Elementary Spectrum Analysis

Before we do anything in the field of spectral modeling, we must be able to competently compute the spectrum of a signal. Since the spectrum is given by the Fourier transform of a signal, Chapter 2 begins with a review of elementary Fourier theory and the most generally useful Fourier theorems for practical spectrum analysis.

In Chapter 3, we look at a number of FFT^2.1windows used in practice. The Fourier theorems give us a good thinking vocabulary for understanding the properties of windows in the frequency domain. In addition to a tour of well known windows, optimal custom window design is addressed.

Chapter 4 is concerned with spectrum analysis of tonal signals, that is, signals having narrowband peaks in their spectra. It turns out the ear is especially sensitive to spectral peaks (which is the basis for MPEG audio coding), and so it is often important to be able to accurately measure the amplitude and frequency of each prominent peak in the spectrum. (Sometimes we will also measure the phase of the spectrum at the peak.) This chapter also discusses resolution of spectral peaks, and how the choice of FFT window affects resolution.

Chapter 5 is concerned with spectrum analysis of noise, where, for our purposes, ``noise'' is defined as any ``filtered white noise,'' and white noise is defined as any uncorrelated sequence of samples. (These terms are defined in detail in Chapter 5.) Unlike the ``deterministic'' case, such as when analyzing tonal signals, we must average the squared-magnitude spectrum over several time frames in order to obtain a statistically ``stable'' estimate of the spectrum. This average is called a power spectral density, and the method of averaging is called Welch's Method. It is noteworthy that the power spectral density is a real and positive function, so that it contains no phase information.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.