Deterministic Spreading Sequences for the Reverse Link of DS-CDMA With Noncoherent -ary Orthogonal...

354 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 1, JANUARY 2008

Deterministic Spreading Sequences for the ReverseLink of DS-CDMA With Noncoherent M -ary

Orthogonal Modulation: Impact and OptimizationQinghua Shi and Q. T. Zhang, Senior Member, IEEE

Abstract—M -ary orthogonal modulation with noncoherentdetection is an attractive scheme for the reverse link of direct-sequence code-division multiple access (DS-CDMA). The perfor-mance analysis of such DS-CDMA systems with random spreadingsequences has been thoroughly studied. However, little satisfactorywork has been done for systems with deterministic spreadingsequences, regardless of their importance from the viewpoint ofmultiple-access interference reduction. To fill this void, we con-sider deterministic spreading sequences in this paper. We firstpresent a detailed error performance analysis for the reverse linkof DS-CDMA with M -ary orthogonal modulation on both addi-tive white Gaussian noise (AWGN) and multipath Rayleigh-fadingchannels. Then, we formulate the design of spreading sequencesfor the DS-CDMA system as a nonlinear discrete optimizationproblem. Two steps are taken to complete the task of sequenceoptimization: 1) A large code space is generated by permutatinga single binary code matrix and imposing a Kronecker productstructure on candidate code matrices, and 2) an evolutionary al-gorithm is applied to efficiently perform optimization. Numericalexamples show that the optimized sequences considerably improvethe system performance, especially when the number of users isrelatively small or an AWGN channel is considered. In addition,our proposed approach can make a good tradeoff between codeperformance and search complexity.

Index Terms—Direct-sequence code-division multiple access(DS-CDMA), evolutionary algorithm (EA), optimization, spread-ing sequences.

I. INTRODUCTION

A UNIQUE feature of direct-sequence code-division mul-tiple access (DS-CDMA) is the use of spreading se-

quences/codes to separate different users. The selection ofspreading sequences is therefore of great importance becausespreading sequences govern, to a large extent, the resultingmultiple-access interference (MAI) [1], which is a fundamentallimitation to the performance of DS-CDMA systems. Accord-ing to their distinct design philosophies [2], the spreadingsequences used in DS-CDMA systems fall into two categories,

Manuscript received February 16, 2004; revised July 19, 2005, September 8,2006, and January 9, 2007. This work was supported by the City University ofHong Kong under Strategy Research Grant 7001772. The review of this paperwas coordinated by Prof. H. Leib.

Q. Shi was with the Department of Electronic Engineering, City Universityof Hong Kong, Kowloon, Hong Kong. He is now with the Nanyang Technolog-ical University, Singapore 639798 (e-mail: [email protected]).

Q. T. Zhang is with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2007.904516

i.e., short versus long codes. Although long codes, particularlym-sequences, have been adopted in the reverse link of IS-95systems, short codes are still subject to extensive investigations[3]. The basic reason is that it is intuitively clear, and, as shownin [4], that appropriately designed short codes can outperformlong codes in terms of MAI. Moreover, the nature of finitedimensional signaling of DS-CDMA systems with short codesfacilitates the incorporation of multiuser detection techniquesto further reduce MAI. Recently, it has been shown that well-designed multidimensional short codes can even completelyeliminate MAI [5].

The classical model for the reverse link of DS-CDMAassumes BPSK modulation and coherent correlation receiver[6]. However, this receiver structure may not be attractive inpractice for two reasons. First, coherent detection in the reverselink is costly compared to noncoherent detection schemes.Second, a correlation receiver shows poor performance in thepresence of MAI. In contrast, M -ary orthogonal modulationwith noncoherent maximum likelihood (ML) detection cangreatly improve the performance of the reverse link of DS-CDMA while retaining reasonable implementation complexity[7]. Consequently, DS-CDMA with M -ary orthogonal modula-tion and noncoherent ML detection has received considerableattention, and in fact, this scheme has been adopted in thereverse link of IS-95 systems. So far, nearly all performanceanalysis of such DS-CDMA systems has been conducted underthe common assumption of random sequences [8]–[10]. Theonly exception is that in [11], where the influence of determin-istic spreading sequences on system performance is addressed.However, the analysis in [11] does not fully characterize theimpact of spreading sequences on performance besides thesimplest additive white Gaussian noise (AWGN) channel con-sidered therein.

In this paper, we first present a detailed bit error rate (BER)performance analysis for the reverse link of a DS-CDMAsystem, which employs short spreading sequences, M -ary or-thogonal modulation, and noncoherent ML detection operatingon both AWGN and multipath Rayleigh-fading channels. Inparticular, we take into account the real effect of spreadingsequences on system performance. The BER is then used asthe criterion for sequence optimization in the context of binaryorthogonal spreading sequences. The selection of spreadingsequences turns out to be a discrete nonlinear optimizationproblem. The challenge is, therefore, how to efficiently solvesuch a discrete optimization problem over a possibly very

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SHI AND ZHANG: REVERSE LINK OF DS-CDMA WITH NONCOHERENT M -ARY ORTHOGONAL MODULATION 355

Fig. 1. Transmitter block diagram for user k.

large code space. We tackle this problem in two steps. First, aKronecker product structure is imposed on the spreading codes,which directly determines the size of a code space. As such, atradeoff between search complexity and performance of spread-ing codes can be made depending on practical considerations.Second, we choose an evolutionary algorithm (EA) [12]–[14]to perform the optimization. By operating on a population ofpotential solutions, EAs select the best at each step to producesuccessively better approximations to an optimal solution. Theyare applicable to any case, as long as the objective function (i.e.,performance measure) can be numerically evaluated, and theyare less likely to be trapped in the local optima due to theirglobal search capability. In the literature, similar approaches,but based on random search, have been used for the selectionof spreading sequences for asynchronous DS-CDMA [15] andquasi-synchronous DS-CDMA [16] systems over an AWGNchannel. Random search is a totally stochastic method thatcannot yield even a local optimum apart from that it is rathertime consuming.

The rest of this paper is organized as follows: Section IIpresents a detailed performance analysis of DS-CDMA withnoncoherent M -ary orthogonal modulation on an AWGN chan-nel. In Section III, a more realistic multipath Rayleigh-fadingchannel is considered. Then, the sequence optimization strategyis elaborated in Section IV. Section V provides numericalresults and discussions, followed by conclusions in Section VI.

II. PERFORMANCE ANALYSIS ON AN AWGN CHANNEL

A. System Model

Consider the reverse link of a K-user DS-CDMA system. Asillustrated in Fig. 1, the transmitter of user k (k = 1, 2, . . . , K)employs M -ary orthogonal modulation [11], i.e., every log2 Mdata bit is mapped to one of M Walsh symbols of duration T .Let PT (t) denote a unit rectangular pulse defined in [0, T ]. TheM -ary orthogonal modulator of user k generates a waveform

W (k)(t) =+∞∑

q=−∞W (k)

q (t)PT (t − qT )

where W(k)q (t) can take on one of the M Walsh symbols

{W0(t),W1(t), . . . ,WM−1(t)} (0 ≤ t ≤ T )

with equal probability, and a Walsh symbol Wm(t) (m =0, 1, . . . ,M − 1) can be further expressed in terms of its chipswm,j ∈ {+1,−1} as

Wm(t) =M−1∑j=0

wm,jPTw(t − jTw)

with Tw (Tw = T/M) signifying the duration of a Walsh chip.The Walsh-modulated signal is next spread by a user-specificsequence. Let A(k)(t) denote the spreading waveform of user kgiven by

A(k)(t) =∞∑

h=−∞a(k)h PTc

(t − hTc)

where we assume that the kth user’s spreading sequence {a(k)h }

is binary (i.e., a(k)h ∈ {+1,−1}) with period N and chip dura-

tion Tc(Tc = T/N).1 The transmitted signal from user k maybe written as

sk(t) =√

2PW (k)(t)A(k)(t) cos(ωct + θk) (1)

where P is the signal power identical for all users, ωc is thecarrier frequency, and θk is the initial carrier phase of user k.Since an AWGN channel is assumed in this section, the receivedsignal at the base station is given by

r(t)=K∑

k=1

√2PW (k)(t−τk)A(k)(t−τk) cos(ωct+φk)+n(t)

(2)

where φk = θk − ωcτk, τk is the delay for the signal of userk, and n(t) is the white Gaussian noise with zero mean andtwo-sided power spectral density N0/2. As shown in Fig. 2,the receiver processes r(t) with noncoherent square-law RAKEcombining and ML detection. Without loss of generality, weassume that user i is the intended user, and a Walsh symbolWλ(t) is sent by user i in the interval [0, T ). The output ofthe pth branch of the receiver, which corresponds to a Walshsymbol Wp(t) (p = 0, 1, . . . ,M − 1), is given by D

(i)λ,p =

|Z(i)Iλ,p |2 + |Z(i)Q

λ,p |2, where we use I and Q to denote in-phaseand quadrature branches, respectively, and

Z(i)Iλ,p =

√2T

T+τi∫τi

r(t)A(i)(t − τi)Wp(t − τi) cos(ωct)dt

=√

PTδλ,p cos φi + nI

+

√P

T

K∑k=1,k �=i

[Rk,i(τ̃k) + R̂k,i(τ̃k)

]cos φ̃k

Z(i)Qλ,p = −

√2T

T+τi∫τi

r(t)A(i)(t − τi)Wp(t − τi) sin(ωct)dt

=√

PTδλ,p sin φi + nQ

+

√P

T

K∑k=1,k �=i

[Rk,i(τ̃k) + R̂k,i(τ̃k)

]sin φ̃k (3)

where δλ,p = 1 for λ = p, and δλ,p = 0, otherwise. The phaseand delay are relative to those of user i. The noise components

1Usually, N and M are chosen such that N/M = Tw/Tc is an integer.


Fig. 2. Receiver block diagram for user i.

nI and nQ are mutually independent and have the same distrib-ution as n(t). The continuous-time partial correlation functionsRk,i(τ) and R̂k,i(τ) are defined by

Rk,i(τ)=

τ∫0

[Wµ(t−τ)A(k)(t−τ)

][Wp(t)A(i)(t)

]dt

R̂k,i(τ)=

T∫τ

[Wν(t−τ)A(k)(t−τ)

][Wp(t)A(i)(t)

]dt (4)

where Wµ(t−τ)=W(k)−1 (t−τ) and Wν(t−τ)=W

(k)0 (t−τ)

represent two consecutive Walsh symbols. Assuming that0 ≤ lTc ≤ τ ≤ (l + 1)Tc ≤ T , (4) can be expressed as

Rk,i(τ) = Ck,i(l − N)Tc + [Ck,i(l + 1 − N) − Ck,i(l − N)]

· (τ − lTc)

R̂k,i(τ) = Ck,i(l)Tc + [Ck,i(l + 1) − Ck,i(l)] (τ − lTc). (5)

Here, the discrete aperiodic cross function Ck,i(l) is com-plicated by three independent Walsh symbols Wµ(t − τ),Wν(t − τ), and Wp(t) as in (6), shown at the bottom ofthe page, where n = N/M is an integer, and �x� denotesthe integer part of a real number x. It can be seen from (6)that spreading is unavoidably coupled with M -ary orthogonalmodulation, which makes an MAI analysis rather involved.

B. MAI Analysis

By using the standard Gaussian approximation [6], the MAIon both I and Q branches has the same variance

σ2i,p =

P

2T

K∑k=1k �=i

E

{[Rk,i(τ̃k) + R̂k,i(τ̃k)

]2}

(7)

where E{·} denotes expectation. Specifically, averaging (7)over τ̃k yields

E{

R2k,i(τ̃k)+R̂2

k,i(τ̃k)}

=T 2

3N3

N−1∑l=1−N

[2C2

k,i(l)+Ck,i(l) · Ck,i(l+1)]

(8)

E{

2Rk,i(τ̃k)R̂k,i(τ̃k)}

=T 2

3N3

N−1∑l=0

[2Ck,i(l)Ck,i(l−N)+Ck,i(l)Ck,i(l+1−N)

+ Ck,i(l+1)Ck,i(l−N)

+ 2Ck,i(l+1)Ck,i(l+1−N)]. (9)

Then, averaging (8) and (9) over Wµ(t − τ̃k) and Wν(t − τ̃k)can be numerically performed. When random codes are em-ployed, (7) reduces to

σ2 =(K − 1)S2

3N(Random Codes) (10)

Ck,i(l) =

∑N−1−l

j=0

[wν,�j/n�a

(k)j

] [w

(i)p,�(j+l)/n�a

(i)j+l

], 0 ≤ l ≤ N − 1∑N−1+l

j=0

[wµ,�(j−l)/n�a

(k)j−l

] [w

(i)p,�j/n�a

(i)j

], 1 − N ≤ l ≤ 0

0, |l| ≥ N

(6)


where S2 = PT is the symbol energy, and the subscripts iand p are omitted as they are not needed. Although a simplerexpression for the variance of MAI is provided in [11], it is notaccurate enough according to the following arguments.

• The cross term detailed by (9) is not considered in [11].• The variance of MAI in [11] is expressed in terms of au-

tocorrelation functions rather than cross-correlation func-tions because the involved orthogonal Walsh modulationsymbols are not correctly averaged.

• When deterministic spreading sequences are employed,the variance of MAI is dependent on the subscripts i andp, as shown in (6) and (7)–(9). This is not taken intoconsideration in [11].

C. BER

When a signal is present at the pth correlator (i.e., p = λ),the correlator’s output is distributed as a noncentral chi-squarevariable with probability density function (pdf) [17]

fλ(x) =1

2σ̃2i,λ

exp

(−x + S2

2σ̃2i,λ

)I0

(√xS

σ̃2i,λ

)(11)

where σ̃2i,p = σ2

i,p + (N0/2) denotes the variance of the totalinterference including MAI and Gaussian noise. When a signalis absent, the output at correlator p follows the central chi-square distribution with pdf given by [17]

fp(x) =1

2σ̃2i,p

exp

(− x

2σ̃2i,p

)(p �= λ). (12)

By letting y = x/(2σ̃2i,λ), it follows that the probability of a

correct decision for user i is equal to

Pi,λ =

∞∫0

M−1∏

p=0,p�=λ

x∫0

fp(t)dt

fλ(x)dx

= exp

(− S2

2σ̃2i,λ

) ∞∫0

M−1∏

p=0,p�=λ

[1 − exp

(−y

σ̃2i,λ

σ̃2i,p

)]· exp(−y)I0

(√2y

S2

σ̃2i,λ

)dy. (13)

The final BER of user i can be readily obtained after averagingPi,λ over the transmitted Walsh symbol Wλ(t). Following [18],one may represent (13) as a sum of elementary functions.However, the number of terms in the sum is 2M−1, which leadsto prohibitive computational complexity even for moderatevalues of M . We therefore use numerical integration instead.

III. PERFORMANCE ANALYSIS ON MULTIPATH

RAYLEIGH-FADING CHANNEL

A. System Model

We proceed to consider an L-path Rayleigh-fading channel.For mathematical tractability, we assume, as in [10], a uniform

multipath profile. The equivalent baseband channel for user kmay be represented by

L∑l=1

αk,l exp(jϕk,l)δ(t − τk,l)

where αk,l, ϕk,l, and τk,l are the gain, phase, and delay ofthe lth path of user k, respectively, the path gain αk,l has theaverage power E{α2

k,l} = 1/L, and the delay τk,l is uniformlydistributed in [0, T ) for any k or l. The received signal at a basestation now becomes

r(t) =K∑

k=1

L∑l=1

αk,l

√2PW (k)(t − τk,l)A(k)(t − τk,l)

· cos(ωct + φk,l) + n(t) (14)

where φk,l = θk,l + ϕk,l − ωcτk,l. The receiver has a structuresimilar to that shown in Fig. 2, except for the replacement ofcorrelators by L-finger RAKEs with equal gain combining.The output of the pth RAKE of user i is given by D

(i)λ,p =∑L

l=1[|Z(i)Iλ,p,l|2 + |Z(i)Q

λ,p,l|2] with

Z(i)Iλ,p,l =

√PTδλ,pαi,l cos φ̃i,l+nI

+

√P

T

L∑l′=1,l′ �=l

[Ri,i(τ̃i,l′)+R̂i,i(τ̃i,l′)

]αi,l′ cos φ̃i,l′

+

√P

T

K∑k=1,k �=i

L∑l=1

[Rk,i(τ̃k,l)+R̂k,i(τ̃k,l)

]· αk,l cos φ̃k,l,

Z(i)Qλ,p,l =

√PTδλ,pαi,l sin φ̃i,l+nQ

+

√P

T

L∑l′=1,l′ �=l

[Ri,i(τ̃i,l′)+R̂i,i(τ̃i,l′)

]αi,l′ sin φ̃i,l′

+

√P

T

K∑k=1,k �=i

L∑l=1

[Rk,i(τ̃k,l)+R̂k,i(τ̃k,l)

]· αk,l sin φ̃k,l (15)

where all phases and delays, which are measured relative to thedesired lth path of user i, are uniformly distributed in [−π, π)and [0, T ), respectively.

B. BER

Based on the standard Gaussian approximation, self-interference and MAI (in both I and Q branches) can bemodeled, respectively, as zero-mean Gaussian random variableswith variance

σ2(ISI)i,p,l =

P

T· L−1

2LE

{[Ri,i(τ̃i,l′)+R̂i,i(τ̃i,l′)

]2}

(l′ �= l)

σ2(MAI)i,p,l =

P

2T

K∑k=1,k �=i

E

{[Rk,i(τ̃k,l)+R̂k,i(τ̃k,l)

]2}

. (16)


When the pth RAKE of user i matches the transmitted signalWλ(t) (i.e., p = λ), the output of the pth branch follows anoncentral chi-square pdf [17]

fλ(x|γ) =1

2σ̃2i,λ

(x

γ

)L−12

exp

(−x + γ

2σ̃2i,λ

)IL−1

(√xγ

σ̃2i,λ

)(17)

where σ̃2i,p

∆= σ2(ISI)i,p,l + σ

2(MAI)i,p,l + (N0/2) (l = 1, 2, . . . , L) is

independent of the subscript l, and γ = S2ΣLl=1α

2i,l. When no

desired signal is present (i.e., p �= λ), the output of the pthbranch follows a central chi-square pdf [17]

fp(x) =1(

2σ̃2i,p

)L Γ(L)xL−1 exp

(− x

2σ̃2i,p

)(p �= λ). (18)

Averaging (17) over γ leads to (see [10, eq. (27)])

fλ(x) =xL−1

Γ(L)(

S2

L + 2σ̃2i,λ

)Lexp

(− x

S2

L + 2σ̃2i,λ

). (19)

Letting y = x/(S2/L + 2σ̃2i,λ) and substituting (18) and (19)

into (13), we obtain the probability of a correct decision foruser i as in (20), which allows us to determine the final BER

Pi,λ =1

Γ(L)

∞∫0

{M−1∏

p=0,p�=λ

[1 − exp

(−y

[S2/L

2σ̃2i,p

+σ̃2

i,λ

σ̃2i,p

])

·L−1∑l=0

1l!

(y

[S2/L

2σ̃2i,p

+σ̃2

i,λ

σ̃2i,p

])l ]}yL−1 exp(−y)dy.

(20)

IV. SEQUENCE OPTIMIZATION USING EA

As will be shown later, the well-studied spreading sequencessuch as Gold and Kasami codes [1] only offer insignificant ad-vantage over random codes when applied to the reverse link ofDS-CDMA with noncoherent M -ary orthogonal modulation. Inthis section, a new method is proposed to find better spreadingsequences in terms of BER performance.

A. Problem Formulation

The variance of the MAI is a good performance measure fora set of spreading codes and indeed widely used in sequenceoptimization. However, since the BER is governed not onlyby the total MAI but also by the distribution of MAI acrossM branches (due to ML detection), a code set that minimizesthe total MAI is not necessarily optimal in terms of BER. Thismotivates us to directly use the average BER as our objectivefunction for sequence optimization. Let C denote a K × Ncode matrix composed of K binary spreading sequences. Thesearch for a set of K optimal sequences can be formulated as

Copt = arg minC∈C

BER(C) (21)

where C stands for a candidate code space. It is clear fromprevious sections that the average BER is dependent on K,N , M , and L; therefore, the obtained optimal code sets aredetermined by these parameters. Since N , M , and L are usuallyfixed, the optimal code sets only change with K. On the otherhand, K also has a significant impact on the convergence ofthe search algorithm because both the code matrix C and theaverage BER heavily depend on K.

Two issues remain to be addressed. First, we need to system-atically construct a large yet manageable candidate code space.Next, a powerful means should be used to tackle the nonlineardiscrete optimization problem.

B. Construction of Code Space

Given an N × N Hadamard matrix, it is shown [19] thatone can generate equivalent Hadamard matrices by changingthe signs of and/or permuting the columns of the originalmatrix. In this paper, we apply this technique to an arbitrary(not necessarily orthogonal) N × N matrix C0 to produce aspace of candidate code sets. In particular, only the permutationmanipulation is adopted since permutations yield a much largercode space than sign changes (clearly, N ! 2N for N ofinterest). The construction of a code space by permutations maybe illustrated by

P : C0 → C (22)

where C is the generated code space whose size is N !, andthe permutation operator P is characterized by a sequence{π1, π2, . . . , πN} that is a permuted version of the sequence{1, 2, . . . , N}.

While the above construction method (22) is very effectivein producing a large code space, its drawback is that as Nincreases, the size of the generated code space will soon becometoo large to be efficiently handled. This combinatorial explosionproblem can be managed by imposing a Kronecker productstructure on the original matrix C0, i.e.,

C0 = W ⊗ C̃0 (23)

where W and C̃0 are N1 × N1 and N2 × N2 matrices, re-spectively, ⊗ denotes the Kronecker product, and N1N2 = N .The motivations of using (23) are twofold. First, the Kroneckerproduct is mathematically well defined and thus widely used incode constructions. Second, combinatorial optimization prob-lems generally suffer from the “curse of dimensionality.” Tomake our optimization problem tractable for a code lengthof practical importance, searching over a code subspace isnecessary.

In this paper, W is chosen to be a Walsh–Hadamard matrix.We only need to consider C̃0. The permutation operator is nowgiven by

P̃ : C̃0 → C̃ (24)

which clearly indicates that the size of a candidate code spaceis reduced from N ! to N2! by using (24) and (23) insteadof (22).


C. EAs

We are facing a discrete combinatorial optimization prob-lem that is generally difficult to solve. Traditional methodsfor continuous functions are no longer applicable; hence, ex-haustive or random search [15], [16] is often used. Here, weadopt EAs not simply because of their increasing popularityacross numerous disciplines, but rather due to their particularsuitability to the discrete nature of our optimization problem.As a stochastic search procedure in essence, EAs [12]–[14]apply the Darwinian principle of survival of the fittest toproduce successively improved approximations to a solution.In comparison with the commonly used random search, EAscan be viewed as improved or smart random search, sincerandom search is totally blind with no control in the searchprocess, whereas EAs are guided to obtain better (no worseat least) solutions at each iteration. Moreover, the strategyof parallel search embedded in EAs makes it less likely toget stuck in the local optima, although a global optimum isnot always guaranteed. The interested reader is referred to[20]–[27] for more applications in communications includingsequence search, channel estimation, multiuser detection, andnetwork design.

Assume that the parameters (i.e., {π1, π2, . . . , πN} for thepermutation operator P) to be optimized can be described bya vector −→x . The basic procedure for implementing our EA isoutlined below. More details can be found in [12]–[14] andreferences therein.

Step 1) [Initialization]: Randomly generate a population of Pparent vectors {−→x1,

−→x2, . . . ,−→xP }, and find the optimal

vector −−→xopt in terms of the objective function.Step 2) [Mutation]: Produce Pm offspring vectors {−→y1,

−→y2,. . . ,−−→yPm

} by applying a mutation operator to Pm

vectors, respectively, which are randomly chosenfrom the P available parent vectors.

Step 3) [Crossover]: Produce Pc offspring vectors {−→z1 ,−→z2 ,. . . ,−→zPc

} by applying a crossover operator to Pc/2pairs of vectors, which are randomly chosen from theP available parent vectors.

Step 4) [Selection]: Compare the Pm + Pc offspring vectors{−→y1,

−→y2, . . . ,−−→yPm

;−→z1 ,−→z2 , . . . ,−→zPc} and the optimal

parent vector −−→xopt according to the objective func-tion. The P optimal vectors, which are denoted by{−→x1,

−→x2, . . . ,−→xP }, are selected as parent vectors for

the next iteration, and the best one among them isdenoted by −−→xopt.

Step 5) [Termination]: The process will stop if a prescribediteration number is attained; otherwise, go to Step 2.

In this paper, the mutation and crossover operators are definedas follows.

1) [Mutation for P]: Randomly choose two elementsπi and πj (1 ≤ i, j ≤ N , i �= j) in the sequence{π1, π2, . . . , πN} that is permuted from {1, 2, . . . , N},and exchange their positions to form a new sequence.

2) [Crossover for P]: We use the so-called order crossover[28, p. 286]. Randomly choose two cutpoints. The firstparent vector’s elements between the two cutpoints arecopied, while the remaining elements are taken from the

Fig. 3. BER of M -ary DS-CDMA on an AWGN channel when differentspreading sequences are employed.

Fig. 4. BER of M -ary DS-CDMA on a multipath (L = 3) Rayleigh-fadingchannel when different spreading sequences are employed.

beginning of the second parent vector respecting theirrelative ordering.

V. NUMERICAL RESULTS AND DISCUSSION

We first investigate the BER performance of M -ary DS-CDMA with Gold, Kasami,2 and random codes. Each of thesecodes stands on its own merit. Specifically, 1) Gold codes arefamous for their good periodic correlation properties and largenumber of available codes [17]. 2) Kasami codes are optimal inthe sense that their maximum periodic cross-correlation valuesachieve the Welch lower bound [17]. 3) Random codes aresimple to analyze and can characterize m-sequences, which areactually used in the reverse link of IS-95 systems. To ensurethat n = N/M is an integer, each code in the Gold and Kasamifamilies is appended by a “+1.” Note that in this way, the Goldcodes have become the so-called orthogonal Gold codes [29].

The average BER performance comparison of the Gold,Kasami, and random codes on AWGN and multipath Rayleigh-fading channels is shown in Figs. 3 and 4, respectively, where

2A small set of Kasami codes is considered in this paper.


the Gold codes of length 63 are generated from a preferred pairof primitive polynomials 103 and 147 in octal and the Kasamicodes from the polynomial 147 [30]. Simulation results arealso provided to examine the accuracy of the standard Gaussianapproximation method. It is observed that the difference amongthese sequences is in general small, especially when the numberof users is relatively large. This phenomenon can be explainedas follows: 1) The Gold and Kasami codes are considered tobe good or even optimal only from the perspective of periodiccorrelation function. This criterion alone however does notjustify their use in asynchronous cases such as the reverse linkof DS-CDMA, where a more appropriate criterion for spread-ing sequence selection is the aperiodic correlation functionamong spreading sequences [6]. 2) Spreading sequences arecomplicated and fundamentally changed by modulation. Forasynchronous M -ary DS-CDMA, there are M2 − M differentcorrelation functions, depending on which Walsh functions aretaken over two adjacent symbols. In contrast, for asynchronousBPSK DS-CDMA, we only need to consider two, i.e., theperiod (even) correlation function and the odd correlation func-tion. 3) The DS-CDMA system under consideration utilizesnoncoherent ML detection, which is far more complex thanthe correlation receiver. For these reasons, the conventionalsequence design strategy is not suited for the system studiedhere due to its complicated modulation and receiver structure.In addition, it can be seen from Figs. 3 and 4 that the theoreticalanalysis based on standard Gaussian approximation agrees wellwith the simulations when the number of users is of practicalinterest. Finally, we notice that clear error floors at high SNRcan be observed. This is because a single-user noncoherentdetection is used at the receiver side, which results in largeresidue MAI.

In view of this situation, we next use EA to search forbetter spreading sequences. We fix N = 128, M = 8, L = 3,and P = Pm = Pc = 10 in the sequel, and two combinations{N1 = 8, N2 = 16} and {N1 = 4, N2 = 64} will be consid-ered. For the case of {N1 = 8, N2 = 16}, the original codematrix C̃0 is given by Gold codes of length 15 derived froma pair of primitive polynomials 23 and 31 in octal,3 while forthe case of {N1 = 4, N2 = 64}, C̃0 is associated with Goldcodes of length 63 generated from a preferred pair of primitivepolynomials 103 and 147 in octal. To get SNR independentsequences, AWGN is neglected when performing sequenceoptimization. We plot in Fig. 5 two typical convergence curvesas a function of iteration number to illustrate the efficiency ofthe proposed method. Only the optimal value at each iterationand the results of the first 500 iterations are shown in Fig. 5.We can see that EA converges very fast when searching overa relatively small code space (N2 = 16, N2! ≈ 2.09 × 1013),whereas its convergence speed becomes very slow for a largecode space (N2 = 64, N2! ≈ 1.27 × 1089). Nevertheless, asexpected, searching over a larger code space leads to betterspreading codes, and EA is found to be able to successfullyovercome many local optima (reflected by the many plateaus).Therefore, our proposed approach can make a good tradeoff

3Strictly speaking, they are not Gold codes, but it does not matter for ourpurpose.

Fig. 5. Typical BER convergence curves of M -ary DS-CDMA on an AWGNchannel as a function of iteration number.

Fig. 6. Comparison of BER performance of M -ary DS-CDMA on an AWGNchannel: random versus optimized codes (K = 32).

between code performance and search complexity by selectinga proper value of N2 or N1.

Figs. 6 and 7 compare the BER performance of randomand optimized codes on an AWGN channel for K = 32 andK = 48, respectively. Compared with random codes, the opti-mized codes yield substantial performance improvement, espe-cially when the number of users is relatively small. On the otherhand, we can see that the performance gain of the optimizedcode over random codes will increase as N2 increases butwill decrease as K increases. In Figs. 8 and 9, we consider amultipath (L = 3) Rayleigh-fading channel where the optimalcodes (N1 = 8, N2 = 16) for K = 32 and K = 48 are givenby P̃opt : [10 12 6 2 7 11 5 8 16 14 13 4 9 1 15 3] andP̃opt : [8 7 11 13 12 6 14 10 3 15 1 5 2 16 4 9], respectively.We have results similar to those shown in Figs. 6 and 7.The only difference is that the performance gain due to se-quence optimization becomes less significant compared to theAWGN case.

Finally, we discuss the possible gain achieved from sequencedesign. For the DS-CDMA system investigated in this pa-per, there are three sources of randomness that destroy the


Fig. 7. Comparison of BER performance of M -ary DS-CDMA on an AWGNchannel: random versus optimized codes (K = 48).

Fig. 8. Comparison of BER performance of M -ary DS-CDMA on a multipath(L = 3) Rayleigh-fading channel: random versus optimized codes (K = 32).

orthogonality among the spreading sequences of different users:random time delays (due to asynchronous transmission); ran-dom path gains, phases, and delays (due to multipath fading);and M -ary orthogonal modulation. A heuristic conclusion isthat the more sources of randomness, the lesser the gain thatcan be obtained from sequence design, no matter whetherthe conventional (periodic correlation function based) or theproposed approach is used.

VI. CONCLUSION

The reverse link of DS-CDMA systems with M -ary orthog-onal modulation and noncoherent ML detection has been con-sidered in this paper. We first investigated its BER performanceon both AWGN and multipath Rayleigh-fading channels, takinginto account the real effect of deterministic spreading sequenceson the system performance. The well-known Gold and Kasamicodes were found to offer no obvious advantage over randomcodes. Then we proposed a spreading sequence optimizationscheme, which uses an EA to efficiently search for the optimalsequences over a large systematically generated code space.

Fig. 9. Comparison of BER performance of M -ary DS-CDMA on a multipath(L = 3) Rayleigh-fading channel: random versus optimized codes (K = 48).

Numerical examples demonstrate that noticeable performanceimprovement can be obtained with the proposed sequenceoptimization approach, especially when the number of users isrelatively small or an AWGN channel is considered.

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Qinghua Shi received the Ph.D. degree in wire-less communications from Southeast University,Nanjing, China, in 2000.

From September 2000 to January 2003, he wasa Research Scientist with the Centre for WirelessCommunications, University of Oulu, Oulu, Finland.From February 2003 to January 2005, he was aResearch Fellow with the Department of Electric En-gineering, City University of Hong Kong, Kowloon,Hong Kong. Since February 2005, he has been withNanyang Technological University, Singapore, as a

Research Fellow. His research interests include spread spectrum, CDMA,multicarrier modulation, and space–time modulation/coding.

Q. T. Zhang (S’84–M’85–SM’95) received theB.Eng. degree in wireless communications fromTsinghua University, Beijing, China, and the M.Eng.degree in wireless communications from the SouthChina University of Technology, Guangzhou, China,and the Ph.D. degree in electrical engineering fromMcMaster University, Hamilton, ON, Canada.

After graduating from McMaster University in1986, he held a research position and was an Ad-junct Assistant Professor at the same institution. InJanuary 1992, he joined the Satellite and Commu-

nication Systems Division, Spar Aerospace Ltd., Montreal, QC, Canada, asa Senior Member of Technical Staff, working on the development and man-ufacturing of the Radar Satellite (Radarsat). He joined Ryerson PolytechnicUniversity, Toronto, ON, in 1993 and became a Professor in 1999. In 1999,he took one-year sabbatical leave at the National University of Singapore. Heis currently a Professor with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong. His research interest is onthe transmission and reception over fading channels with a current focus onwireless MIMO and cross-layer design/optimization.

Dr. Zhang is currently an Associate Editor for the IEEE COMMUNICATIONS

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