1832 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004

Diversity Analysis of Space–Time Modulation OverTime-Correlated Rayleigh-Fading Channels

Weifeng Su, Member, IEEE, Zoltan Safar, Member, IEEE, andK. J. Ray Liu, Fellow, IEEE

Abstract—Most space–time codes in the literature were proposed basedon two ideal channel conditions: either quasi-static or rapid fading. How-ever, these codes may suffer performance degradation due to temporal cor-relation caused by the movement of the mobile terminal or imperfect in-terleaving. In this correspondence, we provide a novel analytical frame-work for the diversity analysis of space–timemodulation in time-correlatedfading environment. We show that the space–time signals of square sizeachieving full diversity in quasi-static fading channels also achieve full di-versity in time-correlated fading channels, independently of the time corre-lation matrix. Consequently, various classes of space–time signals designedfor quasi-static fading channels can also be used for full-diversity trans-mission over time-correlated fading channels. Moreover, we show that ifthe time correlation matrix is of full rank, the design criteria for time-cor-related fading channels are the same as those for rapid fading channels.To illustrate the theoretical results, some simulations were also performedunder various temporal fading conditions.

Index Terms—Diversity, interleaving, multiple antennas, space–timemodulation, time-correlated fading.

I. INTRODUCTION

The challenges imposed by the wireless propagation environmentand the need for high data rate communication links have started a questfor methods to exploit a new resource: the spatial dimension. The ideaof equipping the transmitter and the receiver with multiple antennasand developing coding and modulation schemes to improve the perfor-mance of communication systems have gained increasing popularity,which is demonstrated by the abundant literature that has been pub-lished on this topic recently.

However, most results on space–time coding have been proposedbased on two ideal channel conditions: either spatially independentquasi-static or rapid fading [1]–[16]. These codes may suffer perfor-mance degradation if there is spatial or temporal correlation existingin wireless channel. In addition, different mobile stations or a mobilemoving through different geographical locations may experience dif-ferent channel correlations. This motivates the development of robustcoding and modulation methods that can guarantee good performanceover various channel conditions.

For the quasi-static channel model, the authors of [17] investigatedthe achievable diversity order as a function of spatial correlation, takinginto account some physical propagation parameters. The problem of

Manuscript received May 28, 2003; revised August 13, 2003. This work wassupported in part by the U.S. Army Research Laboratory under CooperativeAgreement DAAD 190120011. The material in this correspondence was pre-sented in part at the IEEE International Conference on Communications, An-chorage, AK, May 2003.

W. Su and K. J. R. Liu are with the Department of Electrical and Com-puter Engineering and Institute for Systems Research, University of Maryland,College Park, MD 20742 USA (e-mail: weifeng@eng.umd.edu; kjrliu@eng.umd.edu).

Z. Safar was with the Department of Electrical and Computer Engineeringand Institute for Systems Research, University of Maryland, College Park, MD20742 USA. He is now with the Department of Innovation, IT University ofCopenhagen, Copenhagen, Denmark (e-mail: safar@itu.dk).

Communicated by D. N. C. Tse, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2004.831848

code design for correlated fading channels was addressed in [18], [19],and general performance criteria were derived for space–time-corre-lated Rayleigh-fading channels. In [20], it was assumed that the channelstays constant for a number of channel symbol periods equal to thenumber of transmit antennas. The performance criteria were obtainedfor this channel model, and hand-crafted trellis codes were proposedcombining multiple trellis coded modulation with Alamouti’s scheme[4]. The general performance criteria of [18], [19] were further simpli-fied in [21], assuming that the space–time correlation matrix is of fullrank. In this case, the design criteria were simplified to those for rapidfading channels.In [22], characterizing the performance of space–time codes over

space–time-correlated Rayleigh-fading channels was also considered.The minimum diversity order achieved over all space–time correlationmatrices of a given rank was defined as the measure of robustness.The relationship between the robustness (diversity) and the rank of thespace–time correlation matrix was also established, and space–timecodes designed for the independent fading channel model wereproposed for communication over space–time-correlated fadingchannels.In this correspondence, we analyze the performance of space–time

modulation in time-correlated fading environment. We assume that thewireless channel exhibits temporal correlation, but there is no spatialcorrelation between the transmit and the receive antennas. This as-sumption is true if the transmit and receive antennas are placed farenough from each other. However, temporal correlation may be causedby the movement of the mobile terminal, or imperfect time interleavingdue to the constraint of allowable decoding delay [23], [24]. Thus, it isof interest to design robust space–time signals that can provide goodperformance over all time-correlated fading conditions.First, we derive the performance criteria for the time-correlated

Rayleigh-fading channel model. Then, we show that the space–timesignals of square size achieving full diversity in quasi-static fadingchannels can also achieve full diversity in time-correlated fadingchannels, irrespectively of the time correlation matrix. We find thatin this case, the maximum achievable diversity is only a function ofthe number of the transmit and receive antennas, and is not limitedby the rank of the correlation matrix, in contrast to the results of[22] obtained for spatially nonwhite channels. As a consequence,various classes of space–time signals designed for quasi-static fadingchannels, for example, cyclic codes [10], codes from orthogonaldesigns [4]–[7], parametric codes [13], Cayley codes [12], and soon, can be used for full-diversity transmission over time-correlatedfading channels, providing robust performance over a wide range ofchannel conditions. We also show that if the time correlation matrixis of full rank, the design criteria for time-correlated fading channelsare the same as those for rapid fading channels, in agreement withthe results of [21].The correspondence is organized as follows. Section II will introduce

the channel model and briefly summarize the relevant results from pre-vious work. The design criteria for the time-correlated fading channelmodel will be derived in Section III. Section IV will provide some sim-ulation results under various temporal fading conditions. The conclu-sion will be given in Section V.

II. CHANNEL MODEL AND BACKGROUND

We consider a wireless communication system withM transmit an-tennas and N receive antennas. The space–time modulator divides theinput bit stream into b bit long blocks, and for each block, it selects

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 1833

one space–time signal from the signal set of size L = 2b. The selectedsignal is then transmitted through the channel over theM transmit an-tennas and T time slots. Each space–time signal can be expressed as aT �M matrix

C =

c11 c21 . . . cM1

c12 c22 . . . cM2...

.... . .

...c1T c2T . . . cMT T�M

(1)

where cit denotes the channel symbol transmitted by transmit antennai, i = 1; 2; . . . ;M , at discrete time t, t = 1; 2; . . . ; T . The space–timesignals are assumed to satisfy the energy constraint

EkCk2F = MT (2)

where kCkF is the Frobenius norm 1 of C , and E stands for expecta-tion.

The received signal yjt at receive antenna j at time t is given by

yjt =

�

M

M

i=1

cithi;j(t) + z

jt ; t = 1; 2; . . . ; T (3)

where zjt is the complex additive white Gaussian noise (AWGN) at re-ceive antenna j at time t with zero mean and unit variance, and hi;j(t)is the channel coefficient from transmit antenna i to receive antenna jat time t. If the system has an interleaver, the hi;j(t)’s are the equiv-alent channel coefficients after interleaving. The channel coefficientsare modeled as zero-mean complex Gaussian random variables withunit variance. These coefficients are assumed to be known at the re-ceiver, but unknown at the transmitter. Moreover, we assume that thechannel fading has only temporal correlation, i.e., the channel coeffi-cients hi;j(t) are independent for different indexes (i; j) and depen-dent only in the temporal domain. The factor � in (3) is the averagesignal-to-noise ratio (SNR) per space–time signal at each receive an-tenna, and it is independent of the number of transmit antennas.

The received signal (3) can be rewritten in vector form as [18], [19]

YYY =�

MDDDHHH +ZZZ (4)

whereDDD is an NT �MNT matrix constructed from the space–timesignal matrix C as shown in (5) at the bottom of the page, in which

Di = diag ci1; c

i2; . . . ; c

iT ; i = 1; 2; . . . ;M: (6)

1The Frobenius norm of C is defined as

kCk2F = tr(CHC) = tr(CCH) =

T

t=1

M

i=1

jcitj2:

Each Di in (6) is related to the ith column of the space–time signalmatrix C . The channel vectorHHH of sizeMNT � 1 is formatted as (7)at the bottom of the page, where

hhhi;j = [ hi;j(1) hi;j(2) . . . hi;j(T ) ]T:

The received signal vector YYY of size NT � 1 is given by

YYY = [ y11 . . . y1T y21 . . . y2T . . . yN1 . . . yNT ]T (8)

and the noise vector ZZZ has the form

ZZZ = [ z11 . . . z1T z21 . . . z2T . . . zN1 . . . zNT ]T : (9)

Suppose thatDDD and ~D~D~D are two different matrices related to two dif-ferent space–time signals C and ~C , respectively. Then, the pairwiseerror probability betweenDDD and ~D~D~D can be upper-bounded as [18], [19]

P (DDD! ~D~D~D) � 2K � 1

K

K

i=1

i

�1

�

M

�K

(10)

whereK is the rank of (DDD� ~D~D~D)RRR(DDD� ~D~D~D)H, 1; 2; . . . ; K are thenonzero eigenvalues of (DDD� ~D~D~D)RRR(DDD� ~D~D~D)H, andRRR = EfHHHHHH

Hg isthe correlation matrix ofHHH . The superscriptH stands for the complexconjugate and transpose of a matrix.Based on the upper bound on the pairwise error probability (10), a

general code design criterion has been proposed in [18], [19]: the min-imum rank of (DDD� ~D~D~D)RRR(DDD� ~D~D~D)H should be as large as possible, andthe minimum value of the product K

i=1 i should also be maximized.

This criterion is consistent with the well-known criteria [1], [2] for twoideal cases: the quasi-static and the rapid-fading channel models whichcan be summarized as follows.

• For quasi-static fading channels: The minimum rank of

� (C � ~C)(C � ~C)H (11)

over all pairs of distinct signals C and ~C should be as large aspossible. If� is of full rank for any pair of distinct signalsC and~C , then the diversity product [10], [11] is given by

�static =1

2pM

minC 6=~C

jdet(�)j (12)

which is related to the coding advantage and should also be max-imized.

• For rapid-fading channels: The minimum number of nonzerorows of C � ~C should be as large as possible for any pair ofdistinct signals C and ~C . If for any pair of distinct signals C and~C , there is no zero row in C � ~C , then the diversity product,given by

�rapid =1

2pM

minC 6=~C

T

t=1

kccct � ~c~c~ctk2F (13)

should be maximized. In (13), ccct and ~c~c~ct are the t-th rows of Cand ~C, respectively.

DDD =

D1 D2 . . . DM 0 0 . . . 0 . . . 0 0 . . . 0

0 0 . . . 0 D1 D2 . . . DM . . . 0 0 . . . 0...

.... . .

...0 0 . . . 0 0 0 . . . 0 . . . D1 D2 . . . DM NT�MNT

(5)

HHH = [hhhT1;1 . . . hhhTM;1 hhhT1;2 . . . hhhTM;2 . . . hhhT1;N . . . hhhTM;N ]T (7)

1834 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004

III. DESIGN CRITERIA FOR TIME-CORRELATED FADING CHANNELS

In this section, we derive the design criteria and investigate theachievable performance limits of space–time coded communicationsystems assuming only time-domain correlation. In this case, thechannel correlation matrixRRR of sizeMNT �MNT becomes

RRR = EfHHHHHHHg

= diag R1;1; . . . ; RM;1; R1;2; . . . ; RM;2;

. . . . . . ; R1;N ; . . . ; RM;N

where

Ri;j = E hhhi;jhhhH

i;j

is the time correlation matrix of the channel coefficients from transmitantenna i to receive antenna j. Assuming that all of the time correlationmatrices Ri;j are the same, which is true for the Jakes fading model[26], the correlation matrix can be expressed as

RRR = IMN R (14)

where denotes the tensor product, IMN is the identity matrix of sizeMN �MN , and R is the time-correlation matrix, defined as

R Ri;j =

r1;1 . . . r1;T

. . . . . . . . .

rT;1 . . . rT;T T�T

:

Using (5), (6), and (14), we obtain (15) at the bottom of the page, where� denotes the Hadamard product,2 and � is defined in (11). Substi-tuting (15) into (10), the pairwise error probability between C and ~Ccan be upper-bounded as

P (C ! ~C) �2rN � 1

rN

r

i=1

�i

�N

�

M

�rN

(16)

where r is the rank of��R, and �1; �2; . . . ; �r are the nonzero eigen-values of��R. As a consequence, we can formulate the design criteriafor time-correlated fading channels as follows.

a) Design for diversity advantage: The minimum rank of � � Rover all pairs of distinct signals C and ~C should be as large aspossible.

2Suppose that A = fa g and B = fb g are two matrices of sizem� n.The Hadamard product of A and B is defined as

A �B =

a1;1b1;1 . . . a1;nb1;n

. . . . . . . . .

am;1bm;1 . . . am;nbm;n

:

b) Design for coding advantage: The minimum value of the productr

i=1�i over all pairs of distinct signals C and ~C should be

maximized.

In the sequel, we will discuss the ultimate limits on the maximumachievable diversity imposed by the time-correlated channel model.First, we will provide the general performance limits, and then wewill describe some results obtained for two special cases: squarespace–time signals with an arbitrary time correlation matrix, andnonsquare space–time signals with a full rank time correlation matrix.If the minimum rank of� �R is � for any pair of distinct signals C

and ~C , we say that the set of space–time signals achieves a diversity of�N . For fixed time durationT , the number of transmit antennasM , andtime correlation matrix R, the maximum achievable diversity or fulldiversity is defined as the maximum diversity level that can be achievedby space–time signals of size T � M . For example, for quasi-staticfading channels

R =

1 1 . . . 1

1 1 . . . 1...

.... . .

...1 1 . . . 1

T�T

:

In this case, the maximum achievable diversity is min(M;T )N . Forrapid-fading channels, R = IT , so the maximum achievable diversityis TN .Assume that the time correlation matrix R is of rank �(1 � � �

T ). Since the matrix � � R is of size T � T , its rank is at most T .Furthermore, according to a rank inequality onHadamard products ([27p. 307]), the rank of matrix � �R can be upper-bounded as

rank(� �R) � rank(�)rank(R):

The rank of � cannot be greater than min(M;T ). Therefore, forspace–time signals of size T �M operating in time-correlated fadingenvironment, the maximum achievable diversity is upper-bounded bymin(M�; T )N , where � is the rank of the time-correlation matrix R.For space–time signals of square size, i.e., T = M , the maximum

achievable diversity cannot be greater than MN . Now we will showthat this upper bound can be achieved for any time-correlated fadingchannel. Note that both � and R are nonnegative definite, and all ofthe diagonal entries ofR are nonzero. Therefore, we can apply Schur’stheorem on Hadamard products ([27, p. 309]): if� is positive definite(i.e., of full rank), then��R is also positive definite (i.e., of full rank).As a consequence, we arrive at the following theorem.

Theorem 1: If a set of space–time signals of sizeM �M achievesfull diversity (MN) for quasi-static fading channels, then it alsoachieves full diversity (MN) for any time-correlated fading channel,independently of the time correlation matrix R.

(DDD � ~D~D~D)RRR(DDD � ~D~D~D)H = IN

M

i=1

(Di � ~Di)R(Di � ~Di)H

= IN

M

i=1

jci1 � ~ci1j2r1;1

M

i=1

(ci1 � ~ci1)(ci2 � ~ci2)

�r1;2 . . .M

i=1

(ci1 � ~ci1)(ciT � ~ciT )

�r1;T

M

i=1

(ci2 � ~ci2)(ci1 � ~ci1)

�r2;1M

i=1

jci2 � ~ci2j2r2;2 . . .

M

i=1

(ci2 � ~ci2)(ciT � ~ciT )

�r2;T

......

. . ....

M

i=1

(ciT � ~ciT )(ci1 � ~ci1)

�rT;1M

i=1

(ciT � ~ciT )(ci2 � ~ci2)

�rT;2 . . .M

i=1

jciT � ~ciT j2rT;T

= IN (C � ~C)(C � ~C)H �R

= IN f� �Rg (15)

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 1835

The result in Theorem 1 is very interesting. It is well known thatif a set of space–time signals of square size achieves full diversity forquasi-static fading channels, then it also achieves full diversity for rapidfading channels [3]. However, it is not obvious that it can also achievefull diversity for any time-correlated fading channels. It follows fromTheorem 1 that all of the space–time signals of square size designed forquasi-static fading channels can also be used to achieve full diversityin any time-correlated fading environment.

Theorem 1 does not hold for nonsquare signals (i.e., T 6= M ). How-ever, if the time correlation matrix is of full rank, we can establish an-other result. We observe that the diagonal entries of� are kccct� ~c~c~ctk2F ,t = 1; 2; . . . ; T , where ccct and ~c~c~ct are the tth rows of C and ~C , respec-tively. Thus, we can apply Schur’s theorem again: if all of kccct�~c~c~ctk2F ’sare nonzero for any pair of distinct signals C and ~C , then � �R is offull rank whenever the time correlation matrix R is of full rank. Thisimplies that the maximum achievable diversity is TN for any full-ranktime correlation matrix R. This result is summarized in the followingtheorem.

Theorem 2: If a set of space–time signals of size T �M achievesfull diversity (TN) for rapid fading channels, then it also achieves fulldiversity (TN) for any time-correlated fading channel, provided thatthe time correlation matrix R is of full rank.

From Theorem 1, we observe that the space–time signals of squaresize achieving full diversity in quasi-static fading channels can alsoachieve full diversity in time-correlated fading channels, irrespectiveof the time correlation matrix. Thus, an efficient way to design ro-bust space–time signals for time-correlated fading channels is to con-sider the construction of space–time signals of square size. In this case,the problem of designing robust space–time signals for time-correlatedfading channels is reduced to that of designing space–time signals forquasi-static fading channels. It follows that the abundant classes ofspace–time signals of square size designed for quasi-static fading chan-nels, for example, cyclic codes [10], codes from orthogonal designs[4]–[7], parametric codes [13], Cayley codes [12], and so on, may alsobe used for time-correlated fading channels.

From Theorem 2, we can also see that for channels exhibiting lowtemporal correlation, i.e., the time correlation matrix R is of full rank,the problem of designing space–time signals is equivalent to that ofdesigning space–time signals for rapid fading channels. In this case,one may consider the construction of space–time signals of nonsquaresize to achieve higher diversity order.

IV. SIMULATION RESULTS

To illustrate the above analytical results, we performed some com-puter simulations. We considered downlink transmission from a basestation with multiple transmit antennas to a mobile station. Assumingthat the mobile station is moving at 55mi/h, the carrier frequency is 900MHz, and the bandwidth is 30 kHz, the normalized Doppler frequencyis approximately fD = 0:0025. The fading channels were modeled bythe Jakes fading model [26], in which the temporal correlation is deter-mined by the normalized Doppler frequency fD .

The above channel model results in a very slowly time-varyingchannel. For space–time signals of small size, the channel will be ap-proximately constant, so we used this channel model when simulatingthe quasi-static scenario. The time-correlated case was simulatedby assuming that the temporal correlation was caused by imperfectinterleaving. We used the slowly time-varying channel model with theTakeshita–Constello interleaver [25] given by

�(i) = modi(i+ 1)

2; I ; i = 0; 1; 2; . . . ; I � 1 (17)

and the length of interleaving, I , was set to 128. In this case, we alsocalculated the entries of 2 � 2 and 4 � 4 temporal correlation matricesnumerically. Averaging over 100 000 realizations of the fading channelwith fD = 0:0025 and using the Takeshita–Constello interleaver, themagnitudes of the entries of the 2 � 2 correlation matrix were obtainedas

0:9967 0:8384

0:8384 0:9967(18)

and the magnitudes of the values in the 4 � 4 correlation matrix wereobtained as

0:9967 0:8384 0:8437 0:8310

0:8384 0:9967 0:8384 0:8437

0:8437 0:8384 0:9967 0:8384

0:8310 0:8437 0:8384 0:9967

: (19)

The fast fading channel model was simulated by generating indepen-dent channel coefficients for each discrete-time instant. We comparedthe performance of each space–time modulation scheme over quasi-static channels, shown as solid lines with stars (“�”), time-correlatedchannels, shown as dotted lines with dots (“�”), and rapid fading chan-nels, shown as dashed lines with circles (“�”).In case of two transmit antennas, the following space–time signals

with L = 4 elements are optimal from the viewpoint of maximizingdiversity product in quasi-static fading channels [12], [13]

V0 =2

3

jjj 1� jjj

�1� jjj �jjj ; V1 =2

3

�jjj �1� jjj

1� jjj jjj

V2 =2

3

�jjj 1 + jjj

�1 + jjj jjj; V3 =

2

3

jjj �1 + jjj

1 + jjj �jjjwhere jjj =

p�1. The spectral efficiency of this scheme is 1 bit/s/Hz.Fig. 1 depicts the performance of this scheme with one receive antennaunder the three channel conditions.We can see that all three curves havethe same diversity order, consistent with the results of Theorem 1. Theperformance over the time-correlated channel (with the time correla-tion matrix in (18)) is close to the performance over the quasi-staticfading channel.Fig. 2 provides the simulation results for the Alamouti scheme [4]

with quaternary phase-shift keying (QPSK) for two transmit and one re-ceive antennas, giving a spectral efficiency of 2 bits/s/Hz. We observethat all the bit-error rate curves have approximately the same asymp-totic slope under the three channel conditions. Moreover, the perfor-mance of the code over the quasi-static channel is the best, and the per-formance over the rapid fading channel is the worst. Fig. 3 shows theperformance of the parametric code [13] with L = 16 for two transmitand one receive antennas. The spectral efficiency of this scheme is also2 bits/s/Hz. We observe that the performance of this code is almost thesame for the three different channel conditions.We also simulated a signal set designed for four transmit antennas.

We chose the full-diversity quasi-orthogonal space–time block code [9]given by

x1 x2 x3 x4

�x�2 x�1 �x�4 x�3

x3 x4 x1 x2

�x�4 x�3 �x�2 x�1

(20)

where x1 and x2 were chosen from QPSK constellation A =f�1;�jjjg, and x3 and x4 were chosen from the rotated QPSKconstellation ejjj A=f�ejjj ;�ejjj g. The spectral efficiency of thisscheme is 2 bits/s/Hz. Fig. 4 depicts the performance of this designwith one receive antenna. The figure shows that all the bit-error ratecurves have approximately the same asymptotic slope under the threechannel conditions.

1836 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004

Fig. 1. Bit-error rate performance of the optimal space–time signals with L = 4 elements.

Fig. 2. Bit-error rate performance of the Alamouti scheme with QPSK.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004 1837

Fig. 3. Bit-error rate performance of the parametric code with L = 16.

Fig. 4. Bit-error rate performance of the full-diversity quasi-orthogonal design with QPSK.

1838 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004

Fig. 5. Bit-error rate performance of the repetition of the Alamouti scheme with QPSK.

The simulations in Figs. 1–4 confirm the theoretical results of The-orem 1. The additional observation is that in Figs. 1–4, the perfor-mance of the simulated schemes over quasi-static fading channels isbetter than that over rapid-fading channels. It seems that for these mod-ulation schemes, the worst channel is the rapid-fading channel, notthe quasi-static fading channel, in contrast to the observations fromspace–time trellis codes [2], [20], [21].

From Theorem 2, we also know that in case of rapid-fading chan-nels (low temporal correlation), one may consider the construction ofnonsquare space–time signals to achieve higher diversity order. For ex-ample, we constructed a design for two transmit antennas by repeatingthe Alamouti scheme two times as follows:

x1 �x�

2 x1 �x�

2

x2 x�

1 x2 x�

1

T

(21)

where x1 and x2 are chosen from QPSK constellation. The spectralefficiency of this scheme is 1 bit/s/Hz. Fig. 5 depicts the performanceof this design with one receive antenna. We can see that the bit-errorrate curve of this design over the time-correlated fading channel (withthe time correlation matrix in (19)) has a better diversity order thanthat over the quasi-static fading channel. We observe a 2-dB gain at abit-error rate of 10�4. Compared to the bit-error rate curve over therapid-fading channel, there is an additional 2-dB gain that can be fur-ther obtained if we increase the interleaving size.

V. CONCLUSION

In this correspondence, we provided a new approach to the per-formance analysis of space–time modulation for time-correlatedRayleigh-fading channels. We showed that the space–time signals ofsquare size achieving full diversity in quasi-static fading environment

can also achieve full diversity in any time-correlated fading environ-ment, independently of the time correlation matrix. This result impliesthat various classes of space–time signals of square size designedfor quasi-static fading channels may also be used for time-correlatedfading channels. The simulations confirmed our theoretical results. Inaddition, we observed that in case of transmitting space–time signalsof square size over spatially independent channels, the quasi-staticfading channel model seems to be the best scenario, so the use ofinterleavers does not seem beneficial in this case.We also showed that if the time correlation matrix is of full rank, the

design criterion for time-correlated fading channels is the same as thatfor rapid-fading channels. In this case, one may consider the construc-tion of space–time signals of nonsquare size to achieve higher diver-sity order. The simulation results showed that the performance of thenonsquare design over the rapid fading channel had a better diversityorder than that over the quasi-static fading situation. The interleavednonsquare design also outperformed the noninterleaved (quasi-static)scenario, suggesting that the use of interleavers can improve the per-formance of nonsquare space–time signals.

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New Family of -ary Sequences With Optimal CorrelationProperty and Large Linear Span

Ji-Woong Jang, Young-Sik Kim, Jong-Seon No, Member, IEEE, andTor Helleseth, Fellow, IEEE

Abstract—For an odd prime and integers and such that =(2 + 1) , a new family of -ary sequences of period 1 with op-timal correlation property is constructed using the -ary Helleseth–Gongsequences with ideal autocorrelation, where the size of the sequence familyis . That is, the maximum nontrivial correlation value of all pairsof distinct sequences in the family does not exceed + 1, which meansthe family has optimal correlation in terms of Welch’s lower bound. Thesymbol distribution of the sequences in the family is enumerated. It is alsoshown that the linear span of the sequences in the family is ( + 2)except for the -sequence in the family.

Index Terms—Family of sequences, optimal correlation, -ary se-quences.

I. INTRODUCTION

In the wireless communication systems employing code-divisionmultiple-access (CDMA) scheme, a signature sequence is assignedto each user, which makes it possible to distinguish his signal fromthose of the other users. In design of sequences for CDMA system, themost important properties of the sequences are low periodic correlationbetween all pairs of distinct sequences and large family size. For an oddprime p, families of p-ary sequences of period pn � 1with optimal cor-relation property have been found, where the optimality of correlationvaluesmeans thatmaximummagnitude of out-of-phase autocorrelationand cross-correlation values of any pairs of sequences of period pn � 1in the family is upper-bounded by Rmax = p + 1. Sidelnikovconstructed a family of p-ary sequences with optimal correlation prop-erty and a family of prime-phase sequences with optimal correlationproperty was introduced by Kumar and Moreno [3]. By extending thealphabet size, Liu and Komo [7] constructed p-ary Kasami sequences.The family of p-ary bent sequences also has the optimal correlationproperty. Using the p-ary bent functions given by Kumar and Moreno,a family of balanced p-ary sequences with optimal correlation propertywas constructed by Moriuchi and Imamura [9]. The known families ofp-ary sequences of period pn � 1with optimal correlation property arelisted in Table I. The family size of the sequences due to Sidelnikov andto Kumar and Moreno are larger than that of the others in Table I. Butthe linear span of the sequences due to Sidelnikov and to Kumar andMoreno are much smaller than those of the others.In this correspondence, for an odd prime p and integers n;m; and k

such that n = (2m+ 1)k, a new family of p-ary sequences of periodpn�1 with optimal correlation property is constructed using the p-aryHelleseth–Gong sequences with ideal autocorrelation, where the sizeof the sequence family is pn. That is, the maximum nontrivial corre-lation value Rmax of all pairs of distinct sequences in the family doesnot exceed p + 1, which means the family has optimal correlationwith respect to Welch’s lower bound. The symbol distribution of the

Manuscript received January 9, 2003; revised February 18, 2004. This workwas supported in part by the Korean Ministry of Information and Communica-tions and the Norwegian Research Council.

J.-W. Jang, Y.-S. Kim, and J.-S. No are with the School of Electrical En-gineering and Computer Science, Seoul National University, Seoul 151-742,Korea (e-mail: jsno@snu.ac.kr).

T. Helleseth is with the Department of Informatics, University of Bergen,N-5020 Bergen, Norway (e-mail: Tor.Helleseth@ii.uib.no).

Communicated by K. G. Paterson, Associate Editor for Sequences.Digital Object Identifier 10.1109/TIT.2004.831837

0018-9448/04$20.00 © 2004 IEEE