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Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2020

Blind Channel Equalization

for Shortwave Digital Radio

Communications

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Master of Science Thesis in Electrical Engineering

Blind Channel Equalization for Shortwave Digital Radio Communications: Tomas Busk

LiTH-ISY-EX--20/5279--SE Supervisor: Magnus Malmström

isy, Linköpings universitet

Peter Nagy

Swedish Defence Research Agency

Examiner: Isaac Skog

isy, Linköpings universitet

Division of Automatic Control Department of Electrical Engineering

Linköping University SE-581 83 Linköping, Sweden Copyright © 2020 Tomas Busk

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Abstract

In this thesis the concept ofBlind Channel Equalization has been examined and

algorithms suitable to blindly equalize the channel are presented and evaluated on simulated data. The concept of blind equalization is to equalize a communica-tions channelwithout relying on a training sequence or pilot tone, which may be

either unknown to the receiver or not exist at all. In total, seven blind equaliza-tion algorithms have been implemented, these are the: lms-cma (Constant Modu-lus Algorithm), rls-cma, mma (Multi ModuModu-lus Algorithm), rca (Reduced Constel-lation Algorithm), cna-6 (Constant Norm Algorithm), lms-dfe (Decision Feedback Equalization) and rls-dfe. The equalizers are designed as adaptive fir-filters

that are recursively updated by either an lms- or rls-algorithm, according to a cost function specified by the chosen algorithm with the aim to appoximate the inverse h−1of the communications channel h. Thanks to the recursive update the algorithms can easily be implemented either in offline or online systems.

The results show that the rls-algorithms offer shorter convergence times and over all better performance than its lms counterparts. If the signal constellation is known by the receiver in advance the rls-dfe offers the best channel tracking ability, resulting in the lowest symbol error rate.The rls-cma offers the roughly the same mseR -performance (mean square error from the equalizer output to

the closest radius of the constellation points) but it lacks the ability to handle the doppler shift as well as the rls-dfe does. The results also show that the mma, cna-6 and rca-algorithms do not offer any better performance than the more commonly used and studied lms-cma algorithm.

When the receiver incorrectly assumes the signal constellation, it can identify the correct constellation. Test results show that the rls-cma is especially good at amplitde recovery, while the rls-dfe is suitable to recover the phase of the signal. Lastly the rca is useful to recover psk-4 modulated signals as its cost function match the psk-4 constellation.

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Acknowledgments

I would like to thank foi for the opportunity to do my master thesis in collabora-tion with them. Special thanks to my supervisors Peter Nagy at foi and Magnus Malmström at isy and my examiner Isaac Skog for their time, valuble feedback, and ideas.

Linköping, February 2020 Tomas Busk

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Contents

Notation ix

1 Introduction 1

1.1 Background and purpose . . . 1

1.2 Problem statement . . . 2 1.3 Related Works . . . 2 1.4 Delimitations . . . 2 2 Theory 3 2.1 Problem illustration . . . 3 2.2 Signal representation . . . 4 2.2.1 Signal modulation . . . 5 2.3 Channel modeling . . . 6 2.4 Blind equalization . . . 8

2.4.1 Least Mean Squares . . . 9

2.4.2 Recursive Least Squares . . . 10

2.5 Decision Feedback Equalization . . . 11

2.6 Phase Correction . . . 13

3 Blind algorithms 15 3.1 Basic assumptions . . . 15

3.2 Constant Modulus Algorithm . . . 16

3.2.1 lms-cma . . . 16

3.2.2 rls- cma . . . 18

3.3 Constant Norm Algorithms . . . 18

3.3.1 Constant Norm Algorithm 6 . . . 19

3.3.2 Constant Square Algorithm . . . 19

3.4 Reduced Constellation Algorithm . . . 20

3.5 Multi Modulus Algorithm . . . 21

4 Implementation and Results 23 4.1 Channel Simulations . . . 23

4.2 Performance Metrics . . . 24

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viii Contents 4.3 Algorithm Implementations . . . 25 4.3.1 lmsalgorithms . . . . 25 4.3.2 rlsalgorithms . . . . 25 4.3.3 dfealgorithms . . . . 25 4.3.4 Phase Correction . . . 26 4.4 Test Results . . . 26

4.4.1 Signals with correctly assumed constellations . . . 27

4.4.2 Signals with incorrectly assumed constellations . . . 40

5 Conclusions and future work 49 5.1 Conclusion . . . 49

5.1.1 Signals with correct constellation . . . 49

5.1.2 Signals with incorrect constellation . . . 50

5.2 Future work . . . 51

A Additional figures 55 A.1 Convergence Time . . . 55

A.2 Tracking ability . . . 58

A.2.1 Moderate channel conditions . . . 58

A.2.2 Disturbed channel conditions . . . 65

A.3 Signals with incorrectly assumed constellations . . . 69

B Simulation Data 73 B.1 Moderate Channel Conditions . . . 73

B.2 Disturbed Channel Conditions . . . 76

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Notation

Notations

Notation Meaning

j Imaginary number j2= −1 Re[z] Real part of complex valued z Im[z] Imaginary part of complex valued z

sgn[z] The complex sign-function

XT Transpose of vector X

XH Hermitian transpose of vector X

XComplex conjugate of vector X

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x Notation

Abbreviations

Abbreviation Meaning

cma Constant Modulus Algorithm cna Constant Norm Algorithm cpm Continous Phase Modulation cqa Constant Square Algorithm d.d. Decision Device

dfe Decision Feedback Equalizer fir Finite Impulse Response

isi Intersymbol Interference lms Least Mean Squares los Line of Sight

mma Multi Modulus Algorithm mse Mean Square Error pll Phase Locked Loop psk Phase Shift Keying

qam Quadrature Amplitude Modulation qamc Quadrature Amplitude Modulation -C

rca Reduced Constellation Algorithm rls Recursive Least Squares

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1

Introduction

This thesis will investigate different methods of so called blind equalization, or

blind deconvolution in short wave digital wireless radio communications.

1.1

Background and purpose

In digital comunications over a wireless channel the signal is distorted by effects such as multipath propagation, frequency-selective fading, and doppler shifts. To counter these effects an equalizer is employed at the receiver to restore the signal as close to the original transmitted signal as possible. One way of doing this is to have the transmitter send a so calledpilot signal or training sequence that

is known a-priori to the receiver, which then compares the received signal to the pilot and estimates the channel using some form of adaptive filters. This proce-dure is known aschannel equalization.

However, in many applications the pilot signal is either unknown to the receiver or entirely nonexistent, and hence the traditional method of channel equalization is no longer suitable. This is the problem ofblind equalization, in which the

re-ceiver has to equalize the signal without any pilot signal.

Therefore, the purpose of this thesis is to investigate how a wireless digital com-munication channel can be blindly equalized and to study which algorithms are suitable to use when the channel is severely distorted and time varying. It will also be investigted if the nonlinear Decision Feedback Equalizer is suitable to use as a blind equalizer.

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2 1 Introduction

1.2

Problem statement

This thesis will investigate the following questions:

• What algorithms are available for blind channel equalization and how well do they handle different signal constellations? How well do they perform when there is a limited amount of data available?

• How sensitive are the algorithms regarding (assumptions of) the signal con-stellations?

• What happens when the receiver assumes the wrong modulation type? Is there any algorithm or algorithm type that is suitable to use when the mod-ualtion type is unknown?

1.3

Related Works

Numerous authors have examined blind equalization earlier. The first algorithm was presented by Yoichi Sato for amplitude-modulated systems in 1975 [1]. Do-minique N. Godard presented what later would be known as theConstant Modu-lus Algorithm, cma in 1980 [2] and since then various similar algorithms suitable

for different constellations have been proposed [3], [4]. The Decision Feedback Equalizer has been examined in [5] and [6], where a feedback link from the Deci-sion Device (d.d.) can be added to try to increase equalizer performance.

Blind equalizers are widely used in communications and transmission systems where the goal is to acheive as high data rates as possible. To acheive this the receivers has to have very good channel tracking ability, as well as filters to keep the signal clear.

Most of the published articles focus on time invarant channels and ignore some of the practical aspects, such as that in a truly blind case the signal constellations are also unknown. Therefore, in this thesis it will be studied what happens when the receiver wrongfully assumes the signal constellation.

1.4

Delimitations

This thesis will focus on linear modulation types, i.e psk (Phase Shift Keying) and qam (Quadrature Amplitude Modulation). More complex forms of non-linear modulation such as cpm (Continuous Phase Modulation) will not be considered. Further, though the convergence speed of the equalization algorithms are impor-tant, the convergence speed will be measured in number of samples until conver-gence and not in seconds. Moreover the algorithms presented can implemented in real time systems, this thesis will not focus on real time performance.

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2

Theory

In this chapter the theoretical background needed for blind channel equalization will be presented. The signals considered are generally complex-valued base-band signals passed trough a linear or non-linear channel modulated with some kind of linear modulation type. To begin with, the problem will be illustrated using an example.

2.1

Problem illustration

A time discrete signal xn is to be sent over a wireless radio channel from a

sta-tionary sender to a possibly non-stasta-tionary receiver, as illustrated in figure 2.1. Because of channel distortions such as multipath-propagation and frequency-selective fading, the movement of the receiver and noise present in the environ-ment, the received signal un will not be equal to the sent signal xn. To

compen-sate for this, the receiver needs some way to extract an estimate ˆxnof xnfrom the

received samples un, illustrated in figure 2.2. This could be done by having the

transmitter start by sending a training sequence that is known to the receiver and have the equalizer consist of an adaptive filter that uses the training sequence to adjust its tap weights. However, this approach is not always feasible in practice because the training sequence may be unknown to the receiver, the receiver may unexpectedly drop the connection, or the training sequence is omitted entirely to conserve bandwidth. Therefore, it would be preferable to be able to equalize the channel without the need of a training sequence.

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4 2 Theory

v

Transmitter Receiver

Figure 2.1:Problem illustration with a stationary transmitter and a moving receiver with velocity v.

Transmitter x Channel u Equalizer ^x

Figure 2.2:Equalization principle.

2.2

Signal representation

In this section data transmission over a wireless channel will be explained. A general radio communications system consists of a transmitter, a channel and a receiver. The transmitter modulates the carrier signal and creates a radio signal that is then sent over the channel. The receiver transforms the received signal back into a baseband signal and then demodulates it, obtaining the original sig-nal that was passed to the transmitter.

Figure 2.3 illustrates a block diagram of a simple communication system. The modulator generates the signals xn,Iand xn,Qwhich are refered to as theIn-phase

component and theQuadrature component, respectively. xn,I and xn,Q are then

up converted to the pass band centered around the carrier frequency fcby means

of multiplication with √

2 cos(2πnT ) respectively −

2 sin(2πnT ). The signal

xn= xn,I

2 cos(2πnT ) − xn,Q

2 sin(2πnT ) (2.1)

where T = 1/fcis the sampling period, is then transmitted trough the channel

with impulse response hn. At the receiver the received signal unis demodulated

using un,I= un2 cos(2πnT ) un,Q= −un2 sin(2πnT ). (2.2)

vnin figure 2.3 is white gaussian noise with zero mean and variance σ2.

Because the sine and cosine terms are orthogonal to each other it is possible to de-scribe the system in figure 2.3 by using complex baseband representation instead, which tend to result in more compact descriptions. Define the complex envelope as

˜

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2.2 Signal representation 5 -√2sin(2πnT) ∑ x x h √2cos(2πnT) x x x x u u n,I n,Q n,I n,Q √2cos(2πnT) -√2sin(2πnT) ∑ v n n xn u n Figure 2.3:IQ-modulation.

where xn,I and xn,Q are the in-phase and quadrature components generated by

the modulator as illustrated in figure 2.4. ˜vn is now complex white gaussian

noise with mean 0 and variance σ2and hnis the complex baseband channel. The

output ˜unrelates to un,Iand un,Qthe same was as ˜xn, i.e.

˜ un= un,I+ jun,Q. (2.4) x h ∑ ṽ u n n n n

Figure 2.4:Complex baseband representation.

2.2.1

Signal modulation

The signal xn sent via the channel hn is modulated in a way that is suitable to

transmit over the channel. In this thesis mainly two different types of modula-tion will be considered, psk, which uses pure phase modulamodula-tion and qam, which combines both phase and amplitude modulation. The transmitted message con-sists of symbols that are a set of different values drawn from a symbol alphabet orconstellation, which depend on the type of modulation that is used.

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6 2 Theory

PSK

pskis a modulation form that modulates a signal using the phase of the carrier signal as the information carrier [8]. If the symbol-alphabet is to consist of M, all

M symbols are spaced equally far apart over the argument of a unit circle with

constant modulus. Figure 2.5 shows the corresponding signal constellations for

M = 4 and M = 8. As an example, the symbol alphabet for a psk-4 modulated

symbol may be the values {ejπ/4 ej3π/4 ej5π/4 ej7π/4}in the complex plane. Each of these values are then assigned a corresponding bit sequence, for example {00 01 11 10}. This means that in this example the system transmits 2 bits per symbol.

Figure 2.5: psk-4 and psk-8 modulation.

QAM

qamis a modulation scheme that uses both the phase and amplitude of the car-rier signal. The M constellation points are equally spaced apart from each other on a rectangular grid. Figure 2.6 illustrates the signal constellation for qam-4 and qam-16. As a side note, the constellations for psk-4 and qam-4 are identical up to an eventual phase rotation. A special case of qam is what in this theis is called the qamc, where all constellation points are located on circles with con-stant amplitudes.

2.3

Channel modeling

When a radio wave propagates trough an environment, such as the earth’s atmo-sphere close to ground level it is subject to reflection, diffraction, and scattering. Reflection occurs when a radio wave collides with an object that is large compared to the wave’s wavelength, resulting in a delayed and amplitude scaled version of the wave arriving at the receiver. Diffraction occurs if the path between trans-mitter and receiver is obstructed by an object with sharp edges, giving rise to a bending of waves behind and around the obstacle. Scattering occurs when the wave propagates trough a medium with objects with small dimensions compared to the wave’s wavelength in it, resulting in a scattering of the incoming wave.

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2.3 Channel modeling 7

Figure 2.6: qam-4, qam-16 and qamc-16 modulation.

In digital wireless communication systems the channel is best assumed as time-varying and frequency-selective. In this thesis the channels will be modeled as

Rayleigh-fading channels. A Rayleigh-fading distribution is used when there is no

direct los-path from the transmitter to the receiver. If there exists one dominant los-path from transmitter to receiver, the channel is aRicean-fading channel [9].

v Transmitter Receiver Direct path Indirect path 1 Indirect path 2

Figure 2.7: Typical operating environment, the moving receiver receives multiple time delayed and amplitude scaled versions of the transmitted sig-nal. If the direct path exists, the channel is aRician channel, if not it is a Rayleigh channel.

Consider the system in figure 2.7, where the receiver is receiving three versions of the signal sent from the transmitter. If the direct path is omitted the channel is best modeled as a Rayleigh-channel with two discrete paths with their respective path loss and delay. If there exists a direct path between transmitter and receiver the channel is best modeled as a Rician-channel instead with the addition of the

K-factor defined as the ratio between the signal power in the direct path and the

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8 2 Theory 0 2 4 6 8 10 Time [s] 10-4 10-2 100 Signal strength

Signal strength versus time

Figure 2.8:Rayleigh fading channel with a doppler shift of 5 Hz.

Because of the relative movement between transmitter and receiver, or because of changes in the atmospheric conditions the received signal will be doppler shifted, i.e. the frequency of the signal will change. As a consequence of this, the the path loss for each path is not constant during the transmission, but will vary over time, as shown in figure 2.8. The doppler shift is calculated with

fd=

vd

λ cos θd (2.5)

where vd is the relative velocity between transmitter and receiver, λ the

wave-length of the carrier wave, θdthe angle between transmitter and receiver and fd

the resulting doppler shift in Hz. When the doppler shift arises from changes in the atmospheric conditions, (2.5) cannot be used, but a phase-tracking loop can still be employed to estimate the changes in the received frequency.

2.4

Blind equalization

The algorithms analyzed in this thesis are of the so called Bussgang Blind Equal-ization type of algorithms, which aims to solve the optimEqual-ization problem

W = arg min

W

J J = E[J(yn)]

(2.6)

where the Cost function J(yn) is dependent on the algorithm [4], and E[ · ]

de-notes statistical expectation. This problem can be solved with either aLeast Mean Squares- (lms) or a Recursive Least Squares (rls)-algorithm.

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2.4 Blind equalization 9 Adaptive filter u LMS g(·)

-+

e y

n n n

Figure 2.9:Linear blind equalization.

2.4.1

Least Mean Squares

A block diagram for a linear time discrete lms-blind equalizer is shown in fig-ure 2.9, where n is the sample index, un is the received modulated signal, ynis

the equalizer output which is typically sent to a decision device and en is the

estimation error defined as

en = g[yn] − yn. (2.7)

The term g( · ) is a zero-memory nonlinear estimator used instead of a known training sequence. The adaptive filter consists of a time-varying fir-filter Wn =

[w0,n ... wN −1,n]T with N filter taps. The equalizer is said to be either symbol

spaced or fractionally spaced, depending on when the filter taps are updated. A symbol spaced filter updates its coefficients once per symbol period, and a frac-tionally spaced filter updates its coefficients more than once per symbol period. The optimization of the filter coefficients is done with a stochastic gradient de-cent algorithm, because the expected value is hard to calculate, this approxima-tion where the excact gradient is replaced with an estimate calculated from ynat

each iteration is used instead. With Wn as the filter coefficient vector at time n

the filter update algorithm is

Wn+1= Wnµ

∂J(yn)

∂yn

¯

Un (2.8)

where µ is the step size and

Un= [un un−1 ... un−N +1]T (2.9)

is the vector containing the N latest received samples. The bar ( ¯· ) denotes the complex conjugate. The nonlinearity g( · ) and the error function in figure 2.9 are chosen based on the cost function such that the error function equals the

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10 2 Theory

derivative of the cost function, i.e.

en= g[yn] − yn= ∂J(yn) ∂yng[yn] = ∂J(yn) ∂yn + yn (2.10)

The linear blind equalizer in figure 2.9 is then expressed as

Wn+1= WnµenU¯n

en = g[yn] − yn

yn = UTnWn

(2.11)

with g[ · ] selected according to (2.10). By different choices of J(yn) different

algo-rithms are obtained. These will be examined in chapter 3.

2.4.2

Recursive Least Squares

When the rls-algorithm is used, the expectation in (2.6) is replaced by an expo-nentially weighted average sum:

J =

n

X

k=1

λn−kJ(yk) (2.12)

where λ is the forgetting factor. However, as the cost function is nonquadratic in the filter weights, it has to be modified to be solved by the rls-algorithm. The standard rls-algorithm is;

Wn= Wn−1+ Kn(yn− UTnWn−1) Kn= Pn−1Un λ + UTnPn−1Un Pn= 1 λ(Pn−1Pn−1UnUTnPn−1 λ + UTnPn−1Un ), (2.13)

where Wnis the filter vector at time n, P is related to the covariance matrix, Un

the input vector and λ is the forgetting factor. This is a design parameter usually in the range [0.98 0.999]. The purpouse of the forgetting factor is to "forget" old measurements as time increases. A forgetting factor very close to one results in an algorithm that is less sensetive to new measurements, and a smaller factor leads to an algorithm that relies more on the most recent samples [10].

For the cost function to be quadratic in the filter weights, the filter input Un

is modified into

˜

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2.5 Decision Feedback Equalization 11 Forward Filter u Decision Device y Backward filter x ∑ + -n n ^n

Figure 2.10:Decision feedback equalizer.

The modified rls-algorithm is then

Wn = Wn−1+ Knen Kn = Pn−1U˜n λ + ˜UHnPn−1U˜n Pn = 1 λ(Pn−1Pn−1U˜nU˜HnPn−1 λ + ˜UHnPn−1U˜n ), (2.15)

enis the error term, defined by the cost function of the chosen algorithm, and the

filter output is calculated as

yn= UHnWn. (2.16)

For a more in detail derivation, see [15] or [16].

2.5

Decision Feedback Equalization

The equalization described in section 2.4 uses a linear equalizer without any sort of feedback from the decision device. The drawback of this is the noise enhance-ment and that the equalizer is unable to equalize the channel if the distortions are to severe, which is often the case with radio channels [8]. The idea behind decision feedback equalization, is that the intersymbol interference from past symbols can be completely removed with a feedback link from the decision de-vice, provided that the symbol decisions made by the decision device are correct [8]. The decsion feedback equalizer consists of a feed forward adaptive filter and a feedback adaptive filter which usually are updated with the same algorithm. A block diagram of a decision feedback equalizer is shown in figure 2.10.

The Forward filter WFn consists of N filter taps and is supposed to cancel the

intersymbol interference induced from future symbols, and the backward filter

WBn consists of M filter taps and is supposed to cancel the intersymbol

interfer-ence from past symbols. The equalizer output ynis expressed as

yn = yfyb= UTnWFnXˆ T

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12 2 Theory Decision Device ∑ + -Adaptive Algorithm en ∑ un yn ^xn yn B yn F WB WF +

-Figure 2.11:Decision feedback equalizer with an adaptive algorithm to esti-mate filter coefficients.

where Un is a vector containing the N past received samples and ˆXn−1is a

vec-tor containing the M latest detected symbols. The decision device in figure 2.10 estimates the transmitted symbol xnby mapping the equalizer output yn to the

closest constellation point. This estimate is then subtracted from the output of the forward filter via the feedback loop to cancel the interference from past sym-bols. The advantage of this over the linear equalizer is that it is possible, at least in theory to completely cancel the intersymbol interference from past symbols, provided that the decisions from the decision device are correct. The drawback is error propagation, i.e. that a symbol error made in the decision device might cause more errors in the next symbols because the erroneous symbol enters the feedback loop. This is why the dfe needs a good initiation to converge.

The dfe can be constructed as an adaptive equalizer in a way similar to the linear equalizer, with the difference that the adaptive algorithm estimates the coeffi-cients for two filters instead of one. Se figure 2.11 for an illustration. Similar to the linear case the dfe-algorithm can be implemented using a stochastic gradient decent algorithm, resulting in [11]:

"WF n+1 WBn+1 # ="W F n WBn # −µen " ¯ Un − ¯Xˆn # . (2.18)

enis the error between the equalizer output and detected symbol;

en= ynxˆn. (2.19)

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2.6 Phase Correction 13

The equalizer can also be constructed with an rls-algorithm, resulting in: "WF n+1 WBn+1 # ="W F n WBn # −K en, (2.20)

where the error between the equalizer output and detected symbol is calculated as en="W F n WBn #H U (ynxˆn). (2.21)

The equalizer output is now computed with (2.22)

yn= yfyb= UHnWFnXˆ H

n−1WBn (2.22)

2.6

Phase Correction

If the signal is doppler shifted, a time varying phase shift, ejθn will be added

to the received samples. To correctly demodulate the equalized signals a phase correction is therefore needed. This is achieved with a second order phase locked loop (pll), which esentially tracks the phase variations and compensates for it;

˜ yn= ynej ˆθn−1 ˆ θn= ˆθn−1+ µθ(n+ β n X i=1 i) n= I m[ ˜y( ˆxny˜n) ∗ ]. (2.23) ˜

yn is the loop output i.e. the corrected output, yn the loop input i.e. the doppler

shifted equalizer output, ˆθ the estimated phase, ˆx the estimated transmitted

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3

Blind algorithms

In this chapter a number of algorithms used in blind equalization is presented. The algorithms all use adaptive algorithms to update the filter-taps in the equal-izer filter and can be implemented as linear equalequal-izers, or asDecision Feedback Equalizers (dfe). The algorithms that will be presented are the Constant Modulus Algorithm (cma), the Constant Norm family of algorithms (cna), Reduced Constella-tion Algorithm (rca) and the Multi Modulus Algorithm (mma).

3.1

Basic assumptions

The algorithms presented do not require a training sequence, but some assump-tions about the signal, mainly the statistics and the constellation of the signal has to be made to equalize the signal. The assumptions made for the blind algorithms are as follows:

• The input sequence xn is assumed to consist of independently and

iden-tically distributed symbols, i.e. all symbols in the symbol alphabet are equally probable to occur.

• The signal constellation is known, or is allowed to be assumed by the re-ceiver.

• The channel is not required to be minimum-phase.

• The channel will be described as a complex-valued and slowly time varying fir-filter with unknown impulse response hn.

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16 3 Blind algorithms -1 0 1 In-phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature Signal Constellation -1 0 1 In-phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature

Equalized signal with CMA

Figure 3.1: A psk-8 modulated signal equalized with cma, because cma is unable to recover the phase the equalized signal appears to be spinning.

3.2

Constant Modulus Algorithm

The Constant Modulus Algorithm (cma) was first proposed by Godard [2] and Treichler and Agee [13] in 1980 and 1983, respectively. The cma algorithm is the most studied algorithm and one of the most used in practice [14]. Because it is relatively simple to understand and implement it is the first algorithm to be presented.

As the name describes, the cma-algorithm assumes that the signal has a con-stant modulus, i.e., the amplitude of the sent signal is concon-stant. This makes the algorithm suitable for psk-M modulated signals, but it also work for signals with non-constant modulus, such as qam-M modulated signals. The idea behind cma is to use a cost function that penalizes deviations of the equalizers output from the constant modulus [14]. Because the cost function only penalizes the deviation in the modulus and ignores deviations in the phase, an equalizer with the cma-algortihm is by itself unable to recover the phase of the signal. This may result in an arbitrary phase shift of the equalizer output or a constant "spinning" of the equalizer output if the signal is doppler shifted. See figure 3.1 for an illustration. To recover the phase and stop the spinning a phase locked loop (pll) is needed.

3.2.1

LMS

-

CMA

The lms-cma cost function is defined as

JLMS(yn) = (|yn|2−R2)2 (3.1) where

R2=

E[|xn|4] E[|xn|2]

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3.2 Constant Modulus Algorithm 17 0 1.5 1.5 1 1 0.5 0.5 Im{y n} Re{yn} 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0.1 J

Cost function for CMA

0.2

Figure 3.2: cmacost function.

is a real-valued scalar constant depending only on the source constellation, which of course may be either known or unknown to the receiver. If the constellation is known in advance R2can be readily calculated, if not it can often be assigned the value of 1, since it will only result in a scaled version of the transmitted constel-lation.

The cma cost function in (3.1) is plotted in figure 3.2 and the nonlinear func-tion g( · ) becomes

g[yn] = 4yn(|yn|2−R2) + yn. (3.2)

This in turn gives the error function as

en= 4yn(|yn|2−R2) (3.3) and finally the complete filter algorithm as

Wn+1= WnµynU¯n(|yn|2−R2)

yn = UTnWn

(3.4)

which will here after be referred to as the lms-cma algorithm. The factor 4 in the error function is included in the step size µ in (3.4) for convenience.

Table 3.1 contains some examples of the constant R2for different signal constel-lations. One thing to note is that for constellations with non-constant modulus the error function enis not small even if perfect equalization is achived, and for

this reason the absolute value of the error term is not a good measure for the performace for non-constant modulus constellations.

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18 3 Blind algorithms Signal Constellation R2 psk-2 1 psk-4 1 psk-8 1 qam-4 1 qam-16 1.3200 qam-32 1.3100 qam-64 1.3810 qamc-16 1.2291 qamc-32 1.3232 qamc-64 1.3414

Table 3.1:R2constant for different signal constellations.

3.2.2

RLS

-

CMA

The rls-cma cost function is defined as

JRLS(yn) = n

X

k=1

λn−k|WHnUkUkHWk−1R2|2, (3.5)

which is derived in detail in [15]. The error function enis defined as

en= WHU UHW −R2 (3.6)

where R2is calculated in the same way as for the lms-cma.

3.3

Constant Norm Algorithms

The cma-algorithm is actually a special case of a series of algorithms that are calledConstant Norm Algorithms, or cna-algorithms. The cna-type of algorithms

all rely on some constant norm to form the error function, the cma as example relies on the 2-norm (the euclidean distance) being constant. It is also possible to use different norms, if one for example uses the 1-norm we obtain the original algorithm derived by Sato in 1975 [3] with the cost function

J(yn) =

1

2|ynγ · sgn(yn)|

2. (3.7)

The constant γ is a scalar real valued constant that is dependent on which signal constellation that is used [1]. Sato’s algorithm is only suitable for use with pure amplitude modulated signals, and will therefore not be studied in this thesis. The general expression for a cna cost function

J(yn) =

1

ab|n

a(y

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3.3 Constant Norm Algorithms 19

can be used to derive all algorithms in this family. The constants a and b gives the algorithm two degrees of freedom, but are usually set to a = b = 2. R is as in the cmacase a scalar real valued constant that depends on the signal constellation and n(yn) is the chosen norm function, with the p-norm defined as

np(yn) = ||yn||p= p

p

|Re[yn]|p+ |I m[y

n]|p. (3.9)

To calculate R is complicated, but becomes much easier if b is fixed to the value of 2, in which case

R = E[n

2a(x

n)]

E[na(xn)]. (3.10)

For the general case with b , 2 and a derivation of R, see [3]. As shown in [3] two norms are of special interest, namely the 6-norm and the infinity-norm, yeilding algorithms named cna-6 andConstant Square Algorithm (cqa).

3.3.1

Constant Norm Algorithm 6

The cna-6 algorithm uses the 6-norm and the constants a = b = 2. The cost function JCN A−6(yn) = 1 4E[(||yn|| 2 6−R)2] (3.11)

leads to g( · ) and enas;

g[yn] = (||yn||26−R) Re(yn)5+ I m(yn)5 ||yn||2 6 + yn (3.12) and en= (||yn||26−R) Re(yn)5+ I m(yn)5 ||yn||2 6 . (3.13)

The complete filter update algorithm

Wn+1= Wnµ(||yn||26−R)

(Re[yn])5+ j(I m[yn])5

||yn||4 6

¯

Un. (3.14)

has to be implemented with care because the division with ||yn||46 may cause the algorithm to diverge if ynis close to 0.

3.3.2

Constant Square Algorithm

TheConstant Square Algorithm, denoted cqa (or cna-∞) utilizes a square norm

with the idea that it will be suitable for constellations with square appearances. The norm itself is defined as

||yn||= max(|Re[yn]|, |I m[yn]|) (3.15) and the cost function with a = b = 2 is

JCQA(yn) =

1 4(||yn||

2

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20 3 Blind algorithms

The filter update algorithm is

Wn+1= Wnµ(||yn||2∞−R)F(yn) ¯Un (3.17) with the function F( · ) defined as

F(yn) =       

Re[yn], if Re[yn] > I m[yn]

I m[yn], otherwise

(3.18)

The g( · ) and enfunctions are calculated as

g(yn) = (||yn||2∞−R)F(yn) + yn (3.19)

and

en = (||yn||2∞−R)F(yn). (3.20)

By comparing the cost functions for these two algorithms (figure 3.3) to the cma cost function it is clear that the cna-6 can be seen as the middle ground between the cma and cqa and the relation between the chosen norm and the cost function becomes clear. 0 0.2 1 0.4 0.6 J 1 Cost function for CQA

Im 0.8 0 Re 1 0 -1 -1 0 0.2 1 0.4 0.6 J 1 0.8

Cost function for CNA6

Im 0 1 Re 0 -1 -1

Figure 3.3: cqaand cna-6 cost functions.

3.4

Reduced Constellation Algorithm

TheReduced Constellation Algorithm, or rca-algorithm is an extension of Sato’s

algorithm for a two-dimensional qam modulated signal. The cost function is defined as JRCA(yn) = 1 2[(| Re[yn]| − RR) 2+ (| Im[y n]| − RI)2] (3.21)

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3.5 Multi Modulus Algorithm 21

The cost function forces the equalizer output to four constellation points, which is also the reason for its name. This cost function splits the real and imaginary parts from each other. The constants RRand RI are constellation-dependent and

calculated as

RR=

E[Re[xn]2] E[| Re[xn]|],

RI is calculated with the imaginary parts of an instead of the real. With g( · )

0 0.2 1 0.4 J 0.6 1 Cost function for RCA

Im 0 0.8 Re 0 -1 -1 0 1 1 J 2 1 Cost function for MMA

Im 0 Re 3 0 -1 -1

Figure 3.4: rcaand mma cost functions.

defined as

g(yn) = Re[yn](| Re[yn]| − RR) + j Im[yn](| Im[yn]| − RI) + yn (3.22)

the error function enbecomes

en= Re[yn](| Re[yn]| − RR) + j Im[yn](| Im[yn]| − RI). (3.23)

3.5

Multi Modulus Algorithm

The Multi-Modulus Algorithm, or mma-algorithm is designed with square type

constellations, such as qam in mind. The idea is to use a cost function that penal-izes deviations from piecewise linear contours, which makes it much less likely to create rotated solutions as cma does [17]. Another benefit of the mma-algorithm is its flexibility to accommodate nonsquare qam-type signal constellations. With the cost function

JMMA(yn) =

1

2[(Re(yn) − R

L)2+ (I m(y

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22 3 Blind algorithms

and the constants RLRand RLI

RLR= E[|Re[xn]| 2L] E[|Re[xn]|L] RLI = E[|I m[xn]| 2L] E[|I m[xn]|L]

the function g( · ) for the most common case, L = 2 becomes

g[yn] = Re[yn](Re[yn]2−R2R) + jI m[yn](I m[yn]2−R2I) + yn. (3.25)

Finally this results in the error function

en= Re[yn](Re[yn]2−R2R) + jI m[yn](I m[yn]2−R2I). (3.26)

If L , 2 then (3.25) and (3.26) has to be modifed accordingly, but since L = 2 is the most used value in practice it will be used in this thesis if nothing else is stated.

The real and imaginary parts of the cost function are usually separated to high-light the fact that the algorithm penalizes devations on the real and imaginary axis separately from each other. This property allows the MMA-algorithm to at least up to some extent recover the phase of the transmitted signal. The cost function is illustrated in figure 3.4.

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4

Implementation and Results

In this chapter the practical aspects of the channel simulations and implementa-tions of the algorithms are presented together with the results of the simulaimplementa-tions.

4.1

Channel Simulations

The channels were modelled as Rayleigh fading shortwave channels with differ-ent doppler shifts and two signal paths. The transmitted symbols were generated randomly and transmitted over the simulated channels at a symbol rate of 2 400 symbols per second, which is common for shortwave radio communications. The receiver sampled the received signal with a sampling frequency of 16 kHz, mean-ing that the signal is sampled at a rate of 16000/2400 = 623 samples per symbol. The equalizers require that this rate is an integer number, and hence the signal has to be resampled. The resampling is done by interpolation of the signal to create a new signal sampled at a prechosen rate. This rate is chosen as 2 samples per symbol in this thesis.

The algorithms were tested on channels with three types of conditions and dis-turbances. In this thesis denoted as channels withquiet, moderate or disturbed

conditions, which are summarized in table 4.1. All channel types had two signal paths with different delays between them and with average path gains of zero dB. The doppler shift fdwas chosen in the range [0.0 0.8] Hz, as it reflects

typi-cal values encountered in realistic short wave scenarios, where the doppler shift mainly arises from changes in the atmospheric conditions between sender and receiver.

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24 4 Implementation and Results

Channel conditions Avg. path delays σ

Quiet 0.5 ms 0.05

Moderate 1 ms 0.25

Disturbed 2 ms 0.5

Table 4.1:Parameters for the different channels.

4.2

Performance Metrics

To measure the performance of the implemented algorithms, thesymbol error rate

(ser) and themean square error (mse) to the closest radius of the constellation

points is used as metrics. The symbol error rate is defined as the rate of correctly decoded received symbols divided by the total number of transmitted symbols, i.e.,

ser, P (xnxˆn , 0)

N (4.1)

Here N is the number of transmitted symbols, xnis the true transmitted symbol

and ˆxnis the equalizers estimate of xn. The mean square error around the radius

of the constellations points is defined as

mseR, 1 M M X n=1 min j yn pj 2 . (4.2)

where ynis the equalizer output, {p}Pn=1the set of constellation points (for a total

of P points) and M is the number of the latest transmitted symbols. Because some algorithms are prone to create rotated solutions, the mean square error around the radius of the constellations points is preferred over the mean square error to the closest constellation point. The advantage is that rotated solutions are treated equally to nonrotated solutions.

Another important performance metric that is used is a visual inspection of the equalized signal. While this metric is hard to quantize into some value, it of-fers the advantage that it is easy to visually inspect to see which equalizers ofof-fers good enough performance and which that do not. See figure for 4.8 for an exam-ple of where it is clear from inspection that rls-dfe offers better performance than cna-6.

To determine if an equalizer is able to recover the true constellation or ampli-tude of a transmitted signal when the receiver uses the wrong constellation, the angles and amplitudes of the equalized signals is considered, as well as visual inspection of the equalizer output.

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4.3 Algorithm Implementations 25

4.3

Algorithm Implementations

The algorithms in chapter 3 have been implemented as linear equalizers and tested. The cqa is not included in the presented tests because initial small scale tests showed that it did not perform satisfactory. The dfe-equalizer described in section 2.5 have been implemented to increse the performance of the lms-cma and rls-cma.

4.3.1

LMS

algorithms

The linear lms equalizer is implemented as algorithm 1 with filter length N , step size µ and the error function e(yn) selected accordingly to the algorithms in

Chap-ter 3. The filChap-ter is initialized with all weights set to zero except for the centre weight, which is set to one. The equalizer calculates the output and updates the filter once per received symbol.

Algorithm 1:Linear lms-type Blind Equalizer Result:Equalized output y

W0= h

0 . . . 1 . . . 0i;

n = 0; m = 0 ; i = nr of samples per symbol;

whileUnprocessed samples left do ifmod(n, i) = 0 then U =hun un−1 . . . un−N +1 i ; ym= UTWm; em= e(ym); Wm+1 = WmµemU ;¯ m = m + 1 end n = n + 1; end

4.3.2

RLS

algorithms

The linear cma − rls-equalizer is implemented as algorithm 2 with a filter length of N initialized as before. λ is the rls forgetting factor and I is the identity matrix of dimension N .

4.3.3

DFE

algorithms

The decision feedback equalizer have been implemented with both lms- and rls-algorithms in algorithm 3 and 4. To initialize the dfe the feedforward filter was first initialized by equalizing the signal with a linear lms or rls equalizer in

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26 4 Implementation and Results

Algorithm 2:Linear cma-rls Blind Equalizer Result:Equalized output y

Wm=

h

0 . . . 1 . . . 0i;

P = 100 · I;

n = 0; m = 0; i = nr of samples per symbol;

whileUnprocessed samples left do ifmod(n, m) = 0 then U =hun un−1 . . . un−N +1 i ym= UHWm Q = U UHWm K = λ+QP QHP Q P = 1λ(P − K QHP ) em= WHmQ − R2 Wm+1= WmK ¯em m = m + 1 end n = n + 1 end

reverse, generally resulting in good enough values to allow the equalizer to sta-bilize. The feedback filter is initialized with all weights set to zero. Note that in algorithm 3 and 4 the complex conjugate is represented with an *, and not a bar.

4.3.4

Phase Correction

Because of the doppler shift the phase of the equalized signals have to be esti-mated and compensated for. To correct the phase algorithm 5 is used on the equalized signal with µθ = 0.01 and β = 0.001. To tune the pll properly these

parameters will have to be adjusted for each individual signal and doppler shift, but since it would result in too many different cases to test, these parameters were fixed and as a result there is probably room for improvement in the phase tracking. The phase tracking is run in reverse because the equalizers are more probable to have converged at the end of the transmission.

4.4

Test Results

The algorithms were simulated using the parameters in table 4.1 and different fil-ter lengths, step sizes and forgetting factors. For each test the mse and mseRwas

calculated online in blocks of the M = 50 latest symbols. The ser was calculated offline after the signal has been equalized and phase-corrected.

To limit the number of simulations to run, the number of filter coefficients, step sizes and forgetting factors were limited to 10, 8, and 4. The filter lengths

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4.4 Test Results 27

Algorithm 3:Decision feedback lms-type Blind Equalizer Result:Equalized output y

WF0: Initialize with lms-cma in reverse;

WB0 =h0 . . . 0 . . . 0i;

n = 0; m = 0 ; i = nr of samples per symbol;

whileUnprocessed samples left do ifmod(n, i) = 0 then U =hun un−1 . . . un−N +1 i ; ˆ X =hxˆn xˆn−1 . . . xˆn−M+1 i ; X =hUHXˆHiT; ym= XTWm; ˆ xm= dec(ym); em= ymxˆm; "WF m+1 WBm+1 # ="W F m WBm # −µem"U ˆ X #∗ ; m = m + 1 end n = n + 1; end [7 9 11 13 15 17 19 21 23 25], step sizes µ = [0.02 0.01 0.005 0.003 0.002 0.001 0.0005 0.0001], and forgetting factors λ = [0.9 0.98 0.99 0.999], were used in all simulations. The dfe also had a feedback filter with 4 coefficients used together with the forward filter of said lengths. The number of feedback coefficients were chosen after initial test that indicated it to be a suitable number.

4.4.1

Signals with correctly assumed constellations

In this section the equalizers are tested with the correct signal constellation as-sumed by the receiver.

Convergence time

The simplest method to compare the convergence time for the algorithms is to use a time invariant channel with zero Hz doppler shift, same filter length and same step size or forgetting factor.

Figure 4.1 shows a psk-8 modulated signal transmitted over a channel with dis-turbed conditions. All equalizers used filters with 9 coefficients and µ = 0.02 or λ = 0.98. All algorithms except cna-6 reach their steady state performance within 50 blocks, or 2 500 symbols. The two rls-algorithms are even faster, reach-ing their steady state after about five to ten blocks, or 250 to 500 symbols. The

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28 4 Implementation and Results

Algorithm 4: rlsDecision Feedback Blind Equalizer Result:Equalized output y

WF0: Initialize with rls-cma in reverse;

WB0 =h0 . . . 0 . . . 0i;

P = 100 · I;

n = 0; m = 0; i = nr of samples per symbol;

whileUnprocessed samples left do ifmod(n, m) = 0 then U =hun un−1 . . . un−N +1 i ˆ X =hxˆn xˆn−1 . . . xˆn−M+1 i X =hUHXˆHiT ym= XHWm Q = XXHWm K = λ+QP QHP Q P = 1λ(P − K QHP ) em="W F m WBm #H X(ymxˆm) "WF m+1 WBm+1 # ="W F m WBm # −K em m = m + 1 end n = n + 1 end

Algorithm 5:Second order pll Result:Phase corrected output ˜y θn = 0; n = nr. of received symbols; whilen , 0 do ˜ yn= ynexp{−jθn+1}; θ = θn+1+ µθ(n+ βPni=1i); n= I m[ ˜yn( ˆxny˜n)∗]; n = n − 1; end

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4.4 Test Results 29

steady state peformance are equal for all algorithms except rca which offers a somewhat higher MSER. Figure 4.1b) is a waterfall plot of the received signal,

which is an illustration on how the signal’s frequency content varies over time.

0 100 200 300 400 500 Blocks 10-4 10-3 10-2 10-1 100 MSE R LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE (a) mseR. -4000 0 4000 8000 Frequency [Hz] 2 4 6 8 10 Time [s]

(b)Waterfall plot of signal.

Figure 4.1: MSERfor disturbed channel with zero doppler shift and

psk-8 modulation. Equalizers with filter length 9 (+4 in the feedback filter for dfe-algortihms), µ = 0.02, and λ = 0.98.

0 100 200 300 400 500 Blocks 10-4 10-2 100 MSE R LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE (a) mseR. -4000 0 4000 8000 Frequency [Hz] 2 4 6 8 10 Time [s]

(b)Waterfall plot of signal.

Figure 4.2:MSERfor channel with zero doppler shift. Equalizers with filter

length 9 (+4 in the feedback filter for dfe-algortihms), µ = 0.01, and λ = 0.99. Signal modulated with qam-16.

With a qamc-16 signal transmitted over a quiet channel, a filter length of 9, step size µ = 0.01 or a forgetting factor of λ = 0.99, all equalizers except the rls-dfeexhibit fast convergence times, as seen in figure 4.2. The best steady state performance is those of the dfe-equalizers, while all linear equalizers except the

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30 4 Implementation and Results

rcaexhibit about the same performance. On the rca it is important to note that even though the mseRdoes not decrease after startup, the channel is equalized

because it is still low, as its ser of 3.6 % shows.

Table 4.2 and 4.3 summarizes the convergence time, ser, and mean mse for the test cases presented so far. It is important to note that the results of the two test signals are not directly comparable against each other, as the channel conditions differ greatly. A quiet channel was used with the qamc-16 signal, this because it was not possible to find a set of parameters for which all algorithms converged on the disturbed and moderate channels. The convergence time for the rls-dfe with the qamc-16 signal is notably longer than the other algorithm’s, possibly because a bad initiation of the feedback filter.

Algorithm Mean mseR ser Convergence time [samples]

lms-cma 0.0081 1.8% 2 500 rls-cma 0.0017 0.1% 500 cna-6 0.0249 10.3% 5 000 rca 0.0081 1.8% 2 500 mma 0.0110 2.4% 2 500 lms-dfe 0.0041 1.60% 2 500 rls-dfe 0.0010 0.1% 500

Table 4.2:Performance summary of the psk-8 case.

Algorithm Mean mseR ser Convergence time [symbols]

lms-cma 0.0031 0% 500 rls-cma 0.0074 1.5% 500 cna-6 0.0045 1.1% 500 rca 0.0189 3.6% 500 mma 0.0062 1.7% 500 lms-dfe 0.0015 0.6% 500 rls-dfe 0.0053 6.2% 2 500

Table 4.3:Performance summary of the qamc-16 case.

Step size and forgetting factor Another important aspect is how the filter length and step size or forgetting factor affect the convergence time. To test this the algo-rithms were simulated with the same data and only one of the parameters filter length or step size/ forgetting factor were allowed to change. In this chapter only a few plots will be presented to make the results more readable. See Appendix A for additional figures.

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4.4 Test Results 31 0 100 200 300 400 500 Blocks 10-4 10-3 10-2 10-1 100 MSE R MSE

R for different , LMS-CMA

0.02 0.01 0.005 0.003 (a) lms-cma. 0 20 40 60 80 100 Blocks 10-4 10-3 10-2 10-1 100 MSE R MSE

R for different , RLS-CMA

0.9 0.98 0.99 0.999

(b) rls-cma.

Figure 4.3:MSERfor channel with zero doppler shift. Equalizers with filter

length 9 (+4 in feedback the filter for dfe-algortihms). Signal modulated using psk-8.

Using the same signal as earlier for the psk-8 signal, it is clear that the step size affects the convergence time for all lms-type of algorithms. Figure 4.3 shows how the choice of step size and forgetting factor affect the convergence time for the two cma equalizers. A larger step size results in faster convergence time, for the rls algorithm the chosen forgetting factors does not greatly affect the conver-gence time. The cna-6 is more affected by the stepsize and a smaller stepsize does not increase the steady state performance of any equalizer. Table 4.4 and 4.5 summarize the convergence time for all equalizers. The convergence time for the rls-equalizers are clearly much lower than those of the lms-equalizers. The cna-6 with µ = 0.003 is to slow to converge within the span of the transmission, but as its mseRis steadily decreasing (figure A.1) it is assumed to converge

some-time after 24 000 symbols. As for the lms-dfe and µ = 0.003 it shows no signs of converging within the transmission.

µ lms-cma mma cna-6 rca lms-dfe

0.02 2 500 2 500 5 000 2 500 2 000

0.01 4 200 4 000 10 000 3 000 4 500

0.005 7 500 7 500 20 000 6 500 5 000

0.003 12 500 12 500 24 000 + 11 000

-Table 4.4:Convergence time for lms-equalizers with filter length 9 (+ 4 for dfe) in samples.

Filter length The filter length is also a design parameter that may affect the equalizers performance and convergence time. Using the same psk-8 signal as

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32 4 Implementation and Results λ rls-cma rls-dfe 0.9 150 250 0.98 250 250 0.99 500 300 0.999 1 000 1 000

Table 4.5: Convergence time for rls-equalizers with filter length 9 (+ 4 for dfe) in samples.

earlier the mseRfor 10 different filter lengths of the cma-equalizers is plotted in

figure 4.4. Table 4.6 and 4.7 summarize the convergence time for all algorithms and filter lengths. The steady state performance for each filter length is about the same, except for the rls-dfe with a filter with 15 + 4 (forward and feedback) which resulted in worse performance, and 7 + 4 as for which the equalizer di-verged. As for the convergence time all equalizers are affected, but a longer filter does not have to result in a longer convergence time, but the rls-equalizers again show faster convergence times and the cna-6 exhibit the slowest convergence time. With the lms-equalizers there is a distinct pattern in the convergence times, as every other filter length have roughly double the convergance time. This has been observed troughout the simulations, but no explanation as to why this hap-pens is given in this thesis, but it is an interesting observation that may reduce the scope of filter parameters.

Another important aspect is the rls-dfe equalizer that diverged, becuase it did not diverge as adaptive filters usually do with an exponentially increasing out-put. Instead it diverged by tuning all weights in the forward filter to zero and some in the feedback filter to one, effectively blocking the input and creating a feedback loop with the detected symbols. This means that if the rls-dfe diverge, its mse will drop to zero instead of increasing exponentially, which in turn may be harder to detect.

Tracking ability of time variant channels

As well as a fast convergence time is important, it is also just as, if not even more important for the equalizer to be able to retain the convergence. If an equalizer which has a very fast convergence time loses track of the channel all the time it will still result in poor performance. In order to test how well the algorithms can retain convergence and track the time varying channel, doppler shifts were added to the channel simulations and the algorithms were again tested.

Because the algorithms are affected by their chosen filter length and step size/ forgetting factor, these parameters were not the same for all algorithms, but in-stead the best combination for each algorithm was chosen. This means that the

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4.4 Test Results 33 0 50 100 150 200 Blocks 10-4 10-3 10-2 10-1 100 MSE R

MSER for different filterlengths, LMS-CMA

7 9 11 13 15 17 19 21 23 25 (a) lms-cma. 0 10 20 30 40 Blocks 10-4 10-3 10-2 10-1 100 MSE R

MSER for different filterlengths, RLS-CMA

7 9 11 13 15 17 19 21 23 25 (b) rls-cma.

Figure 4.4:MSERfor channel with zero doppler shift. cma-equalizers with

step size µ = 0.01 and forgetting factor λ = 0.99. Signal modulated using psk-8.

Filter length lms-cma mma cna-6 rca lms-dfe

7 7 500 6 500 10 000 2 000 7 500 9 3 000 3 500 10 000 3 000 5 000 11 1 500 1 500 5 000 2 000 2 500 13 3 000 3 500 10 000 3 000 5 000 15 1 500 1 500 5 000 2 000 2 500 17 3 000 3 500 10 000 3 000 5 000 19 1 500 1 500 5 000 2 000 2 500 21 3 000 3 500 10 000 3 000 5 000 23 1 500 1 500 5 000 2 000 2 500 25 3 000 3 500 10 000 3 000 4 000

Table 4.6: Convergence time for lms-equalizers with step size µ = 0.01 in samples.

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34 4 Implementation and Results

Filter length rls-cma rls-dfe

7 350 Diverged 9 350 300 11 550 500 13 950 2 000 15 750 650 17 750 1 500 19 550 1 100 21 950 850 23 1 200 1 650 25 950 1 450

Table 4.7: Convergence time for rls-equalizers with forgetting factor λ = 0.99 in samples.

best case for each algorithm were used in the comparison in order to minimize the

influence of bad or unlucky choices of parameters. The idea behind this choice is that it will result in a better comparison of the algorihms, and not the choice of filter length, step size or forgetting factor. The best case is defined as the param-eter combination that resulted in the lowest mean mseRfor the entire signal for

the linear equalizers and the combination that resulted in the lowest ser for the decision feedback equalizers.

The algorithms were tested on signals with three constellations (psk-4, psk-8 and qamc-16), different doppler shifts (0.2 Hz, 0.4 Hz and 0.8 Hz) and different envi-ronmental conditions (moderate and disturbed conditions). For each signal con-stellation, doppler shift, and channel condition five simulations were performed. Graphs of the resulting mseRare presented in this chapter togehter with selected

plots of the equalized signals and their corresponding mseR. Signal plots and bar

charts of the resulting ser are found in Appendix A. In Appendix B the resulting mseRfor all simulations are presented.

Moderate channel conditions Table 4.8 contains the resulting mean mseR of

five psk-4 signals transmitted over channels with moderate conditions. The mseR

is also plotted in figure 4.5 and in figure A.3 the ser is plotted. It is clear that the rls-dfe offer the best tracking performance as is has both the lowest ser and mseR. Figure 4.6 shows the resulting mseRthroughout the transmission for sim-ulation number four with doppler shift fd = 0.4 Hz. In figure A.8 parts of the

equalized signal is plotted and it is clear that the rls-algorithms have the best performance.

With the same channel type and psk-8 modulated signals the mean mseRof five

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4.4 Test Results 35 Alg. fd= 0.2 fd= 0.4 fd= 0.8 lms-cma 0.025 0.031 0.054 rls-cma 0.008 0.008 0.013 mma 0.115 0.114 0.120 cna-6 0.045 0.068 0.097 rca 0.122 0.126 0.129 lms-dfe 0.035 0.072 0.131 rls-dfe 0.004 0.005 0.008

Table 4.8:Mean mseRfor five simulations per algorithm and doppler shift.

psk-4 signal and channel with moderate conditions.

MSE

R, fd = 0.2

LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE

0 0.05 0.1

MSE

R, fd = 0.4

LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE

0 0.05 0.1

MSE

R, fd = 0.8

LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE

0 0.1 0.2

Figure 4.5: mseRfor psk-4 signals and moderate channel conditions.

Alg. fd= 0.2 fd= 0.4 fd= 0.8 lms-cma 0.026 0.031 0.051 rls-cma 0.010 0.009 0.014 mma 0.036 0.040 0.060 cna-6 0.048 0.064 0.099 rca 0.035 0.040 0.065 lms-dfe 0.038 0.054 0.099 rls-dfe 0.008 0.009 0.010

Table 4.9:Mean mseRfor five simulations per algorithm and doppler shift.

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36 4 Implementation and Results 0 100 200 300 400 500 Blocks 10-2 100 MSE R LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE (a) mseR. -4000 0 4000 8000 Frequency [Hz] 2 4 6 8 10 Time [s]

(b)Waterfall plot of signal.

Figure 4.6: mseR. psk-4 signal over channel with moderate conditions, sim-ulation 4.

the rls-equalizers again perform better than the lms-equalizers, but with more doppler shift their ser is increasing. With doppler shift fd = 0.8 Hz all

algo-rithms have trouble with the tracking but the lower mseRfor the rls-equalizers

result in better constellation recovery and piecewise equalization of the channels. In figure 4.7 the mseRfor simulation number five with doppler shift fd = 0.8

Hz is plotted and in figure 4.8 parts of the equalized signals are plotted. For the lms-dfe, mma and cna-6 the channels are not equalized.

0 100 200 300 400 500 Blocks 10-2 100 MSE R LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE (a) mseR. -4000 0 4000 8000 Frequency [Hz] 2 4 6 8 10 Time [s]

(b)Waterfall plot of signal.

Figure 4.7: mseR. psk-8 signal over channel with moderate conditions, sim-ulation 5 and doppler shift fd= 0.8 Hz.

As for the qamc-16 modulation and moderate channel conditions all algorithms succeed to converge and recover the constellation when the doppler shift is 0.2

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4.4 Test Results 37 -3 -2 -1 0 1 2 3 In Phase -3 -2 -1 0 1 2 3 Quadrature Received Signal

(a)Received signal.

-2 -1 0 1 2 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature LMS-CMA (b) lms-cma. -1.5 -1 -0.5 0 0.5 1 1.5 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature RLS-CMA (c) rls-cma. -2 -1 0 1 2 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature CNA6 (d) cna-6. -1.5 -1 -0.5 0 0.5 1 1.5 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature MMA (e) mma. -1.5 -1 -0.5 0 0.5 1 1.5 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature RCA (f) rca. -2 -1 0 1 2 In Phase -2 -1 0 1 2 3 Quadrature LMS-DFE (g) lms-dfe. -1.5 -1 -0.5 0 0.5 1 1.5 In Phase -1.5 -1 -0.5 0 0.5 1 1.5 Quadrature RLS-DFE (h) rls-dfe.

Figure 4.8: psk-8 signal with 0.8 Hz doppler shift, channel with moderate conditions, simulation number five. lms-dfe, cna and mma did not equal-ize the channel.

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38 4 Implementation and Results

Hz, but the over all performance is best for the rls-algorithms and worst for cna-6 and rca, as seen in figure A.9. When the doppler shift is increased to 0.4 Hz all equalizers except the rls-dfe are unable to keep the mse low enough to equalize the channel, although it is slow to converge at start up which is seen in figure 4.9. With the doppler shift increased even more to 0.8 Hz the results are similar, only rls-dfe is able to piecewise track the channel but it has slow convergence time as the first 10 000 symbols are unequalized (nearly half of the total) in simulation number one. The rls-cma is better at tracking than its lms counterpart, but as the rls-dfe it loses the convergance after about 20 000 samples, as seen in figure A.10. By inspecting the equalizers output of a part of the signal it is clear that the rls-algorithms perform better than the lms-algorithms (figure A.11), and that the rls-dfe is able too keep the output from rotating, whilst the rls-cma is not. Figures A.6 and A.7 show the resulting mseRand ser. Table 4.10 summarize the

mean mseRof five simulations per algorithm and doppler shift.

0 100 200 300 400 500 Blocks 10-2 100 MSE R LMS-CMA RLS-CMA MMA CNA-6 RCA LMS-DFE RLS-DFE (a) mseR. -4000 0 4000 8000 Frequency [Hz] 2 4 6 8 10 Time [s]

(b)Waterfall plot of signal.

Figure 4.9: Only rls-dfe is able to keep the mse low and track the chan-nel after initial convergence. Signal with qamc-16 modulation and chanchan-nel with moderate conditions and 0.4 Hz doppler shift.

Alg. fd= 0.2 fd= 0.4 fd= 0.8 lms-cma 0.013 0.024 0.029 rls-cma 0.014 0.022 0.021 mma 0.018 0.028 0.035 cna-6 0.021 0.034 0.040 rca 0.026 0.034 0.038 lms-dfe 0.026 0.035 0.038 rls-dfe 0.007 0.010 0.019

Table 4.10:Mean mseRfor five simulations per algorithm and doppler shift.

References

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