A unified treatment of various computational methods for estimating the instantaneous frequency and the group delay of discrete-time signals is given. Direct, phase-based methods are compared with the newer moment methods. Instantaneous frequency and group delay are related to certain moments of the signal or its Fourier transform, and they also are first-order moments of the Wigner-Ville distribution of the signal. Because instantaneous frequency and group delay are mathematical duals of one another, an algorithm suitable for estimating one can be used, with a simple interchange of time and frequency variables, to estimate the other. The performance of these several different methods is assessed by applying them to a wide variety of representative test signals such as, for example, chirps, tone bursts, as well as more complicated signals, such as linear phase bandpass, minimum-phase low-pass, and all-pass filter responses. In general the moment methods are superior to the phase-based methods because they avoid phase unwrapping errors and approximate digital differentiations. Instantaneous frequency and group delay time can be useful in identifying the moment-to-moment frequency content of an evolving signal and the arrival time of frequency components of the signal's spectrum, respectively.
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