Iterative methods for equalization, joint source/channel coding and decoding

New iterative equalization and decoding algorithms have been developed for communication over intersymbol interference channels [Iter.1-Iter.4]. These algorithms pass soft information in the form of priors over the transmitted symbols between soft-input soft-output devices, such as an MMSE-based equalizer and a MAP decoder. Our results indicate that such iterative strategies can significantly outperform traditional methods, such as those which separate the decoding and equalization steps by only passing soft information from the equalizer to the decoder. The analysis of such joint iterative equalization and decoding strategies is a rather difficult problem. However, with a few key simplifications, i.e. very long codelength, a density evolution analysis can be applied. Applying such an analysis, we were able to identify good operating conditions for very simple equalization schemes. In particular, derive very low complexity hybrid schemes with excellent performance for joint equalization and detection [Iter.5]

When two channels are available for transmission, soft-input/soft-output MMSE equalizers can exchange soft-information until agreement is reached. Such an approach has been demonstrated to dramatically improve receiver performance over traditional joint MMSE equalization [Iter.6-Iter.9]. We have also investigated the suitability of iterative algorithms for joint source-channel coding and decoding [Iter.10]. Iterative techniques for the first time provide a relatively simple mechanism for jointly optimizing a wide variety of constraints, as are typically found in joint image and channel coding. Preliminary results indicate that such iterative methods to joint decoding can significantly improve the overall performance compared to a separate decoding. The generic framework provided by such iterative methods have also proven potentially useful for a variety of related problems [Iter.11-Iter.12]. Another application in which iterative decoding has shown promise for significant gains, is the problem of non-coherent multi-symbol detection over Rayleigh fading channels. One of the central themes of this research is the design of codes that are robust against fading. Such codes, when concatenated with turbo-like codes and coupled with a suitable differential demodulation scheme can eliminate the overhead of pilot symbols required for coherent detection [Iter.13]. The SNR transfer characteristic band of the extrinsic value in the turbo decoding is proposed in [Iter.14]. The proposed method can explain the bimodal Gaussian average histogram of the extrinsic value in the waterfall region of the finite length turbo decoding. The proposed method is also useful to discuss the probabilistic convergence behavior of the finite length turbo decoding as well as the wide waterfall region in the BER performance curve. In [Iter.15] we propose a SNR transfer characteristic band of the extrinsic values in the finite length turbo decoding. It is able to describe the density evolution of the extrinsic value, which has a bimodal Gaussian histogram on the average in a wide waterfall region of the BER performance. Then, we discuss the relationship between the information blocklength and the width of the waterfall region in the BER performance.

[Iter.1] M. Tuechler, R. Koetter, and A. Singer, ``Iterative correction of intersymbol interference via equalization and decoding with priors,'' Proceedings of the 2000 International Symposium on Information Theory, p. 100, June 25-30, 2000, Sorrento, Italy.

[Iter.2] M. Tuechler, R. Koetter, and A. Singer, ``Turbo equalization'': principles and new results. IEEE Transactions on Communications, vol. 50, no. 5, pp. 754-767, May 2002.

[Iter.3] M. Tuechler, A. Singer and R. Koetter, ``Minimum mean squared error equalization using a priori information,'' IEEE Transactions on Signal Processing, vol. 50, no. 3, pp. 673-683, March 2002.

[Iter.4] M. Tuechler, ``Iterative correction of intersymbol interference via equalization and decoding with priors,'' M.\ S. Thesis, Dept. of Electrical and Computer Engineering, UIUC, January 2000.

[Iter.5] M. Tuechler, A. Singer, R. Koetter, "Hybrid equalization strategies for iterative equalization and decoding", Proceedings IEEE International Symposium on Information Theory, Washington D.C., June 2001.

[Iter.6] A. Singer, J. Nelson, and R. Koetter, ``Linear iterative turbo equalization (LITE) for dual channels,'' Proceedings of the Thirty-third Asilomar Conference on Signals, Systems, and Computers, October 24-27, 1999, Monterey, CA.

[Iter.7] J. Nelson, R. Koetter, and A. Singer, ``Evolution of priors in the LITE,'' Proceedings of the 34th Conference on Information Sciences and Systems, March 15-17, 2000, Princeton, NJ.

[Iter.8] J. Nelson, R. Koetter, and A. Singer "Evolution of Prior Information in SISO Equalization," IEEE International Symposium on Information Theory, 2001.

[Iter.9] J. Nelson, A. Singer, and R. Koetter, "Linear Iterative Turbo-Equalization (LITE) for Dual Channels," submitted June 2001 to the IEEE Transactions on Communications.

[Iter.10] I. Kozintsev, R. Koetter, and K. Ramchandran, ``A Factor Graph Framework for Joint Source-Channel Decoding of Images,'' Proceedings of the Thirty-third Asilomar Conference on Signals, Systems, and Computers, October 24-27, 1999, Monterey, CA.

[Iter.11] M. R. Naphade, I. Kozintsev, T. S. Huang, and K. Ramchandran ``A Factor Graph Framework for Semantic Indexing and Retrieval in Video,'' Proceedings of the IEEE Workshop on Content-based Access of Image and Video Libraries, June 20, 2000, Hilton Head SC.

[Iter.12] I. Kozintsev, ``Signal processing for joint source-channel coding of digital images,'' Ph.\ D. Thesis, Department of Electrical and Computer Engineering, UIUC, May 2000.

[Iter.13] R.-R. Chen, D. Agrawal, and U. Madhow, ``Noncoherent detection of factor graph codes over fading channels,'' Proceedings of the 34th Conference on Information Sciences and Systems, 2001. March 15-17, 2000, Princeton, NJ.

[Iter.14] Jeong W. Lee and Richard E. Blahut, "A Note on the Analysis of Finite Length Turbo Decoding," IEEE International Symposium on Information Theory, July 2002, p. 83.

[Iter.15] Jeong W. Lee and Richard E. Blahut, "Analysis of the Extrinsic Values in the Finite Length Turbo Decoding," Conference on Information Sciences and Systems, Princeton, 2002.

[Iter.16] Jeong W. Lee and Richard E. Blahut, "Asymptotic bit error rate analysis of binary turbo codes," in preparation.

Universal methods for linear prediction and adaptive filtering

We recently developed a new approach to linear prediction and adaptive filtering, based on recent developments in theory of universal coding and computational learning. This development provides a novel perspective on the adaptive filtering problem, and represents a significant departure from traditional adaptive filtering methodologies. In this context, we demonstrate a sequential algorithm whose accumulated squared prediction error, for every possible sequence, is asymptotically as small as the best fixed linear predictor for that sequence. Some of the statistical properties of the universal linear prediction algorithm are described in the following papers which also extend the framework to include a variety of multi-stage adaptive filtering algorithms [Univ.1-Univ.3].

[Univ.1] S.S. Kozat and A.C. Singer, ``Multi-stage adaptative signal processing algorithms'' Proceedings of the First IEEE Sensor Array and Multichannel Signal Processing Workshop, pp. 380-384, March 16-17, 2000, Cambridge, MA.

[Univ.2] S.S. Kozat and A.C. Singer, ``On universal linear prediction of Gaussian data,'' Proceedings of the 2000 IEEE Conference on Acoustics, Speech, and Signal Processing, pp. 13-16, Istanbul, Turkey.

[Univ.3] S. Kozat and A. Singer, "Multi-stage adaptive algorithms,'' Submitted to the IEEE Transactions on Signal Processing.

[Univ.4] S. S. Kozat, A. C. Singer, "A Lower bound on the Performance of Sequential Prediction", Proc. International Symposium on Information Theory, Loussanne, June 2002.

[Univ.5] S. S. Kozat, A. C. Singer, " Further results in Multistage Adaptive Filtering", Proc. Int. Conf. Acous. Speech and Sig. Process, Orlando, 2002. To appear.

[Univ.6] A. C. Singer, S. S. Kozat, M. Feder, " Universal linear least squares prediction: upper and lower bounds," IEEE Trans. Information Theory, vol. 48, no. 8, Aug. 2002, pp. 2354-2362.

[Univ.7] D. Kozat and M. Feder, "A sequential probability approach to universal linear least-squares prediction," Accepted to IEEE Transactions on Information Theory.

Causal/anticausal methods for equalization

One aspect of our recent work in equalization is the development of a bidirectional communications operating at data-rates and signal to noise ratios comparable to GSM or EDGE. The BAD algorithm uses an MMSE-DFE structure to process a block of data in both the causal and anti-causal directions, thus generating symbol estimates whose error events due to error propagation are roughly uncorrelated. As a result, we are able to improve the operating range of the system by roughly 3dB, or more than half the gap between the conventional DFE and optimal MAP equalization, for some channels of interest [CAC.1-CAC.2]. This approach may be particularly well-suited to EDGE and other GSM variants, where equalization is performed on a block by block basis using only a mid-amble for channel estimation.

[CAC.1] C. McGahey, A. Singer and U. Madhow, ``BAD: A bi-directional arbitrated decision feedback equalizer,'' Proceedings of the 34th Conference on Information Sciences and Systems, March 15-17, 2000, Princeton, NJ.

[CAC.2] C. McGahey, U. Madhow, and A. Singer, ``Bidirectional arbitrated decision feedback equalization,'' IEEE Transactions on Communications, vol. 53, Feb. 2005, pp. 214-218.

Multi-access capacity and wireless network issues

Our research into networking has led us to consider the capacity of time-slotted ALOHA systems in which multiple users synchronously send packets which may collide at the receiver. Specific coding for ALOHA systems had previously been proposed to avoid complete loss of packets involved in collision, but the capacity of ALOHA systems had not been previously determined. We considered capacity in terms of reliably received rate rather than transmitted rate. We considered capacity-achieving strategies under AWGN for transmission of a single packet that is long enough to achieve capacity over the duration of the packet. By combining concepts from multi-access channels and broadcast channels, we determined the capacity region for a single transmission of a packet in an ALOHA system. The coding for each user takes into account the possibility of collisions with other users in order to establish a capacity region. We derived, according to the users' energies and their probabilities of transmission, the coding and decoding schemes which achieve the capacity region. We also considered the case of multiple packet transmissions under a channel model where users receive the right to transmit the package according to independent Bernoulli processes. We=20 then apply the single-packet coding strategies in order to maximize the expected reliably received rate [Multi.1]. Other restrictions on transmission strategies, coding and decoding policies, and the stability problems for such systems are under active consideration [Multi.2]. Other progress in allocation of wireless network resources is reported in [Multi.3].

We explored optimization of paging and registration policies in cellular networks [Multi.6, Multi.7]. Motion is modeled as a discrete-time Markov process, and minimization of the discounted, infinite-horizon average cost is addressed. The structure of jointly optimal paging and registration policies is investigated through the use of dynamic programming for partially observed processes. It is shown that there exist policies with a certain simple structure that are jointly optimal, though the dynamic programming approach does not directly provide an efficient method to find the policies.

An iterative algorithm for policies with the simple form is proposed and investigated. The algorithm alternates between paging policy optimization and registration policy optimization. It finds a pair of individually optimal policies, but an example is given showing that the policies need not be jointly optimal.

[Multi.1] M. Medard, S. Meyn, J. Huang, and A. Goldsmith, ``Capacity of time-slotted ALOHA system,'' Proceedings of the 2000 International Symposium on Information Theory, June 25-30, 2000, Sorrento, Italy.

[Multi.2] J. Huang, ``Capacity and stability of time-slotted ALOHA system,'' M.S. Thesis, Department of Electrical and Computer Engineering, UIUC, May 2000.

[Multi.3] R.-R. Chen and U. Madhow, ``Admission control and resource allocation for DS-CDMA networks with multiple traffic classes,'' Proceedings of the 2000 International Symposium on Information Theory, June 25-30, 2000, Sorrento, Italy.

[Multi.4] J. Huang, I. Kontoyiannis, and S.P. Meyn, "The ODE Method and Spectral Theory of Markov Operators", Proceedings of the Kansas Workshop on Stochastic Theory and Control, To be published by the Springer-Verlag in the series of Lecture Notes in Control and Information Sciences.

[Multi.5] M. Medard, J. Huang, A. Goldsmith, S. Meyn, T. Coleman, "Capacity of time-slotted ALOHA packetized multiple-access system," IEEE Transactions on Wireless Communications, vol. 3, no. 2, March 2004, pp. 486-499.

[Multi.6] B. Hajek, "Jointly optimal paging and registration for a symmetric random walk," Proc. 2002 IEEE Information Theory Workshop, pp. 20-23, October 2002.

[Multi.7] B. Hajek, K. Mitzel, and S. Yang, Paging and registration in cellular networks: jointly optimal policies and an iterative algorithm, Proc. IEEE INFOCOM 2003, March 2003.

[Multi.8] A. Erylimaz, R. Srikant, and J. Perkins, "Stable scheduling for fading wireless channels," To appear in IEEE/ACM Transactions on Networking.

VLSI Algorithms and Architectures

We began exploring the development of systolic architectures for factor-graph based decoding algorithms. We also studied systolic architectures for the more conventional Reed-Solomon decoding algorithms, and recently devised such an architecture based on the Berlekamp-Massey algorithm. The throughput bottleneck that exists in conventional implementations has been eliminated via a series of algorithmic reformulations that result in a fully systolic architecture. It is estimated that the new decoder architecture can decode a Reed-Solomon code over GF$(2^8)$ at more than double the speed achievable by a conventional decoder (for noninterleaved codes), whereas interleaved codes can be decoded at rates that are an order of magnitude higher than can be achieved by previously published methods. Details can be found in [AlgArch.1-AlgArch.2].

More recently, we have developed a programable universal Reed-Solomon decoder architecture that can decode any Reed-Solomon code of any length over a field of any size up to GF(2M) where M is the length of the registers in the chip. This decoder needs only the block length, the error-correction capability desired, and the coefficients of the irreducible polynomial defining the field as its control inputs. Furthermore, these control inputs can be changed from codeword to codeword as desired, so that the same chip can act as, say, a decoder for a (15,7) 4-error-correcting code over GF(24) for one codeword, as a decoder for a (255,223) 16-error-correcting code over GF(28) using the NASA standard irreducible polynomial) for the next codeword, and as a decoder for a (255,223) 16-error-correcting code over GF(28) using a non-standard polynomial for the codeword immediately following. We are also exploring applications of this concept to multimedia communications systems (the same decoder chip can be used to decode different data streams encoded with different codes with varying levels of performance) to privacy applications (by using different irreducible polynomials, different user pairs can maintain a degree of privacy in publically shared channels), and to packet communications (dropped packets can be treated as symbol erasures in an interleaved high-rate Reed-Solomon code).

[AlgArch.1] D. V. Sarwate and N. R. Shanbhag, ``Very high-speed Reed-Solomon decoders,'' Proceedings of the 2000 International Symposium on Information Theory, p. 419, June 25-30, 2000, Sorrento, Italy.

[AlgArch.2] D. V. Sarwate and N. R. Shanbhag, ``High-speed architectures for Reed-Solomon Decoders,'' IEEE Transactions on VLSI Systems, vol. 9, pp. 641-655, October 2001.

[AlgArch.3] B. Shim and N. R. Shanbhag, ``Complexity analysis of multicarrier and single-carrier systems for very high-speed digital subscriber line" IEEE Transactions on Signal Processing, vol. 51, pp. 282-292, January 2003.

[AlgArch.4] Z. Yan and D. V. Sarwate, ``Modified Euclidean algorithms for decoding of Reed-Solomon codes,'' Proceedings of the 2002 Conference on Information Sciences and Systems, Princeton University, Princeton NJ, March 20-22, 2002.

[AlgArch.5] Z. Yan and D. V. Sarwate, ``Modified Euclidean algorithm for finite field inversions,'' Proceedings of the IEEE 2002 International Symposium on Information Theory, Lausanne, Switzerland, June 30 - July 5, 2002.

[AlgArch.6] Z. Yan and D. V. Sarwate, ``Systolic architectures for finite field inversion and division,'' Proceedings of the IEEE International Symposium on Circuits and Applications, Phoenix AZ, May 20-24, 2002.

[AlgArch.7] Z. Yan and D. V. Sarwate, ``New Systolic Architectures for Inversion and Division in GF(2^m),'' IEEE Transactions on Computers, Vol. 52, Nov. 2003, pp. 1514-1519.

Energy-Efficient Communication Systems

We are exploring the limits of achievable energy-efficiency of wireless communication systems in future deep submicron technologies. Achieving energy-efficiency and reliability in future technologies becomes a challenge due to the emergence of deep submicron noise, process variations, and increased defect densities. We will employ an information-theoretic approach for obtaining bounds on energy-efficiency of wireless microsystems in the presence of noise and other non-idealities. Further, a comprehensive design approach will be developed that is based upon the principle of noise-tolerance that will enable us to approach these bounds. This approach will span algorithmic, architectural and circuit domains.

[Energy.1] L. Wang and N. R. Shanbhag, ``Adaptive error cancellation for low-power digital filtering," Proc. of Asilomar Conference on Signals, Systems and Computers, Oct. 2000, Asilomar, CA.

[Energy.2] R. Hegde and N. Shanbhag, ``VLSI implementation of a low-power soft DSP filter," Proc. of IEEE Workshop on Signal Processing Systems, Oct. 2000, Lafayette, LA.

[Energy.3] S. Sridhara and N. Shanbhag, ``Low-power FFT via reduced precision redundancy," 2001 IEEE Workshop on Signal Processing Systems, Belgium.

[Energy.4] B. Shim and N. Shanbhag "Low-power digital signal processing via reduced precision redundancy," IEEE Transactions on VLSI Design, vol. 12, May 2004.

[Energy.5] B. Shim and N. Shanbhag "Low-power OFDM system using iterative peak cancellation" submitted in ISLPED 2002

[Energy.6] S. Lee, N.R. Shanbhag, and A. Singer, "Low power turbo equalizer architecture," 2002 IEEE Workshop on Signal Processing Systems, San Diego, CA, October 2002.

[Energy.6] S.-J. Lee, N.R. Shanbhag, and A. Singer, "Energy efficient VLSI architecture for linear turbo equalizer," Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology, v 39, no. 1-2 SPEC.ISS., January/February, 2005, p 49-62.

Reduction of peak-to-average transmitted power

Orthogonal frequency division multiplexing and multi-code/multi-carrier modulations are presently used or proposed for a variety of high-frequency, high-bandwidth wireless systems. One of the greatest drawbacks of these methods, in the context of power efficiency, is their high peak-to-average power ratio (PAR). We have recently developed three promising approaches for PAR reduction: the first is a waveform modification approach applied to data-bearing channels and which offers approximately 6 dB of=20 PAR reduction with no rate or performance penalty [Pow.1]; the second applies multi-dimensional constellation shaping to obtain even greater PAR reductions while maintaining rate and performance [Pow.2-Pow.3]. The third uses optimization techniques to efficiently design peak cancelling signals in an allocated bandwidth not transmitting data [Pow.4]. The high peak-to-average power ratio (PAR) in OFDM systems can significantly limit performance. Tone reservation techniques use unavailable or reserved tones to design a peak-cancelling signal that lowers the PAR of a transmit data block. These techniques have been shown to reduce the PAR for real-baseband OFDM signals, such as for ADSL. Our fast-converging real-baseband tone-reservation approach based upon active-set methods is extended to reduce PAR in complex-baseband signals [Pow.5]. This new technique converges very quickly toward a minimum-PAR solution at a low cost.

[Pow.1] Jones, D.L., ``Peak power reduction in OFDM and DMT via active channel modification,'' {\it Proc. 33st Asilomar Conference on Signals, Systems, and Computers} , Pacific Grove, CA, Oct. 24--27, 1999, pp. 1076--1079.

[Pow.2] H. K.-C. Kwok and D.L. Jones, ``PAR Reduction for Hadamard Transform-Based OFDM,'' Proceedings of the 34th Conference on Information Sciences and Systems, March 15-17, 2000, Princeton, NJ.

[Pow.3] H. K.-C. Kwok and D.L. Jones, ``PAR Reduction via Constellation Shaping,'' Proceedings of the 2000 International Symposium on Information Theory, June 25-30, 2000, Sorrento, Italy.

[Pow.4] B. S. Krongold and D. L. Jones, ``A New Method for PAR Reduction in Baseband DMT Systems,'' to appear in the Proceedings of the 35th Asilomar Conference on Signals, Systems, and Computers, Nov. 4-7, 2001, Pacific Grove, CA, pp. 502-503.

[Pow.5] B. S. Krongold and D. L. Jones, "A New Tone Reservation Method For Complex-Baseband PAR Reduction in OFDM Systems", Proceedings of ICASSP 2002, pp. 2321-2324.

[Pow.6] B. S. Krongold and D. L. Jones, ``PAR Reduction in OFDM Via Active Constellation Extension'', IEEE Transactions on Broadcasting, vol. 49, Sept. 2003, pp. 258-268.

[Pow.7] B. S. Krongold and D. L. Jones, ``An Active-Set Approach for OFDM PAR Reduction Via Tone Reservation'', IEEE Transactions on Signal Processing, vol. 52, Feb. 2004, pp. 495 - 509.

[Pow.8] B. S. Krongold and D. L. Jones, ``A Study of Active Constellation Extension for PAR Reduction in OFDM'', Proceedings of the 7th International OFDM Workshop, Hamburg, Germany, September 2002, pp. 107--111.

[Pow.9] B. S. Krongold and D. L. Jones, ``PAR Reduction in OFDM Via Active Constellation Extension'', to appear in the Proceedings of the 2003 International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003.

Models, Capacity and Coding for Wireless Channels

Medard and Gallager recently showed that very large bandwidths on certain fading channels cannot be effectively used by direct sequence or related spread spectrum systems. Our work [Channels.1] complements the work of M\'edard and Gallager. First, it is shown that a key information theoretic inequality of M\'edard and Gallager can be directly derived using the theory of capacity per unit cost, for a certain fourth-order cost function, called fourthegy. This provides insight into the tightness of the bound. Secondly, the bound is explored for a wide-sense stationary uncorrelated scattering (WSSUS) fading channel, which entails mathematically defining such a channel. In this context the fourthegy can be expressed using the ambiguity function of the input signal. Finally, numerical data and conclusions are presented for direct-sequence type input signals. A technically simpler way to constraint a signal to be "spread" is to impose a hard constraint on the amount of energy in each time-freqency "bin." This led to our investigation [Channel.2] of the capacity and the reliability function as the peak constraint tends to zero are considered for a discrete-time memoryless channel with peak constrained inputs. Prelov and van der Meulen earlier showed that under mild conditions the ratio of the capacity to the squared peak constraint converges to one-half the maximum eigenvalue of the Fisher information matrix and if the Fisher information matrix is non-zero, the asymptotically optimal input distribution is symmetric antipodal signaling. Under similar conditions it is shown in the first part of the paper that the reliability function has the same asymptotic shape as the reliability function for the power-constrained infinite bandwidth white Gaussian noise channel. The second part of the paper deals with Rayleigh fading channels. For such channels the Fisher information matrix is zero, indicating the difficulty of transmission over such channels with small peak constrained signals. Asymptotics for the Rayleigh channel are derived and applied to obtain the asymptotics of the capacity of the Marzetta and Hochwald fading channel model for small peak constraints, and to obtain a result of the type of M\'edard and Gallager for wideband fading channels.

A cellular wireless communication system that maximizes its throughput by scheduling service to the user with the best channel is said to be harnessing multiuser diversity. For the same mean conditions, channels showing more fluctuations result in lower delays, and in some cases increased throughput, with the use of minimal feedback. A scheme for inducing fluctuations is "adaptive" if the induced fluctuations are dependent on the feedback received. In a cellular wireless system with one antenna at the base station and SNR feedback from the mobiles, fluctuations are induced by the addition of another antenna at the base station that transmits the same signal as the first antenna but with a (variable in time) phase shift. An adaptive scheme for inducting fluctuations is one which controls this phase shift using the SNR feedback. We investigate [Channels.3] the performance of a proposed adaptive scheme, and compare it with the performance of non-adaptive schemes and the upper bound of perfect beamforming. The adaptive scheme shows throughput and delay gains to be had in scenarios of Rayleigh or Ricean channels, infinitely backogged or finite-queue users and high or low SNR regimes.

The required energy per bit for transmission over a fading channel is certainly limited by bandwidth. It is also reasonable to assume a constraint on peak power, or peak-to-average power, on the transmitted signal. In recent work we found the capacity per unit energy of flat, time-varying Rayleigh fading channels subject to a peak constraint [Channels.4]. The use of a peak constraint is especially significant for fading channels because a peak constraint, which is reasonable from a physical point of view, limits the ability of the reciever to estimate a time-varying channel.

While the design of coding and modulation schemes for the AWGN channel has reached a mature state, the corresponding theory for fading channels is not nearly as advanced. In [Channels.5] we use recent results from linear operator theory and harmonic analysis to study coding and modulation design for underspread time-varying fading channels. Using the fact that underspread channels are approximately diagonalized by biorthogonal Weyl-Heisenberg bases, we arrive at a 'canonical' formulation of modulation and code design. For a coherent receiver employing maximum likelihood decoding, we derive the coded design criteria as a function of channel's scattering function. We find that the resulting criteria are markedly different from those in the purely time-selective and the purely frequency-selective cases which follow as special cases of our theory. Finally, we provide expressions for the maximum achievable diversity order as a function of the channel's scattering function.

It is well-known that independent and identically distributed Gaussian inputs, when scaled appropriately based on the Signal-to-Noise Ratio (SNR), achieve capacity on the Additive White Gaussian Noise (AWGN) channel at all values of SNR. In [Channels.6] consider whether such good input distributions exist for frequency non-selective Rayleigh fading channels, assuming that neither the transmitter nor the receiver has a priori knowledge of the fading coefficients. We show that for a Gauss-Markov channel, the Gaussian input distribution is one of a large class of input distributions that generate bounded mutual information as the SNR gets large. This motivates the search for better choices of fixed input distributions for high-rate transmission over rapidly varying channels. Necessary and sufficient conditions are derived to characterize such distributions for the worst-case scenario of memoryless fading. Examples of both discrete and continuous distributions that satisfy these conditions are presented.

In [Channels.7] we investigate practical coding strategies for noncoherent fading channels that is guided by explicit comparisons with information-theoretic benchmarks. Noncoherent decoding is interpreted as joint data and channel estimation, assuming that the channel is time-varying and a priori unknown. We consider turbo-like iterative coding strategies based on serial concatenation of an outer channel code with an inner modulation code amenable to noncoherent detection. For an information rate of about 1/2 bits per channel use, the proposed schemes are within 1.6-1.7 dB of Shannon capacity for the block fading channel, and are 2.5-3 dB superior to standard differential demodulation in conjunction with the same outer channel code.

In [Channels.8], we summarize some recent results on information-theoretic limits for outdoor wireless mobile channel, with the aim of deriving transceiver design guidelines for fourth generation cellular systems. We consider two main themes: (a) The cost and accuracy of channel estimation must be factored into the derivation of information-theoretic limits for time-varying channels. To this end, we consider noncoherent communication, interpreted as joint channel and data estimation, over a flat Rayleigh fading channel. Results on both block and continuous fading channel models are presented. (b) An investigation of information-theoretic limits for the outdoor wireless channel must begin with channel models that reflect the outdoor environment accurately, while yielding analytical insight. We consider an idealized channel model, neglecting time variations, for wideband, frequence-selective, outdoor channels that account for rapidly decaying power-delay profiles for such channels, and provide results on outage rates and diversity.

[Channels.1] V.G. Subramanian and B.Hajek, ``Broad-band Fading Channels: Signal Burstiness and Capacity'', IEEE Transactions on Information Theory, vol. 48, pp. 809-827, 2002.

[Channels.2] B. Hajek and V.G. Subramanian, ``Capacity and reliability function for small peak signal constraints," IEEE Transactions on Information Theory, vol. 48, pp. 828-839, 2002.

[Channels.3] S. Sanghavi and B. Hajek "Adaptive induced fluctuations for multiuser diversity", Proceedings of the International Symposium on Information Theory, Lausanne, p. 450, 2002.

[Channels.4] V. Sethuraman and B. Hajek, "Capacity per unit energy of fading channels with a peak constraint," Proc. IEEE International Symposium on Information Theory, Yokahamo, June 2003.

[Channels.5] H. Boelsckei, R. Koetter and S. Mallik, 'Coding and modulation for underspread fading channels', Proceedings of the Internation Symposium on Information Theory, Lausanne, June 2002.

[Channels.6] J. Huang and S. Meyn, "Characterization and computation of optimal distributions for channel coding," To appear in IEEE Transactions on Information Theory, 2004.

[Channels.7] Rong-Rong Chen, Ralf Koetter, Dakshi Agrawal, and Upamanyu Madhow, "Joint demodulation and decoding for the block fading channel: a practical framework for approaching Shannon capacity", IEEE Transactions on Communications, Vol. 51, Oct. 2003. pp. 1676 - 1689.

[Channels.8] Upamanyu Madhow, Gwen Barriac, and Rong-Rong Chen, "Transceiver design for the outdoor wireless channel: an information-theoretic perspective". Invited paper, IEEE International Symposium on Advances in Wireless Communication (ISWC 2002), Victoria, Canada, July 2002.

[Channels.9] Y. Jiang, R. Koetter, and A. Singer, "On the separability of demodulation and decoding for communication over multiple-antenna block fading channels," IEEE Transactions on Information Theory , vol. 49, no. 10, pp. 2709-2713, October 2003.

[Channels.10] Rong-Rong Chen, B. Hajek, R. Koetter, and U. Madhow, On fixed input distributions for noncoherent communication over high-SNR Rayleigh-fading channels," IEEE Trans. Information Theory, Vol. 50, Dec. 2004, pp. 3390 - 3396.

Soft Decoding Methods

In [Soft.1] we present exponential upper bounds on the performance of algebraic soft decision decoding algorithm in the decoding of Reed-Solomon codes for a symmetric channel. The bound is a Chernoff-type bound as a function of the length of the code.

[Soft.1] N. Ratnakar and R. Koetter, "Exponential error bounds for algebraic soft-decision decoding of Reed Solomon codes, " To appear in IEEE Transactions on Information Theory.