5.4 Amplitude and phase spectra

Our goal at the start of the chapter was to answer the question “how much of a given frequency is present in some unknown sound?” The Fourier transform almost gives us this, but its output is a complex number rather than a plain amplitude. How do we extract the amplitude?

The answer is simple: convert from rectangular to polar form. The magnitude of \(X(\omega)\) is the amplitude at frequency \(\omega\), and its angle is the phase. These define the amplitude spectrum and phase spectrum:

\[ |X(\omega)| = \sqrt{R^2(\omega) + I^2(\omega)}, \qquad \angle X(\omega) = \tan^{-1}\!\left(\frac{I(\omega)}{R(\omega)}\right). \]

Both are real-valued functions of frequency. The amplitude spectrum \(|X(\omega)|\) is always non-negative, and represents the answer to our original question (“how much of frequency \(\omega\) is in \(x\)?”). The phase information is also preserved and lives in the phase spectrum \(\angle X(\omega)\), taking on values in the range \((-\pi, \pi]\).

Recall from our discussion of initial phase in Chapter 3 that our ear is far more sensitive to amplitude than to phase. For this reason, the amplitude spectrum is by far the more commonly used of the two. The phase spectrum becomes important mainly when we want to reconstruct a signal from its frequency-domain representation, using the inverse Fourier transform that we will meet later.

You may already have encountered the amplitude spectrum if you have ever opened a “spectrum analyzer” in a digital audio workstation:

Fig. 20 An amplitude spectrum of a rich musical tone, displayed the way a DAW spectrum analyzer would show it (amplitude in decibels, frequency on a logarithmic axis). Each harmonic appears as a peak. Notice that the peaks are not the infinitely narrow spikes our idealized analysis predicted, but have a finite width. We will learn why this happens when we study practical frequency analysis using the discrete Fourier transform later in the book.