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

23where 9/ is the Hilbert transform. For signal analysis, it is convenient to work with acomplex signal. When a complex signal is needed when only a real signal is available,the analytic signal may be used.2.3.1.2 Discrete Form of the STFT: For practical application of the STFT, adiscrete form of Equation 2.8 is needed. Samples of the STFT at equidistant timefrequencygrid points (nT, lc F) where T > 0 and F > 0 are the sampling periods forthe time and frequency variables are taken, resulting inThe time function in Equation 2.13, s(t), may also be sampled. The discreteversion of the STFT is then,where g (m) is a sampled window function of finite length M.2.3.2 Gabor ExpansionThe signal s(t) may also be represented as a superposition of time-frequency-shiftedversions of an elementary signal γ(t),In this form, s(t) has been expanded into the basis functionsWhen γ(t) is centered around t 0 and ƒ 0, the basis signal is centered around thetime-frequency point (t', ƒ`). Therefore, the function Tx (t', ƒ') in Equation 2.15 willshow how the 'neighborhood' around the time-frequency point (t', ƒ ' ) contributes tos(t). The coefficient function Tx (t' , ƒ') may be chosen as the STFT,provided that the window ,