Neal C. Gallagher

On the design of nonlinear discrete-time predictors (Corresp.)

The problem of minimum mean-squared error prediction of a discrete-time random process using a nonlinear filter consisting of a zero-memory nonlinearity followed by a linear filter is studied. Classes of random processes for which the best predictor is realizable using a nonlinear filter of the above form are discussed. For those random processes for which the best predictor is not realizable using the above nonlinear filter, an iterative procedure is presented for finding a suboptimal nonlinear filter; special attention is directed to the case where the nonlinearity is a polynomial. Also, a noniterative approach based on nonlinear regression is presented.

Two-dimensional quantization of bivariate circularly symmetric densities

The problem of quantizing a two-dimensional random variable whose bivariate density has circular symmetry is considered in detail. Two quantization methods are considered, leading to polar and rectangular representations. A simple necessary and sufficient condition is derived to determine which of these two quantization schemes is best. If polar quantization is deemed best, the question arises as to the ratio of the number of phase quantizer levels to that or magnitude quantizer levels when the product of these numbers is fixed. A simple expression is derived for this ratio that depends only upon the magnitude distribution. Several examples of common circularly symmetric bivariate densities are worked out in detail using these expressions.

On spherically invariant random processes (Corresp.)

It is shown that a random process is spherically invariant if and only if it is equivalent to a zero-mean Gaussian process multiplied by an independent random variable. Several properties of spherically invariant random processes follow in a simple and direct fashion from this representation.

Stochastic analysis for the recursive median filter process

Vector probability measure functions (density function) for recursively median filtered signals are found when the underlying input binary sequences are either independent identically distributed (i.i.d.) or Markov chains. The results are parametric in the window size of the filter and in the probability distribution of the input sequence. Using statistical threshold decomposition, the same results are found for discrete alphabet random sequences that are either i.i.d. or Markov chains. Some examples illustrating the efficacy of the recursive median filter relative to the nonrecursive implementation are presented. In particular, the breakdown probabilities are tabulated for both recursive and nonrecursive median filters. >

Quantization schemes for bivariate Gaussian random variables

The problem of quantizing two-dimensional Gaussian random variables is considered. It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation. Applications of the results to a transform coding scheme known as spectral phase coding are discussed.

A note on optimal quantization (Corresp.)

For a general class of optimal quantizers the variance of the output is less than that of the input. Also the mean value is preserved by the quantizing operation.

Some properties of uniform step size quantizers (Corresp.)

Some properties of the optimal mean-square error uniform quantizer are treated. It is shown that the mean-square error (mse) is given by the input variance minus the output variance. Furthermore \lim_{N \rightarrow \infty}mse/(\Delta^{2}/12) \geq 1 , where N is the number of output levels and \Delta (a function of M ) is the step size of the uniform quantizer, with equality when the support of the random variable is contained in a finite interval. A class of probability densities is given for which the above limit is greater than one. It is shown that \lim_{N \rightarrow \infty}N^{2} \cdot mse =(b-a)^{2}/12 , where (b-a) is the measure of the smallest interval that contains the support of the input random variable.

Quantizing schemes for the discrete Fourier transform of a random time-series

The problem of quantizing a large-dynamic-range, possibly nonstationary signal after it has been transformed via the discrete Fourier transform (DFT) is investigated. It is demonstrated that, for purposes of d, the polar-form representation for these DFT coefficients is preferable to the Cartesian-form when fixed-information-rate quantization schemes are considered. A technique called spectral phase coding (SPC) is described for transforming the DFT coefficients into a bounded sequence \{\psi_{p}\} , where - \pi . In most cases, the terms \psi_{p} are uniformly distributed over this range. The results indicate that SPC is a robust suboptimum procedure for coding nonstationary or large-dynamic-range signals into digital form.

Properties of minimum mean squared error block quantizers (Corresp.)

Two results in minimum mean square error quantization theory are presented. The first section gives a simplified derivation of a well-known upper bound to the distortion introduced by a k -dimensional optimum quantizer. It is then shown that an optimum multidimensional quantizer preserves the mean vector of the input and that the mean square quantization error is given by the sum of the component variances of the input minus the sum of the variances of the output.

The design of two-dimensional quantizers using prequantization

The theoretical advantages of two-dimensional quantization over univariate quantization have been studied in the literature. However, in many cases there is no known implementation for the two-dimensional quantizer that can operate in real time. A new approach to the design of two-dimensional quantizers is presented. This technique, called prequantization, is used to design two-dimensional quantizers that operate in real time. The importance of prequantization is demonstrated by the design of the optimum uniform two-dimensional (hexagonal) quantizer. Additional examples are given to illustrate the flexibility of this design approach.