### 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^{2.1}*windows* 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.

**Next Section:**

The Short-Time Fourier Transform (STFT) and Time-Frequency Displays