Adaptive robust detection for CDMA using a geneticalgorithm

X. Wu, T.C. Chuah, B.S. Sharif and O.R. Hinton

Abstract: A novel adaptive robust multi-user detector for code-division multiple access (CDMA)using a real-coded genetic algorithm (GA) is proposed. The GA implicitly implements Huber’sM-estimator and is robust against heavy-tailed impulsive noise. The novel feature of this GA-baseddetector lies in joint symbol detection and adaptive estimation of the cut-off parameter of theM-estimator’s object function through a GA optimisation strategy. In particular, the GA-baseddetector carries out a multipoint search by manipulating and maintaining a population ofcandidate solutions for different values of the cut-off parameter to encourage informationformation and exchange. Since the GA treats the cut-off parameter as one of its optimisationparameters, the need for a separate channel estimator is thus eliminated. Simulation results areprovided to examine the evolutionary behaviour and the detection performance of the proposedGA-based detector. It is shown that the GA approach provides good adaptive impulsive noiserejection capability.

1 Introduction

To meet the ever-growing demand for wireless services,multi-user communication techniques that make efficientuse of precious resources are of the utmost importance forsystem design. Direct-sequence code-division multiple-access (DS-CDMA) techniques are among the mostpromising candidates to fulfil this requirement. However,due to non-orthogonal signalling, multiple access inter-ference (MAI) created by other users limits the capacity ofCDMA systems. Multi-user detection [1] presents effectivesignal processing solutions for combating MAI andeliminating the near–far problem.

Despite considerable success in multi-user detectionresearch over the last two decades, this work has beenlargely dominated by widespread use of the additiveGaussian noise assumption. However, it is well known thatin many practical communication channels, such as indoorand urban radio channels, the ambient noise is largelyimpulsive in nature, which can have an adverse effect on theperformance of conventional receivers optimised forGaussian noise [2, 3]. Therefore practical implementationof multi-user detectors necessitates more accurate modellingof the ambient noise statistics along with the developmentof robust detection techniques [4].

Various multi-user detectors based on different perspec-tives and methodologies have been proposed in theliterature. In this paper, we propose a robust multi-userdetector based on a genetic algorithm (GA) that canperform reliably under both Gaussian and non-Gaussiannoise. GAs were first introduced in the early 1970s [5] as afamily of computational tools inspired by phenomenaobserved in natural evolution: genetic inheritance and

Darwinian strife for survival. These algorithms encode apotential solution to a specific problem on a simplechromosome-like data structure and apply probabilisticgenetic-like operators to modify the population, seeking anear-optimum solution to the given task. To date, the rangeof problems to which GAs have been successfully appliedhas far surpassed the initial expectation. The continuingprice and performance improvements of computationalunits have made them attractive for some search andoptimisation problems. In particular, GAs can handleproblems with constraints; and other that are multi-objective, multi-modal, non-differentiable, non-continuousor even NP-hard combinatorial optimisation problems.They are less likely to be ‘stuck’ at local optima thantraditional hill-climbing or gradient search methods. How-ever, being a Monte Carlo technique, they tend to becomputationally expensive.

Several GA-based multi-user detectors based on Gaus-sian noise models have also been reported. For example,Juntti et al. [6] conducted computer simulation of a GA forsuboptimal multi-user detection. They showed that a goodinitial guess vector such as the output of the lineardecorrelating detector (LDD) yields performance close tothat of the single user system. Wang et al. [7] proposed aGA-based multi-user detector based on the maximum-likelihood decision rule combined with a modified Viterbialgorithm for sequential symbol detection. Ergun andHacioglu [8] investigated a class of hybrid detectors thatcombine a GA and a multistage detector. Abedi andTafazolli [9, 10] studied GA implementation of theoptimum detector, and Yen and Hanzo [11] consideredthe problems of joint estimation of transmitted symbols andfading channel coefficients using GAs.

The GA-based detector proposed in this paper is robustin a minimax sense as it utilises an objective function typicalof Huber’s M-estimator [12]. The basic concept is to utilise anon-least-squares type penalty function, which attempts toassign a lesser weight to a small portion of outlyingresiduals that deviate considerably from the normal fit toavoid a catastrophic influence on the estimate. The

The authors are with the Department of Electrical and Electronic Engineering,University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK

r IEE, 2003

IEE Proceedings online no. 20030921

doi:10.1049/ip-com:20030921

Paper received 20th March 2002

IEE Proc.-Commun., Vol. 150, No. 6, December 2003 437

proposed GA-based detector performs joint symbol detec-tion and adaptive estimation of the cut-off parameter forthe M-estimator’s objective function by searching through aspace of potential solutions using a GA optimisationstrategy. It should be noted that the proposed GA-baseddetector implicitly implements a robust version of the LDD[13] based on Huber’s M-estimator proposed in [14], or itsefficient recurrent neural network (RNN) solution [15]. Inthe sequel, we refer to the robust decorrelating detector asM-decorrelating detector (MDD). The MDD studied in[14, 15] uses a fixed cut-off parameter for the M-estimator’sobjective function, which is determined by a rough estimateof the noise scale. Clearly, this requires a separate channelestimator, and as will be seen later, fixing the cut-offparameter also imposes a constraint on the attainableperformance. We will demonstrate how these problems canbe overcome altogether to achieve better performance usinga GA approach.

2 System description

In the following, we consider a synchronous DS-CDMAsystem shared by K users. Each user is assigned a differentspreading code of length N-chips. The modulation schemeis binary phase shift keying (BPSK) and demodulation isassumed to be coherent. The baseband output of the chip-matched filter corresponding to the jth chip of the ithsymbol is given by

rjðiÞ ¼XKk¼1

AkbkðiÞskj þ njðiÞ; j ¼ 1; � � � ;N ð1Þ

or in vector form

rðiÞ ¼XKk¼1

AkbkðiÞsk þ nðiÞ

¼ SA bðiÞ þ nðiÞ ð2Þwhere, for the kth user, Ak, bk(i)A{71} and sk denote,respectively, the received amplitude, ith information bitand the normalised signature code. S ¼ ½s1j � � � jsK � 2 <NxK

is the matrix whose columns are the normalised spreadingcodes of all users. A¼ diag{A1,?, Ak} and bðiÞ ¼b1ðiÞ; � � � ; bKðiÞ½ �T . n(i) contains the sequence of i.i.d. noise

samples with a common univariate probability distributionfunction fn( � ) of zero mean. In this paper, we model thenoise as symmetric a-stable random processes [16]. Due tosynchronous assumption, the symbol index i can bedropped and we define xk ¼Ak bk, and (1) can then bewritten as

rj ¼XKk¼1

skj xk þ nj; j ¼ 1; :::;N ð3Þ

and (2) can be expressed as a linear regression model

r ¼ Sx þ n ð4Þwhere x¼ [x1, ?, xK]T¼Ab.

2.1 Linear decorrelating detector (LDD)The conventional receiver consists of a matched filter (MF)bank with each filter matched to the signature waveform ofa specific user followed by a threshold device to produce thesymbol estimate

b̂bMF ¼ signðST rÞ ¼ signðST Sx þ STnÞ ð5ÞThe normalised crosscorrelation matrix for all signaturecodes is given by R¼STS. The LDD [13] applies the inverse

R�1 to the output of the MF bank

b̂bLDD ¼ signðR�1ST rÞ ¼ signðx þ R�1ST nÞ ð6ÞIt is seen from (6) that the LDD does not depend on thesignal powers of the interfering users and hence is near–farresistant.

2.2 Robust M-decorrelating detector usingrecurrent neural networkLet d¼ [d1, ?, dN]T denote the residual vector where

dj ¼ rj �XKk¼1

skj xk; j ¼ 1; . . . ;N ð7Þ

The M-estimate [12] for the vector x corresponds tominimising the following energy function:

EðxÞ ¼XNj¼1

r rj �XKk¼1

skj xk

!ð8Þ

where r( � ) represents a specific penalty function that isgenerally convex [12]. Hence,

x̂x ¼ arg minx2<K

XNj¼1

r rj �XKk¼1

skj xk

!ð9Þ

To make the minimisation problem meaningful rð�Þ shouldbe non-negative with its minimum at zero. The resultinginferential technique is referred to as the M-estimator,where M refers to the maximum likelihood (ML) typeestimator, where the choice rML(dj)¼�log fn(dj) gives theordinary ML estimate of x. For example, the LDD uses theleast squares (LS) penalty function, i.e. rLS(dj)¼ (dj)

2/2,which is optimal when the residuals dj follow a Gaussiandensity. In the absence of this assumption the LS estimate isoften vulnerable to outliers since each residual entersquadratically.

Huber aimed to achieve robustness in a minimax senseand considered the following penalty function that increasesless rapidly than LS [12]:

rHðdj; hÞ ¼12 ðdjÞ

2; for jdjj � hhjdjj � 1

2h2; for jdjj4h

�ð10Þ

with derivative

cHðdj; hÞ ¼dj; for jdjj � h

h signðdjÞ; for jdjj4h

�ð11Þ

where h40 is a problem-dependent parameter for tuningthe robustness of the estimator. The basic principle of (10) isto use LS and penalise outliers with L1 norm. The RNN-based MDD in [15] uses the logistic function as a smoothapproximation to (10). By applying the gradient method itis easy to show that the solution to (9) is equivalent tosolving the following equation [15]:

STcðr � SxÞ ¼ 0K ð12Þwhere c( � )r0( � ) and 0K denotes a K-vector zero. With theexception of the most elementary forms of c-function suchas the LS, (12) constitutes a set of non-linear equations, andhence iterative methods are required. Several iterativeschemes which have been developed in the statisticsliterature are summarised and compared in [17–20].Basically they can be broadly categorised into three families:Newton’s method, iteratively reweighted least squares, andthe Huber’s method as was used in [14]. In [15], it wasshown that a RNN proposed in [21] could be used toimplement the MDD without the need of the matrixinverse [STS]�1 with a consequent saving in computa-tional complexity, especially for a highly dynamic CDMA

438 IEE Proc.-Commun., Vol. 150, No. 6, December 2003

network. Mathematically, the RNN-based MDD iterativelyestimates x as follows:

x̂xnþ1 ¼ x̂xn þ mSTcðr � Sx̂xnÞ ð13Þwhere n is the iteration index and m is the step size thatcontrols the speed of convergence. The final symbolestimates are taken as the signs of the converged x̂x in(13). Unlike most of the multi-user detectors that processthe matched filter bank output STr, the RNN-based MDDin (13) and the GA-based detector proposed in this papertake r, which is obtained from a chip-matched filter, as theirobservation vector.

3 Genetic algorithm based robust M-decorrelat-ing detector

Our investigation of a GA approach to tackle the robustdetection problem formulated in (9) was motivated byseveral factors. GAs are adaptive and robust optimisationand search techniques which borrow the ideas of naturalselection and ‘survival of the fittest’ from natural evolution.In GAs, the potential solution of a problem is encoded intoa form that is analogue to the chromosomes of biologicalsystems. Each chromosome is made up of a collection ofgenes that represent the optimisation parameters. Thechromosome is typically represented in the GA as a stringof bits. However, integers and floating-point numbers caneasily be used. The simulation process begins with an initialpopulation of random chromosomes, which iterativelyevolves over multiple reproductive cycles (called generationsor iterations) through the application of three probabilisticgenetic operators: selection, crossover (recombination), andmutation.

Therefore, by simulating natural evolution in this way, aGA can easily search for potential solutions to solvecomplex problems in a general, representation-independentmanner. Such a search is not guided by stringentmathematical formulation but often requires balancingtwo conflicting objectives: exploiting the best solutions andexploring the search space [22]. Conventional directed orgradient search methods such as the three existingalgorithms for the M-estimator [17–20] and the RNNsolution in (13) are examples of strategies which exploit thebest solution for a given cut-off parameter h of theM-estimator’s objective function. They are governed bydifferential/difference equations that use the iterativeimprovement technique. The technique is applied to asingle point (the current point n) in a limited search spacedetermined by h. During a single iteration, a new point isselected from the neighbourhood of the current point.Obviously, they are local search techniques in the sense thatthey provide local optimum values that are dependent onthe selection of the cut-off parameter h. For example, theMDDs implemented in [14, 15] fix the cut-off parameter hto a value that is proportional to the noise variance/dispersion and only a single point in the search space isprocessed. Clearly, this requires a separate channelestimator and the fixed cut-off parameter h imposes aconstraint on the attainable performance, since it does notexplore the search space for other possible values of h.

On the other hand, the principle of GAs is neithergoverned by the use of differential/difference equations nordoes it behave like a continuous function, which makes itmore capable of solving difficult problems encountered byconventional gradient-based or hill-climbing techniques[23]. In particular, GAs can perform multi-directionalsearch by manipulating and maintaining a population ofpotential solutions for different values of h and encouraging

information formation and exchange between these direc-tions. At each generation, the population undergoes asimulated evolution in which better chromosomes repro-duce while inferior ones perish. Because of this, GAs canstrike a remarkable balance between exploration andexploitation of the search space by combining elements ofdirected and stochastic search, and subsequently lead tobetter solutions [24]. More remarkably, in addition tosearching for the estimate of x, the proposed GA-basedmultiuser detector also treats h as its optimisation parameterand thus eliminates the need for the channel estimator.

The standard GAs introduced by Holland [5], whichemploy binary chromosomes and binary genetic operators,are limited in several aspects when tackling complicatedproblems. As a result, problem-specific modificationsappropriate for the given environment have been developedto suit the design requirement. In fact, there are many facetsof operational modification that can be introduced onchromosome representation, genetic operators, replacementscheme, and implementation parameters [23]. In thefollowing subsections, we outline the development of theGA-based MDD based on a real-coded GA, whosestructure is depicted in Fig. 1. For more details on theprinciples of GAs, the reader is referred to [24–26].

3.1 Chromosome representationThe encoding scheme of chromosomes may vary accordingto the nature of the problem and has a major impact on theperformance because it can severely limit the search spaceobserved by the system. The most commonly used binarychromosomes are sometimes difficult and unnatural whenapplied to multidimensional, high precision optimisationproblems and the need for richer data structures has beenrecognised for some time [27]. Alternatively, real-codedchromosomes [28, 29], which have received considerableinterest, produce better solutions in certain engineering

Return to step 1 and proceed with the next symbol

step 2maxGG ≤

yes

no

Randomly generate forpc

Evaluate the fitness

Perform roulette wheel selection

Create offspring by crossover and mutation

Apply incremental and elitism replacement toproduce next generation

step 1

step 2

step 3

step 4

step 5

Hard limit the best individual cp for symbol detection

Pp ,,1 …=

forpc Pp ,,1 …=f ( )

Fig. 1 Flowchart depicting structure of proposed GA-based robustmultiuser detector

IEE Proc.-Commun., Vol. 150, No. 6, December 2003 439

problems [24, 30–32]. The real-valued representation iscapable of representing variables over continuous domainswith higher precision and efficiency [24]. For the GA-basedMDD where the user signal components in x and the cut-off parameter h are variables in real space, a real-valuedencoding scheme is used in order to move the representationcloser to the problem domain. Each chromosome isrepresented by a (K+1)-dimensional real-valued vectorc¼ [c1, ?, cK+1], where the first K components correspondto the signal components of all users fx̂x1; � � � ; x̂xKg, and the(K+1)th component is reserved for the estimator cut-offparameter h.

3.2 InitialisationThe population of real-coded chromosomes fcp ¼ ½c1

p; � � � ;cKþ1p �; p ¼ 1; � � � ; Pg is initialised randomly, where P is

known as the population size. The lower and upper boundfor each variable ckp8p in the chromosome is denoted by ak

and bk, respectively. The purpose of using randomgeneration is to distribute the initial trial solutions to ahighly diversified search space (exploration).

3.3 Fitness evaluationThe objective function in (8) provides the mechanism forevaluating the fitness of each chromosome. By convention,the fitness function should be a positive value; otherwise wecan use a scaling mechanism [24, 25]. We assume that theGA-based detector employs the Huber penalty functionrH( � ). Since rH( � ) is non-negative, the fitness function(which is to be minimised) of each chromosome f(cp) isevaluated directly by substituting its elements into (8):

f ðcpÞ ¼PNj¼1

12

rj �PKk¼1

skj ckp

� �2

; for jdjj � cKþ1p

PNj¼1

cKþ1p rj �

PKk¼1

skj ckp

��������� 1

2ðcKþ1

p Þ2; for jdjj4cKþ1p

8>>>><>>>>:

ð14Þ

Therefore, the GA only requires the objective functionassociated with each chromosome for implementation. Bycontrast, the aforementioned conventional gradient techni-ques need the derivatives of r( � ) (calculated analytically ornumerically) in order to be able to climb the next point,which could become infeasible if the derivatives do not exist.

3.4 Genetic operatorsBased on the fitness function defined above, three basictypes of genetic operators are required to modify thepopulation: selection, crossover, and mutation. In thissection, we present some genetic operators for real-valuedGAs used in the proposed detector. More discussion aboutreal-valued genetic operators can be found in [24, 29].

3.4.1 Selection: Selection is a process used for choos-ing parent chromosomes to participate in reproduction forthe next generation. It models the survival-of-the-fittestmechanism observed in nature. The reproductive opportu-nity of an individual parent is normally granted in directproportion to its fitness value so that highly fit chromo-somes contribute more copies (with reselection permitted) tothe mating pool than do those poor ones. Therefore, thebest parents generate more copies, the average stay medium,and the worst die out. Among the many selection schemesavailable, we use the roulette wheel sampling [5].

3.4.2 Crossover operators: Crossover is a crucialoperator that combines two or more parent chromosomesto produce new offspring chromosomes. The key ideabehind crossover is to generate new offspring by integratinggood genetic material from highly fit parents. There aremany ways to implement the crossover operation. Asuitably designed crossover can significantly accelerate thesearch process. In the proposed GA-based multiuserdetector, we examine the use of three crossover operators:simple one-point crossover, arithmetic crossover, andheuristic crossover [24]. As will be seen later, the combineduse of these crossover operators can lead to enhancedperformance.

Let cp and cq be two selected parent chromosomes. One-point crossover creates two offspring chromosomes ~ccp and~ccq by randomly selecting a crossover point and exchangingsegments after that point. Arithmetic crossover producesoffspring as follows:

~ccp ¼ zcp þ ð1 � zÞcq ð15Þ

~ccq ¼ zcq þ ð1 � zÞcp ð16Þ

where zBU(0,1). The rationale of using arithmetic cross-over for the proposed GA detector is to reduce detectiondelays, since some previous studies have shown thatarithmetic crossover could increase the convergence rate[24].

On the other hand, heuristic crossover [24, 29] is distinctfrom both simple and arithmetic crossover because ituses values of the fitness function in determining thesearch direction. It either produces only one new offspringor no new offspring at all. Assume that the parent cp isbetter than cq in term of fitness value, i.e. f(cp)Zf(cq) formaximisation and f(cp)rf(cq) for minimisation problems,then heuristic crossover generates a new offspring ~ccp ¼cp þ zðcp � cqÞ and set ~ccq ¼ cp. If ~ccp is not feasible, i.e.,

~cckpoak or ~cckp4bk for all k¼ 1, ?, K+1, then another

random draw of z is performed to produce a new offspring~ccp. If no new offspring satisfies the constraints after Qattempts, then the operator halts and produces no offspring.The main reasons for incorporating heuristic crossover areto assist in fine local tuning and search in the mostpromising direction [24].

It should be noted that crossover is not always invoked.After selecting a pair of parents, the algorithm implementscrossover only if a random variable z (zBU(0,1)) is greaterthan the crossover rate pc. Otherwise, the parents remainunaltered. Typical values of pc lie within the range 0.6–0.9[25].

3.4.3 Mutation operators: The mutation operatorrandomly alters some genes, values in a chromosome with aprobability determined by the mutation rate pm. This canresult in entirely new offspring chromosomes. Mutation ispotentially useful in restoring genetic diversity that may belost in a population due to premature convergence.Mutation is used very sparingly in most GAs. Typically,the mutation rate pm is less than 0.1 [25].

The common binary mutation operator where a selectedbit is flipped cannot be applied for real-coded chromosomesbecause the concept of complementary value is inconsistent.In the proposed real-coded GA-based detector, we considermulti-non-uniform mutation. Non-uniform mutation [24] isa dynamic (population dependent) mutation operatoraimed at improving single-element tuning and reducingthe drawback of random mutation. It is defined as follows:for a parent cp, randomly select an element ckp and perform

440 IEE Proc.-Commun., Vol. 150, No. 6, December 2003

mutation to yield ~cckp as

~cckp ¼ckp þ D½G; ðbk � ckpÞ�; if z1 � Uð0; 1Þo0:5

ckp � D½G; ðckp � akÞ�; if z1 � Uð0; 1Þ � 0:5

(

ð17Þwhere G denotes the current generation number, and thefunction D[G, y] is defined as [24]:

D½G; y� ¼ yð1 � zð1�G=GmaxÞb2 Þ ð18Þ

where z2BU(0,1), Gmax is the maximum number ofgenerations, and b is a shape parameter which determinesthe degree of dependency on generation number (we useb¼ 3 in the following). The function D[G, y) returns a valuein the range [0, y] such that the probability of D[G, y] beingclose to 0 increases as G increases. This causes the operatorto search the space uniformly at the beginning (when G issmall), and very locally at later generations [24]. The multi-non-uniform mutation applies the non-uniform mutationoperator to all the elements in cp.

3.5 ReplacementAfter a predefined number of offspring has been producedthrough the above genetic operators, a replacement strategyis required in order to modify the old population with thenew generation. Several replacement strategies have beenproposed in the literature, and a good review can be foundin [33]. In the proposed GA-based detector, we use the so-called incremental replacement in which children will havethe chance to compete with some of the parent individuals.Each offspring generated after reproduction is added to thenew generation only if it has a better fitness value than bothof its parents. If the offspring is better than one of theparents, then a random selection is made between the betterparent and the offspring. If the fitness value is worse thanboth parents, then one of the parents is selected randomlyto the new population. Besides, a so-called elitist strategycould be also used to improve algorithm performance [25].We also incorporate this strategy into our replacement. Theelitist strategy appends the best performing chromosome ofa previous generation to the current population and thusensures that the chromosome with the best fitness valuealways survives to the next generation.

3.6 TerminationTermination is the criterion by which the GA decideswhether to continue searching or stop the search. Typicaltermination criterion of the GA involves either satisfying aproblem-specific success indicator or completing a specifiednumber of generations to be run, Gmax. Since in theproposed GA-based detector, the number of iterationsrequired to reach a predefined cost function is not known inadvance, we adopt the latter strategy to avoid excessivelyhigh complexity and detection delay. The final symbolestimate is taken as the sign of the best chromosome cp inthe final generation, i.e.

b̂bk ¼ signðckpÞ; k ¼ 1; � � � ;K ð19Þ

4 Simulation results

In this section, computer simulation results are presented totest the various aspects of the proposed GA structure forrobust multiuser detector. A summary of the various GAparameters used is given in Table 1. Throughout thesimulations, a synchronous CDMA system with 10 equal-

power users (K¼ 10) is considered. Gold spreading codes oflength 31 (N¼ 31) are assigned to all users. For the RNN-based MDD, the converge rate m is chosen as 0.8 and thecut-off parameter h is set to h¼ 3g, where g is the noisedispersion [16]. Due to the infinite-variance property ofa-stable distributions leading to the inconsistency ofstandard signal-to-noise ratio (SNR), we use an alternativemeasure known as the geometric SNR (GSNR) [34]. Beforepresenting the bit error rate (BER) results, it is of interest toexamine the evolutionary behaviour of the GA detector inresponse to different noise scenarios.

4.1 Evolutionary performanceFigures 2a and 2b demonstrate the evolution of energyfunction with respect to the number of generations for theRNN and GA-based MDDs in Gaussian (a¼ 2) andimpulsive noise (a¼ 1.5), respectively. Two differentGSNRs, 0 and 4 dB, are considered. For the GA-baseddetectors, we use a population size of P¼ 60 and the bestchromosomes are used in energy evaluation. Note that wealso compare GAs with and without the arithmeticcrossover operator. It is observed that all GA detectorswere able to reach a lower energy as compared to the RNN,particularly under hostile channel conditions (low GSNRand impulsive noise scenario). This is because the RNNimplements a single point search using a gradient approach,whereas the GA detector evolves in multidimensionalsearch while jointly adapting to find optimal values of thecut-off parameter h. Therefore it is not surprising to findthat the GA methods have a slower convergence rate ascompared to the RNN. Moreover, we also notice that theGA system without arithmetic crossover converges slower.In contrast, the system with arithmetic crossover convergesfaster and provides better solutions than those withoutarithmetic crossover.

In order to examine the response of the GA-baseddetector to different noise environments, Figs. 3–5 illustratethree-dimensional plots between the cut-off parameter h,residual dj (only the first chip d1 is considered), and rH(dj, h)given in (10). The three crossover operations: simple,arithmetic, and heuristic crossover are used in combination.During the first generation (G¼ 1), the (P¼ 60) candidatesolutions are randomly generated and distributed over thesolution space as shown in Fig. 3. Hence, it is highlyexplorative at the beginning. By evolving this population ofchromosomes over successive generations via probabilisticgenetic operations, the GA detector quickly identifies andexploits the subspaces in its search for the best solution, andat the same time maintains the exploration of other parts ofthe solution space. This is illustrated in Figs. 4 and 5 after

Table 1: Summary of GA parameters used for simulation

Parameter Value/Type

Population size P 60, 80

Generation Gmax 50

Representation real-valued

Initialisation random

Generation selection roulette wheel

Crossover operators simple, arithmetic, heuristic

Crossover probability pc 0.88

Mutation operator multi-non-uniform

Mutation probability pm 0.08

Replacement incremental+elitism

IEE Proc.-Commun., Vol. 150, No. 6, December 2003 441

50 generations in Gaussian and impulsive noise, respec-tively. The GSNR is 4 dB. It is interesting to observe thatwhen the noise is Gaussian, the cut-off parameter h isreasonably large (Fig. 4). This is because the noise contains

no impulsive components and ideally, h should approachinfinity to yield the LS estimator (LDD), which is optimal inGaussian noise. On the other hand, when the noise isimpulsive (Fig. 5), the parameter h has been reduced to asmall value in an attempt to prevent the entry of impulsivenoise. Therefore, the GA detector is able to adapt tostatistical variations in noise distribution to achieve robustperformance.

4.2 Symbol detection performanceUltimately, it is required to evaluate the BER performanceof the proposed GA-based MDD. Comparison is madeagainst the matched-filter receiver, the LDD, and the RNN-based MDD. For the GA detector, systems with populationsizes of 60 and 80 are presented. The BER results of all 10users are averaged and shown in Figs. 6 and 7, respectively,in Gaussian (a¼ 2) and impulsive noise (a¼ 1.5). Under thefavourable Gaussian noise channel, it is observed that theRNN and GA-based MDDs have similar performance andthey only incur negligibly small performance loss withrespect to the LDD, which is the optimal estimator for x inGaussian noise [1].

10 20 30 40 500

5

10

15

generation

generation

ener

gy fu

nctio

nen

ergy

func

tion

RNN-0dBRNN-4dBGA-0dB (simple & heuristic crossover)GA-4dB (simple & heuristic crossover)GA-0dB (simple & heuristic & arithmetic crossover)GA-4dB (simple & heuristic & arithmetic crossover)

10 20 30 40 500

5

10

15

20

25RNN-0dBRNN-4dBGA-0dB (simple & heuristic crossover)GA-4dB (simple & heuristic crossover)GA-0dB (simple & heuristic & arithmetic crossover)GA-4dB (simple & heuristic & arithmetic crossover)

a

b

Fig. 2 Evolution of energy function of the RNN and GA-baseddetector with respect to the number of generations at two differentGSNR (0 and 4 dB)a In Gaussian noise channelb In impulsive noise channel (a¼ 1.5)All GAs use population size of P¼ 60 and the best chromosomes areused in energy evaluation

20

10

06

6

44

22

00

−2−2

−4 −4

−6 −6

�

residualh

Fig. 3 Three-dimensional plot of the relationship between cut-offparameter h, residual dj, and rH(dj, h) during initialisation (G¼ 1)Total population P¼ 60

20

10

06

6

4

4

22

00

−2−2

−4−4

−6 −6

�

residualh

Fig. 4 Three-dimensional plot of the relationship between cut-offparameter h, residual dj, and rH(dj, h) in the last generation(G¼ 50)Total population P¼ 60, Gaussian noise at GSNR of 4 dB

20

10

06

6

44

22

00

−2−2

−4 −4

−6 −6

�

residualh

Fig. 5 Three-dimensional plot of the relationship between cut-offparameter h, residual dj, and rH(dj, h) in the last generation(G¼ 50)Total population P¼ 60, impulsive noise (a¼ 1.5) at GSNR of 4 dB

442 IEE Proc.-Commun., Vol. 150, No. 6, December 2003

However, in impulsive noise channels, GA-based detec-tors clearly outperform all other detectors due to itsmultidimensional search and adaptive outlier rejectioncapability. Also, the performance of GA detectors improvesas the population size P increases, since the search spacebecomes larger. However, as P is set too large, an excessiveamount of processing time is required for each generationand the BER improvement becomes unjustified. Hence, forthe system under consideration, population size of 60provides a good compromise between performance andcomplexity. Furthermore, the GA system can often benefitfrom an effective use of parallelism to speed up computa-tion [25].

Finally, it should be noted that the proposed GA-basedMDD eliminates the need for a separate channel estimatorfor the noise parameter. This clearly presents a majoradvantage because it is well known that the channelestimation problem is a challenging issue, particularly in adynamic time-varying channel typical of a mobile radioenvironment.

5 Conclusions

In this paper, a GA-based adaptive robust detectionmethod for CDMA systems is proposed. The GA implicitlyrealises the MDD by simulating natural evolution through

the application of three genetic operators: selection,crossover, and mutation. Unlike the recently proposedRNN-based MDD which performs local, single-pointsearch for a given cut-off parameter h of the M-estimator’sobjective function, the GA approach carries out a multi-point search by manipulating and maintaining a populationof candidate solutions for different values of h andencourages information formation and exchange betweenthese directions. In addition to this, the GA-based detectortreats h as its optimisation parameter and thus eliminatesthe need for the channel estimator. As a result, it couldstrike a remarkable balance between exploration andexploitation of the search space.

Simulation results demonstrate that the GA-based MDDis able to adapt h in a noise-changing environment. It alsooutperforms the RNN-based MDD in non-Gaussian noise,albeit at the expense of added complexity. However, withthe advances in parallel architecture, it can be expected thatthe GA-detector will present a promising alternative to theRNN solution. Also of importance is the fact that theGA-based MDD totally overcomes the challenging pro-blem of channel estimation, particularly in a time-varyingwireless channel.

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0 1 2 3 410−4

10−3

10−2

10−1

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aver

age

bit e

rror

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matched filterLDDRNN-based MDDGA-based MDD (population=60)GA-based MDD (population=80)

Fig. 6 Average BER performance comparison in Gaussian noise

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10−4

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Fig. 7 Average BER performance comparison in impulsive noise(a¼ 1.5)

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