ADAPTIVE FILTER ALGORITHMS FOR CHANNEL EQUALIZATION

Transcription

1 ADAPTIVE FILTER ALGORITHMS FOR CHANNEL EQUALIZATION Omprakash Gurrapu Supervisors: Adepu Nagedra Sr. desig Egieer AN SoftwareTechologies,Idia Examier : Jim Arlebrik Uiversity College Of Borås,Swede The thesis work comprises 30 credits ad is a compulsory part i the Master of Sciece with a major i Masters s Programme i Electrical Egieerig-Commuicatio ad Sigal Processig, 1/009

2 Abstract Equalizatio techiques compesate for the time dispersio itroduced by commuicatio chaels ad combat the resultig iter-symbol iterferece ISI effect. Give a chael of ukow impulse respose, the purpose of a adaptive equalizer is to operate o the chael output such that the cascade coectio of the chael ad the equalizer provides a approximatio to a ideal trasmissio medium. Typically, adaptive equalizers used i digital commuicatios require a iitial traiig period, durig which a kow data sequece is trasmitted. A replica of this sequece is made available at the receiver i proper sychroism with the trasmitter, thereby makig it possible for adjustmets to be made to the equalizer coefficiets i accordace with the adaptive filterig algorithm employed i the equalizer desig. This type of equalizatio is kow as No-Blid equalizatio. However, i practical situatios, it would be highly desirable to achieve complete adaptatio without access to a desired respose. Clearly, some form of Blid equalizatio has to be built ito the receiver desig. Blid equalizers simultaeously estimate the trasmitted sigal ad the chael parameters, which may eve be time-varyig. The aim of the project is to study the performace of various adaptive filter algorithms for blid chael equalizatio through computer simulatios. i

3 Ackowledgemet I would like to thak my guide, Mr.Adepu Nagedra, Sr.Desig Egieer for providig me with the required guidace i the project work. It was so grateful of him to share his kowledge with me. It is hard to imagie the completio of the project without his guidace ad support. I also wish to thak my guide i college, Jim Arlebrik, Lecturer, for extedig his support i my project work as well as course work also. I would especially thak Mr.Sriivas aluguvelli, Sr.Desig Egieer for providig me with the opportuity to be associated with Sr. Software Professioals. I also wish to express my gratitude to the staff of AN SoftwareTechologies for their kid support ad co-operatio durig the teure of the project work. I the ed, I would surely like to thak God ad my parets for teachig me the values of life, which are precious. ii

4 iii Cotets Ackowledgemet ii 1 Itroductio Need for Chael Equalizatio Itersymbol Iterferece i Digital Commuicatio Filters for Chael Equalizatio Adaptive Filter Project Outlie Purpose of the Project Thesis Outlie Chapter Summary Adaptive Filters 8.1 Itroductio Types of Filters Liear Optimum Filters Adaptive Filters Types of Adaptive Filters Factors determiig the Choice of Algorithm How to Choose a Adaptive Filter Applicatios of Adaptive Filters

5 iv.7 Chapter Summary Chael Equalizatio Itroductio Adaptive Chael Equalizatio Types of Equalizatio Techiques Liear Equalizatio Decisio Feedback Equalizatio No-Blid Equalizatio Blid Equalizatio Chapter Summary Simulatio Models Itroductio Geeral Mathematical Model Chael Modelig Geeratig Data Geeratig AWGN Blid Algorithms Loss Fuctio Model Godard Algorithm Sato Algorithm Chapter Summary

6 v 5 Simulatio Results Itroductio Effect of ISI o Eye Patter Learig Curves Steepest Descet LMS Algorithm Trasfer Fuctio of Combiatio of Chael ad Equalizer Other Modulatio Schemes Variatio of BER, or SER with SNR Chapter Summary Coclusios 71 Appedix 7 Bibliography 79

7 Chapter 1 Itroductio By Adaptive sigal processig, we mea i geeral, adaptive filterig. I usual eviromets where we eed to model, idetify, or track time-varyig chaels, adaptive filterig has bee prove to be a effective ad powerful tool. As a result, this tool is ow i use i may differet fields. Sice, the ivetio of oe of the first adaptive filters, the so called least-mea square, by Widrow ad Hoff i 1959, may applicatios appeared to have the potetial to use this fudametal cocept. While the umber of applicatios usig the adaptive algorithms has bee flourishig with time, the eed for more sophisticated adaptive algorithms became obvious as real-world problems are more demadig. Oe such applicatio, Adaptive chael equalizatio, has bee discussed i this thesis. I this chapter, we will discuss the eed for chael equalizatio by cosiderig the problem of Itersymbol-iterferece ISI i digital commuicatio systems. We will also discuss the adaptive filters used for performig chael equalizatio. We fiish this chapter with a outlie of the project ad the thesis.

8 1.1 Need for Chael Equalizatio Itersymbol Iterferece i Digital Commuicatio 1 May commuicatio chaels, icludig telephoe chaels, ad some radio chaels, may be geerally characterized as bad-limited liear filters. Cosequetly, such chaels are described by their frequecy respose C f, which may be expressed as C f jϕ f = A f e 1.1 where, A f is called the amplitude respose ad ϕ f is called the phase respose. Aother characteristic that is sometimes used i place of the phase respose is the evelope delay or group delay, which is defied as 1 dϕ f τ f = 1. π df A chael is said to be o distortig or ideal if, withi the badwidth W occupied by the trasmitted sigal, A f = costat ad ϕ f is a liear fuctio of frequecy [or the evelope delay τ f = costat]. O the other had, if A f ad τ f are ot costat withi the badwidth occupied by the trasmitted sigal, the chael distorts the sigal. If A f is ot costat, the distortio is kow as amplitude distortio ad if τ f is ot costat, the distortio o the trasmitted sigal is kow as delay distortio. As a result of the amplitude ad delay distortio caused by the o ideal chael frequecy respose characteristic C f, a successio of pulses trasmitted through the chael at rates comparable to the badwidth W are smeared to the poit that they are o loger distiguishable as well-defied pulses at the receivig termial. Istead, they overlap ad, thus, we have Itersymbol iterferece ISI. As a example of the effect of

9 delay distortio o a trasmitted pulse, p t fig.1.1 a illustrates a bad limited pulse havig zeros periodically spaced i time at poits labeled ± T, ± T, ± 3T, etc. pt t 5T 4T 3T T T 0 T T 3T 4T 5T a pt 5T 4T T T T T 3T 4T 5T t b

10 3 pt t 5T 4T 3T T T 0 T T 3T 4T 5T c Figure 1.1: Effect of ISI o the Chael: a Chael Iput, b Chael Output, ad c Equalizer Output. If iformatio is coveyed by the pulse amplitude, as i pulse amplitude modulatio PAM, for example, the oe ca trasmit a sequece of pulses, each of which has a peak at the periodic zeros of the other pulses. Trasmissio of the pulse through a chael modeled as havig a liear evelope delay characteristic τ f, however, results i the received pulse show i fig.1.1 b havig zero crossigs that are o loger periodically spaced. Cosequetly a sequece of successive pulses would be smeared ito oe aother, ad the peaks of the pulses would o loger be distiguishable. Thus, the chael delay distortio results i itersymbol iterferece. However, it is possible to compesate for the o ideal frequecy respose characteristic of the chael by the use of a filter or equalizer at the receiver. Fig 1.1c illustrates the output of a liear equalizer that compesates for the liear distortio i the chael.

11 4 Itersymbol iterferece is a major source of bit errors i the recostructed data stream at the receiver output. Thus to correct it ad, allow the receiver to operate o the received sigal ad deliver a reliable estimate of the origial message sigal, give at the iput, to a user at the output of the system, chael equalizatio is performed. Besides telephoe chaels, there are other physical chaels that exhibit some form of time dispersio ad, thus, itroduce itersymbol iterferece. Radio chaels, such as short-wave ioospheric propagatio HF, tropospheric scatter, ad mobile cellular radio are three examples of time dispersive wireless chaels. I these chaels, time dispersio ad hece, itersymbol iterferece is the result of multiple propagatio paths with differet path delays. 1. Filters for Chael Equalizatio I order to couter itersymbol iterferece effect, the observed sigal may first be passed through a filter called the equalizer whose characteristics are the iverse of the chael characteristics. If the equalizer is exactly matched to the chael, the combiatio of the chael ad equalizer is just a gai so that there is o itersymbol iterferece preset at the output of the equalizer. As metioed, the equalizer is a filter which is kow as Adaptive filter Adaptive Filter I cotrast to filter desig techiques based o kowledge of the secod-order statistics of the sigals, there are may digital sigal processig applicatios i which these statistics caot be specified a priori. The filter coefficiets deped o the characteristic

12 5 of the medium ad caot be specified a priori. Istead, they are determied by the method of Least squares, from measuremets obtaied by trasmittig sigals through the physical media. Such filters, with adjustable parameters, are usually called adaptive filters, especially whe they icorporate algorithms that allow the filter coefficiets to adapt to the chages i the sigal statistics. The equalizers, thereby usig adaptive filters are called adaptive equalizers. O chaels whose frequecy respose characteristics are ukow, but time ivariat, we may measure the chael characteristics ad adjust the parameters of the equalizer; oce adjusted, the parameters remai fixed durig the trasmissio of data. Such equalizers are called preset equalizers. O the other had, adaptive equalizers update their parameters o a periodic basis durig the trasmissio of the data ad, thus, they are capable of trackig time-varyig chael respose. The adaptive filters will be discussed, i detail, i the ext chapter. However, at this poit of time, oe eeds to uderstad that the equalizer used to couter itersymbol iterferece effect of the chael is to be adaptive i ature. This is because of the reaso that, there is o priori iformatio available to the filter but oly the icomig data, depedig o which the filter parameters have to adapt. 1.3 Project Outlie Purpose of the Project The mai purpose of this project is to examie the performace of various adaptive sigal processig algorithms for chael equalizatio through computer simulatios.

13 1.3. Thesis Outlie 6 The first chapter gives a geeral backgroud ad itroductio to the problem of chael equalizatio ad filters used i solvig the problem. Chapter covers various types of filters used for chael equalizatio, icludig the adaptive filters. Chapter 3 provides the detailed versio of the problem of chael equalizatio ad various equalizatio techiques used. Chapter 4 discusses the simulatio model ad about the various algorithms used i the project. Chapter 5 presets the results of the simulatio models. Also cosidered are the various factors ivolved i the equalizatio process, alog with the study of variatio of BER, or SER with that of SNR. Fially, the thesis is cocluded by givig the coclusios of the project. 1.4 Chapter Summary I this chapter, we have extesively covered the problem of itersymbol iterferece ad its effects o commuicatio chaels. Later, a itroductio ad the rudimets of adaptive filter have bee preseted. Fially, the purpose of the project alog with the outlie of the thesis is preseted.

14 7 Chapter Adaptive Filters.1 Itroductio I this chapter, we make a compariso of the adaptive filters with other filters ad discuss the comparative advatages. We also study the adaptive filter theory i detail, their types ad applicatios. The chapter also icludes the factors that determie the choice of a algorithm.

15 . Types of Filters 8..1 Liear Optimum Filter We may classify filters as liear or o-liear. A filter is said to be liear if the filtered, smoothed or, predicted quatity at the output of the filter is a liear fuctio of the observatios applied to the filter iput. Otherwise, the filter is o-liear. I the statistical approach to the solutio of the liear filterig problem, we assume the availability of certai statistical parameters i.e., mea ad correlatio fuctios of the useful sigal ad uwated additive oise, ad the requiremet is to desig a liear filter with the oisy data as iput so as to miimize the effects of oise at the filter output accordig to some statistical criterio. A useful approach to this filter-optimizatio problem is to miimize the mea-square value of the error sigal, defied as the differece betwee some desired respose ad the actual filter output. For statioary iputs, the resultig solutio is commoly kow as the Wieer filter, which is said to be optimum i the mea-square error sese. The Wieer filter is iadequate for dealig with situatios i which o statioery of the sigal ad /or oise is itrisic to the problem. I such situatios, the optimum filter has to assume a time varyig form. A highly successful solutio to this more difficult problem is foud i the Kalma filter, which is a powerful system with a wide variety of egieerig applicatios. Liear filter theory, ecompassig both Wieer ad Kalma filters, is well developed i the literature for cotiuous time as well as discrete-time sigals... Adaptive Filters As see i last sectio, Wieer ad Kalma filters are the mostly used filters. But, both

16 9 of them have costraits, i.e., they require some priori iformatio. Wieer filter requires kowledge of sigal covariace, ad Kalma filter requires kowledge of state-space model goverig sigal behavior. I practice, such a priori iformatio is rarely available; what is available is the data sequece of umbers. Moreover, all the data is ot available at a time; the data is comig i sequetially. This is where adaptive processig comes ito play. The basic idea is to process the data as it comes i i.e., recursively, ad by a filter which is oly data depedet, i.e., the filter parameters adapt to the comig data. Such filters are referred to as adaptive filters. By such a system we mea oe that is self-desigig i that the adaptive algorithm, which makes it possible for the filter to perform satisfactorily i a eviromet where complete kowledge of the relevat sigal characteristics is ot available. The algorithm starts from some predetermied set of iitial coditios, represetig whatever we kow about the eviromet. Yet, i a statioary eviromet, we fid that after successive iteratios of the algorithm it coverges to the optimum Wieer solutio i some statistical sese. I a o statioary eviromet, the algorithm offers a trackig capability, i that it ca track time variatios i the statistics of the iput data, provided that the variatios are sufficietly slow. As a direct cosequece of the applicatio of a recursive algorithm whereby the parameters of a adaptive filer are updated from oe iteratio to the ext, the parameters become data depedet. This, therefore, meas that a adaptive filter is i reality a o liear system, i the sese that it does ot obey the priciple of superpositio. Notwithstadig this properly, adaptive filters are commoly classified as liear or o liear. A adaptive filter is said to be liear if its iput-output map obeys the priciple of superpositio wheever its parameters are held fixed. Otherwise, the adaptive filter is said to be o liear.

17 10.3 Types of Adaptive Filters The operatio of a liear adaptive filterig algorithm ivolves two basic processes; 1 a filterig process desiged to produce a output i respose to a sequece of iput data ad a adaptive process, the purpose of which is to provide a mechaism for the adaptive cotrol of a adjustable set of parameters used i the filterig process. These two processes work iteractively with each other. Naturally, the choice of a structure for the filterig process has a profoud effect o the operatio of the algorithm as a whole. The impulse respose of a liear filter determies the filter s memory. O this basis, we may classify filters ito fiite-duratio impulse respose FIR, ad ifiite-duratio impulse respose IIR filters, which are respectively characterized by fiite memory ad ifiitely log, but fadig, memory. Although both IIR ad FIR filters have bee cosidered for adaptive filterig, the FIR filter is by far most practical ad widely used. The reaso for this preferece is quite simple; the FIR filter has oly adjustable zeros; hece, it is free of stability problems associated with adaptive IIR filter, which have adjustable poles as well as zeros. However, the stability of FIR filter depeds critically o the algorithm for adjustig its coefficiets..4 Factors determiig the choice of Algorithm A importat cosideratio i the use of a adaptive filter is the criterio for optimizig the adjustable filter parameters. The criterio must ot oly provide a meaigful measure of filter performace, but it must also result i a practically realizable algorithm. A wide variety of recursive algorithms have bee developed i the literature for the operatio of liear adaptive filters. I the fial aalysis, the choice of oe algorithm over

18 11 aother is determied by oe or more of the followig factors: Rate of covergece. This is defied as the umber of iteratios required for the algorithm, i respose to statioary iputs, to coverge close eough to the optimum Wieer solutio i the mea-square error sese. A fast rate of covergece allows the algorithm to adapt rapidly to a statioary eviromet of ukow statistics. Misadjustmet. For a algorithm of iterest, this parameter provides a quatitative measure of the amout by which the fial value of the mea-square error, averaged over a esemble of adaptive filters, deviates from the miimum mea-square error produced by the Wieer filter. Trackig. Whe a adaptive filterig algorithm operates i a o statioary eviromet, the algorithm is required to track statistical variatios i the eviromet. The trackig performace of the algorithm, however, is iflueced by two cotradictory features: a rate of covergece, ad b steady-state fluctuatio due to algorithm oise. Robustess. For a adaptive filter to be robust, small disturbaces ca oly result i small estimatio errors. The disturbaces may arise from a variety of factors, iteral or exteral to the filter. Computatioal requiremets. Here the issues of cocer iclude a the umber of operatios required to make oe complete iteratio of the algorithm b the size of memory locatios required to store the data ad program, ad c the ivestmet required to program the algorithm o a computer. Structure. This refers to the structure of iformatio flow i the algorithm, determiig the maer i which it is implemeted i hardware form. Numerical Properties. Numerical stability is a iheret characteristic of a adaptive filterig algorithm. Numerical accuracy, o the other had, is determied by the umber of bits used i the umerical represetatio of data

19 1 samples ad filter coefficiets. A adaptive filterig algorithm is said to be umerically robust whe it is isesitive to variatios i the word legth used i its digital implemetatio..5 How to choose a Adaptive Filter Give the wide variety of adaptive filters available to a system desiger, the questio arises how a choice ca be made for a applicatio of iterest. Clearly, whatever the choice, it has to be cost effective. With this goal i mid, we may idetify three importat issued that require attetio: computatioal cost, performace, ad robustess. Practical applicatios of adaptive filterig are highly diverse, with each applicatio havig peculiarities of its ow. Thus, the solutio for oe applicatio may ot be suitable for aother. Nevertheless, be successful, we have to develop a physical uderstadig of the eviromet i which the filter has to operate ad thereby relate to the realities of the applicatio of iterest..6 Applicatios of Adaptive Filter The ability of a adaptive filter to operate satisfactorily i a ukow eviromet ad track time variatios of iput statistics makes the adaptive filter a powerful device for sigal processig ad cotrol applicatios. Ideed, adaptive filters have bee successfully applied i such diverse fields as commuicatios, radar, soar, seismology, ad biomedical egieerig. Although these applicatios are quite differet i ature, evertheless, they have oe basic feature i commo: A iput vector ad a desired respose are used to compute a estimatio error, which is i tur used to cotrol the

20 13 values of a set of adjustable filter coefficiets. The adjustable coefficiets may take the form of tap weights, reflectio coefficiets, or rotatio parameters, depedig o the filter structure employed. However, the essetial differeces betwee the various applicatios of adaptive filterig arise i the maer i which the desired respose is extracted. I this cotext, we may distiguish four basic classes of adaptive filterig applicatios, as follows: I. Idetificatio: I. a. System Idetificatio. Give a ukow dyamical system, the purpose of system idetificatio is to desig a adaptive filter that provides a approximatio to the system. I. b. Layered Earth Modelig. I exploratio seismology, a layered model of the earth is developed to uravel the complexities of the earth s surface. II. Iverse Modelig: II. a. Equalizatio. Give a chael of ukow impulse respose, the purpose of a adaptive equalizer is to operate o the chael output such that the cascade coectio of the chael ad the equalizer provides a approximatio to a ideal trasmissio medium. III. Predictio: III. a. Predictive codig. The adaptive predictio is used to develop a model of a sigal of iterest; rather tha ecode the sigal directly, i predictive codig the predictio error is ecoded for trasmissio or storage. Typically, the predictio error has a smaller variace tha the origial sigal, hece the basis for improved ecodig. III. b. Spectrum aalysis. I this applicatio, predictive modelig is used to estimate the power spectrum of a sigal of iterest. IV. Iterferece cacellatio: IV. a. Noise cacellatio. The purpose of a adaptive oise caceller is to subtract oise from a received sigal i a adaptively cotrolled maer so as to improve

21 14 the sigal-to-oise ratio. Echo cacellatio, experieced o telephoe circuits, is a special form of oise cacellatio. Noise cacellatio is also used i electrocardiography. IV.b.Beamformig. A beamformer is a spatial filter that cosists of a array of atea elemets with adjustable weights coefficiets. The twi purposes of a adaptive beamformer are to adaptively cotrol the weights so as to cacel iterferig sigals impigig o the array from ukow directios ad, at the same time, provide protectio to a target sigal of iterest. The applicatio of adaptive filter cosidered i this project is Equalizatio, belogig to the Iverse modelig class of adaptive filterig applicatio. Cosider fig..1, which illustrates the iverse modelig class of adaptive filterig applicatio. The followig otatio is used i the figure: u = iput applied to adaptive filter; y = output of the adaptive filter; d = desired respose; ad e = d y = estimatio error. Figure.1: Iverse Modelig Class of Adaptive Filterig Applicatios. System u System Iput 15 Adaptive Output Plat Filter I iverse modelig, the fuctio of the adaptive filter is to provide a iverse model that represets the best fit to a ukow oisy plat. Ideally - y i the case of a liear system, the iverse model has a trasfer fuctio equal to the reciprocal iverse of the plat s trasfer fuctio, such that the combiatio of the two costitutes a ideal e + d trasmissio medium. A delayed versio of the plat system iput costitutes the Delay desired respose for the adaptive filter. I some applicatios, the plat iput is used without delay as the desired respose. The chael equalizatio applicatio will be dealt i detail i the ext chapter.

22 .7. Chapter Summary I this chapter, we have covered various types of filters as well as types of adaptive filters. We first made a compariso of adaptive filters with that of Wieer ad Kalma filters ad cocluded that it is the most suitable filter for chael equalizatio. Later, the various factors that affect the choice of a algorithm ad the choice of adaptive filter have bee discussed. Fially, the various applicatios of the adaptive filters have bee preseted, ad the applicatio cosidered i this project itroduced. 16 Chapter 3 Chael Equalizatio

23 3.1 Itroductio I this chapter, we will discuss the problem of chael equalizatio i detail. We will also study the various types of equalizatio techiques used for this purpose, alog with the blid equalizatio techique used i this project. 3. Adaptive Chael Equalizatio 17 Fig.3.1 shows a block diagram of a digital commuicatio system i which a adaptive equalizer is used to compesate for the distortio caused by the trasmissio medium chael.

24 a Data Sequece Trasmitter Filter Chael Time Variat Filter Noise Receiver Filter Sampler Decisio Device aˆ Adaptive Equalizer Referece Error Sigal d Adaptive Sigal Algorithm Figure 3.1: Applicatio of Adaptive Filter to Adaptive Chael Equalizatio. The digital sequece of iformatio symbols a is fed to the trasmittig filter whose output is s t = k= 0 a k p t kt s where, pt is the impulse respose of the filter at the trasmitter ad T s is the time 18 iterval betwee iformatio symbols; that is, 1 Ts is the symbol rate. For the purpose of this discussio, we may assume that a is a multilevel sequece that takes o values from the set ± 1, ± 3, ± 5,..., ± k 1, where k is the umber of possible values. Typically, the pulse p t is desiged to have the characteristics illustrated i fig.3..

25 pt 1 t 5T s 4Ts 3T s Ts Ts 0 Ts Ts 3T s 4Ts 5Ts Figure 3.: Pulse Shape for Digital Trasmissio of Symbols at a Rate of per Secod. 1 T s Symbols Note that p 0 = 1 at t = 0 ad p T s = 0 at t = Ts, =± 1, ±,... As a cosequece, successive pulses trasmitted sequetially every T s secod do ot iterfere with oe aother whe sampled at the time istats t = Ts. Thus, a = s Ts. The chael, which is usually modeled as a liear filter, distorts the pulse ad, thus, 19 causes itersymbol iterferece. For example, i telephoe chaels, filters are used throughout the system to separate sigals i differet frequecy rages. These filters causes frequecy ad phase distortio. Fig.3.3 illustrates the effect of chael distortio of pulse p t as it might appear at the output of the telephoe chael.

26 s = + = k = 0 s s s + + k = 0 k s s s qt 5T s 4Ts 3T s Ts Ts Ts T s 3T s 4Ts 5Ts t Figure.3.3: Effect of Chael Distortio o the Sigal Pulse i fig.3.. Now, we observe that the samples take every T s secod are corrupted by iterferece from several adjacet symbols. The distorted sigal is also corrupted by additive oise, which is usually wide bad. At the receivig ed of commuicatio system, the sigal is first passed through a filter that is primarily desiged to elimiate the oise outside of the frequecy bad occupied by the sigal. We may assume that this filter is a liear phase FIR filter that limits the badwidth of the oise but causes egligible additioal distortio o the chael-corrupted sigal. Samples of the received sigal at the output of this filter reflect the presece of itersymbol iterferece ad additive oise. If we igore, for the momet, the possible time variatios i the chael, we may express the sampled output at the receiver as 0 x T a k q T kt w T a q0 a k q T kt w T 3.1

27 where, w t represets the additive oise ad q t represets the distorted pulse at the output of the receiver filter. To simplify the discussio, we assume that the sample q 0 is ormalized to uity by meas of a automatic gai cotrol AGC cotaied i the receiver. The, the sampled sigal give i the equatio 3.1 may be expressed as x = a + a k q k + w 3. k = 0 k where, x x Ts, q q Ts, ad w w Ts. The term a i equatio 3. is the desired symbol at the th samplig istat. The secod term, k = 0 k a k q k costitutes the itersymbol iterferece due to the chael distortio, ad w represets the additive oise i the system. I geeral, the chael distortio effect embodied through the sampled values q is ukow at the receiver. Further more, the chael may vary slowly with time such that 1 the itersymbol iterferece effects are time-variat. The purpose of adaptive equalizer is to compesate the sigal for the chael distortio, so that the resultig sigal ca be detected reliably. Let us assume that the equalizer is a FIR filter with M adjustable coefficiets, h. Its output may be expressed as

28 3.3 a ˆ = M 1 k= 0 h k x + D k where D is some omial delay i processig the sigal through the filter ad aˆ represets a estimate of the traied by trasmittig a kow data sequece d ˆ th iformatio symbol. Iitially, the equalizer is. The, the equalizer output, say d, is compared with d ad a error e is geerated that is used to optimize the filter coefficiets. This is illustrated i fig.3.4. Traiig Phase d Trasmitter Chael x Equalizer v e Figure 3.4: Chael Equalizatio usig Traiig Sequece. If we agai adopt the least squares error criterio, we select the coefficiets h k to miimize the quatity. ξ M = N = 0 [ d aˆ ] = N = 0 [ d M 1 k= 0 h k x + D k] 3.4

29 The result of the optimizatio is a set of liear equatios of the form 3.5 M 1 k= 0 h k r xx l k = r dx l d, l = 0,1,,..., M 1 where, r xxl is the autocorrelatio of the sequece x ad r dx l is the cross correlatio betwee the desired sequece d ad the received sequece x. Although the solutio of the equatio 3.5 is obtaied recursively i practice, i priciple, we observe that these equatios result i values of the coefficiet for the iitial adjustmets of the equalizer. After the short traiig period, which usually last less tha oe secod for most of the chaels, the trasmitter begis to trasmit the iformatio sequece a. I order to track the possible time variatios i the chael, the equalizer coefficiets must cotiue to be adjusted i a adaptive maer while receivig data. As illustrated i fig.3.1, this is usually accomplished by treatig the decisios at the output of the decisio device as correct, ad usig the decisios i place of the referece d to geerate the error sigal. This approach works quite well whe decisio errors occur frequetly for example, less tha oe decisio error per hudred symbols. The occasioal decisio error cause oly small misadjustmets i the equalizer coefficiets. 3.3 Types of Equalizatio Techiques Liear Equalizatio

30 Cosider the liear equalizer structure i fig 3.5. The liear filter tries to ivert the chael dyamics ad the decisio device is a static mappig, workig accordig to the earest eighbor priciple. ut Chael B q yt Equalizer C q zt Decisio ˆ ut D Figure 3.5: Structure of Liear Equalizer. A liear equalizer cosists of a liear filter C q followed by a decisio device. The equalizer is computed from kowledge of a traiig sequece ofu t. uderlyig assumptio is that the trasmissio protocol is such that a traiig sequece, kow to the receiver is trasmitted regularly. This sequece is used to estimate the iverse chael dyamics C q accordig to the least squares priciple. The domiatig model structure for both chael ad liear equalizer is FIR filters. The FIR model for chael is motivated by physical reasos; the sigal is subject to multi-path fadig or echoes, which implies delayed ad scaled versios of the sigal at the receiver. The FIR model for equalizer structures, where the equalizer cosists of a liear filter i series with the chael, is motivated by practical reasos. A equalizer of order, C, is to be estimated from L traiig symbols a total time delay of D. Itroduce the loss fuctio The u t aimig at 4 V L L 1 C, D = u L t= 1 t D C q y t

31 The least squares estimate of the equalizer is ow Cˆ D = arg mi V C L C, D The desiger of the commuicatio system has three degrees of freedom. The first, ad most importat, choice for performace ad spectrum efficiecy is the legth of the traiig sequece, L. This has to be fixed at a early desig phase whe the protocol, ad for commercial systems the stadard, is determied. The the order ad delay D have to be chose. This ca be doe by comparig the loss i the three dimesioal discrete space L,, D Decisio Feedback Equalizatio Fig. 3.5 shows the structure of a decisio feed back equalizer. The upper part is idetical to a liear equalizer with a liear feed-forward filter, followed by a decisio device. The differece lies i the feed back path from the o-liear decisios. Oe fudametal problem with a liear equalizer of FIR type is the may taps that are eeded to approximate a zero close to the uit circle i the chael. With the extra degree of freedom we ow have, these zeros ca be put i the feedback path, where o iversio is eeded. I theory, D q = B q ad C q = 1 would be a perfect equalizer. However, if the oise iduces a decisio error, the there might be a recovery problem for the DFE equalizer. There is the fudametal trade-off: split the dyamics of the chael betwee C q ad D q so few taps ad robustess to decisio errors are achieved. 5

32 H q et u t B q v t y t C q z t ˆ ut D D q Figure 3.6: Structure of Decisio Feedback Equalizer. I the desig, we assume that the chael is kow. I practice, it is estimated from a traiig sequece. To aalyze ad desig a o-liear system is geerally very difficult,. A simplifyig assumptio, that domiates the desig described i theliterature, is oe of so called Correct Past Decisios CPD. The assumptio implies that we ca take the iput to the feedback filter from the true iput ad we get the block diagram i

33 fig.3.7.the assumptio is soud whe the Sigal-to-Noise Ratio SNR is high, so a DFE ca be oly assumed to work properly i such systems. H q et 6 u t B q v t y t C q z t ˆ ut D D q Decisio errors Figure 3.7: Modified Structure of DFE uder CPD. We ca see from fig.3.7 that, if there are o decisio errors, the z t = C q B q + D q ut + C q H q et. For the CPD assumptio to hold, the estimatio errors must be small. That is, choose C q ad Dq to miimize D u ~ t D ut D zt q C q B q D q u = = t C q H q e t.

34 There are two priciples described i the literature: The zero forcig equalizer. Neglect the oise i the desig ad choose C q ad D q so that q D C q B q D q = 0. The miimum variace equalizer. 7 Cˆ, Dˆ = arg mi C q, D q E u~ t D = arg mi C q, D q e iwd C e iw B e iw D e iw Φ e u iw + C e iw H e iw Φ e e iw ad e. Here, we have used Parseval s formula ad a idepedece assumptio betwee u I both cases, a costrait of the type c = 0 1 is eeded to avoid the trivial miimum for C q = 0, i case the block diagram i fig.3.6 does ot hold. The advatage of DFE is a possible cosiderable performace gai at the cost of a oly slightly more complex algorithm, compared to a liear equalizer. Its applicability is limited to cases with high SNR. As a fial remark, the itroductio of equalizatio i very fast modems, itroduces a ew kid of implemetatio problem. The basic reaso is that the data rate comes close to the clock frequecy i the computer, ad the feedback path computatios itroduce a sigificat time delay i the feedback loop. This meas that the DFE approach collapses, sice o feedback delay ca be accepted No-Blid Equalizatio

35 I the previous two sectios, we have see that the adaptive equalizers used, require a iitial traiig period, as illustrated i fig 3.4, durig which a kow data sequece is trasmitted. A replica of this sequece is made available at the receiver i proper sychroism with the trasmitter, thereby makig it possible for adjustmets to be made to the equalizer coefficiets i accordace with the adaptive filterig algorithm employed 8 i equalizer desig. Whe the traiig is completed, the equalizer is switched to its decisio directed mode, ad ormal data trasmissio may commece. This type of equalizatio, i which the traiig period is available to the receiver, is kow as No- Blid equalizatio Blid Equalizatio Whe traiig sigals are etirely abset, the trasmissio is called blid, ad adaptive algorithms for estimatig the trasferred symbols ad possibly estimatig chael or equalizer iformatio are called blid algorithms. Sice, traiig iformatio is ot available; a reliable referece is missig, leadig to a very slow learig behavior i such algorithms. Thus, blid methods are typically of iterest whe a large amout of data is available ad quick detectio ot importat. The structure of blid equalizer is show i fig 3.8. ut Chael B q yt Equalizer C q zt Decisio ût Figure 3.8: Structure of Blid Equalizer.

36 Their major applicatio is thus, broadcastig for digital radio or T.V. However, recetly the cocept of basis fuctio to describe time-variat chaels has bee icorporated, provig that blid techiques also have potetial for time-varyig chaels. The best kow applicatio of blid equalizatio is to remove the distortio caused by the chael i digital commuicatio systems. The problem also occurs i seismological ad uderwater acoustics applicatios. The chael of a blid equalizer is as usual modeled as a FIR filter, as show i fig B q = b 1 1 b... b t q + bt q + + bt q, 3.6 ad the same model structure is used for the blid equalizer 1 t t 1 C q = c q + c q c c c t q. 3.7 The impulse respose of the combied chael ad equalizer, assumig FIR models for both, is h = b c t t where deotes covolutio. Agai, the best oe ca hope for is ht mδ t D, where D is a ukow time delay, ad m with m = 1 is a ukow modulus. For istace, it is impossible to estimate the sig of the chael. The modulus ad delay do ot matter for the performace, ad ca be igored i applicatios. Assume biary sigal u t BPSK. For this special case, the two most popular loss fuctios defiig the adaptive algorithm are give by:

37 3.8 V 1 = E[1 z ] Modulus restoral Godard 3.9 V 1 = E[ sig z z ] Decisio directed Sato The modulus restoral algorithm also goes uder the ame Costat Modulus Algorithm CMA. Note the covetio that decisio-directed meas that the decisio is used i a 30 parameter updates equatio, where as decisio feedback meas that the decisios are used i a liear feedback path. Algorithms usig the costat modulus CM property of the trasmitted sigal are historically probably the first blid algorithms. The costat modulus algorithm is the most well-kow procedure ad depeds o the o-liear fuctio i may variatios ad applicatios. While, the covergece aalysis of such algorithms is limited to very few cases the aalysis of its trackig behavior, i.e., its steady-state performace has made progress. 3.4 Chapter Summary I this chapter, we have covered, i detail, the problem of chael equalizatio. We have also covered various techiques of equalizatio, icludig Liear ad DFE equalizer, which are o-blid equalizatio scheme. Fially, the blid equalizatio techique used i this project is discussed, alog with the CMA ad decisio-directed algorithms.

38 31 Chapter 4 Simulatio Models 4.1 Itroductio

39 I this chapter, we describe the simulatio modelig used for the results geerated i this project work. We first discuss the geeral mathematical model of the chael equalizatio process. After that, we look at the various algorithms used i this project work, ad their implemetatio. 4. Geeral Mathematical Model 3 The equalizer structure is show i fig.4.1. Noise, Symbol Source si Chael ui xi Equalizer sˆ i Figure 4.1: A Represetatio of a geeral Mathematical Model for Adaptive Equalizatio.

40 Symbols s i are geerated usig a symbol source ad trasmitted through the chael. The output of the chael is the corrupted by additive white Gaussia oise AWGN, v i. The received sigal, x i is expressed as x i = s i h + v i where, h is the trasfer fuctio of the commuicatio chael, ad * deotes covolutio. The received sigal, x i is processed by the equalizer ad a estimate of the iput, s ˆ i is geerated. 4.3 Chael Modelig 33 I this sectio, we explai the method used to geerate data, ad the geeratio of additive white Gaussia oise. We cosider the chael H z 1 = 0.3+ z z ad proceed to desig adaptive equalizers for it Geeratig Data The symbols are geerated usig a symbol source ad are cosidered radomly. The differet modulatio schemes used are PAM, QPSK ad QAM.

41 4.3. Geeratig AWGN I all the simulatios, zero-mea additive white Gaussia oise is added to simulate the effect of oise i the receivers. A oise sigal havig a flat power spectral desity over a wide rage of frequecies is called white oise by aalogy to white light. The power spectral desity PSD of white oise is give by η o PSD = watts/hz where, η o is a fuctio of the oise power spectral desity, ad is icluded to idicate that the PSD is a two-sided PSD. 34 We will use zero-mea Gaussia white oise as the model for the oise that accompaies the sigal at the receiver iput. 4.4 Blid Algorithms As discussed i sectio 3.3.4, the equalizatio used i this project is a blid equalizatio, i.e., the equalizer is supposed to operate blidly. I blid equalizatio, the equalizer is supposed to operate without a referece sequece ad therefore without a traiig mode. The structure of blid adaptive equalizatio is a modified versio of fig 4.1 ad is show i fig 4..

42 Noise, Symbol Source si Chael Equalizer zi sˆ i ei Fuctio Figure 4.: A Represetatio of a Geeral Structure for Blid Adaptive Equalizatio. I blid equalizatio, the received sigal x i is processed by equalizer, whose output, z i is fed ito a decisio device to geerate { s ˆ i }. These sigals are delayed decisios ad the value of delay,, is determied by the delay that the sigals 35 udergo whe travelig through the chael ad equalizer. Most blid algorithms use the output of the equalizer, z i, to geerate a error sigal e i equalizer coefficiets accordig to the rule, which is used to adapt the w i = µ * wi 1 + e i s i which is kow as the LMS algorithm. 4.5 Loss Fuctio Model

43 For calculatig the loss fuctio values i this project, we have used the well-kow costat modulus algorithm CMA, or Godard algorithm, ad Sato algorithm. As discussed i sectio 3.3.4, the loss fuctios are give by: V V 1 = E[1 z ] Godard 1 = E[ sig z z ] Sato Godard Algorithm Godard 1980 was the first to propose a family of costat-modulus blid equalizatio algorithms for use i two-dimesioal digital commuicatio systems e.g., M-ary QAM sigals. The Godard algorithm miimizes a ocovex cost fuctio of the form J = E[ y R p ], 4.1 where, p is a positive iteger ad 36 E[ x R p = E[ x p p ] ] 4. is a positive real costat. The Godard algorithm is desiged to pealize deviatios of the blid equalizer output x from a costat modulus. The costat R p is chose i such a way that the gradiet of the cost fuctio J is zero whe perfect equalizatio [i.e., x ˆ = x ] is attaied.

44 The tap-weight vector of the equalizer is adapted i accordace with the stochastic gradiet algorithm w ˆ + 1 = wˆ + µ u e *, 4.3 where µ is the step-size parameter, u is the tap-iput vector, ad e = y y p p R p y 4.4 is the error sigal. From the defiitio of the cost fuctio J i equatio 4.1 ad from the defiitio of the error sigal e i equatio 4.4, we see that the equalizer adaptatio, accordig to the Godard algorithm, does ot require carrier phase recovery. The algorithm therefore teds to coverge slowly. However, it offers the advatage of decouplig the ISI equalizatio ad carrier phase recovery problems from each other. Two cases of the Godard algorithm are of specific iterest: 37 Case 1: p=1 The cost fuctio of equatio 4.1 for this case reduces to J = E[ y R 1 ], where

45 E[ R 1 = E[ x ] x ] Case 1 may be viewed as a modificatio of the Sato algorithm. Case : p= I this case, the cost fuctio of equatio 4.1 reduces to J = E[ y R ], where E[ R = E[ x x 4 ] ] Case is referred to i the literature as the costat-modulus algorithm CMA. The Godard algorithm is cosidered to be the most successful blid equalizatio algorithms. I particular, we must say the followig about Godard algorithm: 38 The Godard algorithm is more robust tha ay other algorithm with respect to the carrier phase offset. This importat property of the algorithm is due to the fact that the cost fuctio used for its derivatio is based solely o the amplitude of the received sigal. Uder steady-state coditios, the Godard algorithm attais a mea-square error that is lower tha ay other algorithm. The Godard algorithm is ofte able to equalize a dispersive chael, such that the eye patter is opeed up whe it is iitially closed for all practical purposes Sato Algorithm

46 The idea of blid equalizatio i M-ary PAM systems dates back to the pioeerig work of Sato The Sato algorithm cosists of miimizig a ocovex cost fuctio 4.5 J = E[ xˆ y ], Where y is the trasversal filter output ad x ˆ is a estimate of the trasmitted datum x formula. This estimate is obtaied by a zero-memory oliearity described by the xˆ = αsg[ y ] The costat E[ x ] α = E[ x ] sets the gai of the equalizer. 39 The tap weight vector of the equalizer is adapted i accordace with the stochastic gradiet algorithm 4.6 w ˆ + 1 = wˆ + µ e u where µ is the step-size parameter, u is the tap-iput vector, ad e = sg y y

47 is the error sigal. The Sato algorithm for blid equalizatio was itroduced origially to deal with oedimesioal multilevel M-ary PAM sigals, with the objective of beig more robust tha a decisio-directed algorithm. Iitially, the algorithm treats such a digital sigal as a biary sigal by estimatig the most sigificat bit; the remaiig bits of the sigal are treated as additive oise isofar as the blid equalizatio process is cocered. The algorithm the uses the results of this prelimiary step to modify the error sigal obtaied from a covetioal decisio-directed algorithm. 4.6 Chapter Summary I this chapter, we have discussed i detail about the geeral mathematical model for adaptive chael equalizatio. The, we have preseted the methods to geerate data ad additive white Gaussia oise. Fially, the well-kow blid algorithms, Godard ad Sato algorithms, used i the project are thoroughly discussed. 40 Chapter 5

48 Simulatio Results 5.1 Itroductio I the previous chapter, we discussed about the algorithms used i this project. I this chapter, we preset the results of the simulatios. I the followig sectios, we discuss about the various factors ivolved i equalizatio process. The results for various modulatio schemes used i the project will be preseted. Fially, the variatio of the most importat factor i determiig the equalizer s performace Bit Error Rate BER or Symbol Error Rate SER with SNR is cosidered. 5. Effect of ISI o Eye Patter 41 I a digital commuicatio system, chael distortio causes Itersymbol Iterferece ISI, as illustrated i chapter1. The amout of itersymbol iterferece ad oise i a digital commuicatios system ca be viewed o a oscilloscope. For PAM sigals, we ca display the received sigals o the vertical iput with the horizotal sweep rate set at 1, where T is the sigal T iterval. The resultig oscilloscope display is called a eye patter, because of its

49 resemblace to huma eye. For example, fig. 5.1 illustrates the eye patters for biary PAM modulatio. Figure 5.1: Example of Eye Patter for Biary PAM Modulatio. 4 The effect of itersymbol iterferece is to cause the eye to close, thereby reducig the margi for additive oise to cause errors. For PSK ad QAM, it is customary to display the eye patter as a two-dimesioal scatter diagram. Cosider, for example, a case of 8-PSK sigal. I the absece of itersymbol iterferece ad oise, the superimposed sigals at the samplig istats would result i eight distict poits correspodig to the eight trasmitted sigal phases. Itersymbol Iterferece ad oise result i a deviatio of the received samples from the desired 8-PSK sigal. The larger the itersymbol iterferece ad oise, the larger the scatterig of the received sigal samples relative to the trasmitted sigal poits.

50 The followig plots show the eye patters for the Sato ad Godard Algorithms, for PAM modulatio. 43

51 J 0.9 Sato:Eye patter Figure 5.: Eye Patter for Sato Algorithm, for PAM modulatio. Iteratios 44

52 J 0.9 Godard:Eye patter Figure 5.3: Eye Patter for Godard Algorithm, for PAM modulatio. Iteratios 5.3 Learig Curves Steepest Descet

53 I desigig a FIR adaptive filter, the goal is to fid the vector miimizes the quadratic fuctio { e } J = E w at time that Although the vector that miimizes J may be foud by settig derivative of J with respect to w * equal to zero, aother approach is to search for the solutio usig method of steepest descet. A plot of J versus the umber of iteratio, is referred to as the learig curve ad idicates how rapidly the adaptive filter lears the solutio to the Wieer-Hopf equatios give by lim w = R 1 x r dx Where, R x is autocorrelatio matrix ad r dx is cross correlatio. I Steepest Descet, the statistics are assumed to be kow beforehad. The weight vector update equatio i this case therefore is give by w * { e x } + 1 = w + µ E 5.3. LMS Algorithm 46 A practical limitatio with Steepest Descet algorithm is that the expectatio { e x * } E is geerally ukow. I order to overcome this limitatio, we adopt

54 the LMS algorithm. I this algorithm, the expectatio { e x * } a estimate such as the sample mea E is replaced with 5.1 E ˆ L * 1 { } e x = L l= 1 0 e l x * l Icorporatig this estimate ito steepest descet algorithm, the update for w becomes 5. w µ L 1 * + 1 = w + e l x l L l= 0 A special case of equatio 5. occurs if we use a oe-poit sample mea L=1, * { e x } e x ˆ * E = I this case, the weight vector update equatio assumes a particularly simple form w * 1 = w + e x + µ ad is kow as LMS algorithm. LMS algorithm is also called stochastic gradiet algorithm, sice a estimate of the gradiet vector is used. As the statistics are ot kow, the covergece rate i LMS algorithm is very slow. 47 For joitly wide-sese statioary processes, the LMS algorithm coverges i the mea if

55 0 < µ < λ max Where, µ is the step-size parameter ad λ max is the maximum eigevalue. Although based o a very crude estimate of E { e x * }, the LMS adaptive filter ofte performs well eough to be used successfully i a umber of applicatios. Sice, this project assumes the statistics to be ukow, we cosider the LMS algorithm. The followig plots show the learig curves of Sato ad Godard algorithm for PAM modulatio. 48

56 0.1 Sato:Learig curve J Iteratio Figure 5.4: Learig curve for Sato Algorithm Correspodig to the Cost Fuctio J = E[ xˆ y ]. 49

57 0. Godard:learig curve J Iteratio Figure 5.5: Learig curve for Godard Algorithm Correspodig to the Cost Fuctio p J = E[ y R ]. p Trasfer fuctio of Combiatio of Chael ad Equalizer

58 Give a chael of ukow impulse respose, the purpose of a adaptive equalizer is to operate o the chael output such that the cascade coectio of the chael ad the equalizer provides a approximatio to a ideal trasmissio medium. Ideally, i the case of a liear system, the equalizer model has a trasfer fuctio equal to the reciprocal of the chael s trasfer fuctio, such that the combiatio of two costitutes a ideal trasmissio medium. This is illustrated i fig.5.6. Chael Equalizer Figure 5.6: A represetatio of the Combiatio of Chael ad Equalizer. Here, the trasfer fuctio of the chael is give by H c f, whereas the equalizer trasfer fuctio is reciprocal of chael s, give by 1 H c f. The combiatio of the chael ad equalizer therefore should be a uit impulse. Therefore, the chael ad equalizer should be chose, so as to satisfy the above criterio. The followig plots show that this criterio is followed i the algorithms used i this project. 51

59 1.4 Sato:Impulse respose Amplitude Tap idex Figure 5.7: Impulse Respose of the Chael Blue, Equalizer Red ad Combiatio of Chael ad Equalizer Gree, for Sato Algorithm. 5

60 1.4 Godard:Impulse respose Amplitude Tap idex Figure 5.8: Impulse Respose of the Chael Blue, Equalizer Red ad Combiatio of Chael ad Equalizer Gree, for Godard Algorithm Other Modulatio Schemes

61 I the previous sectios, we have see the results of the Sato ad Godard algorithms for the case of PAM modulatio. As discussed i sectio 4.3.1, alog with PAM modulatio, we ca also use PSK ad QAM modulatio schemes. The followig plots will show the results for the case of PSK ad QAM, respectively For PSK Modulatio: 3.5 PSK:Learig curve 3.5 J Iteratio Figure 5.9: Learig Curve for the Case of PSK Modulatio. 54

62 5 PSK:Learig curve i db J Iteratio Figure 5.10: Learig Curve i db for the Case of PSK Modulatio. 55

63 Received Sequece Im Re Figure 5.11: Scatter Diagram of the Received Sequece for PSK Modulatio. 56

64 1.5 Equalizer Output Im Re Figure 5.1: Scatter Diagram of the Equalizer Output for PSK Modulatio. 57

65 1 Impulse respose 0.5 Amplitude Tap idex Figure 5.13: Impulse Respose of the Chael Blue, Equalizer Red ad Combiatio of Chael ad Equalizer Gree, for PSK Modulatio. For QAM Modulatio: 58

66 4500 QAM:Learig curve J Iteratio Figure 5.14: Learig Curve for the Case of QAM Modulatio. 59

67 38 QAM:Learig curve i db J Iteratio Figure 5.15: Learig Curve i db for the Case of QAM Modulatio. 60

68 Figure 5.16: Scatter Diagram of the Received Sequece for QAM Modulatio. 61

69 Figure 5.17: Scatter Diagram of the Equalizer Output for QAM Modulatio. 6