By Hans G. Feichtinger, Thomas Strohmer
The utilized and Numerical Harmonic research (ANHA) e-book sequence goals to supply the engineering, mathematical, and medical groups with major advancements in harmonic research, starting from summary har monic research to easy purposes. The identify of the sequence displays the im portance of purposes and numerical implementation, yet richness and relevance of functions and implementation rely essentially at the constitution and intensity of theoretical underpinnings. hence, from our perspective, the interleaving of concept and functions and their artistic symbi otic evolution is axiomatic. Harmonic research is a wellspring of rules and applicability that has flour ished, built, and deepened over the years inside many disciplines and via inventive cross-fertilization with various parts. The complicated and primary dating among harmonic research and fields comparable to sig nal processing, partial differential equations (PDEs), and snapshot processing is mirrored in our state-of-the-art ANHA sequence. Our imaginative and prescient of recent harmonic research comprises mathematical components reminiscent of wavelet thought, Banach algebras, classical Fourier research, time frequency research, and fractal geometry, in addition to the varied subject matters that impinge on them.
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3) The quadratic expressions Af := A(j, f) and W f := W(j, f) are called the (radar) ambiguity function and the Wigner distribution of f. The quadratic expressions, in particular the Wigner distribution, are often praised as ideal time-frequency representations, because they do not depend on the choice of a window and thus seem to exhibit the pure timefrequency behavior of a function. In practice, however, the non-linearity of Af and W f makes these time-frequency representations less useful and difficult to interpret because of the existence of so-called cross terms.
When 1 :S e:S 2 - N-l and t E [0,1), at least one of the numbers e - t - liN, l = 0,1, ... , N - 1, falls in the interval (e, 2](mod 1). 4) ;::: 1. When 2 - N- 1 < e < 2 and t E (1 N-l,e-l), all points e-t-lIN, l = 0, ... ,N -1, fall in the interval [1,e). M. Janssen vanishes at l/ = ~. 4) = O. Case lcJ > 1. 15) ensuring that the series stays away from 0 when l/ E [0,1). This leads then to the necessary and sufficient condition c - lcJ E [N-l, 1- N- 1] for (X[O,c) , N- 1 , 1) to be a frame. 1. )'~' 1) is a frame fo1' M = 1,2, ....
Under a weak continuity assumption it can be shown rigorously that all such time-frequency representations are of the form C(f, f) = W f * ell for some tempered distribution ell E S' [18, Thm. 1]. According to Metatheorem C the uncertainty principles discussed so far can be adapted to yield uncertainty principles for time-frequency representations of the form C(f, f) = W f * ell. At least for "nice" convolution terms ell this should be possible by analyzing the effect of the convolution on the uncertainty principles for the Wigner distribution.
Advances in Gabor Analysis by Hans G. Feichtinger, Thomas Strohmer