Recovery of transit times and frequencies of multiple pulses via the short-time Fourier transform

In this work, we present a study to determine the transit times and frequencies of pulses by using the Short-Time Fourier Transform (STFT). We consider the case of an acoustic signal composed of five Gaussian pulses that have a high overlapping in time but oscillate at different frequencies. We proceeded in three steps. First, we illustrate how the STFT calculated through a sliding window produces a spectrogram where transit time is on one axis and frequency on the other. Second, we derive an exact analytical solution of the STFT to develop an intuitive vision of the mathematical technique. Finally, in a third step, we present an experiment to demonstrate that the STFT is a useful technique to characterize a complex acoustical signal.