Redundancy Allocation in Turbo-Equalizer Design - IEEE Xplore

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Abstract—This paper considers properties of the extrinsic infor- mation transfer (EXIT) functions of turbo equalized intersymbol interference channels and ...
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 2, FEBRUARY 2005

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Redundancy Allocation in Turbo-Equalizer Design Matthew J. M. Peacock, Student Member, IEEE, and Iain B. Collings, Senior Member, IEEE

Abstract—This paper considers properties of the extrinsic information transfer (EXIT) functions of turbo equalized intersymbol interference channels and finite-impulse response precoders. An analytic expression is derived for the maximum value of the EXIT function of the equalizer. Using this parameter, a design strategy is proposed for allocating redundancy between the equalizer and the decoder. The key quantities are pilot-symbol rate and code puncturing, for fixed overall data and symbol rates. Index Terms—Extrinsic information transfer (EXIT) charts, turbo equalization.

I. INTRODUCTION

T

URBO equalization [1], [2] has been shown to yield remarkable bit-error rate (BER) performance over frequency-selective channels [3]. Similar to closely related turbo-coding techniques, the approach is to iterate between equalization and decoding stages in order to converge to a vastly improved data estimate [4]. Unfortunately, turbo-equalization schemes are notoriously difficult to analyze, and have tended to be investigated using Monte Carlo BER simulations. Recently, in the field of turbo coding, extrinsic information transfer (EXIT) functions have been proposed [5] as a new analysis tool. EXIT functions have been used to determine a stopping criterion for iterative decoding [6], and have also been applied to turbo-equalization schemes over fixed multipath channels [7], [8]. It has been observed that the maximum value of the EXIT function for the equalizer stage in a turbo equalizer is less than one, for general finite-impulse response (FIR) channels and signal-to-noise ratios (SNRs) of interest. In some cases, this value is significantly less than one. This implies that for the turbo equalizer to have a low BER, the outer code must be strong enough such that its EXIT function intersects the equalizer EXIT function on the right-hand axis of the EXIT chart. Clearly, this leads to an interesting design problem. In this paper, we derive an analytic expression for the maximum value of the equalizer EXIT function for fixed known intersymbol interference (ISI) channels. This value is related to the matched-filter bound [9], and as such, is a function of only the channel SNR (i.e., independent of the particular values of the ISI taps). We show that this value is independent of the pilot-symbol rate, and that the expression is also applicable to

Paper approved by K. Chugg, the Editor for Signal Processing and Iterative Design of the IEEE Communications Society. Manuscript received August 29, 2002; revised March 1, 2004. This work was supported in part by CSIRO Telecommunications and Industrial Physics, Australia. The authors are with the Telecommunications Laboratory, School of Electrical and Information Engineering, University of Sydney, NSW 2006, Australia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.841984

FIR precoders. It is also directly extendable to fading channels by averaging over the channel SNR distribution. An important design issue is the choice of appropriate codepuncturing and pilot-symbol rates. Pilot bits are added to aid equalizer performance, while code-parity bits are added to aid decoder performance. For practical systems with fixed dataand modulation-symbol rates (i.e., fixed data-rate services employing hardware with a fixed transmit bandwidth and modulation format), for each SNR, a design tradeoff exists between the pilot-symbol rate and the code (puncturing) rate. In this paper, we develop an analytically-based design strategy for optimizing the choice of pilot rate and code puncturing, in order to provide the best initial iteration for the turbo equalizer, while still meeting a BER requirement. The design is based on our analytic expression for the maximum value of the equalizer EXIT function, coupled with our observation that adding pilot symbols increases the area under the EXIT function. The analysis also allows us to determine the SNR range for which the equalizer will converge to a target BER for particular code-puncturing rates. II. TRANSMISSION SYSTEM MODEL In this paper, we assume a standard model for data communications when using a turbo equalizer, as shown in Fig. 1 (e.g., as used in [9] and others). Binary data is encoded with a binary to obtain a sequence of coded convolutional encoder of rate indicates the rate of the unpunctured outer code). Opbits ( tionally, the coded sequence is punctured at a rate , that is, th bit from the coded sequence is removed. This every sequence is then interleaved and interspersed with pilot bits with , that is, a pilot symbol is inserted after every frequency data symbols. We will call this sequence , which, for simplicity, we assume is modulated using binary phase-shift keying , where and (BPSK) to obtain the sequence . The data is processed in blocks of length , i.e., . Since we are interested in optimizing the allocation of redundancy (code puncturing and pilot symbols) in the system, all analysis and simulations in this paper are for a fixed data rate and a fixed channel-symbol rate. In other words, we are optimizing the choice of and for a fixed energy per bit. Therefore, the channel-symbol SNR is also fixed. Note that if this approach is not taken, then the channel-symbol SNR would need to be scaled in order to account for the waste of energy resulting from using pilots, when comparing different pilot frequencies. Assume the channel symbols are transmitted with symbol period , and denote the discrete-time equivalent impulse response of the combined channel and transmit/receive filters by

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Fig. 1.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 2, FEBRUARY 2005

Block diagram of transmitter, channel, and iterative receiver.

rate to obtain

. The received signal is sampled at the symbol , where (1)

where each is independent, identically distributed (i.i.d.) additive white Gaussian noise (AWGN) with zero mean and . In this paper, we two-sided power spectral density primarily consider fixed ISI channels; however, the results extend to rate-1 FIR precoders and fading channels in a straightfor, where ward manner. Receive SNR is defined as . The receiver we are considering contains a typical turboequalization loop, consisting of a soft-input soft-output (SISO) a posteriori probability (APP) equalizer, de/interleavers, and a SISO APP decoder. Pilot-symbol insertion and removal may also be included within the loop. We consider the optimal APP equalizer [10], and assume full channel-state information (CSI) is available, but again this can be extended. This equalizer computes the APPs , given both the a priori probabili, and the received sequence . ties The output of the equalizer is the extrinsic log-likelihood ratio (LLR), given by the difference between the a posteriori and a priori LLRs, i.e., (2) where . During the first is zero for all (except in iteration of turbo equalization, positions which contain pilot symbols). According to the turbo is given by the interprinciple, in succeeding iterations, leaved LLRs from the extrinsic output of the decoder. III. MAXIMUM-OUTPUT MUTUAL INFORMATION Let denote the average mutual information (MI) between the data sequence and a priori inputs going into the equalizer. denote the average MI between the data seSimilarly, let quence and the extrinsic information coming out of the equalizer. That is (3)

(4) is the MI between the scalar random variables where and . The EXIT function for the equalizer is then found by plotting as a function of . An EXIT chart [5] is composed of the EXIT functions of the component decoder and equalizers. For large block lengths, it has been demonstrated that EXIT charts accurately predict the behavior of the iterative equalizer, and as such, in this paper we assume is large. For a serially concatenated system, if the a priori information is modeled as a binary erasure channel (BEC) with erasure probability tuned to , it has been shown that the area under the EXIT function is equal to [11] (5) where and . Also denotes the rate of the inner code, which for an ISI channel . When pilot symbols are or rate-1 FIR precoder is employed in the manner described in Section II, then . In fact, it is generally accepted [12], [13] that (5) is true, or at least an excellent approximation, for the EXIT functions of any serially concatenated system employing the turbo principle. That is, it not necessary that the a priori information is the output of a BEC channel. Hence, (5) is applicable for the EXIT function of the inner decoder of the turbo-equalization system depicted is the uniform input capacity of in Fig. 1, for which the ISI channel or FIR precoder. Fig. 2 shows the EXIT function of the APP equalizer for several values of SNR and pilot bit rates for the Proakis Channel-C . [14, p. 631], i.e., obtained does Note that for high , the maximum value of not approach one. It is also interesting to note that for pilot-aided equalization, the pilot rate only affects the EXIT function for is independent of low . In addition, the maximum value of the pilot rate, and depends only on the received SNR. In this section, we derive an analytical expression for the maximum value of , in terms of the channel SNR (these analytical points are shown with stars in Fig. 2), and use (5) to conclude some important properties of the EXIT charts when pilot symbols are used.

PEACOCK AND COLLINGS: REDUNDANCY ALLOCATION IN TURBO-EQUALIZER DESIGN

Fig. 2. EXIT function of equalizer for Proakis Channel-C with various pilotsymbol rates and SNRs.

Recall that

is computed using the received sequence and , and, in particis not used. In [9, Sec. V.E], it was observed ular, note that implies that as for all that the absence of , the ISI disturbing a received symbol can be removed completely. Therefore, under these conditions, the output of the APP equalizer becomes equivalent to the output of a matched filter with all ISI removed. This observation can be further used to derive an explicit expression for the extrinsic LLRs, based on a typical weight-1 error path in the equalizer trellis. Note that this is not a high SNR approximation, as was considered for ISI precoders in [15]. Regardless of SNR, highly confident priors effectively mean that , every bit in the trellis is known, except when calculating for all , the exfor . Therefore, as trinsic LLR of the th symbol, given in (2), becomes exactly (6) (7) Examining

terms in (6), the random variable is zero mean. Moreover, when condi. By further tioned on , it is Gaussian with variance conditioning on , it follows that is Gaussian with mean and variance . We can now write an explicit expression for the MI . Firstly, consider , where , and is zero-mean Gaussian noise with variance , and . define the LLR In [16, App.], the following expression is given for the MI , for which they use the symbol :

(8)

Fig. 3. Simulated and analytic maximum-output MI I symbol SNR for Proakis Channel-C.

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versus channel-

as We can now directly apply (8) to obtain for all , which we define as , since we , as follows: previously determined that (9) in (4) Note that the result is independent of , which means . is exactly Therefore, we now have an analytic expression for the maximum amount of MI between the data sequence and the extrinsic outputs of the APP equalizer. It is interesting to note that we can is, in fact, independent of the parnow see from (9) that ticular channel, and only depends on the total power of channel taps. This is, of course, related to the so-called “matched-filter bound,” i.e., it is a measure of the best possible performance as if the channel had no ISI and was just AWGN. Graphically, (9) gives the top right-hand-side point of the EXIT function for the equalizer. For example, consider Proakis Channel-C at 5 dB SNR. Using (9), we have . This number has previously been evaluated by simulations, as in [8, Fig. 3], which shows for “Measure . More 4.” Our resimulation of this result gave generally, Fig. 3 shows the empirical and analytic values of versus channel symbol SNR . The empirical values have been obtained by Monte Carlo simulation of the system using Proakis Channel-C, but, of course, this figure applies to any fixed ISI channel with the same SNR, as discussed above. Note that (9) can also be directly extended to fading channels, by averaging over the fading power distribution of , e.g., the exponential distribution for Rayleigh fading channels. For cases where the channel is unknown, approximations can be used for the channel taps, and further averaging over the channel errors can be performed. An interesting additional point is that in [9], it was observed that the maximum-output MI values for both the APP equalizer and a SISO minimum mean-square error (MMSE) linear equalizer were identical. Therefore, it is reasonable to expect that (9) is also valid for the SISO MMSE linear equalizer.

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Note that the derivation of (9) also applies to rate-1 FIR pre(where addition is modulo-2). Howcoders, e.g., , and therefore, (9) applies with ever, in this case equal to the weight of the precoder (e.g., for ), although note that this does not imply that the transmitted power has increased. In the case of a serial concatenais tion of a precoder and an ISI channel, the calculation of not as straightforward as (9), however, the same principles apply. This explains why the addition of a rate-1 precoder to a turboequalized ISI channel improves the high-input MI performance (e.g., as in [9], [15], and [17]). However, we can see that even with the inclusion of a precoder, it is still the case that for suffiof the equalizer will be less than one. ciently low SNR, Focusing on the pilot rates, another interpretation of the argument used in deriving (6) is that every channel symbol can be for all considered to be a pilot symbol when . This explains why (9) is independent of the pilotsymbol rate, , as indicated by Fig. 2, discussed previously. The pilot rates do, however, affect the total area under the , as seen in the figure, and indicated by (5). By recurve, is reduced, since placing a data bit with a pilot bit, the pilot symbols are known at both transmitter and receiver. Of course, if all the channel symbols were pilot symbols, then . Note that in (5) is also decreased by adding pilot symbols. It is not obvious which of these effects will have , and therefore, whether the area under a greater impact on the EXIT function will increase or decrease. Simulation-based , e.g., approaches have been proposed for calculating [18], however, no analytic expression exists. Clearly, from Fig. 2, is not as great we can at least observe that the decrease in , since the area under the EXIT functions as the decrease in is increased by adding pilot symbols. We have calculated EXIT charts for a wide range of channels, SNRs, and pilot-symbol rates, and this observation has always been seen to hold. Since is independent of the pilot rate, and the area under the EXIT function increases with pilot rate, this, therefore, implies that the left-hand (low-input ) end of the EXIT function is raised by adding pilots (since the EXIT function is strictly nondecreasing). This observation will be used later in our design strategy. A final observation regarding the equalizer EXIT function is that (5) may be generalized to the case of nonzero a priori information (i.e., with feedback from the decoder in a turbo-equalization loop), by incorporating the a priori information into the channel model. We now compute a new uniform input capacity for this augmented channel model by using the technique described in [18], but by also feeding in a priori LLRs generated in the usual way when generating an EXIT function value for a par, each a priori LLR input is computed ticular (i.e., for , where is i.i.d. zero-mean according to ). We denote Gaussian noise with variance , and this new uniform input capacity for the augmented channel as . By doing this, we have observed that (5) can be extended to

(10)

Fig. 4. Simulated area and uniform input capacity versus input MI for Proakis Channel-C.

To illustrate this result, the shaded region under the EXIT dB and no pilot symbols, function in Fig. 2 (for SNR ) has area . In comparison, . More generally, Fig. 4 shows both the empirical values (dashed lines) and (solid lines) for a range of of the area channel-symbol SNRs, as a function of for the Proakis Channel-C in the absence of pilot symbols. Similar plots for Proakis Channel-B, and these channels including pilot symbols, indicate that (10) is at least a good approximation. IV. DESIGN STRATEGY We now aim to use the results from the previous section to allocate redundancy within the turbo-equalizer system. Since we are considering a fixed channel-symbol rate and a fixed total information rate, there is a corresponding fixed amount of redundancy. Of course, if we add more pilot symbols to aid the equalizer, then we will need to use a higher rate code in order to maintain a constant overall data rate, and in doing so, the decoding will be less powerful. The overall rate of the transmitter , so, for example, if is and we desire , then we are restricted . The allocation of this redundancy is an important to tradeoff in turbo-equalization systems. In this section, we propose a design strategy for fast convergence of the iterative receiver for a given threshold SNR. We first consider the relationship between the BER and the convergence point of the EXIT chart, denoted (i.e., the first intersection of the equalizer and decoder EXIT functions). This was considered for parallel concatenated turbo codes in [5]. Using the same argument as in [5, Sec. III-C], we have BER where

where

. Note that (10) is identical to (5) when

.

(11) is the functional inverse of (8), and . This equation can be used to find BER contours

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TABLE I SNR THRESHOLDS FOR TURBO EQUALIZER TO ACHIEVE BER = 10 EXAMPLE PUNCTURED CONVOLUTIONAL CODE

Fig. 5. Upper right-hand corner of EXIT chart, for punctured convolutional code and Proakis Channel-C at 2 dB channel symbol SNR.

in the EXIT chart. Note that for BERs of interest, the contour lines are very close together in the top right-hand corner of the EXIT chart. To achieve a given target minimum BER, clearly, the turbo is above equalizer must be designed such that and to the right of the corresponding target minimum BER contour on the EXIT chart. Note that for BERs of interest, the convergence points are extremely close to the right-hand axis. In , given in (9). This then allows us to this region, for this target use (11) to find the corresponding value of minimum BER, which we will denote . Any decoder EXIT when evaluated at , will function which is greater than converge to a BER less than the target minimum. Fig. 5 shows the upper right-hand corner of an EXIT chart. For this example, the code is a punctured, constraint length 7, rate-1/2, nonsystematic, nonrecursive, convolutional code with maximal minimum free distance from [19, Table 11-5], given , where is the impulse response by of the th output of the encoder in common octal notation. The equalizer EXIT functions are for the Proakis Channel-C with dB. Curves are shown for various puncturing rates SNR and pilot-symbol rates ( , ). Note that all the curves for the different pilot-symbol rates lie on top of each other in this high-input MI region. Also note that equalizer EXIT functions are close to horizontal, justifying the approximation in the previous paragraph. The analytic value for this SNR is marked with a star in the figure. The of for BER are shown for each punctured values of code, marked with open boxes in the figure. We can, therefore, , the converged BER will be see that for puncturing rates . less than In Table I, we show the intersection point of the decoder EXIT contour for each code-puncturing rate functions and the (determined via simulation), i.e., the pairs from Fig. 5. Using the approximation , we have , given in (9), and found the inverted the expression for corresponding SNR threshold for each puncturing rate, which is shown in the final column of the table. This can be used in

FOR

the design process to eliminate codes or puncturing rates which will not be able to achieve a target BER at a specified SNR. We are now led to propose the following design strategy. First determine the worst-case SNR under which the system as described in is expected to operate. Then solve for Section III. We start the design here, since this value is independent of pilot rate. Then pick the highest rate punctured code (lowest ) for which the decoder EXIT function is at least for an input MI of . Picking the highest rate punctured code corresponds to allocating the minimum acceptable amount of redundancy to the decoder. Then the pilot rate should be chosen as frequently as possible (smallest ), while still satisfying the desired end-to-end data and symbol rates. As discussed in Section III, this will raise the left-hand side of the EXIT function for the equalizer, and therefore provide the best start for the turbo-equalizer iteration. Of course, the EXIT functions of the decoder and equalizer may still cross on the EXIT chart at low-input MI values, indicating that the particular combination of code and channel will not operate at the desired BER and SNR. Although (9) only applies to known channels, for cases where the channel is unknown, approximations can be used to generate , as noted previously. We have observed that estimates of the above design strategy also applies in the unknown-channel case. By using the highest acceptable puncturing rate, and therefore, freeing up redundancy for pilot symbols, in this unknownchannel case, there is a dual benefit of not only assisting equalization, but also assisting channel estimation. We have observed that pilot symbols are especially crucial in the early iterations when channel estimation is being performed based on unreliable data. In this region, the pilots are critical for starting the turbo-equalizer iteration. V. CONCLUSION We have considered properties of the EXIT functions of turbo-equalized ISI channels and FIR precoders. An analytic expression was derived for the maximum value of the EXIT function of the equalizer. The effect of pilot symbols on the EXIT functions was discussed. Design strategies were proposed for allocating redundancy between the equalizer and the decoder. The key quantities are pilot-symbol rate and code puncturing, for fixed overall data and symbol rates. REFERENCES [1] C. Douillard, M. Jezequel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference: Turbo-equalization,” Eur. Trans. Telecommun., vol. 6, pp. 507–511, Sep./Oct. 1995. [2] L. Hanzo, T. H. Liew, and B. L. Yeap, Turbo Coding, Turbo Equalization and Space–Time Coding. New York: Wiley, 2002.

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[3] G. Bauch and V. Franz, “A comparison of soft-in/soft-out algorithms for turbo detection,” in Proc. Int. Conf. Commun., Jun. 1998, pp. 259–263. [4] K. M. Chugg, A. Anastasopoulos, and X. Chen, Iterative Detection, Adaptivity, Complexity Reduction, and Applications. Norwell, MA: Kluwer, 2001. [5] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, pp. 1727–1737, Oct. 2001. [6] I. Land and P. A. Hoeher, “Using the mean reliability as a design and stopping criterion for turbo codes,” in Proc. ITW, Sep. 2001, pp. 27–29. [7] J. Hagenauer. Log-likelihood ratios, mutual information and EXIT charts—A primer. presented at 12th Joint Conf. Commun., Coding. [Online]. Available: http://www.lnt.e-technik.tu-muenchen.de/veroeffentlichungen/ [8] M. Tüchler, S. ten Brink, and J. Hagenauer, “Measures for tracing convergence of iterative decoding algorithms,” in Proc. 4th Int. ITG Conf. Source, Channel Coding, Berlin, Germany, Jan. 2002, pp. 53–60. [9] M. Tüchler, R. Koetter, and A. C. Singer, “Turbo equalization: Principles and new results,” IEEE Trans. Commun., vol. 50, pp. 754–767, May 2002. [10] R. W. Chang and J. C. Hancock, “On receiver structures for channels having memory,” IEEE Trans. Inf. Theory, vol. 12, pp. 463–468, Oct. 1966. [11] A. Ashikhmin, G. Kramer, and S. ten Brink, “Extrinsic information transfer functions: A model and two properties,” in Proc. Conf. Inf. Sci., Syst., Mar. 2002, pp. 742–747. [12] M. Tüchler and J. Hagenauer, “Exit charts of irregular codes,” in Proc. Conf. Inf. Sci., Syst., Mar. 2002. [13] M. Tüchler, “Design of serially concatenated systems for long or short block lengths,” in Proc. IEEE Conf. Commun., vol. 4, May 2003, pp. 2948–2952. [14] J. Proakis, Digital Communications, 4th ed. New York: McGraw-Hill, 2001. [15] K. R. Narayanan, “Effect of precoding on the convergence of turbo equalization for partial response channels,” IEEE J. Sel. Areas Commun., vol. 19, pp. 686–698, Apr. 2001. [16] S. ten Brink and G. Kramer, “Design of repeat-accumulate codes for iterative detection and decoding,” IEEE Trans. Signal Process., vol. 51, pp. 2764–2772, Nov. 2003. [17] I. Lee, “The effect of a precoder on serially concatenated coding systems with an ISI channel,” IEEE Trans. Commun., vol. 49, pp. 1168–1175, Jul. 2001. [18] D. Arnold and H. A. Loeliger, “On the information rate of binary-input channels with memory,” in Proc. IEEE Conf. Commun., vol. 9, Jun. 2001, pp. 2692–2695. [19] S. B. Wicker, Error Control Systems for Digital Communications and Storage, 1st ed. New York: Prentice-Hall, 1995.

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Matthew J. M. Peacock (S’01) is from Willaura, Australia, and was born in 1978. He received the combined B.E. degree in electrical engineering and B.Sc. degree in mathematics/computer science from the University of Melbourne, Melbourne, Australia, in 2001. He is currently working toward the Ph.D. degree in electrical engineering at The University of Sydney, Sydney, Australia. He is also with the Telecommunications and Industrial Physics group at Australia’s Commonwealth Scientific and Industrial Research Organization (CSIRO). His current research interests include turbo equalization, large-system analysis, information theory, and random matrix theory. Mr. Peacock received the University Medal in Electrical Engineering upon graduating from the University of Melbourne. He also received a postgraduate scholarship from CSIRO.

Iain B. Collings (S’92–M’95–SM’02) was born in Melbourne, Australia, in 1970. He received the B.E. degree in electrical and electronic engineering from the University of Melbourne, Melbourne, Australia, in 1992, and the Ph.D. degree in systems engineering from the Australian National University, Canberra, in 1995. In 1995, he was a Research Fellow in the Australian Cooperative Research Center for Sensor Signal and Information Processing, Adelaide, Australia, where he worked in the area of radar signal processing. From 1996 to 1999, he was a Lecturer at the University of Melbourne, and since 1999, he has been with the School of Electrical and Information Engineering, University of Sydney, Sydney, Australia, where he is currently an Associate Professor. His current research interests include synchronization, channel estimation, equalization, and multicarrier modulation, for time-varying and frequency-selective channels. Dr. Collings currently serves as an Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He also served on the Technical Program Committees for the IEEE GLOBECOM Conference, Taipei, Taiwan, 2002, and for the IEEE Vehicular Technology Conference, Orlando, FL, 2003. Additionally, he served on the Organizing Committees for the IEEE International Symposium on Spread Spectrum Techniques and Applications, Sydney, Australia, 2004, the IEEE Information Theory Workshop, Cairns, Australia, 2001, and was a Founding Member for the Australian Communications Theory Workshop 2000–2004.