Signal Description By Means Of A Local Frequency Spectrum

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Abstract
The description of a signal by means of a local frequency spec-trum resembles such things as the score in music, the phase space in mechanics, and the ray concept in geometrical optics. Two types of local frequency spectra are presented: the Wigner distribution function and the sliding-window spectrum, the latter having the form of a cross-ambiguity function. The Wigner distribution function in particular can provide a link between Fourier optics and geometrical optics; many properties of the Wigner distribution function, and the way in which it prop-agates through linear systems, can be interpreted in geometric-optical terms. The Wigner distribution function is linearly related to other signal representations like Woodward's ambiguity function, Rihaczek's complex energy density function, and Mark's physical spectrum. An advantage of the Wigner distribution function and its related signal representations is that they can be applied not only to deterministic signals but to stochastic signals as well, leading to such things as Walther's generalized radiance and Sudarshan's Wolf tensor. On the other hand, the sliding-window spectrum has the advantage that a sampling theorem can be formulated for it: the sliding-window spectrum is completely determined by its values at the points of a certain space-frequency lattice, which is exactly the lattice suggested by Gabor in 1946. The sliding-window spectrum thus leads naturally to Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal, which is again another space-frequency signal representation, and which is related to the degrees of freedom of the signal.

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