Spectrum Analysis Windows

In spectrum analysis of naturally occurring audio signals, we nearly always analyze a short segment of a signal, rather than the whole signal. This is the case for a variety of reasons. Perhaps most fundamentally, the ear similarly Fourier analyzes only a short segment of audio signals at a time (on the order of 10-20 ms worth). Therefore, to perform a spectrum analysis having time- and frequency-resolution comparable to human hearing, we must limit the time-window accordingly. We will see that the proper way to extract a ``short time segment'' of length $N$ from a longer signal is to multiply it by a window function such as the Hann window:

$$w(n) = \cos^2\left(\frac{n}{N}\pi\right), \quad n = -\frac{N-1}{2}, \ldots, -1,0,1,\ldots,\frac{N-1}{2}$$ (4.1)

We will see that the main benefit of choosing a good Fourier analysis window function is minimization of side lobes, which cause ``cross-talk'' in the estimated spectrum from one frequency to another.

The study of spectrum-analysis windows serves other purposes as well. Most immediately, it provides an array of useful window types which are best for different situations. Second, by studying windows and their Fourier transforms, we build up our knowledge of Fourier dualities in general. Finally, the defining criteria for different window types often involve interesting and useful analytical techniques.

In this chapter, we begin with a summary of the rectangular window, followed by a variety of additional window types, including the generalized Hamming and Blackman-Harris families (sums of cosines), Bartlett (triangular), Poisson (exponential), Kaiser (Bessel), Dolph-Chebyshev, Gaussian, and other window types.