# Sound in the Frequency Domain

So far all the discussion of sound has centered on its description as fluctuations in air pressure over time. The representation of sound in the time domain is important to understand, but in some ways it is also awkward. For instance, the frequency of a sound is one of its most important physical properties, but determining frequency from a waveform requires making measurements of time intervals and then doing arithmetic. Indeed, for many complex waveforms, where multiple sinusoids of various frequencies are simultaneously present, it is often unclear where the intervals to be measured begin and end. The frequency domain provides an alternative description of sound in which the time axis is replaced by a frequency axis. In the frequency domain, sounds are represented in a frequency by amplitude and/or phase diagram.

## Line Spectra, Harmonic Spectra, and Continuous Spectra

The next figure is a frequency domain representation of the 200 Hz sine wave we saw in the first figure. In the frequency domain, this sound is represented by a line at a point on the frequency axis corresponding to 200 Hz and with a length corresponding to its amplitude. Figures like this are called line spectra.

There are several things to note in this figure. First, the Y axis is labeled Amplitude rather than pressure because the axis now provides a measure of the strength of the pressure changes: neither absolute pressure, nor the direction of relative pressure change is represented. In fact, pressure need not be the physical measure on which amplitude is based here. With sound, we often measure the voltage fluctuations produced by a microphone rather than pressure per se. Consequently, amplitude is a better, more general, term. Second, note that the Amplitude axis has no values less that zero. In this spectral representation, called a magnitude spectrum amplitudes cannot be less than zero--it is not possible to have negative amounts of sound energy. A third feature to note is the labeling of the Frequency axis which is in units of Kilohertz or thousands of cycles per second.

One of the most convenient features of frequency domain representations of sound is that sounds of many different frequencies can be plotted simultaneously on the same figure. This figure, for instance, shows all of the components we used above to start an approximation to a square wave. In this figure, each line is one of the harmonics of the 200 Hz fundamental frequency of the square wave. The height of each harmonic line indicates the amplitude of the sinusoid at that frequency. This figure does not show us anything about the phase relationships among the harmonics which were obvious in the time-domain figures earlier. Try clicking on each line in the line spectrum; if you're careful where you click, you should hear a sine wave at the appropriate frequency and amplitude. Next click in the figure but not on one of the spectral lines; you should hear the complex sound which results from summation of the four spectral components in the figure. See if by listening carefully you can hear any of the individual tones in the complex sound.

Notice that the amplitude reduces very quickly with each successive harmonic in this spectrum. In fact, the apparent differences in amplitude are actually much larger than the differences we would hear when listening to each of these tones. In this next figure, amplitude is expressed in dB rather than in linear units. The amplitude relations among the harmonics expressed in dB are much closer to the loudness relations we hear among the harmonics. This figure doesn't play any tones: they'd be exactly the same as the last figure--only the scaling of the Amplitude axis is different--that's the point.

Line spectra exactly represent periodic signals like sine waves and square waves, but these are a special case in that sounds we encounter in nature are never truly periodic. First, most sounds we encounter are bounded in time and/or may be periodic only within certain temporal bounds. Further, many important sounds like the voiced sounds of speech are only approximately periodic since they vary slightly from one period to the next. We refer to these sounds as quasiperiodic. Let's take another look at the spectrum of the four-harmonic square wave approximation we've been using, but this time treating it in the way sounds are most often actually handled for study in the laboratory. First, because we are normally interested in looking at the spectrum of a sound at a particular point in time, we will apply what's called an analysis window to the sound. This makes the sound fade in and back out again gradually. When we first window the sound and then determine its frequency components, we get this kind of a figure. The axes and frequency scale are the same as the previous figure, but the amplitude scale is different in this figure. Previously, the amplitude scale was set to arbitrary units, but now, amplitude is based on the units used in the digitized and windowed sound.

The most important (and probably most obvious) difference between this figure and the last however is that the harmonic lines now look like pointed bars. These are still called harmonics, but they no longer represent pure tones, instead, they represent the presence of sound energy at many frequencies quite close to the true harmonic frequencies. If you listen to the sounds underlying this figure by clicking on the harmonics or outside the harmonics to hear the complex tone, you'll hear the way the tones fade in and out rather than starting and ending abruptly. We call spectra like this harmonic spectra rather than line spectra.

The difference between line spectra and the broader bars of harmonic spectra illustrates an important general difference between sounds represented in the time domain and in the frequency domain. Sounds which extend for long times and with great consistency in the time domain have very narrow profiles in the frequency domain. A sinusoid extending forever at a fixed frequency has the narrowest possible profile (a line) in the frequency domain. On the other hand, sounds which are narrowly defined in time, that is, have a brief temporal extent, exhibit a broader frequency profile. Thus, sinusoids which fade in and out as in the last example, have a broader distribution in frequency.

Carrying this trend to its logical conclusion, the shortest possible sound (a single pressure spike; like a hand clap but even shorter in duration) would have the broadest possible frequency profile. In fact, a pure impulse sound (i.e., a sound that is of zero amplitude at all times except for one infinitesimal instant when its amplitude is non-zero) would spread out in frequency to the point of having a perfectly flat spectrum. Of course, this would no longer be called a harmonic spectrum, it would be a continuous spectrum. Continuous spectra are associated with sounds that are not periodic, that is, with aperiodic sounds. An impulse is the paradigm exemplar of an aperiodic sound, but other more commonly encountered aperiodic sounds are the hissing sounds of fricatives in speech, and generally any sounds which do not have an identifiably tonal quality.

To summarize, we have discussed three kinds of spectra:

Line Spectra
Associated with strictly periodic signals or sounds that are (at least theoretically) unbounded in time.
Harmonic Spectra
Associated with quasiperiodic sounds or signals that are bounded in time.
Continuous Spectra
Associated with aperiodic sounds.

Before finishing with this discussion of sound represented in the frequency domain, let's look at two more spectra. These are associated with actual speech sounds.

## Spectra of Speech Sounds

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