An analysis of the effect of chip timing jitter in MC-CDMA systems

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 6, NOVEMBER 1999 1889

An Analysis of the Effect of Chip TimingJitter in MC-CDMA Systems

Luciano Tomba,Member, IEEE,and Witold A. Krzymien,Senior Member, IEEE

Abstract— In this paper, we investigate the effect of chipsynchronization error (or chip timing jitter) in a multicarrier-codedivision multiple access (MC-CDMA) scheme. The chip timingjitter is modeled by a stationary random process with a knownstatistic, and the error produced by the jitter is reduced to anadditive source of noise. This decomposition is useful because itallows for the use of analytical methods for the computation ofthe error probability. First, the analysis has been applied to thesimple Gaussian channel; afterwards, it has been extended to amultipath fading channel typical of large indoor areas. Use ofa semi-analytical approach has been the key factor in obtainingreliable results at moderate computational complexity.

Index Terms—MC-CDMA, OFDM, timing jitter.

I. INTRODUCTION

T HE INCREASING demand for mobile radio services,especially for indoor applications (e.g., offices, shop-

ping malls, and factories), is a major driving force behindthe development of new, highly efficient, wireless networks.Beside the TDMA and CDMA multiple access techniques, amulticarrier CDMA (MC-CDMA) scheme based on the jointuse of CDMA and OFDM [1]–[4] has been recently proposedfor high bit rate transmission (applicable to data, images, andATM packets). For similar applications DS-CDMA may beimpractical because of the difficulty of implementing a RAKEreceiver [5] capable of taking advantage of the path diversityintroduced by the multipath channel. The OFDM scheme,instead, is able to combat the frequency selectivity of thechannel by using a simple one-tap equalizer [4], [6]. Sincethe OFDM symbol duration is usually very large comparedto that of single carrier modulation, the radio channel canbe viewed as frequency nonselective. However, to get thesame data rate with longer symbols, many symbols must betransmitted in parallel without interference, and this is possiblewith multicarrier modulated signals which are orthogonal toeach other over the symbol duration. The resulting compositesignal is the sum of all modulated subcarriers. The subchannelscorresponding to different subcarriers overlap spectrally, but ifthe channel distortion is mild, the signals of each subchannel

Manuscript received August 6, 1997; revised August 26, 1998. This workwas supported by the Telecommunications Research Laboratories (TRLabs),Edmonton, Alta., Canada, the Natural Sciences and Engineering ResearchCouncil of Canada, and the “Consiglio Nazionale delle Ricerche,” Rome,Italy.

L. Tomba is with the Universita degli Studi di Padova, Padova, Italy (e-mail: [email protected]).

W. A. Krzymien is with the University of Alberta and TRLabs, Edmonton,Alta., Canada (e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(99)07389-2.

can be demodulated with negligible penalty because of theorthogonality.

In principle, the performance of a MC-CDMA systemis very good in a nearly ideal system, i.e., with a linearpower amplifier at the transmitter, a linear channel, perfectcarrier recovery, and sampling instant regeneration. Whenthese assumptions are relaxed, as it happens in practicalsystems, performance degrades severely. This is the case in thepresence of a phase offset and/or noise in the carrier frequencyrecovery loop [7]–[9], or in radio channels with delay spreadlonger than an estimated maximum value [10].

Here, our attention is focussed on the effect ofchip syn-chronization erroror chip timing jitter (hereafter jitter) in thedownlink receiver of a MC-CDMA system. In fact, one ofthe key issues in an OFDM-based system is the estimation ofthe sampling instants used to obtain the samples which buildan OFDM symbol. Whatever is the cause of a timing error,its consequence is a loss of orthogonality between subcarrierswhich leads to interchannel interference.

In this paper, we present an accurate technique to studythe sensitivity of the downlink of an MC-CDMA system topossible errors in the receiver chip timing. A similar analysismay be applied to the uplink too; however, because of itscomplexity, the study of an MC-CDMA uplink [11] deservesa separate treatment and it may be a subject of future work.

The analysis is applied both to an additive white Gaussiannoise (AWGN) channel and to a general Rice multipathfading channel. In the latter case, the use of a semi-analyticalapproach is essential to obtain reliable results at the expenseof reasonable computational complexity. We note that thedetermination of the system performance degradation due tothe timing jitter can also be achieved by brute-force simulation,once the chip timing jitter statistic is known. However, weneed to emphasize the fact that such simulation for eachtrial requires execution of an inverse fast Fourier transform(IFFT), filtering, and an FFT, and becomes unwieldy whenthe multipath fading channel is considered.

The paper is organized as follows. A brief overview ofthe MC-CDMA system is presented in the next section. InSection III, we give the parameter values of the radio channelmodel that we consider. In Section IV we present spectralanalysis of the timing jitter and a suitable model of the errorintroduced by the jitter. All the mathematical details (which arequite cumbersome) are deferred to the Appendixes. Section Vdescribes the performance evaluation methodology and inSection VI some numerical results are discussed. Conclusionsare in Section VII.

0018–9545/99$10.00 1999 IEEE

1890 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 6, NOVEMBER 1999

(a)

(b)

Fig. 1. Simplified block diagrams of the MC-CDMA system for the downlink [(a) base-station transmitter, (b)kth receiver;_: add virtual carriers, :remove virtual carriers;%: add cyclic prefix,&: remove cyclic prefix].

II. OVERVIEW OF THE MC-CDMA SCHEME

The transmitter and the receiver block diagrams are shownin Fig. 1. At the time ( is an integer and is the symbolperiod), the th user ( is the number ofsimultaneously active users) transmits a quaternary symbol,

. The sequenceis assumed stationary, with independent and identically

distributed symbols. Each symbol, , is spread by a PNWalsh sequence, , of length . The valuesof are drawn from the set ; in this way the energyof the sequences is normalized to unity. We have chosenWalsh sequences because in the downlink the base station canensure synchronous transmission of data belonging to differentusers although they may arrive asynchronously. The resultingdiscrete-time sequence is composed of multilevel complexsymbols, , given by

(1)

which are equally spaced at intervals . For reasons wellexplained in [7], each array is trans-formed into an array of components before transmission.The array

(2)

composed of complex values is sent to the inputof the OFDM coder. This procedure implies insertion ofvirtual carriers in the rolloff region of the transmit filter. Thecomponents of are denoted as .1 Theyare transmitted by an OFDM system capable of transmitting

complex numbers at the same time. It is interesting to notethat the symbols can be obtained in the same way as

1We use the notationc0mi

to indicate explicitly themth value at the inputto the IDFT for theith OFDM symbol. However, instead ofc0

mi, a simpler

notation,cn, wheren = m+ iN , is also possible.

by defining the following modified Walsh sequences:

(3)

hence

(4)

In principle, a multicarrier signal is the superposition ofsinusoidal subcarriers with frequency spacing[12]; each subcarrier is modulated by symbols of duration

. The modulated subcarriers overlap spectrally, but becauseof their orthogonality within the symbol duration (thethcarrier’s frequency is , where is somereference frequency and ), the signals modulatingeach tone can be recovered as long as the channel does notdestroy the orthogonality of modulated subcarriers. In practice,the samples of the OFDM signal are generated by taking theinverse discrete Fourier transform (IDFT) of the discrete-timeinput sequence and passing the transformed samples through apulse-shaping interpolating filter [see Fig. 1(a)]. As it is wellknown [12], [13], in order to (ideally) eliminate intersym-bol and interchannel interference produced by channels withnonzero delay spread, a guard time containing acyclic prefixisinserted before each symbol. Let be the th value of theth OFDM symbol at the output of the IDFT processor, namely

(5)

Then, the complex envelope of the continuous time transmittedsignal is

(6)

TOMBA AND KRZYMIEN: CHIP TIMING JITTER IN MC-CDMA SYSTEMS 1891

Fig. 2. Equivalent scheme for the analysis of the chip timing jitter error(tn = nT + �n).

TABLE IPARAMETER VALUES OF THE INDOOR

RADIO CHANNEL MODEL

where corresponds to the sequence completed withthe prefix of length .2 We assume to have a square-root raised cosine spectral characteristic with bandwidth

, where is the rolloff factor [5]. At the receiver, afilter matched to the transmitted pulse is used. Then, afterremoving the cyclic prefix, DFT is applied to the usefulsamples of each OFDM symbol. At the output of the DFT thenull symbols corresponding to the virtual carriers are removed.For multipath fading channels, a one-tap linear equalizer [4],[6] is applied to each sample at the output of the DFT to reducethe impact of the random amplitude and phase fluctuations.Finally, the equalized block of samples is despread to give thedecision variable from which an estimate of the transmitteddata is retrieved.

III. RADIO CHANNEL

The MC-CDMA transmission scheme considered in thispaper is designed to be used in channels with a delay spreadshorter than the duration of the cyclic prefix. A typicalapplication may be for a data network in a large indoorenvironment (e.g., a shopping mall, an airport, etc.). Hence, weconsider a channel with the characteristics presented in Table I,which is taken from [15, Table 6B.2, Chn. B]. This channelcorresponds to a typical indoor commercial environment, asspecified by the Joint Technical Committee (JTC) on wirelessaccess.

The impulse response of the channel can be expressed as[16], [17]

(7)

where is the number of paths, each with random amplitude,phase and delay , respectively; is the Diracimpulse. Often is assumed Rician distributed, areassumed Rayleigh with the second moment given in Table I,

2The following analysis does not take into account the presence of the cyclicprefix which, in any case, is removed before the digital signal processing atthe receiver side. Hence, as also done in [14, footnote 1], we ignore the cyclicprefix in the remainder of the paper.

while are assumed uniform. However, in our numericalexamples we assume that is also Rayleigh distributed.

IV. JITTER ANALYSIS

A. Equivalent Model of the Nonideal Sampler

To evaluate the impact of the chip timing jitter we use theequivalence illustrated by Fig. 2. It represents the MC-CDMAscheme from the input to the transmit filter to the output ofthe nonideal sampler at the receiver (the AWGN has beenomitted). The transmit filter, the channel, and the receive filterare combined into one single filter with frequency response

, where and are the Fouriertransforms of and , respectively [we call thefrequency response of the composite channel]. The signalat the output of the composite channel [denoted by ] issampled by a nonideal sampler everys. This means that thesampling instants are not , but [14]

(8)

where is the chip timing jitter of the th samplinginstant; therefore, the sampled signal is insteadof . (Without loss of generality, in the following weassume .) Since the jitter sequence is random,the error produced by the jitter on the sampled values has anoise-like nature; this error propagates through the subsequentoperations and it causes degradation of the system. Thestatistic of depends on the timing recovery mechanism inuse in the receiver. Here, we model the sequenceby astationary Gaussian random process statistically independentof the input signal with zero mean and variance. Usingthe decomposition , the nonideal sampler isreplaced with an ideal sampler and an additive noise source

(see Fig. 2).

B. Spectral Analysis of the Equivalent Jitter Error

The evaluation of the jitter power at the decision pointinvolves evaluation of the autocorrelation of the jitter sequence

. Unfortunately, the sequence is not stationary because,although the sequences are assumed wide-sense stationary,the signal processing performed on them (spreading, IDFT)causes nonstationarity; to be more precise,is a cyclosta-tionary process of period [however, when the cyclic prefixis present, the period of cyclostationary becomes ]. Thisimplies that correlation functions depend on two variables, i.e.the reference instant and the shift.

The evaluation of the mean value and the autocorrelationfunction of is possible in terms of:

• the spectral density of the sequence (seeAppendix B);


• the first- and second-order characteristic function [18] ofthe jitter, namely ( stands for expectation)

(9)

• the frequency response .

From the above decomposition, the error sequenceis

(10)

The mean value and the autocorrelation function ofare,respectively (see Appendix C for the mathematical details)

(11)

where

(12)

In the case of zero-mean Gaussian jitter, the functionsand are [18]

(13)

where is the jitter autocorrelation function. As anexample, we assume the following form of :

(14)

which allows us to consider jitter with different amountsof correlation; corresponds to uncorrelated jitter ondifferent samples.

V. PERFORMANCE EVALUATION

The sensitivity of the BER to a possible error in thesampling instant represents an important information for thedesign of the synchronization circuit. This section presents themethodology to determine such sensitivity. It is convenient tosplit the evaluation of the BER considering first the AWGNchannel and then the multipath fading channel, because thelatter analysis is based on the former.

A. Equivalent Model of the Multicarrier Scheme

Thanks to the decomposition (10), each sample at the outputof the DFT is related to the data at the input to the IDFTas [7]

(15)

where we have the following.

• is the gain of the th subchannel within thethOFDM symbol [19]. We assume the channel is knownand not-varying within at least one OFDM symbol. Whenthe AWGN channel is considered, .

• is a sample of the thermal noise. Each sample isdrawn from a zero-mean white Gaussian process becauseof the linearity of the transformations involved and theparticular shape of the receive filter.

• is a sample of the noise dueto the timing jitter. In general, it is neither white norGaussian.

B. Error Probability for AWGN Channels

At the output of the DFT the virtual carriers are discarded.Hence, we let ; a similar definitionholds for . For AWGN channels the decisionvariable, , corresponding to the data transmitted by thethuser is3

(16)

which can be rewritten as

(17)

where the orthogonality of the Walsh sequences has been usedand we have the following.

• is a complex Gaussian random vari-able with , where isthe variance (var) of each sample. The real and theimaginary parts4 of are assumed independent and iden-tically distributed; hence,

.• is a complex random variable with

independent and identically distributedLaplacian com-ponents (see Appendix D), with variance each.

To evaluate the variance of , let us consider the columnvector of dimension containing the jitter errors on thesubcarriers of theth OFDM symbol at the input to the DFT.As can be verified watching Fig. 3, in general the errors arecorrelated. The corresponding vector at the DFT output is

being the matrix of components, where and are the row and column

indexes, respectively. Let be the th row of the modifiedWalsh matrix. Hence, for theth user we get and

(18)

where is the matrix with the autocorrelations of the jittersamples ( denotes transposition and complex conjugate). Inparticular, the element at location in is

(19)

3For the cyclostationarity of periodN , the system performance is indepen-dent of the time indexi.

4The real and the imaginary part ofa are denoted by<[a] and =[a],respectively.


Fig. 3. Average spectral density of the sequence"n: comparison betweenanalysis and simulation (AWGN channel is assumed).

where the averaging over samples is justified by the factthat the process is cyclostationary of period. On the basisof the above assumptions and taking into account [5, 5-2-59],it can be shown that the symbol error rate is

SER

(20)

where is the normal integral.

C. Error Probability for Multipath Fading Channels

In the presence of multipath fading it is necessary to equal-ize the received samples before the despreading operation.Here, we consider the minimum mean-square error (MMSE)equalizer [5] which includes, as a particular case, the zero-forcing equalizer. This means that theth useful sample at theoutput of the DFT must be multiplied by [5], [7]

(21)

where is the ratio between the variance of the additivenoise and the variance of the transmitted data symbols. Hence,the equalized decision variable corresponding to the datatransmitted by the th user is

(22)

where . Replacing in (22) as given by (1), weget

(23)

Let

(24)

For any realization of the channel and for a fixed number ofactive users and are constant. Moreover thequantities

(25)

are random variables with distribution similar to and ,respectively (see Appendix D). Hence, we may rewrite (23) as

(26)

and find the error probability conditional on a given channeltransfer function by an analytical procedure. This approach ispossible because we know the statistics of the symbols, of

and . In conclusion, we get the following result for theconditional SER SER :

SER (27)

where

(28)

(29)

is the set and is given by (18)once is replaced with . It is apparent that thecomputational complexity of (27) grows exponentially with.However, even for moderately high values of(e.g., )we may approximate by an independent Gaussian variable;hence, is still a zero-mean Gaussian variable.In conclusion, for large , the last approximation yields

SER (30)

where

(31)

(32)

and .

VI. COMMENTS ON THE RESULTS

The objective of this section is to evaluate the sensitivity ofthe system under investigation to possible errors in the sam-pling instants. The error may be introduced through variousmechanisms, in which we have no particular interest here. Weassume to know the statistics of the jitter error; moreover, weassume that the synchronization error is constant within eachOFDM symbol, which implies in (14) ( hasbeen assumed in our computations). Both the simple AWGN


(a) (b)

Fig. 4. SER versus the SNR/bit for different values of��F andK in the AWGN channel [(a):K = L=2; (b): K = L].

(a) (b)

Fig. 5. SER versus the SNR/bit for different values of��F andK in the multipath Rayleigh fading channel [(a):K = L=2; (b): K = L].

channel and an indoor multipath Rayleigh fading channel areconsidered. The symbol period has been assumed equal to100 ns.

To validate our simplified model, we have compared theresults of the analysis with the simulations of the completesystem (we note again that the simulations of the completesystem are very time-consuming). The first result is presentedin Fig. 3 where we compare the average spectral density of

5 with the one calculated by well-known spectral estimationtechniques [22]. The agreement shown in Fig. 3 is good.

Fig. 4(a) and (b) shows the SER versus the SNR/bit fordifferent values of the normalized jitter standard deviationand a set of system parameters. In particular, two situations arecompared, namely the system with users and the fully

5From the results of the Appendixes, it is not difficult to show that theaverage spectral density of"n is

2

N

1

n=�1

1

�1jG(�)j2�(0; �;n)�

K�1

k=0

N�1

h;m=0e�j2��m

�Hm �� hN

�Wk �� hN

2ej2�(��f)n d�:

loaded system . The length of the spreading codeshas been limited to 64 to make possible a comparison with sim-ulation results (simulation time increases with). It is apparentthat, as the number of users increases, the noise produced bythe jitter increases to the extent that unacceptable performanceis offered at with a fully loaded system. Similarresults are shown in Fig. 5 for the multipath fading channeldescribed in Section III. Although the error probability valuesare in this case different from the AWGN channel, it is stillapparent that performance degrades severelyexceeds .(These results have been obtained with a one-tap MMSEequalizer; the semi-analytical method has been used for thecomputation of the SER.) Hence, we have an idea about theprecision required of the chip timing synchronization circuit.

Furthermore, to illustrate the dependence of the SER on, in Fig. 6 we have shown the SER versusat 10 dB of

SNR/bit. Similar results are reported in Fig. 7 for the Rayleighfading channel. In particular, we note that the SER degradationis not linear with , at least for for the AWGNchannel. The SER depends strongly on the number of active


Fig. 6. SER versus the number of active usersK for different values of��F in the AWGN channel.

Fig. 7. SER versus the number of active usersK for different values of��F in the multipath Rayleigh fading channel.

users, which severely limits the system performance.

VII. CONCLUSION

In this paper we have investigated the effect of chip-timing jitter in a MC-CDMA system using a (semi)-analyticalapproach and assuming a simple AWGN channel, as well as amultipath fading channel typical of large indoor areas. Theresults of this study provide an insight into the sensitivityof a MC-CDMA system to errors in the timing recoverycircuit. The use of the semi-analytical approach has resultedin reduction of the prohibitive computational complexity ofbrute-force simulations caused by the need to run the FFT andthe IFFT algorithms.

APPENDIX ATHE SPECTRALLY EQUIVALENT SCHEME

For the purposes of the next Appendixes, we need anequivalent scheme to represent the part of Fig. 1(a) composedof the S/P converter, the IDFT, and the P/S converter. Theappropriate method for finding the equivalent scheme is the

multirate decomposition [20] which gives the solution rep-resented by Fig. 8. The detailed derivation of the results isreported in [21]. For convenience, we reiterate here the mainresults. The th branch, , of the spectrallyequivalent scheme is composed of: 1) a filter with frequencyresponse

(A1)

2) an -fold decimator; 3) an -fold expander; and 4) a delayunit of samples (denoted by ) [20].

When the input process is nonstationary its autocorre-lation function depends both on

and . By definition [18], the spectral density is the two-dimensional discrete-time Fourier transform of ,namely [22]

(A2)

where and are real variables. The spectral density ofis

(A3)

where is the cross spectral density of and .To evaluate (A3), first we determine the relationship betweenthe and of and , respectively, i.e., [20], [22]

(A4)

Then, we evaluate the product and, becauseof the linearity of the scheme of Fig. 8, we can replace

with andwith [18]. Hence, we get

(A5)

which gives the desired result when substituted in (A3).

APPENDIX BSPECTRAL DENSITY OF

Let us consider the sequence where, according tofootnote ( and are integers numbers in theranges and ). The autocorrelationfunction of is defined as

(B1)


Fig. 8. The spectrally equivalent scheme.

By using (1), the assumption of independence of and, we obtain

(B2)

where and is the Kroneckerfunction. Now, taking into account that is periodic ofwith respect to and the equalities ,we get

(B3)

We observe that integer, whichmeans that the sequenceis cyclostationary of period . Byusing this property, its spectral density is

(B4)

Moreover, by using (B3), we further obtain

(B5)

where . In conclusion

(B6)

The spectral density of is given in terms ofby using (A3) and (A5). Replacing in (A5) withgiven by (B6) and with in (A3), after some algebraicmanipulations we get

(B7)

APPENDIX CPROOF OF (11)

The mean value of is

(C1)

where is the mean value of which is given interms of by

(C2)


Assuming identically distributed users’ symbols,and we get

(C3)

Hence, which implies . The correlationfunction of is

(C4)

and conditioning with respect to the sampling errors

(C5)

The inner expectation in (C5) can be expressed as

(C6)

where ( real) is the autocorrelation function of ,which is given in terms of the spectral density by

(C7)

Hence, the second line of (C6) becomes

(C8)

Lastly, the average with respect to and yields

(C9)

where the function is defined by (12). In addition,because of the linearity of the composite channel,

.6

6To be precise, the spectral density of the sequence�dn including the cyclicprefix should be used instead of the spectral density ofdn. However, as notedpreviously, to limit the complexity of the analysis we neglect the presence ofthe prefix. In practice, this approximation is sufficiently accurate as long asthe length of the cyclic prefix is limited to a relatively small fraction of thelength,N , of the OFDM symbol.

Fig. 9. Normalized histogram of the real part of�k and its comparison withthe Laplacian and Gaussian pdf for the AWGN channel.

Fig. 10. Normalized histogram of the real part of~�k and its comparison withthe Laplacian and Gaussian pdf for the multipath Rayleigh fading channel ofTable I.

APPENDIX DON THE STATISTICS OF AND

It is not easy to determine the probability density function(pdf) of and . Hence, with the help of simulations wehave found its approximation which we believe is sufficientlyaccurate. In particular, we have created histograms of thereal and imaginary parts of the error and for manydifferent jitter and system parameter values. The histogramsshow a reasonably good agreement with Laplacian distribution,as demonstrated by examples presented in Figs. 9 and 10.It is not surprising that the statistics of the real and theimaginary parts of are not Gaussian, because (as wellas ) is a linear combination of random variables which,in general, are neither Gaussian, nor independent. Hence, theimportant assumption of independence required by the centrallimit theorem [18] is not satisfied. Furthermore, BER valuesobtained by simulation are in a better agreement with theresults of the analysis when the Laplacian rather than theGaussian distribution is used. In conclusion, in this paper


the real and imaginary parts of are modeled bytwo independent and identically distributed Laplace randomvariables. Concerning the independence assumption of the realand imaginary parts of (or ), we can only attest that formany simulations their correlation coefficient has been foundalmost negligible.

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Luciano Tomba (S’87–M’91) received the Laureadegree in electronics engineering in 1987 and thePh.D. degree in communication systems in 1992,both from the University of Padova, Italy.

From September 1991 to June 1992, he was avisiting scholar at the University of California, LosAngeles (UCLA). From July to December 1995 hewas a Visiting Researcher at the Telecommunica-tions Research Laboratories (TRLabs), Edmonton,Canada, working in the area of multicarrier CDMAwireless systems. Since May 1992 he has been with

the “Dipartimento di Elettronica e Informatica” (DEI) of the University ofPadova as a Researcher. In addition, since 1996 his has been a Lecturerof Telecommunication Systems at the “Dipartimento di Ingegneria Elettrica,Gestionale e Meccanica” (DIEGM) of the University of Udine, Italy. His mainrecent research interests are in the area of wireless communication systemsand networks.

Witold A. Krzymien (M’78–SM’93) received theM.Sc. (Eng.) and Ph.D. degrees (both in electri-cal engineering) in 1970 and 1978, respectively,from the Poznan University of Technology, Poznan,Poland. He received the Minister’s of Science, Tech-nology and Higher Education Award for his Ph.D.thesis.

From 1970 to 1978, he was a Research Engineerand Teaching Assistant, and from 1978 to 1980an Assistant Professor of Electrical Engineering, atthe Poznan University of Technology. In 1980, he

received a Research Fellowship at the Twente University of Technology inEnschede, the Netherlands, for 1980/1981. In the following year, he was aResearch Assistant Professor there. From 1982 to 1986, he was an AssistantProfessor of Electrical Engineering at Lakehead University in Thunder Bay,Ontario, Canada. In 1986, he joined the University of Alberta and AlbertaTelecommunications Research Centre (now Telecommunications ResearchLaboratories, or TRLabs) as an Associate Professor of Electrical Engineering.Presently, he is a Professor of Electrical and Computer Engineering atthe University of Alberta and a TRLabs Scientist. His current researchinterests are in signal processing and medium access control techniques forefficient digital transmission in multiservice wireless cellular and personalcommunication systems employing code division multiple access. He spentthe 1993/94 academic year as a Senior Industrial Fellow at Bell-NorthernResearch in Montreal, Qu´ebec, Canada, working on effective synchronizationand discontinuous transmission schemes for code division multiple accessmicrocellular systems.

Dr. Krzymien is a licensed Professional Engineer in the Province of Ontario,Canada. He received the 1991/92 A.H. Reeves Premium Award in digitalcoding from the Institution of Electrical Engineers (U.K.) for a paper publishedin the IEE Proceedings, Part I.

Documents

An analysis of the effect of chip timing jitter in MC-CDMA systems