Space/time/frequency methods in adaptive radar - New Jersey ...

26Figure 2.12 Gabor transform of a multicomponent signalIn the context of the Gabor expansion, there are two important issues relatingto the Gabor basis, gmn(t ): completeness and linear independence [12, 36]. Thecompleteness of the Gabor basis guarantees that any finite-energy signal may berepresented by a linear combination of a set of Gabor basis functions given in theGabor expansion (2.18). A necessary condition for completeness of the Gabor basisis TF < 1; this condition is a bound on the density of the "time-frequency samplinggrid". In the case of "critical sampling" , where TF = 1, the number of Gaborcoefficients equals the number of signal samples (assuming a band-limited signalsampled with minimum sampling rate) meaning the Gabor coefficients G x (n,m) donot contain any redundancy. Redundancy in the coefficients may be introduced byoversampling (TF < 1) to some degree. This is recommended so that the coefficientshave some numerical stability, at the cost of the Gabor basis signals not being linearlyindependent.