Audio Engineering

456 Chapter 15

the “ ringing ” on the waveform is simply the consequence of band-limiting a square wave,
that simple, minimum phase systems will produce an output rather like Figure 15.11(b),
and there is not much evidence to show that the effect is audible. The concept of building
up complex wave shapes from simple components is used in calculating the shape of

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Figure 15.11(a) : Composing a square wave from the harmonics is an elementary example of

a Fourier series. For the square wave of unit amplitude the series is of the form.

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Figure 15.11(b) : In practice, a truly symmetrical shape is rare, as most practical methods

of limiting the audio bandwidth do not exhibit linear phase, but delay progressively the

higher frequency components. Band-limiting niters respond to excitation by a square wave

by revealing the effect of the absence of higher harmonics and the so-called “ ringing ” is thus

not necessarily the result of potentially unstable fi lters.