Name
Abstract | ||
|---|---|---|
ne of the most funda- mental identities in the radar and sonar litera- ture is the Sussman identity for ambiguity functions. In the quantum mechanics literature, Moyal's formula establishes the value of the inner product between two scattering functions. In the time- frequency literature, Janssen's formula establishes identities for mixed inner products between waveforms and Gabor wavelets. The casual reader of these dif- ferent literatures might not realize that there is a connection between these three fundamental identities in that they all follow from an elementary con- volution, or linear systems, identity. Moreover, Sussman's identity appears to be the most fundamental of the three, following as it does from the convolution identity as a Fourier trans- form consequence. Then from Sussman follows Moyal as an initial value theo- rem of Fourier analysis, and then from Moyal follows Janssen as a sampled, or Poisson sum, version of Moyal. Our aim in this short lecture note is to derive these identities, starting with an easily derived convolution, or linear systems, identity. Then Sussman, Moyal, and Janssen follow from this identity with the aid of Fourier trans- form properties and a four-corners dia- gram. The four-corners diagram keeps thinking straight, visually codes the connection between the various identi- ties, and supports intuition, which is the aim of any good diagram. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1109/MSP.2006.1628888 | Signal Processing Magazine, IEEE |
Keywords | DocType | Volume |
Fourier analysis,Poisson equation,convolution,radar signal processing,sonar signal processing,wavelet transforms,Fourier analysis,Gabor wavelets,Janssen formula,Moyal formula,Poisson sum formula,Sussman identity,ambiguity functions,initial value theorem,radar literature,scattering functions,sonar literature | Journal | 23 |
Issue | ISSN | Citations |
3 | 1053-5888 | 3 |
PageRank | References | Authors |
0.46 | 3 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| David C. Farden | 1 | 32 | 32.04 |
| Louis L. Scharf | 2 | 2525 | 414.45 |