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

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.