Question

ot Utb mode to r tu follouin ustows plk -2 t굿 2 2) A Gosal doort LTI Gitn has the trste unction (2+1)" C2-1)2 H(2): 0.976 d) (Use ta locations 아 the poles and 2005 to kteh tae b) plot the ma tude and frase repens (Zp2tf) fretz)

Answer 1

SOLUTION:

1.

(a)The transfer function

H(z)=

2 + ~-1-0.5~--+ 0.25:-3

multiplying the numerator and denominator by $z^{3}$ we have

H(z)=

2~3 之2-0.5: 0.25

MATLAB code for pole zero map

%%-----------CODE -------------%%

num=[1 0 1 0];
den=[2 1 -.5 .25];
zplane(num,den)

%%----------- END --------------%%

plot:

5 5 18642024681 2 ued lueubeul

(b)the transfer function

H(z) =

multiplying the numerator and denominator by $z^{3}$ we have

23 21.5z 0.25 231.522 +0.52 H(z) =

%%----------- MATLAB CODE ---------------%%

num=[1 1 1.5 0.25];
den=[1 1.5 0.5 0];
zplane(num,den)

%%------------------ END ----------------------%%

plot:

5 5 0 5 18642024681 ued lueubeul

(2)

(a)the gain is k=0.976

zeros are at z=1,1,-1,-1

poles are at p=0.3575+0.5889j, 0.3575-0.5889j, 0.7686+0.3338j, 0.7686-0.3338j

%%--------- MATLAB CODE -------------%%

%zeros
z=[1;1;-1;-1]
%poles
p=[0.3575+0.5889j; 0.3575-0.5889j; 0.7686+0.3338j; 0.7686-0.3338j]
%gain
k=0.976
%numerator and denominator from pole,zero and gain
[num,den]=zp2tf(z,p,k)
%taking the time period = 1s
T=1
%transfer function
H=tf(num,den,T)
%frequency response
freqz(num,den)

%%--------------- END -----------------%%

the transfer function

   0.976 z^4 - 1.952 z^2 + 0.976
H(z)= --------------------------------------------------------------
   z^4 - 2.252 z^3 + 2.276 z^2 - 1.232 z + 0.3333

PLOT:

50 o -50 -100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 Normalized Frequency (xI rad/sample) 200 100 -100 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 Normalized Frequency (xI rad/sample)

N.B:Copy the code,save in a mfile and run in matlab simulator to have the best results.