Ibrahim C. Abou-Faycal

The capacity of discrete-time memoryless Rayleigh-fading channels

We consider transmission over a discrete-time Rayleigh fading channel, in which successive symbols face independent fading, and where neither the transmitter nor the receiver has channel state information. Subject to an average power constraint, we study the capacity-achieving distribution of this channel and prove it to be discrete with a finite number of mass points, one of them located at the origin. We numerically compute the capacity and the corresponding optimal distribution as a function of the signal-to-noise ratio (SNR). The behavior of the channel at low SNR is studied and finally a comparison is drawn with the ideal additive white Gaussian noise channel.

Binary adaptive coded pilot symbol assisted modulation over Rayleigh fading channels without feedback

Pilot symbol assisted modulation (PSAM) is a standard approach for transceiver design for time-varying channels, with channel estimates obtained from pilot symbols being employed for coherent demodulation of the data symbols. In this paper, we show that PSAM schemes can be improved by adapting the coded modulation strategy at the sender to the quality of the channel measurement at the receiver, without requiring any channel feedback from the receiver. We consider performance in terms of achievable rate for binary signaling schemes. The transmitter employs interleaved codes, with data symbols coded according to their distance from the nearest pilot symbols. Symbols far away from pilot symbols encounter poorer channel measurements at the receiver and are therefore coded with lower rate codes, while symbols close to pilot symbols benefit from recent channel measurements and are coded with higher rate codes. The performance benefits from this approach are quantified in the context of binary signaling over time-varying Rayleigh fading channels described by a Gauss-Markov model. The spacing of the pilot symbols is optimized to maximize the mutual information between input and output in this setting. Causal and noncausal channel estimators of varying complexity and delay are considered. It is shown that, by appropriate optimization for the spacing between consecutive pilot symbols, the adaptive coding techniques proposed can improve achievable rate, without any feedback from the receiver to the sender. Moreover, channel estimation based on the two closest pilot symbols is generally close to optimal.

A fast maximum-likelihood decoder for convolutional codes

The lazy Viterbi decoder is a maximum-likelihood decoder for block and stream convolutional codes. For many codes of practical interest, under reasonable noise conditions, the lazy decoder is much faster than the original Viterbi decoder. For a code of constraint length 6, the lazy algorithm is about 50% faster than an optimized implementation of the Viterbi decoder whenever SNR>6 dB. Moreover, while the running time of the Viterbi decoder grows exponentially with the constraint length, under reasonable noise conditions, the running time of the lazy algorithm is essentially independent of the constraint length. This paper introduces the lazy Viterbi decoder and shows how to make it efficient in practice.

Using Hermite Bases in Studying Capacity-Achieving Distributions Over AWGN Channels

This paper studies classes of generic deterministic, discrete time, memoryless, and “nonlinear” additive white Gaussian noise (AWGN) channels. Subject to multiple types of constraints such as the even-moment and compact-support constraints or a mixture, the optimal input is proved to be discrete with finite number of mass points in the vast majority of the cases. Only under the even-moment constraint and for special cases that emulate the average power constrained linear channel, capacity is found to be achieved by an absolutely continuous input. The results are extended to channels where the distortion is generally piecewise nonlinear where the discrete nature of the optimal input is conserved. These results are reached through the development of methodology and tools that are based on standard decompositions in a Hilbert space with the Hermite polynomials as a basis, and it is showcased how these bases are natural candidates for general information-theoretic studies of the capacity of channels affected by AWGN. Intermediately, novel results regarding the output rate of decay of Gaussian channels are derived. Namely, the output probability distribution of any channel subjected to additive Gaussian noise decays necessarily “slower” than the Gaussian itself. Finally, numerical computations are provided for some sample cases, optimal inputs are determined, and capacity curves are drawn. These results put into question the accuracy of adopting the widely used expression 1(1+ SNR) for computing capacities of Gaussian deterministic channels.

On the capacity of additive white alpha-stable noise channels

Many communication channels are reasonably modeled to be impaired by additive noise. Recent studies suggest that many of these channels are affected by additive noise that is best explained by alpha-stable statistics. We study in this work such channel models and we characterize the capacity-achieving input distribution for those channels under fractional order moment constraints. We prove that the optimal input is necessarily discrete with a compact support for all such channels. Interestingly, if the second moment is viewed as a measure of power, even when the channel input is allowed to have infinite second moment, the optimal one is found to have finite power.

On the performance of peaky capacity-achieving signaling on multipath fading channels

We analyze the error probability of peaky signaling on bandlimited multipath fading channels, the signaling strategy that achieves the capacity of such channels in the limit of infinite bandwidth under an average power constraint. We first derive an upper bound for general fading, then specialize to the case of Rayleigh fading, where we obtain upper and lower bounds that are exponentially tight and, therefore, yield the reliability function. These bounds constitute a strong coding theorem for the channel, as they not only delimit the range of achievable rates, but also give us a relationship among the error probability, data rate, bandwidth, peakiness, and fading parameters, such as the coherence time. They can be used to compare peaky signaling systems to other large bandwidth systems over fading channels, such as ultra-wideband radio and wideband code-division multiple access. We find that the error probability decreases slowly with the bandwidth W; under Rayleigh fading, the error probability varies roughly as W/sup -/spl alpha//, where /spl alpha/>0. With parameters typical of indoor wireless situations, we study the behavior of the upper and lower bounds on the error probability and the reliability function numerically.

A cauchy input achieves the capacity of a Cauchy channel under a logarithmic constraint

In this work, we consider a discrete-time memoryless communication channel where the input is subjected to an independent additive Cauchy noise. We find the input constraint under which a Cauchy input is capacity achieving. The constraint is logarithmic and depends on a scalar parameter k which we interpret as a power measure. We draw a parallelism between this setup and that of the Gaussian channel under the second moment constraint. In fact, a Cauchy input yields a Cauchy output over this channel and achieves a capacity value of “log(1 + SNR)”.

On the capacity of some deterministic non-linear channels subject to additive white Gaussian noise

We consider a variety of memoryless discrete-time noisy communication channels where the noise is modeled as an additive white Gaussian noise process, and where the input of the channel is distorted according to a deterministic function f(X) of the form i) f(X) = αXⁿ ii) f(X) = α|X|ⁿ iii) f(X) = α|X|^1/q iv) f(X) = αsgn(X)|X|^1/q, for all α ∊ R^⋆, n ∊ N^⋆\{1}. Subject to an average power constraint, we study the capacity-achieving input distributions of these classes of channels and prove them to be discrete except for the linear case f(X)=αX, where the optimal input is Gaussian distributed. Furthermore, we prove that these optimal input distributions have a finite number of mass points for classes iii) and iv). The results are reached through the development of a methodology and tools that are based on standard decompositions in a Hilbert space with the Hermite Polynomials as a basis, and we conjecture that these bases are natural candidates for general information-theoretic studies of the capacity of channels affected by additive white Gaussian noise.

On the Finiteness of the Capacity of Continuous Channels

Evaluating the channel capacity is one of many key problems in information theory. In this work, we derive rather-mild sufficient conditions under which the capacity of continuous channels is finite and achievable. These conditions are derived for generic, memoryless, and possibly nonlinear additive noise channels. The results are based on a novel sufficient condition that guarantees the convergence of differential entropies under point-wise convergence of probability density functions. Perhaps surprisingly, the finiteness of channel capacity holds for the majority of setups, including those where inputs and outputs have possibly infinite second-moments.

The capacity of average power constrained additive non-Gaussian noise channels

It is well known that a Gaussian input achieves the capacity of the linear additive white Gaussian noise channel under the average power constraint. While the continuity of an optimal input of a continuous channel might be expected, the Gaussian noise presents the only such scenario. In fact, we study in this paper the capacity-achieving inputs for the linear additive noise channel where the noise is not necessarily Gaussian. We impose an average power constraint on the input and prove that, except for a Gaussian channel, the optimal input is of a discrete nature.