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

47Consequently, when the slope S, is measured from the WVD, the chirp rate can befound from the relation:As can be seen from Equations 2.42 and 2.44, both the STFT and the Gaborexpansion reach maxima along the line w Using the frequency and timeand the slope S, measured from theSTFT and Gabor expansion respectively, the chirp rate p, can be calculated usingEquation 2.55.The instantaneous frequency of a simulated linear-FM signal (chirp) wasestimated by each of the TF analysis techniques considered. For each method, theinput signal was a linear FM signal of 256 points sampled at 1000 Hz. The chirp ratewas 8.84 Hz/sec. The SNR of the input signal was varied from 15 dB to —15 dB in1 dB steps and 250 trials were averaged for each SNR level. The data was analyzedas follows: for the WVD a K=64 point square window, for the STFT a 64 pointhamming window, for the Gabor expansion, a 32.x16 TF grid that was oversampledby 4, and the scalogram (CWT) was analyzed with a window of 32 points at thecoarsest scale. For each transform, and in order to minimize frequency quantizationerrors, the peak response at each time sample was found from an interpolation alongthe frequency axis. After the peaks for each time sample were determined the chirprate was found from the least-squares linear fit through these maxima. The errorbetween this line and the actual IF of the input was found and is plotted for eachTF analysis technique in Figure 2.23. The CRB for N 64 is plotted on the figurealso. It can be seen that the WVD approaches this bound at high SNR. The STFTis closer to the bound than the Gabor expansion and the scalogram. All methodsexhibit a threshold. This figure confirms the optimality of the WVD [7] and therelative advantages of the STFT over the Gabor expansion [11].