Kishan Shenoi

Quadratic residues: Application to chirp filters and discrete Fourier transforms

A complete solution is given to the problem of finding the number of distinct quadratic residues for a composite modulus. Two specific applications of this result are described. The first one concerns the efficient implementation of chirp filters. It is shown that by an optimum choice of the number of taps, the number of multiplications required to realize a transversal chirp filter can be greatly reduced. Secondly, an algorithm for the computation of DFT, based on chirp filtering, is discussed. It has the potential of being faster than the FFT in certain cases and, in addition, requires less storage for the sine-cosine values.

On the design of recursive lowpass digital filters

The magnitude-squared characteristic of an ideal lowpass filter is approximated, over the finite interval [-1,1], by the ratio Φ(x)/Φ(x)+P(x) of two n-th-degree polynomials. The polynomials Φ(x) and P(x) are chosen such that the ratios P(x)/ Φ(x) and Φ(x)/P(x) approximate, in a Chebyshev sense, the zero function over the passband [x p ,1] and the stopband [-1,x s ], respectively. The passband and stopband form two disjoint intervals. The polynomials are iteratively determined by repeated applications of Darlingtons technique for obtaining rational function generalizations of Chebyshev polynomials. The efficacy of the iterative method is demonstrated by examples.

The Arcsine transform and its applications in signal processing

By initially transforming a signal into its Arcsine value, the multiplications required in the subsequent processing of the signal can be replaced by additions and table look-ups. With the advent of large read-only memories, this may be an attractive method to reduce computation time and simplify hardware in signal processing systems. An elegant method of obtaining the Arcsine transform and its application to several important signal processing problems are discussed. Among these are computation of discrete Fourier transforms and correlation functions, realization of digital filters, modulation and detection of signals, and construction of frequency synthesizers.

M-Adic invariant filters

A technique is described to approximate a linear time-invariant (LTI) system by a linear m-adic invariant (LMI) system or, equivalently, approximate a circulant matrix by a supercirculant matrix. This approximation reduces the number of multiplies required for computing cyclic convolution. Furtermore, the concepts of LMI systems are presented in a tutorial fashion. Examples are included to illustrate the efficacy of the approximation technique.

Specification-based design of ΣΔM for A/D and D/A conversion

The paper establishes a correspondence between performance requirements, mathematical parameters, and circuit parameters of a sigma-delta modulator. The sigma-delta modulator is viewed as a device which distributes the noise power over a much broader band, compared to signal bandwidth, shapes and amplifies it, and allows filtering of the out-of-band noise. The shaping and amplification are quantified by two parameters, F and P, whose product is analogous to the square of step size of a uniform coder. These two parameters are related, on one hand, to the time constants or location of zero and poles. On the other hand, inequalities are set up between performance parameters, like signal-to-noise ratio and dynamic range, and F and P.