An impulse-based framework for signal functional representations

Signal transformations such as Fourier, Laplace, and z are used frequently in signal processing to obtain functional representations of signals. Usually those pairs of transform and its inverse are defined independently. This paper introduces a framework for defining functional representations of signals, by defining signals as distributions of impulses. Within this framework, various transformations can be derived from various definitions of impulses. This impulse distribution term is used in an inverse formula, and then rearranged such that kernel term can be identified. This kernel term is then selected for the transform formula. When the newly obtained transform formula is reapplied into the integral superposition, we have a simplified final form of the inverse formula. In this paper we have applied the framework to derive various well known transforms, such as Fourier, Laplace and z. We should be able to rediscover other transforms such as Hilbert, Mellin, and Wavelet using a similar approach. In fact it is our hope that our framework can trigger discoveries of new transform pairs in the future.