals can be resolved, since the combined single delaysare not resolvable. One possible explanation is that the delays chosen somehowcancel 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 ispossible to resolve multipath components to a certain threshold. The case with 5combined delays in the r +nr case, Figure C.12 is not resolvable within the 5000bits of convergence. In some cases, having multiple long delays with high powerseems to cause a severe isi which cannot be resolved.

6.1 Results 41

NR

RE

R

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 inFigure 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 combina-tion of re and nr. It would thus seem reasonable to believe that minimizing thearea where the equalizer is unable to resolve the single delay multipath signalsshould result in better performance of the equalizer.

It is important to note that not all combinations can be tested, these results arethus, once again, only basis for a general perception of how good the equalizerworks. 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 distor-tion. It is, however, important to have an Automatic Gain Controller (agc) beforedetection to prevent large power deviations. The equalizer will work better if theinput power is closer to 1 since it calculates the deviation from one and tries tocounteract that deviation. A power 4 times more or less than the original signalwould imply a large deviation and cause the equalizers cost function to grow to-wards infinity.

The problem with larger power was encountered when adding multiple multi-path 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 beforerunning the equalizer. This can not be seen as a real agc but it does have thesame 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 givereasonable results both in the single and the multiple delay case. Looking at theoutline of the apron in Figure 1.1, it is possible to get a rough idea of the mostlikely delays that the receiver will encounter. The aircraft is most often placedas 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 transmis-sion sideways and straight forward, the power of the transmission behind theaircraft can be neglected due to the symmetry of the aircraft. When placed atthe apron two antennas are transmitting directly into the fuel tanks which willcause shorter but stronger delays if the signal is reflected once. The signal can,however, bounce multiple times between the tanks causing longer delays. Thereceiver starts having a hard time resolving multipath signals at around one bitdelay. Assuming close to complete reflectivity where the reflection has a powerof 0.99 (input is 1) one bit delay would require roughly 46 bounces with a remain-ing power of around 0.6. Furthermore, surrounding buildings will cause around250 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 reflectionfrom one or more fuel tanks, and the long delays caused by multiple reflections,should be resolvable without any equalizer. The combination with longer reflec-tions from buildings could result in erroneous detection even after equalizer. Inthe stationary case, reflections from the fence should be less likely than reflec-tions from surrounding houses due to the positioning of the transmitters. Thereflectivity of a normal brick house is much lower than the reflectivity of a metalfence, meaning that the long delays should have power low enough to be cor-rectly equalized. Looking at the results and the surroundings on the apron, thereshould not be many multipath combinations that the equalizer is unable to re-solve. When taxiing to the runway on the other hand, one of the transmitters willbe transmitting directly into the fence which will create multipath signals thatare 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 signalsare not resolvable by the receiver, 5 out of the 9 signals can be resolved com-pletely with the equalizer whilst 3 out of 9 produce 2 or 3 errors and only one isnon-resolvable. These results follow the reasoning of the apron outline in Section6.1.4 since only one out of the 15 datasets is far from reaching error free recep-tion with the equalizer. These tests are, however, not exactly like the the real casewhen the aircraft is made ready at the apron. There will be both individuals andvehicles moving around causing other multipath signals than the ones recordedin the test. It is reasonable to assume that the reflections caused will be short andstrong. However, it is not investigated how a time varying channel affects theequalization. Once the channel changes the equalizer has to converge again andthat 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 conver-gence time there will, most likely, be erroneous detection. It is also unknown howthe channel will change when an object enters the signal path. The case could bethat there is more or less complete fading, where the singal is unable to reach thereceiver, a severe reflection might be faded making detection possible or it couldcause 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 tochange slowly allowing the equalizer to converge and have a constant channel fora short while.

The data sets from the taxiway also reflects the reasoning in Section 6.1.4, whereonly 2 out of the 10 data sets are resolvable without any equalizer and the cmamanages to resolve half of the remaining signals. It is therefore difficult to re-solve any multipath signals from the taxiway. The data sets are furthermore takenwhen 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. Whentaxiing there are usually no moving objects around, limiting the movement to theaircraft 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 thechannel at the taxiway proves to be very difficult, and more or less impossiblewith the blind cma equalizer.

6.2 Method

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

The parameters tested are separated with relatively large step sizes, meaning thatthere could exist parameters that provide a better equalization. The number oftaps 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 arenot explored that could potentially provide a better equalization. The area plotscould be generated with smaller power and delay steps, as well as with a largerdata vector. There are millions of combinations of multiple delays that are, forobvious reasons, not tested. The way to test the equalizer in a more detailed man-ner 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 thatnewer, more modern versions might exist. The main idea behind the algorithm isstill 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 thatare dependent on phase, for example Quadrature Phase Shift Keying (qpsk). Thearticle chosen, [18], only focuses on the original cma algoritm and not on findinga correct phase. There are many articles that provide test results where cma hasbeen used to equalize telemetry signals. Unfortunatley there is little informationabout the cma algoritm used and parameter settings. The articles that providenecessary information have not performed the tests in such severe environmentas on the apron, making the settings different.

6.3 A wider perspective

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

7Conclusion

In this thesis, an equalizer based on the constant modulus algorithm has beentested. Different combinations of parameters were compared to find a combina-tion that would provide sufficient enough equalization. Initial tests on only singlemultipath delays showed that using a µ value of 2−13, 300 taps and an update in-terval of 23 samples provided good performance in these cases. The parameterswere then used to test the equalizer on multiple delays generated from 3 differ-ent single delay combinations, where the receiver is either able to resolve themultipath with or without the equalizer as well as the case where the equalizeris unable to contribute to better reception. The equalizer can completely solve orsignificantly reduce the ber in 8 out of 9 cases at the apron or 4 out of 8 cases onthe taxiway by using the proposed parameters. The proposed parameters seemsto be able to provide a sufficient enough equalization when the aircraft is station-ary at the apron.

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

7.1 Future work

To use the equalizer with a time variant channel caused by a moving aircraftneeds 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 frequencyis a decreased time resolution, a deeper study into how the sampling frequencyaffects the equalization performance is needed. There are many areas that areinteresting to analyze deeper, for example how increasing the bit rate would af-fect the results or how a different modulation technique with constant moduluswould have to be tuned.

Appendix

ACommunication Model Listings

Listing A.1: NCO Code

funct ion [ b i tS tar tVec , val , scrambled_sig ] = NCO( Fs ,t l e n )

%Fs= Sampling frequency%t l e n = length of time vector

f = 2163000;NrOfBitChanges = 0 ;M=2^32* f / Fs ;idx = 0 ;b i t S t a r t V e c = zeros ( 1 , t l e n ) ;nonscrambled_sig = zeros ( 1 , t l e n ) ;scrambled_sig = zeros ( 1 , t l e n ) ;scrambler = comm. Scrambler ( 2 , [ 0 −14 −1 5 ] , . . .[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ) ;bitchange = ’ f a l s e ’ ;c u r r s c r = 1 ; %assuming f i r s t b i t i s 1

while ( idx < t l e n )idx = idx +1;s = rememb ; %Save the previous pointrememb = mod(rememb+M, 2^32) ; %Count up one step

i f ( rememb < s ) %I f overflowb i t S t a r t ( idx ) = mod( nonscr_s ig ( idx ) +1 ,2) ;

49

50 A Communication Model Listings

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

% Create the FF00 sequencei f (mod( NrOfBitChanges , 1 6 ) < 8)nonscrambled_sig ( idx ) = 1 ;e l s enonscrambled_sig ( idx ) = −1;end%I f b i t change , s t a r t adding the samples of%that scrambled value u n t i l next b i t%change / overflowi f strcmp ( bitchange , ’ true ’ )c u r r s c r = scrambler ( ( nonscr_s ig ( idx ) +1) /2) ;bitchange = ’ f a l s e ’ ;endscrambled_sig ( idx ) = 2* currscr −1;end

Listing A.2: Transmitter Code

funct ion y = modufm( x , Fc , Fs , freqdev , t )int_x = cumsum( x ) / Fs ; % I n t e g r a l of messagey = cos (2* pi *Fc * t ( 1 : length ( x ) ) + 2*pi * freqdev * in t_x ) ;

%Phase noise only operates on baseband s i g n a l s[ baseband , ~ , ~] = IQdemod( y ’ , t , Fc ) ;pnoise = comm. PhaseNoise ( ’ Level ’ , −8 0 , . . .’ FrequencyOffset ’ , 5 0 0 , . . .’ SampleRate ’ , Fs ) ;y = IQmod( pnoise ( baseband ) , t , Fc ) ;

Listing A.3: Multipath Generator

funct ion [ MultiPathSig ] = . . .MultipathGen ( Original , DelayVec , AmplitudeVec )

MultiPathSig = Orig inal ;len = length ( Orig inal ) ;

51

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

Listing A.4: Receiver Code

funct ion z = demodufm( y , Fc , Fs , freqdev , t )yq = h i l b e r t ( y ) . * exp(−1 i *2* pi *Fc * t ( 1 : length ( y ) ) ) ;z = ( 1 / ( 2 * pi * freqdev ) ) * . . .[ zeros ( 1 , s i z e ( yq , 2 ) ) ; d i f f ( unwrap ( angle ( yq ) ) ) *Fs ] ;end

BSimulation Results for Single delays

53

54 B Simulation Results for Single delays

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

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

55



57



59



61



63



65



67


CSimulation Results for Multiple Delays

69

70 C Simulation Results for Multiple Delays

Figure C.1: BER vs SNR of short resolvable multipath combinations withoutequalizer

Figure C.2: BER vs SNR of short resolvable multipath combinations withequalizer

71

Figure C.3: BER vs SNR of long resolvable multipath combinations withoutequalizer

Figure C.4: BER vs SNR of long resolvable multipath combinations withequalizer


Figure C.5: BER vs SNR of combination of Resolvable with equalizer multi-path combinations without equalizer

Figure C.6: BER vs SNR of combination of Resolvable with equalizer multi-path combinations with equalizer

73

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

Figure C.8: BER vs SNR of non-resolvable with equalizer multipath combina-tions with equalizer


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

Figure C.10: BER vs SNR of resolvable with equalizer and resolvable multipathcombinations with equalizer

75

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

Figure C.12: BER vs SNR of non-resolvable with equalizer and resolvable mul-tipath combinations with equalizer


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

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

77

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

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

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