Digital filters - Continuous and Discrete Time Signals and Systems - page 657

In Chapter 13, we introduced a M ATLAB M-file residuez for the partial
fraction expansion of a given rational function. Similarly, the M-file tf2zp
was introduced to calculate the location of poles and zeros for a given transfer
function. These M-files can also be used to derive the cascaded and parallel
forms of the transfer function. We illustrate the application of these M-files by
deriving the cascaded and parallel forms for the transfer function,

z 3 − 2 z 2 + z
z 3 − 0 . 1 z 2 − 0 . 07 z − 0 . 065

− 2
1 − 0 . 1 z − 1 − 0 . 07 z − 2 − 0 . 065 z − 3 ,

− 1 + z

1 − 2 z

H ( z ) = =

considered in Example 14.5.

14.9.1 Parallel form

The M ATLAB code to determine the partial fraction expansion is given below.
The explanation follows each instruction in the form of comments.

>>B=[1 − 2 1 0]; % Coefficients of the
% numerator of H(z)
>>A=[1 − 0.1 -0.07 − 0.065]; % Coefficients of the
% denominator of H(z)
>> [R, P, K] = residuez(B, A); % Calculate partial
% fraction expansion

The returned values are given by

R = [0.4310 0.2845+3.3362j 0.2845 − 3.3362j]
P = [0.5000 − 0.2000+0.3000j − 0.2000 − 0.3000j] andK=0.

The transfer function H ( z ) can therefore be expressed as follows:

0 . 4310
1 − 0 . 5 z − 1

0 . 2845 + j3 . 3362
1 − ( − 0 . 2 + j0 . 3) z − 1

0 . 2845 − j3 . 3362
1 − ( − 0 . 2 − j0 . 3) z − 1 .

H ( z ) = + +

To eliminate complex-valued coefficients, we combine the complex poles as
follows:

− 1
1 + 0 . 4 z − 1 + 0 . 13 z − 2 .

0 . 4310
1 − 0 . 5 z − 1

0 . 5690 − 1 . 8879 z

H ( z ) = +

The partial fraction expansion is then implemented using the parallel form as
shown in Fig. 14.25.