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Effect of Chip Waveform Shaping on the Performance of Multicarrier CDMA Systems Ha H. Nguyen, Member, IEEE
Abstract—This paper studies the effect of chip waveform shaping on the performance of band-limited multicarrier direct-sequence code-division multiple-access (MC-DS-CDMA) systems. The performance criterion is the average multiple access interference at the output of a correlation receiver. A criterion based on the elementary density function is introduced for the performance comparison of various chip waveforms. It is demonstrated that the performance of MC-DS-CDMA systems is quite insensitive to the chip waveform shaping. Moreover, the optimum chip waveform for MC-DS-CDMA systems is practically the same as that of a single-carrier DS-CDMA system. Index Terms—Chip waveform, direct-sequence code-division multiple access (DS-CDMA), multicarrier DS-CDMA, multiple-access interference.
I. INTRODUCTION
R
ECENTLY, a number of muticarrier code-division multiple-access (MC-CDMA) systems have been proposed as an alternative to the classical single-carrier CDMA (SC-CDMA) systems [1]–[5]. Among these systems, multicarrier direct-sequence CDMA (MC-DS-CDMA) combines time-domain spreading and multicarrier modulation, as opposed to the combination of frequency-domain spreading and multicarrier modulation of other systems [5]. In MC-DS-CDMA systems, the available channel bandwidth is divided into a set of equal-width (possibly overlapped) subchannels, and narrow-band CDMA waveforms are transmitted over the subchannels. Because of this configuration, an MC-DS-CDMA system is capable of supporting high data rate services over hostile radio channels. Another interesting property of MC-DS-CDMA systems is that the modulation and demodulation can be implemented with the aid of the fast Fourier transform. As with the conventional SC-CDMA systems, the performance of an MC-DS-CDMA system degrades mainly due to the multiple-access interference (MAI). Accordingly, it is important to identify the system parameters that characterize the MAI. Loosely speaking, when the channel bandwidth and the total number of users are fixed, the following system parameters affect the MAI in MC-DS-CDMA systems: i) the number of carriers and the spacing between two adjacent carriers, ii) the Manuscript received January 28, 2002; revised May 12, 2003; October 27, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. The review of this paper was coordinated by Prof. T. Lok. The author is with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2005.844688
spreading sequences, and iii) the shape of the chip waveform employed. Different carrier spacings have been used for MC-DS-CDMA systems. For systems using time-limited chip waveforms in [2], the distance between the two adjacent carriers equals the bit rate (after serial-to-parallel conversion), whereas it equals the chip rate for the systems considered in [1], [3], and [6]. For MC-DS-CDMA systems using a band-limited chip waveform, there is yet another popular choice of carrier spacing such that the spectrums of two adjacent subchannels do not overlap [4]. More general carrier spacing for MC-DS-CDMA systems using a rectangular chip waveform is also considered in [7] and [8], where it is shown that the optimal carrier spacing is very close to the chip rate. Moreover, in terms of the signal model, there is no difference between MC-DS-CDMA systems and multiple SC-CDMA systems with spectral overlapping. Thus the studies of spectral overlap for multiple SC-CDMA systems in [9]–[14] can also be applied for MC-DS-CDMA systems. Regarding the effect of the number of carriers on MAI, it was generally observed that increasing the number of carriers reduces the MAI level and there is a saturation in the level of MAI that can be reduced by increasing the number of carriers [1], [2], [8]. Although the search for good signature sequences could improve the performance of MC-DS-CDMA systems [15], random binary signature sequences have been widely used to analyze the MAI in MC-DS-CDMA systems [2]–[8]. Reasons for using random signature sequences are as follows [16]. First, random signature sequences are often used in an attempt to match certain characteristics of extremely complex signature sequences with a very long period. Second, random signature sequence models may serve as substitutes for deterministic models when there is little or no information about the structure of the signature sequences to be used. Finally, for a system with a large number of users and very long signature sequences, the use of random signature sequences helps to obtain computable closed-form expressions for the system analysis. The chip waveform has been noted to be an important system parameter for conventional SC-CDMA systems. The effect of both time-limited and band-limited chip waveforms on MAI level in SC-CDMA systems has been investigated in [17]–[22]. In contrast, for most of the MC-DS-CDMA systems found in the literature, either a time-limited rectangular waveform or a bandlimited raised cosine waveform is generally employed [2]–[8]. The exceptions are in [23] and [24], where the use of several time-limited chip waveforms is considered together with bandlimiting transmit and receive filters.
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NGUYEN: EFFECT OF CHIP WAVEFORM SHAPING ON THE PERFORMANCE OF MULTICARRIER CDMA SYSTEMS
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Fig. 1. The PSD of a band-limited MC-DS-CDMA system.
The focus of this paper is to investigate the effect of chip waveform shaping on MAI level in MC-DS-CDMA systems. This paper concentrates on band-limited chip waveforms as most practical systems are essentially band-limited. This also means that, different from [23] and [24], no band-limiting filters are needed. A performance index based on average MAI level is established so that the comparison amongst various chip waveforms can be made. The optimum band-limited chip waveform is also identified. This paper is organized as follows. Section II describes the model of multicarrier DS-CDMA systems under consideration. Multiple-access interference analysis is carried out in Section III. Numerical examples and discussions are given in Section IV. Conclusions are drawn in Section V.
. This implies that factor corresponding to general expression for
and a is as follows:
(2)
where tions:
is any function that satisfies the following condifor and for
(3) and
II. MULTICARRIER DS-CDMA SYSTEMS Consider an MC-DS-CDMA system with users. At the is transmitter, each user’s bit stream with bit duration serial-to-parallel converted into lower rate streams. The . Let new bit duration on each lower rate stream is be the th user’s lower rate bit stream that will be transmitted through the th carrier during the th bit duration. The bit stream is spread by a random signature of chip duration . Both sequence and are modeled as sequences of independent and identically distributed (i.i.d.) random variables taking values 1 1 with equal probability. Furthermore, for each bit in chips. Thus and duration , there are . The transmitted signal of the th user can be written as follows:
(1) In (1), is the energy per chip, is the carrier frequency is the random phase (uniformly of the th carrier, and distributed over [0,2 ] and independent for all and ). The chip waveform is band-limited and is normalized to have , where denotes a Fourier unit energy. Let is limited to 2 2 transform. It is assumed that and satisfies the Nyquist criterion with a rolloff
For later discussion, the function is called the elementary density function. Fig. 1 plots the generic power spectral density (PSD) of an be the total bandwidth of the MC-DS-CDMA system. Let and are fixed, the chip duration system. When both (and hence the processing gain ) in an MC-DS-CDMA system depends on the number of carriers , the carrier spacing , and of each subchannel. Since the main focus of the bandwidth the study is on the effect of chip shaping, the carrier spacing throughout this paper. The impact of carrier is fixed at 1 spacing on the performance of MC-DS-CDMA systems employing different band-limited chip waveforms can be found in [25]. Thus the processing gain is given by (4) is a parameter characterizing the bandwhere width-bit duration product of the system. The channel model can either be additive white Gaussian noise (AWGN) or slowly varying frequency-selective Rayleigh be the chip pefading with maximum delay spread . Let riod of the single-carrier system, which is related to the system . Then the number of resolvbandwidth by able paths of the fading channel is (5) It was shown in [4] and [6] that if the number of carriers is chosen to be the same as the number of resolvable paths,
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Fig. 2. The lth branch of the k th user’s receiver.
i.e., , then the subchannels for MC-DS-CDMA systems become frequency nonselective. It follows that the complex low-pass impulse response of the th user’s subchannels can be modeled as
In (8),
is
filtering operation; (corresponding to
the
chip matched filter output of , where represents a low-pass is the desired signal component , ), which is given by
(6) and are the fading amplitude and phase of the where th subchannel, respectively. The fading amplitudes are generally correlated, but after the information bits are properly interleaved, they can be assumed to be i.i.d. Rayleigh random vari. Note that if is set to a constant ables with one, then (6) models an AWGN channel. The received signal is given by
(9) The function
(7) and the overall phase shift In (7), the propagation delay are i.i.d. uniform random variables on [0, ] and [0,2 ], is AWGN with two-sided power spectral respectively; and . density of The receiver for user employs a bank of chip matched filters, each of which detects the bits transmitted on a particular carrier. The output of each chip matched filter is sampled at and then correlated with the corresponding spreading every sequence to generate the decision statistic. The th branch of the receiver for user is illustrated in Fig. 2. III. INTERFERENCE ANALYSIS Consider the detection of the bit stream associated with the th carrier of the first user when the carrier, code, and bit are assumed to be perfectly synchronized. Since only relative delays and . The and phases are important, one can set signal at the output of the chip matched filter in Fig. 2 can be written as
in (9) is defined as , where denotes the convolution operation. The other three terms in (8) account for the following types of interference. The interference from other carriers of user one (corre, ) is given by sponding to
(10) 1 The interference from the same carrier of the other , ) can be expressed as users (corresponding to follows:
(11)
(8)
Note that can be treated as the MAI in a single-carrier DS-CDMA (SC-DS-CDMA) system (with carrier frequency ). The third term in (8) is the interference from all other carriers of the 1 users (corresponding to ). This interfer, where ence can be expressed as
NGUYEN: EFFECT OF CHIP WAVEFORM SHAPING ON THE PERFORMANCE OF MULTICARRIER CDMA SYSTEMS
is the interference from the th carrier of the th user. The expression for is as follows:
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are interpreted as intercarrier interferences from the same user and different users, respectively. in (2), it is not hard to verify that With the expression for and can be written in terms of the elementary density as follows: functions
(12) is obThe decision statistic for the transmitted bit tained by sampling at the chip rate and correlating the results with the corresponding signature sequence. This decision statistic can be written as (13)
, for and . Note that property for has been used to obtain the expression for . Conditioned on and (for the case of fading channel), the signal-to-interference ratio (SIR) at the input of the comparator in Fig. 2 is given by
(21) and
where
SIR (14) Since , , , and are uncorrelated and have zero-means, the variance of conditioned on and can be calculated as
(22) It is interesting to see from (22) that, except for the first and , ), the interference from the sigthe last carriers ( nals of the 1 users to the th carrier of the user of interest does not depend on the shape of the chip waveform. Because the interference from the signals of other users is dominant in MC-DS-CDMA systems, this property has a direct influence on the role of the chip shaping to the total MAI level, as discussed later. The numerator of (14) depends on the specific chip waveform through the processing gain . To take this into account, rewrite the SIR as follows:
(15) SIR
Using the results in [4] and [6], one has
var
var
(16)
var
(17)
var
(18)
where
(23)
where is the energy per bit, interference parameter, given by
is the normalized
(24) (19) and
(20)
and is the signal-to-interference ratio corresponding to . an AWGN channel, i.e., when It can be seen from (23) that when the signal-to-background noise ratio and the number of users are fixed, the performance of different chip waveform is determined by , which in turn depends on particular carrier under considerais larger for the inside carriers and smaller tion. Intuitively,
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for the two outer carriers. Thus to take this dependence into account, define the following normalized average interference parameter:
Nevertheless, when is used as the performance index, it follows from (26) that the optimal waveform is the one that maximizes the integral in (26), regardless of the number of carriers . Thus, as long as minimizing is concerned, the optimal chip waveform for multicarrier DS-CDMA systems is practically the same as that of a single carrier DS-CDMA system. The following proposition identifies the spectrum of the chip wave. form that minimizes , Proposition 1: For a given rolloff factor in (26) has the the optimal chip waveform that minimizes following spectrum:
(27)
Proof: The parameter is minimized when the integral in (26) is maximized. Write the integral in (26) as follows: (25)
In general, it is sensible to use the chip waveform that minimizes the above interference parameter in order to improve the SIR performance.1 Furthermore, as the intercarrier interference from the same user can be approximated by the intercarrier interference from an additional user, the first term in (25) can be is large2 and the normalized average interneglected when ference parameter can be well approximated as
(26)
and imply that The above expressions of when both the number of users and the number of subare large enough, then and the carriers shape of the chip waveform has almost no influence on the MAI level. This observation can be intuitively explained as follows. Since the systems under consideration employ Nyquist pulses and the carrier spacing that equals an inverse of the chip rate, the sum of two adjacent PSDs of the same interfering user is flat over each frequency band. Thus, as the number of carriers is large enough, the interfering signal from any interfering user looks like white noise.3 It follows that the chip waveform does not have a significant impact on the interuser interference. The only possible influence of the chip waveform is on the intercarrier interference of the same user. The intercarrier interference of the same user is, however, insignificant compared to the interuser interference when the number of users is large. 1It should be pointed out that minimizing average interference parameter is carriers. not necessarily the same as maximizing the average SIR among the The design of chip waveform to maximize the average SIR, however, appears to be very complicated. 2For example, if 15, then at most 6.25% of intercarrier interference from the same user is neglected. 3Note that the PSD is not exactly flat over the lower frequency band associated with the first carrier and the upper frequency band associated with the last carrier.
M
K
(28) with equality holds when , . in (2) yields (27). Substituting Note also that, since when , the spectrum in (27) is the optimal spectrum for SC-DS-CDMA systems to minimize the average MAI. From this observation, it is not surprising that (27) agrees with the spectrum found in [22] for SC-DS-CDMA systems. An optimal chip waveform can be found as [22]
(29) where . Though optimal, due to the sharp transitions of the spectral 2 , the transmit and receive filters corredensity at 1 sponding to (27) are difficult to implement. Section IV considers several suboptimal chip waveforms that have smoother spectral densities. For the comparison of these chip waveforms, the following lemmas are useful. and be the two elementary denLemma 1: Let and sity functions associated with the two chip waveforms . If , , then the use of chip waveform results in a lower average MAI level than the . use of chip waveform
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Proof: From the lemma’s assumptions and the properties of an elementary density function, one has , . Thus
(30) Combining the above inequality with (26), (24), and (23) justifies the lemma. Lemma 2: Let and be two chip waveforms. If produces a lower average MAI level than using using in SC-DS-CDMA systems, then using also results in a lower average MAI level in MC-DS-CDMA systems. Proof: The proof is trivial from (26). Finally, by approximating the interference as a Gaussian random variable, the bit error rate (BER) performance of the MC-DS-CDMA systems over an AWGN channel can be approximated as
Fig. 3. Plots of the elementary density functions for different chip waveforms. (X (f )—optimal, X (f )—cosine, X (f )—“Better than Nyquist,” X (f )—ellipsoid, X (f )—raised cosine).
(31) For fading channels, the probability of error conditioned on the . Averaging over fading amplitude is and over all the subchannels, the average error the density of probability is given by
(32)
IV. NUMERICAL EXAMPLES Consider the chip waveforms corresponding to the following elementary density functions, defined over [0, 2 ]. 1) Optimal chip waveform . 2) The elementary density function is a cosine function
3)
“Better than Nyquist pulse” [26]
4)
The elementary density function is a straight line (the corresponding spectrum is ellipsoidal)
5)
Frequency raised cosine
Fig. 4. Interference reduction of MC-DS-CDMA systems compared to an SC-CDMA system over an AWGN channel: F = 256, K = 16, = 0:5 (dash curves) and = 1:0 (solid curves).
These elementary functions are plotted in Fig. 3. Note that here the elementary density functions are purposely indexed such that the chip waveform corresponding to the elementary function with smaller index has a better MAI performance. This can be easily verified based on Lemma 1 and Fig. 3. Thus according to this arrangement, it is interesting to observe that among all the chip waveforms considered, the commonly used frequency raised cosine waveform has the worst MAI performance. In particular, it is worthwhile to point out that the “Better than Nyquist” waveform not only has a superior MAI performance but also has smaller distortion, a more open receiver eye, and a smaller symbol error rate in the presence of symbol timing error than the raised cosine waveform [26]. To evaluate the effect of different chip waveforms on the MAI performance over an AWGN channel, Fig. 4 plots the ratios between the normalized interference parameter
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TABLE I BER PERFORMANCE OF MC-DS-CDMA SYSTEMS USING DIFFERENT CHIP WAVEFORMS OVER A FADING CHANNEL: M = 4 AND = 1:0
Finally, Table I provides the BER performance obtained by simulation of the above systems over a fading channel. Here it makes each subchannel experiis assumed that using encing frequency-nonselective Rayleigh fading. As in the case of an AWGN channel, it can be observed again that the relative performance of the chip waveforms agrees with Lemma 1 and that chip waveform shaping has very little influence on the BER performance of MC-DS-CDMA systems. V. CONCLUSIONS
Fig. 5. BER performance of SC-CDMA (dash curves) and MC-DS-CDMA (solid curves, M = 4) systems using different chip waveforms over an AWGN channel: = 1:0.
of MC-DS-CDMA systems and that of the SC-DS-CDMA systems as functions of and for two different values of the rolloff factor, namely, and . Here the parameter is set to 256, while the number of users is . The reference SC-DS-CDMA systems use the raised cosine chip waveform. As can be seen from Fig. 4, the relative performance of different chip waveforms agrees perfectly with that predicted by Lemma 1. For all the chip waveforms considered, the performance of MC-DS-CDMA systems increases as the number of carriers increases. However, there is a saturation for the performance improvement by increasing the number of carriers . It can also be observed that the performance of MC-DS-CDMA systems is less sensitive to the chip waveform shaping than the . In fact, if the number of SC-DS-CDMA systems subcarriers is large enough, then the shape of a chip waveform has almost no influence on the MAI level of an MC-DS-CDMA . system. This is clearly observed in Fig. 4 when Simulation results are presented in Fig. 5 to illustrate the effect of chip shaping on BER performance of both SC-DSCDMA and MC-DS-CDMA systems over an AWGN channel. and three repreThe systems under consideration use sentative chip waveforms, namely, the optimal, the “Better than . Nyquist,” and the raised cosine waveforms, each with Although appropriate chip shaping can improve the error performance, Fig. 5 again confirms that, compared to the SC-DSCDMA systems, chip waveform shaping has very little effect on the BER performance of MC-DS-CDMA systems. In fact, if the is further increased, then there is no number of subcarriers difference in the error performance of MC-DS-CDMA systems by using different chip waveforms.
The effect of band-limited chip waveforms on the performance of MC-DS-CDMA systems has been investigated in this paper. It was demonstrated that, compared to SC-DS-CDMA systems, chip shaping has little influence on the performance of MC-DS-CDMA systems. The use of the elementary density function was also introduced for the comparison of different chip waveforms. Finally, it should be mentioned that only multicarrier CDMA systems with time-domain spreading (i.e., each carrier is bearing a different data bit of each user) have been considered in this paper. It seems interesting to also study the effect of chip waveform shaping on the performance of multicarrier CDMA systems that employ frequency-domain spreading, where all carriers are used to transmit the same data bit of each user. ACKNOWLEDGMENT The author would like to thank the reviewers for their helpful comments, which improved the presentation of this paper. REFERENCES [1] V. M. DaSilva and E. S. Sousa, “Multicarrier orthogonal CDMA signals for quasisynchronous communications systems,” IEEE J. Sel. Areas Commun., vol. 12, pp. 842–852, Jun. 1994. [2] L. Vandendorpe, “Multitone spread spectrum multiple access communications system in a multipath rician fading channel,” IEEE Trans. Veh. Technol., vol. 44, pp. 327–337, May 1995. [3] E. A. Sourour and M. Nakagawa, “Performance of orthogonal multicarrier CDMA in a multipath fading channel,” IEEE Trans. Commun., vol. 44, pp. 356–366, Mar. 1996. [4] S. K. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238–245, Feb. 1996. [5] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., pp. 126–133, Dec. 1997. [6] Y. H. Kim, I. Song, H. G. Kim, and J. Lee, “Design and performance analysis of a convolutionally coded overlapping multicarrier DS/CDMA systems,” IEEE Trans. Veh. Technol., vol. 49, pp. 1950–1967, Sep. 2000. [7] S. M. Elnoubi and A. El-Beheiry, “Effect of overlapping between succesive carriers of multicarrier CDMA on the performance in a multipath fading channel,” IEEE Trans. Commun., vol. 49, pp. 769–773, May 2001. [8] H. H. Nguyen and E. Shwedyk, “On carrier spacing in multicarrier CDMA systems,” in Proc., Can. Conf. Elec. Comp. Eng., Winnipeg, Canada, May 2002, pp. 1216–1220. [9] F. Behbahani and H. Hashemi, “On spectral efficiency of CDMA mobile radio systems,” in Proc. IEEE Int. Conf. Commun., 1994, pp. 505–509.
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[10] J. H. Han and S. W. Kim, “Optimal spectral overlay of DS/CDMA communication systems,” in Proc. Int. Conf. Universal Personal Commun., 1995, pp. 625–629. [11] , “Capacity of DS/CDMA communication systems with optimal spectral overlap,” IEEE Commun. Lett., vol. 2, pp. 298–300, Nov. 1998. [12] A. Taghol and H. A. Aghvami, “Bandwidth efficiency and capacity estimation of a multi-band W-CDMA system with partial spectral overlap,” in Proc. IEEE Veh. Technol. Conf., 2000, pp. 1768–1772. [13] M. Chen, B. Lee, and C. Wu, “Performance evaluation of a direct-sequence spread-spectrum multiple-access communication systems interfered by other CDMA/DSSS systems,” IEICE Trans. Fundamentals, vol. E83-A, pp. 1247–1256, Jun. 2000. , “Performance evaluation for multiple DSSS systems with [14] channel bands overlapped,” IEICE Trans. Fundamentals, vol. E83-A, pp. 1315–1325, May 2001. [15] B. M. Popovic´ , “Spreading sequences for multicarrier CDMA systems,” IEEE Trans. Commun., vol. 47, pp. 918–926, Jun. 1999. [16] E. Geraniotis and B. Ghaffari, “Performance of binary and quaternary direct-sequence spread-spectrum multiple-access systems with random signature sequences,” IEEE Trans. Commun., vol. 39, pp. 713–724, May 1991. [17] E. A. Geranitois and M. B. Pursley, “Error probability for direct-sequence spread-spectrum multiple-access communications—part II: approximations,” IEEE Trans. Commun., vol. COM-30, pp. 985–995, May 1982. [18] P. I. Dallas and F.-N. Pavlidou, “Innovative chip waveforms in microcellular DS/CDMA packet mobile radio,” IEEE Trans. Commun., vol. 44, pp. 1413–1416, Nov. 1996. [19] A. J. Viterbi, “Very low rate convolutional codes for maximum theoretical performance of spread-spectrum multiple-access channels,” IEEE J. Sel. Areas Commun., vol. 8, pp. 641–649, May 1990. [20] M. A. Landolsi and W. E. Stark, “DS-CDMA chip waveform design for minimal interference under bandwidth, phase and envelope constraints,” IEEE Trans. Commun., vol. 47, pp. 1737–1746, Nov. 1999.
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[21] T. F. Wong, T. M. Lok, and J. S. Lehnert, “Asynchronous multiple-access interference suppression and chip waveform selection with aperiodic random sequences,” IEEE Trans. Commun., vol. 47, pp. 103–114, Jan. 1999. [22] J. H. Cho and J. S. Lehnert, “An optimal signal design for band-limited asynchronous DS-CDMA communications,” IEEE Trans. Inform. Theory, vol. 48, pp. 1172–1185, May 2002. [23] J. Lee, R. Tafazolli, and B. G. Evans, “Effect of adjacent carrier interference on SNR under the overlapping carrier allocation scheme in FD/DS-CDMA,” Electron. Lett., vol. 32, pp. 171–172, Feb. 1996. [24] , “Capacity of the overlapped carriers scheme in FD/DS-CDMA,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. Appl., 1996, pp. 375–379. [25] S. Sureshkumar, E. Shwedyk, and H. H. Nguyen, “The impact of carrier spacing on the performance of multicarrier DS-CDMA systems,” in Can. Workshop Information Theory (CWIT), 2003, pp. 222–225. [26] N. C. Beaulieu, C. C. Tan, and M. O. Damen, “A “better than” Nyquist pulse,” IEEE Commun. Lett., vol. 5, pp. 367–368, Sep. 2001.
Ha H. Nguyen (M’01) received the B.Eng. degree from Hanoi University of Technology, Hanoi, Vietnam, in 1995, the M.Eng. degree from the Asian Institute of Technology, Bangkok, Thailand, in 1997, and the Ph.D. degree from the University of Manitoba, Winnipeg, Canada, in 2001. Since 2001, he has been with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, Canada, as an Assistant Professor. His research interests include digital communications, spread spectrum systems, and error control coding.