Z transforms

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).