An introduction to distribution theory for signals analysis.: Part II. The convolution
This article continues the exposition in Part I [D.C. Smith, An introduction to distribution theory for signals analysis, Digital Signal Process. 13 (2003) 201232] of certain key concepts from distribution theory which are essential for understanding signal theory. In this second part, we demonstrate the power of distribution theory by focusing on distributional convolution and some of its applications to signals analysis. The well-known classical convolution theorem (CCT) states that the Fourier transform of a convolution of integrable functions is the product of their Fourier transforms, and is essential in signals processing, by providing a method for removing noise and undesirable spectral artifacts from signals. Unfortunately, signal processing texts routinely apply this theorem nonrigorously, with questionable results, e.g., in attempts to apply it to singular functions in derivations of the Hilbert conjugate and the analytic signal, in attempts to recover an unknown causal impulse response modeling a data channel, and in derivations of the Sampling Theorem [D.C. Smith, An introduction to distribution theory for signals analysis, Digital Signal Process. 13 (2003) 201232]. Fortunately, many inadequacies of the CCT are overcome by its generalizations to the tempered distributions. In this article we discuss three distributional convolution theorems (DCTs), each of which has important theoretical and practical consequences for signal theory. We demonstrate that when the Cauchy principal value distribution is used to properly define the Hilbert conjugate, the first DCT may be applied to rigorously derive the analytic signal. The second DCT is particularly useful for applications involving compactly supported distributions, including the Dirac delta. The third DCT concerns the lesser-known distributional Laplace transform, which is shown by example to be superior to either the classical or distributional Fourier transform for recovering an unknown impulse response modeling a causal linear time-invariant system. This article upholds the style of Part I, by avoiding unnecessary abstraction and supplying very detailed proofs in hopes of appealing to a wider audience.
distribution theory distributional convolution hilbert conjugate analytic signal tempered distributions cauchy principal value distributional laplace transform distributional fourier transform functional convolution