Page 31 - Master Handbook of Acoustics

Speech and music wave shapes depart radically from the simple sine wave, and are considered as
complex waveforms. However, no matter how complex the waveform is, as long as it is periodic, it
can be reduced to sine components. The obverse of this states, that any complex periodic waveform
can be synthesized from sine waves of different frequencies, different amplitudes, and different time

relationships (phase). Joseph Fourier was the first to prove these relationships. The idea is simple in
concept but often complicated in its application to specific speech or musical sounds. Let us see how
a complex waveform can be reduced to simple sinusoidal components.

Harmonics

A simple sine wave of a given amplitude and frequency, f , is shown in Fig. 1-9A. Figure 1-9B
1
shows another sine wave f that is half the amplitude and twice the frequency. Combining A and B at
2
each point in time, the wave shape of Fig. 1-9C is obtained. In Fig. 1-9D, another sine wave f3 that is
half the amplitude of A and three times its frequency is shown. Adding this to the f + f wave shape
2
1
of C, Fig. 1-9E is obtained. The simple sine wave of Fig. 1-9A has been progressively changed as
other sine waves have been added to it. Whether these are acoustic waves or electronic signals, the
process can be reversed. The complex waveform of Fig. 1-9E can be disassembled, as it were, to the
simple f , f , and f3 sine components by either acoustical or electronic filters. For example, passing
1
2
the waveform of Fig. 1-9E through a filter permitting only f and rejecting f and f3, the original f 1
2
1
sine wave of Fig. 1-9A emerges in pristine condition.