Random Coding Error Exponent Research Articles

Error exponents for maximum-likelihood and successive decoders for the Gaussian CDMA channel

Random-coding error exponents are derived for the Gaussian code-division multiple-access (CDMA) channel for the maximum-likelihood and optimum successive decoders. Error exponents not only specify the capacity region of the channel, which is known, but also give lower bounds on the rate of exponential decay of the average probability of error as a function of the block length of random codes. A comparison of the two decoders in terms of their error exponents is included.

Random coding error exponents for flat fading channels with realistic channel estimation

There has been a considerable interest in determining the limits to communications over multipath fading channels. However, most studies have assumed that the channel is perfectly known at the receiver. In this paper, the random coding error exponent for flat fading channels with realistic channel state information (CSI) is studied. It is assumed that the CSI is obtained via some practical technique which utilizes a linear estimation scheme. Two commonly used techniques for channel estimation are considered in this paper, namely pilot tone extraction and pilot symbol transmission. The degradation in the achievable performance due to partial CSI is assessed and comparison of the different channel estimation methods is made. The focus of this paper is on the Jake's mobile Rayleigh flat fading model. Although Jake's model does not have a Markov property, such as that found in the commonly used exponential correlation model, which is usually attractive from the mathematical tractability point of view, Jake's model has a physical basis. Also, this model is considered herein from the standpoint of the random coding exponent. The results in this paper shed light on the amount of degradation in the achievable performance that is expected when the receiver has partial CSI. Finally, the sensitivity of the loss in achievable performance for the various channel estimation techniques with respect to channel parameters, such as Doppler spread and signal-to-noise ratio (SNR), is studied.

On random coding error exponents of watermarking systems

Watermarking codes are analyzed from an information-theoretic viewpoint as a game between an information hider and an active attacker. While the information hider embeds a secret message (watermark) in a covertext message (typically: text, image, sound, or video stream) within a certain distortion level, the attacker processes the resulting watermarked message, within limited additional distortion, in attempt to invalidate the watermark. For the case where the covertext source is memoryless (or, more generally where there exists some transformation that makes it memoryless), we provide a single-letter characterization of the maximin game of the random coding error exponent associated with the average probability of erroneously decoding the watermark. This single-letter characterization is in effect because if the information hider utilizes a memoryless channel to generate random codewords for every covertext message, the (causal) attacker will maximize the damage by implementing a memoryless channel as well. Partial results for the dual minimax game and the conditions for the existence of a saddle point are also presented.

Information theoretic considerations for coded modulation over fading channels

We compute the random coding error exponent for coded modulation transmitted over a flat, memoryless, Rayleigh fading channel. In addition, estimates of code lengths required to achieve a certain error probability are determined and compared to those required for the additive white Gaussian noise channel. Finally, the effect of receiver antenna diversity is also considered as a method to compensate for fading and is shown to have a significant, positive impact on the error exponent. The results we obtain represent an information theoretic view that complements the existing literature on the performance of coded modulation over fading channels with receiver diversity.

Random coding error exponents for two-dimensional flat fading channels with complete channel state information

In this correspondence, we consider the random coding error exponent for nondispersive two-dimensional (2-D) (quadrature) fading channels for which the channel state information (CSI) is perfectly known at the receiver and the input to the transmitter is constrained in its average power. Also, the effect of space diversity on improving the performance is demonstrated. The results obtained in this correspondence shed light on the effect of fading on communications reliability as well as the amount of coding complexity required to achieve a certain decoding error rate. A treatment for the random coding error exponent for time-correlated flat fading channels with Rayleigh fading and Bessel function correlation (the Jake's model) is also provided.

On the error exponent for memoryless flat fading channels with channel-state-information feedback

We derive the random coding error exponent for the time-independent flat fading channel with perfect knowledge of the channel state information (CSI) at both the receiver and the transmitter. That is, the CSI is feedback from the receiver to the transmitter. In such a situation, the transmitter is capable of optimizing the power allocation of the transmitted signal according to the fading state in order to obtain, for our case, the best error exponent. The power scheme obtained here is different from the water-pouring one, which is known to maximize the channel capacity.

On the universality of the LZ-based decoding algorithm

A universal decoder for a family of channels is a decoder that can be designed without prior knowledge of the particular channel in the family over which transmission takes place, and it yet attains the same random-coding error exponent as the optimal decoder tuned to the channel in use. We study Ziv's (1985) decoding algorithm, which is based on Lempel-Ziv (1978) incremental string parsing, and demonstrate that while it was originally proposed as a universal decoder for the family of finite-state channels with deterministic (but unknown) transitions, it is in fact universal for the broader class of all finite-state channels. We also demonstrate that the generalized likelihood decoder may not be universal even for finite families for which a universal decoder always exists.

Universal decoding for channels with memory

A universal decoder for a parametric family of channels is a decoder whose structure depends on the family but not on the individual channel over which transmission takes place, and it yet attains the same random-coding error exponent as the maximum-likelihood receiver tuned to the channel in use. The existence and structure of such decoders is demonstrated under relatively mild conditions of continuity of the channel law with respect to the parameter indexing the family. It is further shown that under somewhat stronger conditions on the family of channels, the convergence of the performance of the universal decoder to that of the optimal decoder is uniform over the set of channels. Examples of families for which universal decoding is demonstrated include the family of finite-state channels and the family of Gaussian intersymbol interference channels.

Universal decoding for memoryless Gaussian channels with a deterministic interference

In [1] universal decoding schemes for finite-alphabet, finite-state channels were proposed end shown to be optimal in the sense of attaining the highest possible random coding error exponent, when the channel input vectors are chosen randomly under a uniform probability distribution. In other words, the average error probability over the ensemble of randomly selected codewords, decays at the fastest exponential rate. In the special case of DMC'S, the proposed universal decoder selects a codeword that minimizes the empirical conditional entropy of the channel input given the channel output. We derive an analogous result for memoryless Gaussian channels with an unknown deterministic interference from a fairly wide class. The empirical conditional entropy of the input given the output is induced by an auxiliary backward channel whose parameters are estimated from the given output vector and each one of the codewords. We also allow a more general class of input distributions by slightly modifying the decoding rule. Similarly to [1], it is shown that the proposed universal decoder attains the same error exponent as that of the ML decoder which is fully informed of the channel and the interfering signal. The proposed decoder is different from an heuristic approach [2], where the channel and message are jointly estimated by the ML method. While the former decoder is based on the backward channel as mentioned earlier, the latter corresponds to the forward channel. For the simple special case where there is no interference and the only uncertainty is in the channel fading parameter, it is demonstrated that the error exponent of the proposed rule might be strictly better than that of the joint ML channel-and-message estimation approach.

Universal decoding for finite-state channels

Universal decoding procedures for finite-state channels are discussed. Although the channel statistics are not known, universal decoding can achieve an error probability with an error exponent that, for large enough block length (or constraint length in case of convolutional codes), is equal to the random-coding error exponent associated with the optimal maximum-likelihood decoding procedure for the given channel. The same approach is applied to sequential decoding, yielding a universal sequential decoding procedure with a cutoff rate and an error exponent that are equal to those achieved by the classical sequential decoding procedure.

Linear codes for sources and source networks: Error exponents, universal coding

For Slepian-Wolf source networks, the error exponents obtained by Korner,Marton, and the author are shown to be universally attainable by linear codes also. Improved exponents are derived for linear codes with large rates. Specializing the results to simple discrete memoryless sources reveals their relationship to the random coding and expurgated bounds for channels with additive noise. One corollary is that there are universal linear codes for this class of channels which attain the random coding error exponent for each channel in the class. The combinatorial approach of Csiszar-Korner-Marton is used. In particular, all results are derived from a lemma specifying good encoders in terms of purely combinatorial properties.