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.

A definition for channels with asymptotically decreasing memory and anticipation is given. The relation to indecomposable finite-state channels and to a certain class of infinite-state channels is established. It is shown that for channels with asymptotically decreasing memory and anticipation every ergodic input source is admissible and that the coding theorem holds.

For an arbitrary band-limited generalized functionH(t)a representation is obtained that is a combination of a Taylor series and a conventional sampling series. A parameterNthat enters can be chosen such as to make the series rapidly convergent. Furthermore, it is shown that, for suitably restricted test functions,H(t)can be represented by a conventional sampling series and by a series of delta functions with weights equal to the sampling values. The results apply also to ordinary band-limited functions.