Beats and Amplitude Modulation

So far, we have been talking about the harmonic content of a single complex tone. What we've discovered is that a complex periodic sound can be decomposed using a Fourier analysis as a superposition of pure tones from the harmonic series. In general, the sounds that we hear are combinations of either pure or complex tones. There are new phenomena, both in the sound waves themselves and also in our perceptions, which occur when we combine two different tones. Today we'll talk about one of these phenomena when two pure tones of nearly the same frequencies are sounded together.


Beats

The figure above shows the pressure variation of two pure tones with slightly different frequencies but the same amplitudes. The lower frequency wave has a longer time between crests than does the higher frequency wave. Because of this, there are periods of time where the two waves are "in phase" and periods of time where the two waves are "out of phase". For the latter, this means that the crest of one wave overlaps with the trough of the other and vice versa. This occurs near t = 10 and 30 ms. The waves are "in phase" when the crests (troughs) of the two waves overlap (at times t = 0, 25 and 50 ms). From what we have seen previously, when the two waves are "in phase" there is constructive interference, meaning the amplitudes of the two waves add to make a wave with twice the amplitude (and hence, four times the intensity) of the individual pure tones. When the two waves are "out of phase" there is destructive interference, meaning there is very little amplitude for the resulting wave formed by superposition.

The superposition of two pure tones of slightly different frequencies (f1 and f2) is shown above. This wave is formed by simply adding together the pressure differences from the two pure tone waves. The consequence of the two pure tones alternating between being "in phase" and "out of phase" is evident by the modulation of the amplitude of the superposed wave. If the frequency difference between the two pure tones is small enough, we can hear the loudness of the superposed wave vary with time. This loudness variation is known as beats. We hear a pure tone with a pitch given by the average of the frequencies of the two tones: (f1 + f2) / 2. The amplitude variation occurs at the beat frequency, given by the difference between the two pure tone frequencies: fbeat = f1 - f2.

Amplitude Modulation

From the above discussion, it should be clear that the superposition of two pure tones with slightly different frequencies results in an amplitude modulation of the superposed wave. Amplitude Modulation (AM) is one of the two basic ways that radio signals are transmitted (the other is frequency modulation or FM). In this case, rather than having sound waves, AM radio uses electromagnetic waves to transmit information. The way AM radio works is that there is a carrier wave of frequency, fc, which is the basic frequency you tune your radio to. Sounds are converted into electric oscillations of the same frequency (fssamplitude of the carrier wave. (This is somewhat similar to turning up and down the volume control of your radio at a frequency fsbeat phenomena, except that amplitude modulation actually corresponds to the superposition of three waves: one at the carrier frequency, fc; one at the frequency fc - fs; and one at the frequency fc + fs. The latter two frequencies are called side bands. The quality of AM radio is limited by the frequency range allocated by the FCC for different radio stations. AM radio is permitted to transmit audio frequencies only up to 5000 Hz. This does not match the full range of audible frequencies (the maximum audible frequency is roughly 20,000 Hz) and so AM radio sound quality is not very good.

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Last updated: 17 Nov 1999
Comments: bland@indiana.edu