Abstract :
It is shown how the Zak transform can be used to find nontrivial examples of functions f, g∈L2(R) with f×g≡0≡F×G, where F, G are the Fourier transforms of f, g, respectively. This is then used to exhibit a nontrivial pair of functions h, k∈L2(R), h≠k, such that |h|=|k|, |H |=|K|. A similar construction is used to find an abundance of nontrivial pairs of functions h, k∈L2 (R), h≠k, with |Ah |=|Ak| or with |Wh|=|W k| where Ah, Ak and Wh, Wk are the ambiguity functions and Wigner distributions of h, k, respectively. One of the examples of a pair of h, k∈L2(R), h≠k , with |Ah|=|Ak| is F.A. Grunbaum´s (1981) example. In addition, nontrivial examples of functions g and signals f1≠f2 such that f1 and f2 have the same spectrogram when using g as window have been found
Keywords :
frequency-domain analysis; information theory; signal processing; time-domain analysis; transforms; Fourier transforms; Wigner distributions; ambiguity functions; nontrivial examples; nontrivial pair of functions; signal analysis Zak transform; spectrogram; time-frequency analysis; Distributed algorithms; Land mobile radio; Optimal scheduling; Packet radio networks; Polynomials; Process design; Routing; Scheduling algorithm; Spectrogram; Time frequency analysis;