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

182.3.1 Short-Time Fourier TransformThe short-time Fourier transform (STFT) has played a large role in the history ofdigital signal processing [30, 31, 32]. It is defined aswhere s(t) is the signal being analyzed and g (t) is a time-shifted windowing function.The squared magnitude of the STFT is known as the spectrogram. A simple interpretationof the STFT is a "sliding-window Fourier transform". The analysis of thefrequency content of s at time t o may be found by sliding the window function, g (t)across the signal until it is centered at t o and then taking the Fourier transform ofthe signal-window product. Each constant-time slice, $(t, ƒ), of the STFT representsthe frequency content of the signal around time t o . Many functions have beenused successfully as STFT windowing functions: Gaussian, Hamming, Hanning, andsquare/rectangular functions. The computation of the time slice at time to is shownin Figure 2.7. The spectrogram of s(t) = sin(2π ƒt) using a hamming window isshown in Figure 2.8.The STFT can also be expressed in the terms of the Fourier transform of thesignal and windowing functions in the formThe STFT of the multicomponent signalcomputed using a hamming window is shown in Figure 2.9. As is shown, the threesignal components are centered at the points (T 1 , ƒ ƒ1), (T2 , ƒ2), and (T3 , ƒ3).