Gabor’s Signal Expansion and Its Relation to Sampling of the Sliding-Window Spectrum

  • Martin J. Bastiaans
Part of the Springer Texts in Electrical Engineering book series (STELE)

Abstract

It is sometimes convenient to describe a time signal ϕ(t), say, not in the time domain, but in the frequency domain by means of its frequency spectrum, i.e., the Fourier transform \(\bar{\varphi}(\omega)\) of the function ϕ(t) which is defined by
$$\bar{\varphi }(\omega ) = \int {\varphi (t){{e}^{{ - j\omega t}}}dt;}$$
a bar on top of a symbol will mean throughout that we are dealing with a function in the frequency domain. (Unless otherwise stated, all integrations and summations in this contribution extend from -∞ to +∞.) The frequency spectrum shows us the global distribution of the energy of the signal as a function of frequency. However, one is often more interested in the momentary or local distribution of the energy as a function of frequency.

Keywords

Convolution Sine Prolate Undersampling 

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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Martin J. Bastiaans

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