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

30reconstruction only takes place in the sense of the signal's energy. The convergenceof Equation 2.29 is related to the numerical robustness of the reconstruction [12].2.3.3.3 Scalograms: The spectrogram was defined in section 2.3.1 as the squaremodulus of the STFT, and is a common tool in signal analysis. The spectrogramprovides a distribution of the energy of the signal in the time-frequency plane. Asimilar distribution can be defined in the case of the wavelet transform. Since theCWT behaves like an orthonormal basis decomposition, it can be shown that it isisometric [37]. Therefore,where Ex f x(t) 2 dt is the energy of the signal x(t). In this manner, thescalogram is defined as the squared modulus of the CWT.The scalogram of s(t) sin(2πƒt) is shown in Figure 2.14 and the scalogramof the multicomponent signal of Equation 2.10 is shown in Figure 2.15.In Figure 2.16, some differences between the scalogram and spectrogram areillustrated. In Figure 2.16(a), a Dirac pulse is analyzed around t to , the CWTshows that the signal's behavior is localized around t o for small scales. In the case ofSTFT shown in Figure 2.16(b), the corresponding region is as large as the extent ofthe analysis window over all frequencies. Since the time-scale analysis of the CWTis logarithmic in frequency, the area of influence in the case of the CWT for a purefrequency ƒo in the signals increases with fo , as shown in Figure 2.16(c). Figure2.16(d) displays how the area remains constant in the case of a spectrogram.2.3.3.4 Discrete Form: The basis functions (wavelets) act as an orthonormalbasis for wavelet analysis. If the basis function, h(t), is chosen correctly, then adiscrete form of wavelet analysis may be used. The theory of wavelet frames {12]provides a framework that allows for choices in sample redundancy and sampling