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

24s(t) may be represented as a combination of Equation 2.16 and Equation 2.15,This equation may be viewed as an expansion of s(t) into time-frequency shiftedversions gm?, (t) of an elementary function g (t) ,where g(t) is the dual frame generating function [12]. The coefficients c„ are givenbyand represent the sampled complex spectrogram of s(t). The squared magnitude ofthe complex spectrogram is the waveform's physical spectrum. This discrete timesignal expansion is known as the Gabor expansion[33, 351. The functions g(t)are known as the Gabor logons and the coefficients c„ are known as the Gaborcoefficients. Gabor originally used time-frequency-shifted Gaussian functions as thelogons due to their concentration in time and frequency[33].The basis signals gmn(t) may be constructed so that they are localized andconcentrated with respect to both time and frequency. If they are, the expansioncoefficients c„, indicate the signal's time-frequency content around the timefrequencylocation (nT , m F) . Since the logon basis signals g(t) are all derivedfrom the elementary function g (t) through simple time-frequency shifts, they can beeasily generated.is shown in Figure 2.11 and theGabor transform of a multicomponent signal, as represented by Equation 2.10, isshown in Figure 2.12. The frequency resolution is dependent on the particular basissignals that are used in the transform.