Understanding the Poisson Sum Formula

Abstract

The presentation is about rather general problems in signal processing the way they occur in every day SAR raw data and image data processing. Usual formulas are often too complicated due to the richness of their technical details. Thereby, they hide their basic message which in turn would be crucial for understanding. In the presentation we derive correspondingly simple formulae in order to promote general understanding of important relationships in signal processing. The technique used is the concept of operators. Operators can be applied to functions in order to produce other functions. This is very suitable for signal processing since "processing" is an operation and "signals" are functions. As an result it is shown that by introducing a sampling operator (in order to digitize signals) and a replication operator (in order to produce infinite many replications of a signal) most problems can be written in simple formulae. Moreover, the introduced symbols allow very easy symbolic calculation in a similar fashion the way we have integration and differentiation rules. As an example, it is shown that the Poisson Sum Formulae state that sampling turns into replicating (and vice versa) if a Fourier transform is applied to a digital signal. Finally, the connection between Discrete and Continuous Fourier Transform is explained as follows: The Discrete Fourier Transform is nothing else than (continuously) Fourier transforming a digital signal.