Why wavelet?

Short-time Fourier transform and Gabor transform

In Chapter 3, we learned that a signal can be represented as either a time function x(t) as the amplitude of the signal at any given moment t, or, alternatively and equivalently, as a spectrum X(f) = F[x(t)] representing the magnitude and phase of the frequency component at any given frequency f. However, no information in terms of the frequency contents is explicitly available in the time domain, and no information in terms of the temporal characteristics of the signal is explicitly available in the frequency domain. In this sense, neither x(t) in the time domain nor X(f) in the frequency domain provides complete description of the signal. In other words, we can have either temporal or spectral locality regarding the information contained in the signal, but never both at the same time.

To address this dilemma, the short-time Fourier transform (STFT), also called windowed Fourier transform, can be used. The signal x(t) to be analyzed is first truncated by a window function w(t) before it is Fourier transformed to the frequency domain. As all frequency components in the spectrum are known to be contained in the signal segment inside this particular time window, certain temporal locality in the frequency domain is achieved.