DTFT, DFT, ZT, and FFT
551
Example 5.16
Create the script fi le pole_zeroes that returns the rational transfer function H(z), the
numerator and denominator coeffi cients of H(z), and the fi rst 11 coeffi cients of their system impulse response h(n) given the following system’s zeros, poles, and gain.
z 1 = 0.21
p 1 = −0.45
z 2 = 3.14
p 2 = 0.67
z 3 = −0.3 + j5
p 3 = 0.81 + j0.72
z 4 = −0.3 − j5
p 4 = 0.81 − j0.72
k = 2.3
p 5 = −0.33
MATLAB Solution
% Script file: pole _ zeroes
z1 = 0.21 ; p1 =-0.45;
z2 = 3.14 ; p2 = 0.67;
z3 =-0.3 + 5j; p3 = 0.81+0.72j;
z4 = -0.3-5j; p4 = 0.81-0.72j;
k =2.3; p5 = 0.33;
zr = [z1 z2 z3 z4];
pl = [p1 p2 p3 p4 p5];
z = zr’; p=pl’;
[num,den] = zp2tf(z,p,k);
n =0:10;
disp (‘*************************************************’)
disp (‘**************** R E S U L T S ****************’)
disp (‘*************************************************’)
tf(num,den,1)
disp(‘********************’)
disp (‘Coefficients of the numerator polynomial of H(z)’); disp(num’);
disp(‘********************’)
disp (‘Coefficients of the denominator polynomial of H(z)’);
disp(den’);L=11; [y,t] = impz(num,den,L);
disp(‘********************’)
disp (‘The first 11 coefficients of the impulse response h(n) are:’);
disp(‘********************’)
disp(‘ coef. num.
h(n)’)
disp(‘********************’)
[n’ y]
disp (‘*************************************************’)
The script fi le pole_zeroes is executed, and the results are as follows:
>> pole _ zeroes
*********************************************************
****************R E S U L T S **************************
*********************************************************
Transfer function:
2.3 z^4 - 6.325 z^3 + 54.6 z^2 - 192.4 z + 38.05
------------------------------------------------------------z^5 - 2.17 z^4 + 1.837 z^3 - 0.1757 z^2 - 0.43 z + 0.1169
Sampling time: 1
***************************
Coefficients of the numerator polynomial of H(z)
CRC_47760_CH005.indd 551
CRC_47760_CH005.indd 551
7/23/2008 1:47:42 PM
7/23/2008 1:47:42 PM