The z-transform - Continuous and Discrete Time Signals and Systems - page 613

Fig. 13.11. Location of poles and
zeros obtained in Example 13.21
using M ATLAB

4

3

2

1

0

−1

−2

−3

−4

−6 −4 −2 0

real part

2 4

13.11.3 Computing the z-transfer function from poles and zeros

M ATLAB provides the built-in function zp2tf to calculate the z-transfer
function from poles and zeros. In terms of Eq. (13.49), the syntax for zp2tf
is given by

>> [B,A] = zp2tf(Z,P,K); % Calculate poles and zeros

where vector Z contains the location of the zeros, vector P contains the location
of the poles, and K is a scalar providing the gain of the numerator. The numerator
coefficients are returned in B and the denominator coefficients in A .

Example 13.22
Consider the poles and zeros calculated in Example 13.21. Using the values of
the poles and, zeros and the gain factor, determine the transfer function H ( z ).

Solution
The M ATLAB code to determine the coefficients of the transfer function is
listed below. The explanation follows each instruction in the form of comments.

>> Z = [0; -5.666667]; % Zeros in a column vector
>> P = [3+4 * j; 3-4 * j; 1]; % Poles in a column vector
>>K=6;

%Gain of the numerator

>> [B,A] = zp2tf(Z,P,K); % Calculate poles and zeros

The returned values are given by

B=[06340]andA=[1-731-25],

which implies that the transfer function is given by

6 z 2 + 34 z
z 3 − 7 z 2 + 31 z − 25 .

H ( z ) =