# Sound in the Time Domain

## Amplitude, Frequency, and Phase

Sound is perceived when fluctuations in air pressure cause structures inside our ears to vibrate. These air pressure fluctuations can be quite small or large and can occur slowly or rapidly. We refer to the rate at which pressure fluctuates cyclically from higher to lower to higher and so forth as its frequency. Typically we express frequency in cycles per second or equivalently Hertz. The following figure is a graph of two "cycles" of fluctuation. This figure shows the Amplitude of air pressure variations relative to mean air pressure (in no particular units) as a function of Time (expressed in milliseconds or thousandths of a second). Thus, 0 on the Pressure scale corresponds to the mean air pressure. In this figure the pressure starts at the average air pressure, increases to a value of 100 at a time corresponding to about 1.25 msec, decreases to -100 at 3.75 msec and returns to zero at 5.0 msec before starting the second cycle. The length of each cycle in time is called the period of the waveform because the shape of the waveform repeats periodically at this interval. Since the period of this waveform is 5.0 msec, there would be 200 periods or cycles in one second. The frequency of this sound is thus 200 cycles per second or 200 Hertz (which we will abbreviate as Hz hereafter). More generally, the frequency of a periodic waveform is the inverse of its period; F = 1/P or in this example, 200 = 1.0 / 0.005. If you would like to hear what this 200 Hz waveform sounds like, click on the graph with your mouse or pointer.

In addition to the frequency of a sound, we can describe its amplitude. In general, small variations in pressure produce weak (or quiet) sounds while large variations produce strong (or loud) sounds. The next figure shows another sound which is lower in amplitude that the previous example because the pressure varies less extremely over time. This figure shows a sound which also differs in frequency from the sound illustrated in the previous figure. Note that frequency and amplitude vary independently. Although the amplitude is lower in this figure, the pressure fluctuations are more rapid than in the previous figure; six cycles occur within ten msec so this tone has a frequency of 600 Hz. Consequently, this sound is higher in frequency but lower in amplitude than the sound depicted in the first figure.

One other property called phase is important in describing the physical properties of sound. To illustrate what is meant by phase, the next figure shows two 200 Hz sinusoids, one drawn with a solid line and the other drawn with a dotted line. The two sinusoids are identical except that they are differently aligned with respect to the time axis. These two sinusoids are said to differ in phase while having the same amplitude and frequency. This is a good moment to point out that the notion of `beginning' and `ending' needs some qualification here. The figures drawn on this page have waveforms which obviously begin and end within the limits of the graph. However, they represent snippets of functions which do not have beginning and ending points. Thus, the phase differences shown in the present figure do not reflect the notion that one function started at a different time than the other. Rather, the phase differences represent the way the two functions are aligned with respect to each other at all times, including those which lie outside the bounds of the present graph.

The physical properties of amplitude, and frequency correspond to the sensory/perceptual qualities of loudness and pitch. It is often useful to maintain a clear distinction between the physical properties of sound and the perceptual correlates of those properties. For one thing, the perceptual domains of pitch and loudness are bounded by the capabilities of our auditory systems whereas the physical properties of sound are not. The normal young human auditory system is sensitive to a range of frequencies from about 20 Hz to about 20,000 Hz. The amplitude range is substantially broader, beginning at a level so low that we can almost "hear" the fluctuations in air pressure due to random motion of air molecules near the ear drum and extending to the threshold of pain at about 10 million times that level.

A second important difference between the perceptual properties of sound and its physical properties is that even within the bounds of the perceptual system, the relationship between the perceived and physical properties of sound is generally non-linear. For example, if we increase the amplitude of a sound in a series of equal steps, the loudness of the sound will increase in steps which seem successively smaller. Similarly, increasing the frequency of a sound in equal steps will lead to perceived increases in pitch that seem to grow smaller with each step. Here's an example. Click on any of the following numbers to hear a tone of the corresponding frequency. Note that as you go through these tones in 25 Hz steps, the steps sound like they are getting closer together. For instance, compare the step between 200 and 225 Hz with the step from 400 to 425 Hz. The step from 200 to 225 sounds larger than the step between 400 and 425 Hz even thought both are exactly 25 Hz.

We often describe sounds using scales that reflect equal perceptual differences. For frequency, one such scale is the Mel scale. Equal Mel steps will correspond to equal changes in pitch, but not equal changes in frequency. Similarly, for loudness, it is most convenient to describe sound over the enormous range of perceptible amplitudes in logarithmic units called Decibels and abbreviated dB. On the decibel scale, 0.0 dB corresponds to about the normal threshold of hearing and 130 dB to the point at which sound becomes painful. Moreover, each 1 dB step corresponds to approximately a Just Noticeable Difference in loudness, that is, the smallest change in loudness that is noticeable about 50% of the time.

The third physical property of sound, its phase is less directly related to perceived sound quality. In most work related to speech perception, phase is entirely disregarded. However, phase is important in describing how complex sounds can be constructed from the simple sinusoidal sounds we've discussed so far.

## Simple versus Complex Sound

Despite their differences in amplitude and frequency, the sounds shown and heard above depict simple sounds because the pressure fluctuations associated with these sounds are sinusoidal. That is, the pressure variations over time follow the form of a trigonometric sine or cosine function. Most sounds in nature are not so simply described; their shape, rather than being sinusoidal, is of some other form, typically one for which we have no name. Fortunately, it turns out that such complex sounds can be described mathematically as combinations of simple sounds. Consider for example, the sound illustrated in the next figure which simply alternates between a region of constant high pressure and a region of constant low pressure. This particular waveform does have a name, it is called a square wave because of its boxy shape. This square wave is very similar to the 200 Hz sine wave shown in the first figure in that it too repeats a single pattern two hundred time a second. Moreover, (if you haven't already listened to it, you should now) it has the same pitch as the 200 Hz sine wave, but a different timbre.

This complex square wave can be described as the summation of a set of simple sinusoids. In other words, the square wave can be formed by adding together sinusoids of the appropriate amplitude, frequency, and phase. Not surprisingly, the first and strongest sinusoid needed to form the square wave in our example is a sine wave of 200 Hz. This first component corresponds to what is called the Fundamental Frequency (hereafter abbreviated as F0) and is the frequency which gives rise to the pitch we normally hear when listening to a complex sound. Thus, the common F0 accounts for the pitch similarity between the 200 Hz sine wave and the 200 Hz square wave. To construct a square wave we need, in addition to a 200 Hz sine wave, a sequence of higher frequency sine wave components. The components in this sequence are called overtones or harmonics, and by definition, can only occur at integer multiples of F0. Since F0 in this example is 200 Hz, the harmonics can only occur at 400 Hz, 600 Hz, 800 Hz, and so forth. However, the square wave is a special case in that all of the even-multiple harmonics (i.e., the ones at 2F0, 4F0, 6F0, etc.) have zero amplitude so they contribute nothing to the shape of the square wave.

Using only the odd-numbered harmonics then, we can construct a square wave by adding sine waves at F0, 3F0, 5F0, and so forth. For our example 200 Hz square wave, this means we need sine waves at 200 Hz, 600 Hz, 1000 Hz, 1400 Hz, and on. In addition to having harmonics of the correct frequencies, the amplitude relations among the harmonics must be correct or we will not construct the desired waveform. For a square wave, the 3rd harmonic (at 600 Hz) should be 1/3 the amplitude of the fundamental. This is exactly the sinusoid shown in the second figure above. Here is the waveform that results from adding a 200 Hz sine wave with a 600 Hz sine wave at 1/3 the amplitude. Already, the combined waveform is beginning to take on some features of the square wave with a more extended portion near its most positive and negative values (albeit still with much fluctuation).

Continuing to build a square wave by adding sinusoids, the third component needed (the fifth harmonic at 1000 Hz) should be 1/5 the amplitude of the fundamental, and the fourth sine wave, corresponding to the 7th harmonic (at 1400 Hz) should be 1/7 the amplitude of the fundamental. These are shown in the next set of figures along with the square wave approximations when we sum all harmonics up to and including the given harmonic.

As you can see, with the addition of each subsequent harmonic, the complex waveform more nearly approaches the shape of a square wave. The addition of each higher frequency harmonic reduces the amplitude (and increases the frequency) of the small ripples in the more stationary parts of the square wave. To achieve the shape of a true square wave with absolutely no ripple in the stationary parts would require the summation of an infinite number of sinusoids. But we don't have time for that in the present tutorial.