Representation of Audio Signals 455Such behavior can produce gross waveform distortion as might be imagined if a 2-kHz
component were to emerge 2 ms later than a 1-kHz signal. In most simple circuits, such
as mixing desk equalizers, the output phase of a signal with respect to the input signal is
usually the ineluctable consequence of the equalizer action. However, for reasons which
we will come to, the process of digitizing audio can require special fi lters whose phase
response may be responsible for audible defects.
One conceptual problem remains. Up to this point we have given examples in which the
output phase has been given a negative value. This is comfortable territory because such
phase lag is converted readily to time delay. No causal signal can emerge from a system
until it has been input, as otherwise our concept of the inviolable physical direction of time
is broken. Thus all practical systems must exhibit delay. Systems that produce phase lead
cannot actually produce an output that, in terms of time, is in advance of its input. Part
of the problem is caused by the way we may measure the phase difference between input
and output. This is commonly achieved using a dual-channel oscilloscope and observing
the input and output waveforms. The phase difference is readily observed and can be
readily shown to match calculations such as that given in Figure 15.10. The point is that
the test signal has essentially taken on the characteristics of a signal that has existed for
an infi nitely long time exactly as it is required to do in order that our use of the relevant
arithmetic is valid. This arithmetic tacitly invokes the concept of a complex signal, which is
one which, for mathematical purposes, is considered to have real and imaginary parts (see
Figure 15.9 ). This invocation of phase is intimately involved in the process of composing,
or decomposing, a signal using the Fourier series. A more physical appreciation of the
response can be obtained by observing the system response to an impulse.
Since the use of the idea of phase is much abused in audio at the present time, introducing
a more useful concept may be worthwhile. We have referred to linear phase systems
as exhibiting simple delay. An alternative term to use would be to describe the system
as exhibiting a uniform (or constant) group delay over the relevant band of audio
frequencies. Potentially audible problems start to exist when the group delay is not
constant but changes with frequency. The deviation from a fi xed delay value is called
group delay error and can be quoted in milliseconds.
The process of building up a signal using the Fourier series produces a few useful insights
[ Figure 15.11(a) ]. The classic example is that of the square wave and it is shown in Figure
15.11(a) as the sum of the fundamental, third and fi fth harmonics. It is worth noting that