Finite-state Channel Research Articles

Concatenated coding performance for FSK modulation on time-correlated Rician fading channels

We present an analytical method for evaluating the performance of noninterleaved concatenated codes over channels modeled as a nonfrequency selective correlated Rician fading channel with a known power spectral density. The main idea is to model the communication system from the modulator input to the demodulator output as a finite state channel (FSC) model, and apply powerful enumeration techniques to such a discrete channel in order to gain useful information on the system performance. The concatenated scheme makes use of two codes; Reed-Solomon codes are employed for the outer code, and binary block codes are used as the inner code. Next, the method is extended to study the effect on the performance when an interleaving with finite depth is incorporated into the communication system. A comparison between symbol and bit interleaving is made. Finally, we study the potential gain produced when channel information is passed on to the outer decoder in the form of an erasure symbol. In all cases, analytical expressions for the probability of the number of error symbols produced by the FSC model were obtained in terms of a coefficient in a formal power series. This is an interesting alternative approach with respect to computer simulations.

Generating Series for Error Statistics of Block Codes on Channels with Memory

We present an analytical method for evaluating the performance of non-interleaved forward error correcting codes on channels that exhibit statistical dependence in the occurrence of errors. We consider a model for such a channel based on the probabilistic function of a Markov chain, also known as finite state channel (FSC) model. The main idea is to apply combinatorial methods to derive simple expressions for the probability of the number of symbol errors produced by the channel in terms of a coefficient in a formal power series. These methods are used to derive expressions for the codeword error probability of various practical coding schemes, including nonbinary block codes and the concatenation of two block codes. The general expressions are specialized for a Gilbert-Elliott channel with known model parameters, and numerical results are derived.

Modeling burst channels using partitioned Fritchman's Markov models

Discrete models based on functions of Markov chains (also referred to as hidden Markov models or finite-state channel (FSC) models) have been used to characterize the error process in communication channels with memory. One important property of these models is that the probability of any observed sequence can be expressed as a linear combination of the probability of a finite set of sequences of finite length, the so-called basis sequences. In this paper, we express the parameters of a class of FSC models as a simple function of the probability of the basis sequences. Based on this approach, we propose a new method for the parameterization of the Fritchman (1967) channel with single-error state as well as the interesting cases of Fritchman channels with more than one error state and the Gilbert-Elliott channel ((GEC) nonrenewal models). To illustrate the method, FSC models for the nonfrequency-selective Rician fading channel are presented. The number of states and the probability of state transitions are estimated for a given set of fading parameters.

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.

Optimal quantization for finite-state channels

Optimal scalar quantizer design for transmission over a finite-state channel is considered. The objective Is to minimize the mean-squared error when the channel is in the normal mode of operation, while guaranteeing a minimum fidelity when the channel is in the bad state. An optimal quantizer design algorithm for the general case where noisy state information is available both at the receiver and at the transmitter is derived. It is shown that using mixed strategies is necessary in order to achieve the optimal performance. Finally, the case where the observation is noisy is considered and it is shown that the optimal scheme in this case is to apply the algorithm for the no observation noise to the mean-squared estimate of the desired random variable from the noisy data.

Adaptive quantization for fading channels with feedback

The effect of the presence of a feedback channel on the transmission of information was first considered by Shannon, who showed that the capacity of a memoryless channel is not increased by the existence of a feedback link even if the feedback link is noiseless. Later it was shown that the information on a feedback channel can be used to improve considerably the performance of channel coding. In this work we study the transmission of an information source through a fading channel with feedback, modeled by a finite-state channel in the Gilbert-Elliot sense. We show that by employing the feedback information in the quantizer design for this finite-state channel, one can achieve lower overall distortion compared to the case where feedback is not available. The feedback channel is used to estimate the channel state using a hidden Markov model, and a quantizer matched to the channel state is chosen based on this information.

Finite-state Markov channel-a useful model for radio communication channels

The authors first study the behavior of a finite-state channel where a binary symmetric channel is associated with each state and Markov transitions between states are assumed. Such a channel is referred to as a finite-state Markov channel (FSMC). By partitioning the range of the received signal-to-noise ratio into a finite number of intervals, FSMC models can be constructed for Rayleigh fading channels. A theoretical approach is conducted to show the usefulness of FSMCs compared to that of two-state Gilbert-Elliott channels. The crossover probabilities of the binary symmetric channels associated with its states are calculated. The authors use the second-order statistics of the received SNR to approximate the Markov transition probabilities. The validity and accuracy of the model are confirmed by the state equilibrium equations and computer simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the generalized symmetric cutoff rate for finite-state channels

Finite-state discrete-time channels with equiprobable M-ary inputs are considered. The generating function bound, which is ubiquitously applied to upper-bound the error probability of uncoded signaling over these channels, is used here to lower-bound the corresponding generalized symmetric cutoff rate, which lower-bounds the practically achievable rates and error exponents in these channels with symmetric signaling. The bound accounts for general additive metrics. For the special case of the optimal maximum-likelihood metric, the corresponding bound is shown to be asymptotically tight in the region where the symmetric cutoff rate approaches its maximum value of log/sub 2/ M (bits-per-channel use). For a finite-state additive white Gaussian noise (AWGN) channel this feature is used to relate the minimum Euclidean distance of an uncoded symmetric system to the corresponding symmetric cutoff rate. The results are demonstrated for AWGN channels corrupted by linear and nonlinear intersymbol interference. They are also used to assess the efficiency of concatenated coding over the 4-ary AWGN channel where the finite-state mechanism is defined by simple rate 1/2, four-state Ungerboeck 4-AM trellis code. >

Convolutional coding for finite-state channels

We propose new decoders for decoding convolutional codes over finite-state channels. These decoders are sequential and utilize the information about the channel state sequence contained in the channel output sequence. The performance of these decoders is evaluated by simulation and compared to the performance of memoryless decoders with and without interleaving. Our results show that the performance of these decoders is good whenever the channel statistics are such that the joint estimate of the channel state sequence and the channel input sequence is good, as, for example, when the channel is bursty. In these cases using even a partial search decoder such as the Fano decoder over the appropriate trellis is nearly optimal. However, when the information between the output sequence and the sequence of channel states and inputs diminishes, the memoryless decoder with interleaving outperforms even the optimal decoder which knows the channel state.

Nonlinear channel models for digital magnetic recording

Methods for identifying the digital magnetic recording channel in which data dependent nonlinear effects are present are described. The models used include the finite state machine (FSM), Volterra series, and channel state dependent transition shifting. Procedures to compute the model parameters are described, and the models are compared with a view to their accuracy and use for studying receiver performance. Results are included from prototype channels with various recording densities with significant nonlinear distortion. >

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.

Uniquely decodable codes for deterministic relay channels

The deterministic relay channel is analyzed and explicit code constructions for all binary and all ternary/binary channels are given. An explicit set of equivalence conditions is used to make a classification of all such relay channels, for which also the capacity is evaluated. The coding problem is then reduced to finding all possible output sequences of a certain finite-state channel determined by the relay coding strategy. The channel states correspond to the possible relay memory contents. For some relay channels capacity is reached by using simple uniquely decodable codes, thus establishing the zero-error capacity of those channels with finite-memory relay strategies. For other relay channels the relay memory must be arbitrarily large to achieve zero-error rates arbitrarily close to capacity. One such code construction is given. It is not known whether there exist relay channels for which the zero-error capacity is strictly smaller than the average-error capacity. The code construction problem for the semideterministic relay channel and for the nonsynchronized relay channel is briefly considered. >

A branching process analysis of the stack algorithm for variable channel conditions

A branching process analysis in a random environment is presented for bounding the average number of computations of sequential decoding over a finite state channel. Closed-form expressions applicable to specific cases are derived and evaluated. These unique bounds substantially reduce the need for lengthy simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Capacity and coding for memories with real-time noisy defect informatiom at encoder and decoder

The paper investigates the problem of information storage in a defective medium where real-time noisy information is available on the defects at both the encoder and the decoder. The problem is modelled as communication over a finite state channel with noisy state information available at both sides, It is shown that this problem can be transformed into Shannon's channel with side information problem. An optimal coding strategy is described and the capacity is derived. In general the capacity is given in terms of strategies. It is further shown that in some cases of interest the capacity can be described without employing Shannon strategies.

Capacity and zero-error capacity of Ising channels

Simple binary symmetric one- and two-dimensional channel systems called Ising channels are investigated. The one-dimensional version is a certain binary finite-state channel with input intersymbol interference. Its zero-error capacity is determined together with lower and upper bounds to its Shannon capacity. For the two-dimensional channel system, a lower bound is obtained for the zero-error capacity, and an efficient zero-error coding scheme is developed. >

Combinatorial structure and capacity of the permuting relay channel

Blackwell's trap-door channel is an interesting example of a finite state channel. Its deterministic version, that is, the permuting channel, has been studied by Ahlswede and Kaspi ina multiterminal information-theoretic framework. They determined the capacities of permuting jammer channels and relay channels for some special cases. The capacity problem for permuting relay channels is completely solved. More specifically, when a is the cardinality of alphabet, and \beta is the number of available storage locations in the channel, the capacity C_{R}(\alpha, \beta) of the permuting relay channel is given by \log \lambda , where \lambda denotes the maximum eigenvalue of a matrix Q derived from the state-transition mechanism associated with the channel.

Ergodicity of Markov channels

A Markov channel is a discrete information channel that includes as special cases the finite state channels and finite state codes of information theory. Kieffer and Rahe proved that one-sided and two-sided Markov channels have the following property: If the input source to a Markov channel is asymptotically mean stationary (AMS), then so is the resulting input-output process and hence the ergodic theorem and the Shannon-McMillan-Breiman theorem hold for the input-output process. Kieffer and Rahe also provided a sufficient condition for any AMS ergodic source to yield an AMS ergodic input-output process. New conditions for a Markov channel to have this ergodicity property are presented and discussed here. Several relations are developed among various classes of channels, including weakly ergodic, indecomposable, and strongly mixing channels. Some connections between Markov channels and the theory of nonhomogeneous Markov chains are also discussed.

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.

An information channel model of a neuron encoder and possible microwave radiation effects on capacity

A popular neuron encoder model is cast into the form of an information channel, and the encoder state is shown to be measured in probability by a Markov chain. Channel capacity of the model is calculated under the assumption that the neuron threshold follows an approximately Gaussian distribution. The model is also shown to be characterized in information theory terms as a finite-state indecomposable channel subject to intersymbol interference and fading. From experiments on snail neurons which were the stimulus for the modelling efforts, the hypothesis is suggested that, to the extent the model captures the actual performance of neurons, microwave radiation could significantly alter the information processing ability of neurons.