454 Chapter 15
Emergence of the 1 is the useful part here because you may recall that analysis of
the simple analogue circuit ( Figure 15.10 ) involving resistors and capacitors produces an
expression for the attenuation and the phase relationship between input and output of that
circuit, which is achieved with the help of 1.
The process that we refer to glibly as the Fourier transform considers that all waveforms
can be considered as constructed from a series of sinusoidal waves of the appropriate
amplitude and phase added linearly. A continuous sine wave will need to exist for
all time in order that its representation in the frequency domain will consist of only a
single frequency. The reverse side, or dual, of this observation is that a singular event,
for example, an isolated transient, must be composed of all frequencies. This trade-off
between the resolution of an event in time and the resolution of its frequency components
is fundamental. You could think of it as if it were an uncertainty principle.
The reason for discussing phase at this point is that the topic of digital audio uses terms
such as linear phase, minimum phase, group delay, and group delay error. A rigorous
treatment of these topics is outside the scope for this chapter but it is necessary to describe
them. A system has linear phase if the relationship between phase and frequency is a
linear one. Over the range of frequencies for which this relationship may hold the systems,
output is effectively subjected to a constant time delay with respect to its input. As a simple
example, consider that a linear phase system that exhibits –180° of phase shift at 1 kHz will
show –360° of shift at 2 kHz. From an auditive point of view, a linear phase performance
should preserve the waveform of the input and thus be benign to an audio signal.
Most of the common analogue audio processing systems, such as equalizers, exhibit
minimum phase behavior. Individual frequency components spend the minimum
necessary time being processed within the system. Thus some frequency components of a
complex signal may appear at the output at a slightly different time with respect to others.
RV 1 C VoFigure 15.10 : The simple resistor and capacitor attenuator can be analyzed to provide us with
an expression for the output voltage and the output phase with respect to the input signal.