In applying Shannon's sampling theorem, evaluation of the sampled signal Fourier spectrum is based on the fact that sampling the continuous‐time signal is the result of multiplying the signal by distributions. If the signal has discontinuities, a multiplication of distributions – an undefined operation – is encountered. Such undefined operation has led to errors in the literature which to date accompany the formulation of sampling of signals containing discontinuities. This paper presents an approach to evaluating the product of distributions as a means of sampling discontinuous signals, eliminating such errors. It is shown that the value of the product of distributions may be found by invoking the Mittag–Leffler expansion. As an illustration of errors that have existed for decades and still exist in the digital signal processing literature whenever discontinuous signals are sampled the approach of impulse invariance provides a case in point. It was already noted that this approach has an inherent error. Yet, impulse invariance is still considered as one of the two main approaches for converting analogue to digital filters. In this study, the true spectra of sampled discontinuous signals are evaluated, and a new approach to the transformations between continuous‐time and discrete‐time systems eliminating the error, is proposed.

A novel generalized spectral analysis approach which applies a weighting to z-transform spectra evaluated on contours in the z-transform plane is proposed. A parallel algorithm and 1D and 2D parallel processor architectures for the estimation of the pole-zero mathematical model of a system from a truncated version of its impulse response by successive parallel evaluations of the proposed weighted z-transform spectra are subsequently presented. The algorithm is applicable to system identification and digital filter synthesis. The proposed weighted z-transform spectra and associated energy spectra make possible the evaluation of the poles and zeros of an infinite impulse response system of which the order is unknown from a truncated version of its impulse response with reasonable accuracy. A parallel dynamic weighting of z-transform spectra that is a function of |z| is shown to overcome the effect of exponential divergence of the z-transform of finite duration sequences as the z-plane center is approached.

A new class of general-base matrices, named sampling matrices, which are meant to bridge the gap between algorithmic description and computer architecture is proposed. "Poles," "zeros," "pointers," and "spans" are among the terms introduced to characterize properties of this class of matrices. A formalism for the decomposition of a general matrix in terms of general-base sampling matrices is proposed. "Span" matrices are introduced to measure the dependence of a matrix span on algorithm parameters and, among others, the interaction between this class of matrices and the general-base perfect shuffle permutation matrix previously introduced. A classification of general-base parallel "recirculant" and parallel pipelined processors based on memory topology, access uniformity and shuffle complexity is proposed. The matrix formalism is then used to guide the search for algorithm factorizations leading to optimal parallel and pipelined processor architecture.>

Algorithms for system identification and the computation of its mathematical model through a ``fast'' Z transformation of its sampled response in the presence of noise are introduced. It is shown that by iteratively applying constant-damping-and constant-frequency contour finite Z transforms a system's mathematical model-in the presence of noise can be efficiently evaluated. On line tracking of the poles and zeros of relatively rapidly time-variant systems such as a space shuttle or a jet aircraft are possible applications. An organization for a high-speed machine including a fast Fourier transform processor for on line identification of relatively rapidly time-variant system is suggested. Applications of the described algorithms include enhancement of poles in spectral analysis of signals, representation of signals by poles and zeros for signal classification, coding and recognition, filter synthesis, adaptive filtering, identification of parameters in curve fitting problems, in addition to system identification in the presence of noise. The procedure presented here is a transform domain approach that is distinct, to the knowledge of the author, when compared to known identification techniques in which a best fitting is made to an assumed mathematical model of the system. In addition to the smoothing obtained here through the computation of spectra in the Z plane of a time series including redundancy, no priori knowledge of the order of the system needs be assumed.

The Walsh-Hadamard transform has recently received increasing attention in engineering applications due to the simplicity of its implementation and to its properties which are similar to the familiar Fourier transform. The transform matrices found so far to possess fast algorithms are the naturally ordered and dyadically ordered matrices, whose algorithms are similar to the Cooley-Tukey algorithm, and to the machine-oriented algorithm of Corinthios [2], respectively.

The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

The design of a class of special-purpose computers for time-series analysis by Fourier transformation is described. The computers are sequential machines which implement machine-oriented fast Fourier transform algorithms obtained by factoring the discrete Fourier transform to an arbitrary radix.

An arbitrary-radix fast Fourier transform algorithm and a design of its implementing signal processing machine are introduced. The algorithm yields an implementation with a level of parallelism proportional to the radix r of factorization of the discrete Fourier transform, allows 100 percent utilization of the arithmetic unit, and yields properly ordered Fourier coefficients without the need for pre- or postordering of data.