Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

The zeros of entire functions: Theory and engineering applications

Abstract: Functions that are analytic over a complex plane are called entire functions. They may be viewed as generalizations of polynomials because they admit power series expansions that converge everywhere. Entire functions are useful models for physical phenomena having finite "spectra," such as important classes of time-varying signals and spatially varying fields. Entire function models may be studied through the linear expansion techniques (e.g., Fourier series and integrals) familiar to most engineers, but product expansions akin to polynomial factorizations provide a less familiar alternative which is often more powerful when phenomena have a nonlinear character. This paper provides a tutorial introduction to relevant aspects of the theory of entire functions of a single complex variable, and also a discussion of exemplary applications. The paper deals specifically with a class of entire functions of exponential type, dubbed B-functions, which contain all functions that are band limited according to the variious extant definitions of bandlimitation. Product expansions are emphasized because they lead one to regard the real and complex zeros of a B-function as its information bearing attributes. This view leads naturally to the study of sampling, clipping, and similar applications, and to more general issues concerned with the sufficiency of zero-based representations and the recovery of a B-function's waveform from a zero-based representation.

Fourier optics described by operator algebra

Abstract: Fresnel diffraction is described by replacing the Fresnel-Kirchhoff integral, the lens transfer factor, and other operations by operators. The resulting operator algebra leads to the description of Fourier optics in a simple and compact way, bypassing the cumbersome integral calculus. Aberration effects and Gaussian beam illumination are also treated as a simple extension of the present theory.

Optimal filtering in fractional Fourier domains

Abstract: The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.