Audio Engineering

Representation of Audio Signals 453

through the analysis of much simpler waveforms. We rely on the straightforward principle
of superposition of waveforms such as the simple cosine wave.

On its own an isolated cosine wave, or real signal, has no phase. However, from a
mathematical point of view the apparently simple cosine wave signal, which we consider
as a stimulus to an electronic system, can be considered more properly as a complex
wave or function that is accompanied by a similarly shaped sine wave ( Figure 15.9 ). It is
worthwhile throwing out an equation at this point to illustrate this:

ft()Re j Imft() ft()

where f ( t ) is a function of time, t , which is composed of Re f ( t ), the real part of the
function and j Imf ( t ), the imaginary part of the function and j is  1.

Re

Re

t

t

t

Im

Im

eJWT Imf (

t)

Ref (

t)

Complex plane

Imaginary plane

Real plane

Figure 15.9 : The relationship between cosine (real) and sine (imaginary) waveforms in the

complex exponential eJWT. This assists in understanding the concept of phase. Note that

one property of the spiral form is that its projection onto any plane parallel to the time axis

will produce a sinusoidal waveform.