A unified framework for the Sussman, Moyal, and Janssen formulas

Abstract

This paper derives three fundamental identities in the radar and sonar literature, namely, Sussman' identity for ambiguity functions, Moyal's formula which establishes the value of the inner product between two scattering functions, and Janseen's formula which establishes identities for mixed inner products between waveforms and Gabor wavelets. Starting from the fundamental convolution identity, we derive Sussman's identity. Following from an initial value theorem of Fourier analysis, we obtained Moyal's formula. Following from Poisson's sum formula and an initial value theorem, we also obtained Janssen's equality. The relationship between these three identities is as follows: Janssen's formula is a sampled-data version of Moyal's formula, and both follow from Sussman's identity. In turn, Sussman's identity is a consequence of the fundamental convolution identity.