An approach to Ungerboeck coding for rectangular signal sets (Corresp.)

A method of finding good Ungerboeck codes for large rectangular [quadrature amplitude modulation (QAM)] signal sets is described. Using the concept of Euclidean weights due to Ungerboeck, we prove that a2^{n}point basic constellation may be employed to determine exactly the free distance for an Ungerboeck-coded rectangular2^{m}point set, whenm-n-1bits are uncoded and the remaining bits pass through a rate(n-1)/nconvolutionai encoder. It is shown that rate2/3encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered.     

A product construction for perfect codes over arbitrary alphabets (Corresp.)

A general product construction for perfect single-error-correcting codes over an arbitrary alphabet is presented. Given perfect single-error-correcting codes of lengthsn, m, andq + 1over an alphabet of orderq, one can construct perfect single-error-correcting codes of length(q - 1)nm + n + mover the same alphabet. Moreover, if there exists a perfect single-error-correcting code of lengthq + 1over an alphabet of orderq, then there exist perfect single-error-correcting codes of lengthn,n = (q^{t} _ 1)/(q - 1), and(t > 0, an integer). Finally, connections between projective planes of orderqand perfect codes of lengthq + 1over an alphabet of orderqare discussed.     

An iterative code construction for the broadcast channel with degraded message sets (Corresp.)

Ahlswede and Dueck introduced an iterative method for constructing codes for the discrete memoryless channel that meet the random coding bound. Their codes are constructed by making a relatively few permutations of a single codeword. In this correspondence their idea is extended to the broadcast channel with degraded message sets. The codes constructed here have average error probabilities below the error bounds derived by Körner and Sgarro.     

Performance bounds for discrimination problems with uncertain statistics (Corresp.)

Decision designs that are insensitive to modeling uncertainty are developed for the Chernoff bounds on the performance of binary hypothesis testing problems. These designs are based on observations with statistical uncertainty modeled by using general classes generated by 2-alternating capacities. The results are illustrated for the two cases of independent identically distributed observations with uncertainty in the probability distribution and discrete-time stationary Gaussian observations with spectral uncertainty, and they are applicable to several other cases as well. For the Chernoff upper bounds on the error probabilities, a "robust" decision design based on the I/kel/hood-ratio test between a least-favorable pair of probability distributions or spectral measures, respectively, is derived. It is then shown that for all elements in the uncertainty class this choice of likelihood ratio guarantees the exponential convergence of the aforementioned Chernoff bounds to zero as the number of observations or the length of the observation interval increases.     

Modified Chernoff bounds for PAM systems with noise and interference (Corresp.)

By using principles of analytic continuation, upper and lower bounds on the error probability of a canonical binary system corrupted by additive interference and independent zero-mean Gaussian noise are derived. The bounds, which are simple functions, require only the evaluation or bounding of the moment generating function of the interference. For large signal-to-noise ratios (SNR) and even for moderately large interference, the bounds are shown to be tight and useful.     

Absolute error rate-distortion functions for sources with constrained magnitudes (Corresp.)

The explicit evaluation of the absolute error rate-distortion function has been obtained previously for a class of i.i.d, sources with probability densities supported on the entire real line. The technique is extended to various i.i.d, sources with densities supported on a proper subset of the real line, such as uniform and one-sided exponential densities. Comparisons of these rate-distortion functions with their Shannon lower bounds are also given.     

Extended Varshamov- Gilbert and sphere-packing bounds for burst-correcting codes (Corresp.)

The study of burst-error-correcting codes has generally been made without weight considerations while the random-error-correcting capabilities of a linear code largely depend upon the minimum weight. In this correspondence we have obtained extensions of the Varshamov-Gilbert and sphere-packing hounds to burst-error-correcting codes.     

Finite-time-averaged power spectra and signal energy distribution in time and frequency (Corresp.)

In response to a recent paper, this correspondence points out the difficulty of applying the concept of the energy distribution in time and frequency of deterministic signals to random signals and finite-time-averaged power spectra. For deterministic signals, an appropriately defined finite-time spectrum will indicate the frequency band within which the signal energy is concentrated at a particular time. However, it does not appear meaningful to apply such concepts to finite-time averages of random processes since taking sample averages would typically destroy any time-frequency distribution of the energy perhaps found in an individual sample of the process.     

Distance measures and estimation performance bounds for continuous-time data (Corresp.)

Recursive expressions for certain distance measures between continuous-time stationary vector-Gaussian processes are derived and used to derive upper bounds to the mean square error performance of the Bayes and maximum-likelihood estimates of a parameter, when only a finite-valued parameter set is used. The question of convergence when the true parameter value does not belong to the finite set is also answered.     

A sequence of upper and lower bounds for the Q function (Corresp.)

A sequence of upper and lower bounds for theQfunction defined asQ(x)= 1/ sqrt{2 pi} int_{x}^{infty} exp[(-y^{2})/2]dyis developed. These bounds are shown to be tighter than those most commonly used.     

A generalization of Huffman coding for messages with frequencies given by upper and lower bounds (Corresp.)

A generalization of the Huffman coding procedure is given for cases in which the source letter probabilities are known only to fall in certain ranges.     

Optimal smoother for discrete time point processes with finite-state Markov rate (Corresp.)

A closed-form optimal nonlinear smoothing algorithm is derived for estimation of signal that is indirectly observed through a discrete time point process (DTPP). A finite-state Markov signal influences the rate of the point process. The smoothers obtained are simple, recursive, and finite dimensional. An illustrative example of the derived estimation scheme is presented.     

A trellis code construction and coding theorem for stationary Gaussian sources (Corresp.)

For stationary discrete-time Gaussian sources and the squared-error distortion measure, a trellis source code is constructed. The encoder consists of a Karhunen-Loeve transform on the source output followed by a search on a trellis structured code, where the decoder is a time-variant nonlinear filter. The corresponding code theorem is proved using the random coding argument. The proof technique follows that of Viterbi and Omura, who proved the trellis coding theorem for memoryless sources. The resultant coding scheme is implementable and applicable at any nonzero rate to a stationary Gaussian source with a bounded and continuous power spectrum. Therefore. for stationary sources, it is more general than Berger's tree coding scheme, which is restricted to autoregressive Gaussian sources in a region of high rate (low distortion).     

On the Cramer- Rao bound for time delay and Doppler estimation (Corresp.)

Using a theorem due to Whittle, simple derivations of the Cramer-Rao lower bound are presented for some delay estimation problems related to a single source, multiple sources, and multipath. The problem of Doppler estimation is briefly discussed.     

A heuristic algorithm for the construction of a code with limited word length (Corresp.)

A depth-first algorithm is presented for the construction of a binary minimum-redundancy variable length code with limited word length. In this algorithm, heuristic information on the mean word length is used for efficient searching. The extension toQ-ary codes is also discussed.     

An adaptive digital filter for improving the signal-to-noise ratio of a signal with additive noise (Corresp.)

An adaptive digital filtering scheme is developed to deal with the problem of improving the signal-to-noise ratio of a signal corrupted by noise when only general {em a priori} assumptions regarding the signal and the noise are possible. Specifically, the noise is assumed to be white zero-mean and uncorrelated; while the signal is considered to be band-limited, possibly with slowly varying spectrum. The proposed adaptive digital filtering scheme is based upon a class of variable wave digital filters. Adaptation of the digital filter multipliers is accomplished through the use of an identification procedure based on an adaptive spectral estimation method.     

Error statistics and construction of error correcting codes for differentially encoded CPSK channels (Corresp.)

A simple Markov chain model for differentially encoded CPSK Channels faciliates the calculation of the probability of the occurrence of several most probable error patterns in a given codeword. Several error correcting codes which can efficiently correct several most-probable error patterns are proposed and their error performances are analyzed.