Generalized harmonic analysis for distributions

Generalized harmonic analysis in the sense of Wiener is extended to the framework of Schwartz distributions. The approach seems mathematically and physically more transparent than the classical scheme, since every distribution possesses a Fourier transform so that the use of integrated Fourier transforms is avoided. A generalized Wiener-Khintchine representation is given which agrees well with the intuitive concept of the power spectrum. The latter is shown to be a tempered measure, in general, whose support is contained in the support of the Fourier transform of the signal. The correlation functional and power spectrum of filtered distributional signals is derived for a class of generalized filter impulse responses, which includes those that have bounded support or correspond to stable rational transfer functions. As an illustration, the form of the correlation functional and power spectrum for periodic and almost-periodic distributions and for delta-pulse trains occurring in sampled-data systems is given, and a deterministic white noise signal is constructed.