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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 FFT2.1windows 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.

In Chapter 4, we apply both the Fourier theorems of Chapter 2 and the FFT windows of Chapter 3 to the topic of FIR digital filter design--that is, the numerical design of finite-impulse response (FIR) filters for linear filtering in discrete time. We will need such filters in Chapter 10 when we implement FIR filters using FFT convolution in the framework of the short-time Fourier transform (STFT).

Chapter 5 is concerned with spectrum analysis of tonal signals, that is, signals having narrow-band 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 6 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 6.) 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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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