The fast Fourier transform for experimentalists. Part VI. Chirp of a bat

Two assumptions underlie the Fourier transform process: stationarity and linearity. When signals deviate from these conditions, the transform outcomes are suspect. A chirp, which by definition has a frequency that varies with time, doesn't satisfy these requirements, and its fast Fourier transform (FFT) doesn't adequately express the changing nature of the signal's frequency content. In this analysis of a bat chirp, I first examine how the FFT handles a chirp and then how we can use a sequence of windows that individually span only a portion of the total time-domain signal to generate a frequency versus time description of the signal. The trade-off in this kind of windowing is between dynamic response and resolution: we obtain improved dynamics if we use shorter windows, whereas we get better resolution with longer windows. This article and this series concludes with a brief look at the Hilbert-Huang transform, which isn't constrained by the same assumptions as the FFT. This transform process consists of two independent sets of operations. The first, called empirical mode decomposition, generates a set of intrinsic mode functions (IMFs), from the data. The second step extracts phase information from each IMF and its Hilbert transform. The derivative of the phase with respect to time yields the instantaneous frequency. The net effect of these operations is to transform the time-domain data to frequency versus time data instead of the amplitude versus frequency the FFT obtains