Performance analysis of a wireless multirate direct-sequence CDMA using fast Walsh transform and decorrelating detection

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000 1405

Performance Analysis of a Wireless MultirateDirect-Sequence CDMA Using Fast Walsh Transform

and Decorrelating DetectionMohamed F. Madkour, Member, IEEE,and Someshwar C. Gupta, Fellow, IEEE

Abstract—In this paper, we analyze the performance of aflexible multirate scheme for direct-sequence code-division mul-tiple-access (CDMA) mobile radio systems. The proposed schemeuses a variable processing gain serial pseudonoise modulation as amultirate strategy. To reduce the interference effects, the CDMAsystem utilizes the coherent fast Walsh transform transmissiontechnique. The proposed scheme can be used in the reverselink (mobile-to-base station) of the upcoming third-generationwide-band CDMA system (has the feature of coherent reverselink). We analyze the system performance with and withoutusing a decorrelating multiuser detector. The uncoded bit-errorprobability (BEP) with and without decorrelating detection on amultipath fading channel is derived analytically. In addition, thevalidity of the analysis results is demonstrated by computer sim-ulations using the IMT-2000 vehicular multipath channel model.In order to make sure that the proposed processing techniques donot distort the soft values at the demodulator output, the proposedmultirate scheme is also simulated in case of using turbo codes.The turbo-coded BEP is calculated for different user data ratesand different number of decoding iterations.

Index Terms—Multiple-access methods, multirate scheme, mul-tiuser detection, spread-spectrum techniques, wide-band CDMA.

I. INTRODUCTION

I T IS well recognized that code-division multiple access(CDMA), based on spread spectrum (SS) techniques,

provides high capacity, low interference averaging andanti-multipath capabilities [1]–[3]. CDMA is also emergingas one of the air interfaces for the third-generation mobileradio systems [4], [5]. This third-generation system requiresdata rates considerably higher than 9.6 kbits/s, and its wirelessair interface must be able to deal with multipe-user data ratesand for a greater number of users in order to meet the servicerequirements. However, the ability to achieve high bit rates atlow error rates over wireless channel is severely limited by thepropagation characteristics of wireless environments wheresignals typically arrive at the receiver via multiple propagationpaths with different time delay, attenuation, and phase [6].This multipath propagation together with the nonuniform

Paper approved by E. S. Sousa, the Editor for CDMA Systems of the IEEECommunications Society. Manuscript received June 29, 1998; revised January20, 2000. This paper was presented in part at the IEEE Sympium on Personal, In-door and Mobile Radio Communication (PIMRC’98), Boston, MA, September1998, and in part at the Wireless’98, Calgary, AB, Canada, July 1998.

M. F. Madkour is with Professional Services and Consulting, Ericsson Inc.,Richardson, TX 75081 USA (e-mail: [email protected]).

S. C. Gupta is with the Electrical Engineering Department, SouthernMethodist University, Dallas, TX 75275 USA (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(00)07107-5.

users’ power levels (because of different data rates) resultin multiple-access interference (MAI) which limits the linkcapacity and leads to an irreducible error floor [7]. There aretwo possible techniques to support higher data rates with highquality of service (QoS). One possible technique, which hasreceived a lot of attention recently, is multicarrier modulation ormultitone modulation in which the transmitted data is dividedinto several interleaved bit streams which are then used tomodulate several subcarriers [8]. However, such an approachoften requires equalization in the frequency domain, which canprove to be quite complex with high transmission rates andtime-varying wireless channel. The other possible technique isto directly spread the data stream to the available bandwidth.Utilizing that, a new scheme, called multicode modulation, hasbeen developed for high-speed data transmission over wirelessenvironment [9]–[11]. In this technique, the incoming high-ratedata stream is divided into a number of parallel low-rate datastreams as in multitone modulation. However, the low ratebit streams are Walsh modulated and direct-sequence (DS)spread using pseudonoise (PN) codes in order to separate thedifferent subchannels and to reduce the MAI, respectively.In order to support several kinds of communication services,including, e.g., voice, images, and even motion picture trans-mission [1], a multirate strategy should be utilized. There areseveral strategies available to design and support multiple datarates, henceforth multirate multiuser communications system.Error correcting codes with variable coding rate can be usedas a means in spreading the spectrum, alone or along withPN codes [12]. In IS-95 [13], a repetition coding is used tosupport different rates, but this is of course only practical insupporting a few data rates. A more conventional way is toalter the processing gain and spread all signals, independentlyof the bit rate, to the same bandwidth [14], [15]. Furthermore,it is possible to alter the chip rate or the modulation formator use parallel channels [16] or maybe combine several ofthese schemes. In our work, we propose a flexible DS-CDMAmultirate scheme that uses a variable size fast Walsh transform(FWT), to match the different data rates, together with variableprocessing gain serial modulation. In variable processing gainserial modulation, an -fold increase in data rate ( is therate multiplier integer) is achieved by modulating subcodes oflength where is the lowest transmitteddata rate and . Additionally, we use a suboptimallow complexity decorrelating multiuser detector [17], [18] toeliminate the MAI between the different users. The work inthis paper is different from our previous work [19], [20] in the

0090–678/00$10.00 © 2000 IEEE

1406 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000

Fig. 1. Transmitter block diagram for user #k.

sense that it includes a detailed performance analysis and acomparison between both theoretical and simulation results.

The rest of the paper is organized as follows. In Section II, wedescribe the proposed conventional multirate scheme using vari-able processing gain serial modulation and orthogonal Walshsubchannels. We present a complete system bit-error probability(BEP) analysis as well. In Section III, we extend the work inSection II by using a decorrelating detection together with theproposed multirate scheme. Section IV presents the theoreticaland the simulation results for both uncoded and turbo-codedschemes. Finally, in Section V, we draw the conclusions.

II. M ULTIRATE SYSTEM MODEL USING ORTHOGONAL

WALSH SUBCHANNELS

In this communication system, we considerusers trans-mitting their information data bits with different data rates overtheir individual multipath fading channel to a single-receiverbase station.

A. Transmitter Model

Each user transmits using a binary phase-shift keying (BPSK)version of DS-SS modulation, however, any modulation formatcan be used in general. Theth user data rate is whereis an integer, assume it is power of 2 for simplicity, andis the lowest user data rate. Assume that the users’ transmittingpowers , for all , are such that all users transmitat same signal-to-noise ratio (SNR) per bit [14]. That makes thehigh data rate users have higher average power than the low datarate users do. For example, the average power of a user with datarate is times that of the single rate user, therefore, an

-rate user can be considered as single rate users (effective users). Theth user transmitter is shown in Fig. 1. Itsdata signal and spreading signal are defined as

(1)

(2)

where the unit rectangular pulse is defined asfor and otherwise. The data sym-bols are considered to be independent and identically dis-tributed (i.i.d.) sequences with

. is the chip period such that whereis the channel bandwidth. The spreading sequence has a

processing gain of . can be considered as a concatena-tion between the user and the cell specific codes. Theth user’sdata signal is converted from serial to parallel data streamsthe data on theth subchannel is defined as

(3)

The data on the subchannels are orthogonalized (lineartransformation) by Walsh codes [13], [19]. This orthogo-nalization process will remove any correlation that might existbetween the different subchannels. Theth Walsh code for theuser whose data rate is is defined as

(4)

According to the user data rate, the spreading waveform,,is partitioned into waveforms, for, henceforth, reducing the processing gain. It is assumed that

these waveforms are normalized so that,. In parallel modulation, every has support in

. The other option is serial modulation, which may be re-alized by modulating subcodes of length . Further,assume that are time-orthogonal, i.e., has supportin some subinterval of length . Using serial modulation,

is defined as

(5)

One advantage of serial transmission [15] is that only one wave-form is transmitted at any given time. Therefore, it is consider-ably easier to maintain constant envelope transmission making itsuitable for application where linear amplifiers are not practicalor economical. The main advantage is that only one spreadingcode is used by the user independent of whatever data rate it haswhich increases the capacity (unlike the supplementary parallelchannel assignment as in IS-95 B and C [13]). The disadvan-tage of serial transmission is that we need a good synchroniza-tion circuit for the RAKE receiver as will be seen in the receiverstructure section. Theth user transmitted signal is

(6)

MADKOUR AND GUPTA: WIRELESS MULTIRATE DS-CDMA USING FWT AND DECORRELATING DETECTION 1407

Fig. 2. Conventional receiver block diagram for user #k.

where

(7)

has a time support of with a processing gainof (independent of the user’s data rate).

B. Channel Model

Assume that the th user physical channel is given by thecomplex low pass-equivalent impulse response

(8)

where is the number of paths , and , ,and are the th user th path gain, time delay, and phase,respectively. We assume that the channel has fixed number ofpaths [6], [7] and introduces an additive white Gaussian noise(AWGN). Assume that the path gain, the delay, and the phase arefixed, and that the channel is stationary over the longest symbolperiod. Each path amplitude has a Rayleigh distributionand each path phase has a uniform distribution over theinterval .

C. Proposed Conventional Receiver Model

Let us assume that the desired receiver can coherently recoverthe carrier phase and delay lock to theth path of the receivedsignal (RAKE receiver). Considering baseband representation,the receiver demodulates the superimposed signal

(9)

where is a white zero-mean complex and circularly sym-metric Gaussian process with a two-sided noise power spectraldensity of at the receiver input. The proposed conven-tional receiver model is shown in Fig. 2. The received signal

passes through an -rate -fingers RAKE, , asshown in Fig. 3. There are parallel integrators per finger torecover consecutive bits at one time. The RAKE receiveroutput vector of the th user for theth symbol ( data bits)is

(10)

Fig. 3. TheM -rate,Q-finger RAKE filter.

where1

(11)

(12)

Finally, after the inverse FWT filter, the estimated data vector is

(13)

where is the inverse of the Walsh matrix [13] defined asfollows:

(14)

for (15)

D. Performance Analysis

As we mentioned earlier, no errors in the estimation ofchannel parameters are assumed in the analysis. Without lossof generality, we will consider the detection of theth user in-formation data signal. We may note that, although the analysisresults in this section drew upon some of the analysis results in[21], they are different in the sense that they are done for userswith different data rates and hence, different number of Walsh

1Matrix superscriptsT , �1, and�T denote transpose, inverse, and inversetranspose, respectively.


subchannels. The output of theth finger of the -rate RAKEon the th subchannel, as shown in (12), can be written as

(16)

where represents the RAKE’s output of theth user

desired signal, the output of its ISI terms, the

output of the interchip interference (ICI) terms, the

output of the MAI terms, and the output of the AWGNwith variance . Specifically

(17)

(18)

(19)

(20)

where

(21)

For perfect time delay estimation, we can show that. This is due to the fact that the subcodes are time orthog-

onal. Therefore, the output decision statistics from thesub-channels are identical. After the inverse FWT filter, (17), (18),and (20) are reduced to (22)–(24) (we will drop the subchannelindex)

(22)

(23)

(24)

where

(25)

(26)

For a given , the th user output signal power onthe th finger is [22] . It follows that the con-

ditional SNR for the th user on the th finger is

where2

(27)

(28)

(29)

Substituting and [22] and whereis the average power of the user with the lowest data rate, we

get

(30)

Thus, th user output conditional SNR from all fingers is

(31)

The BEP can now be obtained by assuming that the interferenceis Gaussian distributed. This may not be necessarily correct, butthis approximation has been shown to be relatively good [21],[22]. Hence, the conditional probability of error is

(32)

where . The unconditionalBEP for the th user can be obtained by integratingwith respect to the density function of[7], [22]. As a result,we get

(33)

where and .

III. M ULTIUSER DETECTION

In this section, we will consider the use of a simple mul-tiuser detector. Multiuser detection, exploiting knowledge ofother users’ spreading sequences to cancel the MAI, has the ca-pability of eliminating the near–far problem and providing a sig-nificant capacity increase in CDMA systems [17], [18].

2The bit indexi has been removed for ease notations.


Fig. 4. Thekth user transmitter with a pre-whitening filter.

A. System Model

The transmitter, shown in Fig. 4, is almost the same as thatin Fig. 1 except that the insertion of a pre-whitening filter withmatrix response (the reason for this filter will be discussedlater). To simplify the analysis in case of multiuser detection, wewill use matrix notations. The received-elements row vectorfor the th symbol (corresponds to information data bits)

is defined as

(34)

where

(35)

(36)

... (37)

for

otherwise(38)

where is a zero row vector of length , is an -el-ements complex Gaussian noise with covariancewhere is the unit matrix of size . The proposed receiveris shown in Fig. 5. The received signal is passed throughpar-allel RAKE receivers. Each RAKE receiver has to be matched toboth the information data rate and its spreading sequence. TheRAKE receiver output vector of theth user for theth symbol( data bits) is

(39)

The composite output vector from the RAKE receivers oflength is defined as

(40)

where

Fig. 5. The proposed multirate DS-CDMA receiver using decorrelator.

(41)

(42)

(43)

where is the composite code matrix defined by, is the composite noise vector

of covariance matrix , and is the unit square matrix ofsize . In [18], a decorrelating detector is introduced to cancelthe MAI via matrix inversion. The output of the decorrelatingfilter is given by

(44)

where is a Gaussian noise with covariance matrix .

B. Performance Analysis

Without loss of generality, consider the output of the first user.Substituting from (35), (41), and (42) in (44) and taking the first

elements yield

(45)

where is a nonwhite Gaussian noise with covariance matrix. The output signal is passed on to a whitening filter.

However, this whitening filter will distort the desired signal.That is why we used a pre-whitening filter at the transmitterto have undistorted data signal at the receiver output. The


whitening filter matrix response can be derived using Choleskydecomposition [18].

(46)

Now using a whitening filter with matrix responseyields a white noise as well as an undistorted data output. Thefinal decision statistic after the inverse FWT filter is given bytaking the sign of where

(47)

It is to be noted that the output noise after the inverse FWTstage is still white with variance because isan orthogonal transformation. The output SNR for the first usercan be written as

(48)

where

(49)

The unconditional BEP for the first user can be derived [7] to be

(50)

where and.

IV. NUMERICAL RESULTS

The uncoded BEP for the proposed variable processing gainserial modulation conventional multirate scheme was calculatedfrom (33) and that for the decorrelating detection case from(50). In addition, computer simulations for both the uncodedand turbo-coded [24] schemes were carried out.

A. Parameters

For the channel parameters, each user’s link was modeledwith the same parameters. The simulations were performed fortwo different channel models: 1) flat fading channel with vari-able Rayleigh amplitude and 2) IMT-2000 vehicle channel A at900 MHz and 100 Km/h [1] with multipath parameters shown inTable I. Channel coefficients are generated using Jake’s model[23]. Some other simulation parameters are as follows. BPSKdata and chip modulation at chip rate 1.2288 M chip/s indepen-dent of the user data rate. Spreading codes of length 127 (a set ofbalanced gold sequences). Lowest symbol rate is kbit/s.Higher symbol rates of up to , with lower processing gain,are also considered on a transmission bandwidth of 1.25 MHzand frame length of 20 ms. Perfect chip synchronization is as-sumed in these simulations.

TABLE IIMT-2000 VEHICULAR CHANNEL A PARAMETERS

Fig. 6. Uncoded BEP as a function of the SNR for a two-user groups system.Group 1: two single-rate users. Group 2: two 4-rate users.

B. Results

Fig. 6 is a plot of the uncoded BEP versus the average SNRwhen users with two different data rates coexist and commu-nicate on a Rayleigh fading channel. The solid lines representthe results from theory, while the dotted lines represent thosefrom simulation. A subset of users, group 1, consists of twosingle-rate (9.6 kbit/s) users and group 2 consists of two userswith 4-fold data rate (38.4 kbit/s). For the users of group 2,we used four Walsh subchannels to decrease the data rate to9.6 kbit/s, while the users of group one use just one channel.We used the conventional one-finger multirate RAKE receivershown in Fig. 2. Simulations were performed by generating100 frames of 192 bits for the low data rate users and 768 bits forthe high data rate users and then repeating for ten times. BEP hasbeen obtained by averaging over the users in each group. As canbe seen from the plots, the high data rate users perform betterthan the low data rate users due to accurate channel estimation,using orthogonal subchannels and the higher transmitted power.Results in Fig. 7 are also for a two-user groups system but fora greater number of users to demonstrate the effect of the mul-tiuser detection. Group 1 has eight single-rate (9.6 kbit/s) users;and group 2 has four 4-fold data rate (38.4 kbit/s) users. Thethin lines are for the proposed conventional receiver, while thethick lines are for the multiuser detection case. As we can see,the multiuser detection has eliminated the error floor at highSNR. Plots in Fig. 8 show the BEP versus the average SNR fordifferent data rates, with fixed information capacity (a total of32 subchannels). We consider three cases where there are two,four, and eight users sending their information data signals withdata rates , and , respectively. The communicationsis on a frequency-selective fading channel with the parametersshown in Table I. We compare the performance in the two cases


Fig. 7. Uncoded BEP as a function of the SNR for a two-user groups systemwith and without multiuser detection (MUD). Group one: eight single-rate users.Group two: four 4-rate users.

Fig. 8. Uncoded BEP as a function of the SNR at fixed information capacityon IMT-2000 channel A using a 2-finger RAKE receiver.

of with and without Walsh subchannels and multiuser detection.The receiver uses a two-finger RAKE. Also, it is clear that thereis a significant performance improvement as a result of usingorthogonal Walsh subchannels and decorrelating detector. Thediscrepancy between the simulation and the theoretical results isdue to the use of Gaussian approximation in modeling the MAI.From the Central Limit Theorem, as the number of the activeusers in the system and the length of the spreading sequence in-crease, the modeling becomes more accurate. Finally, in order tomake sure that the proposed processing techniques do not distortthe soft values at the demodulator output, the proposed multiratescheme (without the decorrelating detector) has also been sim-ulated in case of using turbo codes. Turbo coding is a relativelynew form of channel coding [24], which was reported to ap-proach the Shannon performance predictions, when using a suf-ficiently high number of decoding iterations. A turbo encoderwith two identical recursive systematic codes (RSC) encoderswith matrix response was used. The size of the

Fig. 9. BEP as a function of the SNR for the turbo-coded system.

random interleaver equal to the block length. The data is punc-tured to a rate of . The decoder uses the soft output Viterbialgorithm (SOVA) [25]. The simulation example in Fig. 9 is aplot of a turbo-coded system. Plots of BEP versus the averageSNR using two different data rates (frame sizes) 9.6 kbit/s onone channel; and 76.8 kbit/s on eight Walsh subchannels areshown. As can be seen, turbo coding gives a satisfactory per-formance especially at high user data rates. Also, it is seen thatwe did not lose any information as a result of the proposed pro-cessing techniques.

V. CONCLUSION

Multiple users with different data rates are supported by aflexible multicode multirate DS-CDMA scheme using FWT to-gether with variable processing gain code spreading in a fre-quency-selective Rayleigh fading channel. We considered theuse of a decorrelating multiuser detection together with a vari-able rate coherent RAKE receiver which was able to increase thesystem capacity and robustness at the expense of system com-plexity. The uncoded BEP with and without multiuser detec-tion on a multipath fading channel was derived analytically. Inaddition, the validity of the analysis results was demonstratedby computer simulations. Also, we considered the use of turbo-codes together with the proposed system, which gave a satis-factory performance especially for high user data rates. Bothnumerical and simulation results illustrated the potential of thisapproach for supporting multimedia services in the next gener-ation mobile radio systems.

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Mohamed F. Madkour (S’98–M’00) was born inAlexandria, Egypt, on May 25, 1968. He received theB.Sc. (with honors) and M.Sc. degrees in electricalengineering from Alexandria University, Alexandria,Egypt, in 1991 and 1994, respectively. He is a cur-rently pursuing the Ph.D. degree in the Electrical En-gineering Department, Southern Methodist Univer-sity (SMU), Dallas, TX.

He joined the Department of Electrical En-gineering, Alexandria University, in 1991, as aTeaching Assistant and Lab Instructor. From 1996 to

1999, he was a Research Assistant at the Department of Electrical Engineering,SMU. Currently, he is Member of the Professional Services and ConsultingGoup, Ericsson Inc., Richardson, TX. His current research interests includemobile and spread-spectrum communications.

Dr. Madkour received the SMU Outstanding Graduate Student Award in1999. He also received the Best Graduate Student Paper Award from the IEEECommunications and Vehicular Technology Societies, Dallas Chapter in 1999.

Someshwar C. Gupta (S’61–M’63–SM’65–F’99)received the B.S. (with honors) and M.S. degreesin mathematics from Punjab University, India, theB.S. (with honors) degree in electrical engineeringfrom Glasgow University, U.K., and the M.S.E.E.and Ph.D. degrees from the University of California,Berkeley.

He taught at Carnegie Mellon University andArizona State University prior to joining SouthernMethodist University, Dallas, TX, in 1967. Presently,he is the Cecil H. Green Professor of Engineering.

He has considerable teaching, research supervision, academic administration,and industrial experience. He has been very active in the areas of cellular andpersonal communications for the last 18 years. He is the author or co-authorof three widely used textbooks and a over 100 technical papers. Almost all hisresearch has been supported by federal agencies such as NSF, NASA, USAF,DCA, AFOSR, and industrial companies such as GE, Ericsson, Nortel, LCCI,and Raytheon. He has supervised over 60 Ph.D. dissertations.

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Performance analysis of a wireless multirate direct-sequence CDMA using fast Walsh transform and decorrelating detection