CHAPTER 5

WINDOWING AND LOCALIZATION

**5.1 OVERVIEW: NONLOCALITY OF THE DFT**

Let’s begin with an example.

EXAMPLE 5.1

Consider two different audio signals *f(t)* and *g(t)* defined on 0 ≤ *t ≤* 1, given by

and

Both are composed of the same two basic waveforms, sin(2π(96)*t*) and sin(2π(235)*t*). In *f* the waveforms are present throughout, while in *g* each waveform is present for exactly half of the time interval [0, 1],

Let us sample each at 1000 Hertz to produce sample vectors **f** and **g**, and then compute the DFT of each sampled signal. The magnitude of each is plotted in Figure 5.1, DFT(**f**) on the left and DFT(**g**) on the right, where in each graph the horizontal index *k* corresponds to *k* Hertz. It’s obvious from the plots that each signal contains dominant frequencies near 96 and 235 Hertz, and the magnitude of the DFT’s are otherwise fairly similar. But the signals *f(t)* and *g(t)* are quite different in the time domain, a fact that is difficult to glean from the DFT graphs.

This example illustrates one of the shortcomings of traditional Fourier ...