Audio Engineering

806 Chapter 27

variable that can take on real and imaginary (i.e., complex) values. When the

z -transform is used to describe a digital signal, or a digital process (like a digital

fi lter), the result is always a rational function of the frequency variable z. That’s to

say, the z -transform can always be written in the form:

Xz

Nz

Dz

Kz z

zp

()

()

()

()

()

 ,





1

1

where the z’s are known as “ zeros ” and the ‘ p’s are known as “ poles. ”

A very useful representation of the z -transform is obtained by plotting these poles

and zeros on an Argand diagram; the resulting two-space representation is termed

the “ z -plane. ” When the poles and zeros are plotted in this way, they give us a very

quick way of visualizing the characteristics of a signal or digital signal process.

Problems With Digital Signal Processing

Sampled systems exhibit aliasing effects if frequencies above the Nyquist limit are

included within the input signal. This effect is usually no problem because the input

signal can be fi ltered so as to remove any offending frequencies before sampling takes

place. However, consider the situation in which a band-limited signal is subjected to

a nonlinear process once in the digital domain. This process might be as simple as a

“ fuzz ” -type overload effect, created with a plug-in processor. This entirely digital process

generates a new large range of harmonic frequencies (just like its analogue counterpart),

as shown in Figure F27.6. The problem arises that many of these new harmonic

F

A

Figure F27.6 : Generation of harmonics due to nonlinearity