The *z*-transform is defined by:

The sequence, *x*(*n*) is known and *z* is a complex number. Hence *X*(*z*)
is just a weighted sum. For example, for the sequence:
*x*(0) = 1, *x*(1) = 3, *x*(2) = 3, *x*(3) = 1 and *x*(*n*) = 0 otherwise

and evaluating this at a particular point, e.g. *z* = *i*/2

Only defined for values of *z* where the series converges.

That is, *z*-transform is the general version of the discrete Fourier
transform. To obtain the Fourier restrict *z* to lie on the unit circle
.

There are several ways of obtaining the inverse *z* transform:

- a)
By inspection: if
*X*(*z*) can be written as a simple polynomial in*z*then the time domain sequence is the coeffients of the polynomial - b)
By expansion: expanding
*X*(*z*) as a polynomial in*z* - c)
By decomposition: breaking up
*X*(*z*) into parts whose inverse*z*transforms are known (e.g. see table 3.1 in [4]) - d)
By definition: the inverse transform is defined by:

Where*C*is a closed contour that includes*z*= 0.

The *z* transform is a linear transform, i.e.

So, if *y*(*n*) is the convolution of two signals, *h*(*n*) and *x*(*n*), i.e.:

then

The linear filters of section 2.2 can now be
expressed in terms of *z*-transforms.

The general linear filter is expressed as:

where *H*(*z*) is called the ``system function'' and is the *z*-transform
of the unit sample response.

For the FIR filter of order *q*:

Similarly for the IIR filter:

This is useful as *H*(*z*) can be factored:

From this equation it can be seen that if then the filter will have zero response - these are the ``zeros'' of the linear system.

Similarly, defines the ``poles'' of the linear
system. When *q* = 0, as in linear prediction, we have an ``all pole''
filter.

For a stable system, all the poles must lie within the unit circle.

**Figure 34:** An argand diagram showing a stable pole-pair within the unit circle

An unstable system is one whose output is unbounded in response the unit impulse.

Manipulation of the form of *H*(*z*) allows many different implementations.
For example, as the coefficients and are real, the poles and
zeros occur in complex conjugate pairs. By grouping these together
*H*(*z*) can be expressed in terms of second order sections:

This ``cascade form'' is illustrated in figure 35.

**Figure 35:** The cascade form for a linear filter

It is also possible to expand *H*(*z*) in terms of partial fractions:

This ``parallel form'' is illustrated in figure 36.

**Figure 36:** The parallel form for a linear filter

Both forms are popular in speech synthesis - indeed the Klatt synthesiser has both a parallel and a cascade path (for ease of specifying the coefficients I assume).