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Z transforms

The z-transform is defined by:


 eqnarray621

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


eqnarray627

and evaluating this at a particular point, e.g. z = i/2
eqnarray632

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 tex2html_wrap_inline3001.

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:
eqnarray644
Where C is a closed contour that includes z = 0.

The z transform is a linear transform, i.e.
eqnarray649

So, if y(n) is the convolution of two signals, h(n) and x(n), i.e.:
eqnarray651
then
eqnarray655

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

The general linear filter is expressed as:
eqnarray655
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:
eqnarray660

Similarly for the IIR filter:
eqnarray668

This is useful as H(z) can be factored:
eqnarray676

From this equation it can be seen that if tex2html_wrap_inline3039 then the filter will have zero response - these are the ``zeros'' of the linear system.

Similarly, tex2html_wrap_inline3041 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.

  figure686
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 tex2html_wrap_inline2855 and tex2html_wrap_inline3049 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:


eqnarray693

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

  figure706
Figure 35: The cascade form for a linear filter

It is also possible to expand H(z) in terms of partial fractions:
eqnarray713

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

  figure725
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).


next up previous contents
Next: Linear Prediction analysis Up: Speech Analysis Previous: The Autocorrelation from the

Speech Vision Robotics group/Tony Robinson