Windowing and Localization
S. Allen Broughton, Kurt Bryan · 2018
In this chapter, the authors focus on windowing discrete signals of finite length, but analogous results hold for analog signals. They also focus on two propositions: the first is a counterpart to the convolution theorem in which the roles of the time and frequency domains are reversed, as well as the roles of the discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT). The DFT has prominent side lobes, since the rectangular window is to good approximation a sampled version of a discontinuous function. When the authors use a rectangular window for localization, they end up convolving the “true” spectrum of the signal with the DFT of the window. The authors provide some of the trade-offs illustrated in the Matlab exercises.