Windowed Fourier Transforms
Fourier series are ideal for analyzing periodic signals, since the harmonic modes used in the expansions are themselves periodic. By contrast, the Fourier integral transform is a far less natural tool because it uses periodic functions to expand nonperiodic signals. Two possible substitutes are the windowed Fourier transform (WFT) and the wavelet transform. In this chapter we motivate and define the WFT and show how it can be used to give information about signals simultaneously in the time domain and the frequency domain. We then derive the counterpart of the inverse Fourier transform, which allows us to reconstruct a signal from its WFT. Finally, we find a necessary and sufficient condition that an otherwise arbitrary function of time and frequency must satisfy in order to be the WFT of a time signal with respect to a given window and introduce a method of processing signals simultaneously in time and frequency.
KeywordsUncertainty Principle Instantaneous Frequency Inverse Fourier Transform Window Function Chirp Signal
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