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OT3OS1 19.10.2010.
OT3OS1
19.10.2010.
Z transformacija1
n
nznxzX
jez
j
n
njj
zXeX
enxeX
Veza z transformacije X(z) i
Furijeove transformacije X(ej)
Z transformacija2
N
kk
M
kk
N
kk
M
kk
M
kk
N
kk
knxbknya
knxbknyany
knyaknxbny
00
01
10
M
k
k
k
N
k
k
k
M
k
k
k
N
k
k
k
N
k
k
k
M
k
k
k
za
zb
za
zbzH
zbzXzazY
1
0
0
0
00
1
Z transformacija3
M
k
kM
k
M
N
k
kN
k
N
M
N
M
M
N
M
k
k
k
N
k
k
k
zaz
zb
z
zzH
z
z
z
z
za
zbzH
1
0
1
0
1
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Part
Imag
inar
y P
art
MATLAB
• Funkcija rootsnule=roots(b);
polovi=roots(a);
• Funkcija zplanefigure,zplane(nule,polovi)
figure,zplane(b,a)
nule i polovi – vektori kolone b i a – vektori vrste
Zadatak 1
Nacrtati impulsni i frekvencijki odziv za moving average filtar dužine 4.
Zadatak 1 – rešenje
Zadatak 1 – grafici
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
Real Part
Imag
inar
y P
art
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Primer 1
144
1
4
1
44
11
4
1
44
13
4
12
4
11
4
11
34
12
4
11
4
1
4
14
13
0
nynxnxny
nxnynxny
nxnxnxnxny
nxnxnxnxny
knxnyk
Primer 1 - MATLAB
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
Real Part
Imag
inar
y P
art
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
416
Real Part
Imag
inar
y P
art
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
!!! impz “vraća”kolonu!!!
Primer 1 - MATLAB
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
Real Part
Imag
inar
y P
art
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
Real Part
Imag
inar
y P
art
Zadatak 1
Dat je impulsni odziv sistema:
Odrediti diferencnu jednačinu sistema.
34.0 nunh n
Zadatak 1 - rešenje
1
3
0
131
3
1
3
1
4.01
064.0
4.04.0
4.04.0
34.0
z
z
zz
zz
znhzH
nunh
k
k
n
n
n
nn
n
n
1
33
2
33
2
3
1
21
2
1
3
1
3
3
1
33
4.01
4.0
4.0
3
4.0
4.0
4.0
34.0
34.04.0
34.0
z
z
zHzzH
zHzzH
nhnh
nunh
zHzH
nhh
nunh
nunh
nunh
p
pp
pp
n
p
p
p
n
p
n
n
q
aS
qaSk
k
10
00
Linearnost
Kašnjenje
Zadatak 1 - MATLAB
14.03064.0
4.01
064.0
34.0
1
3
nynxnyz
zzH
nunh n
4.0064.04.01
064.0
2
3
3
1
3
zzzH
z
z
z
zzH
Zadatak 1 - MATLAB
close all, clear;b=[0 0 0 0.064];a=[1 -0.4];figure,zplane(b,a);[H,w]=freqz(b,a);figure,plot(w/pi,abs(H));zx=-1.2:0.001:1.2;zy=-1.2:0.002:1.2;[zxm,zym]=meshgrid(zx,zy);z=zxm+1i*zym;Hz=0.064*z.^(-3)./(1-0.4*z.^(-1));figure,mesh(zxm,zym,20*log10(abs(Hz))),xlabel('x'),ylabel('y'),xlim([-1.2 1.2]),ylim([-1.2 1.2]);figure,mesh(zxm,zym,20*log10(abs(Hz))),xlabel('x'),ylabel('y'),xlim([-1.2 1.2]),ylim([-1.2 1.2]),view([0 90]);
Zadatak 1 - MATLAB
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
2
Real Part
Imag
inar
y P
art
Zadatak 1 - MATLAB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Zadatak 1 - MATLAB
Zadatak 1 - MATLAB
Zadatak 2
Dat je sistem opisan diferencnom jednačinom:
Odrediti impulsni odziv sistema.
18.015.0 nynxnxny
Zadatak 2 - rešenje
?
8.01
5.01
5.018.01
15.018.0
18.015.0
1
1
11
nh
z
z
zX
zYzH
zzXzzY
nxnxnyny
nynxnxny
nunnhz
z
z
z
zzH
n8.0375.08.01
1375.0625.0
8.01
1
25.1
75.01625.0
25.1
75.01625.0
25.1
2
8.0
5.0
1
1
1
1
1
Zadatak 2 - MATLABN=50;n=[0:N-1];x=zeros(N,1);x(1)=1;hML(1)=x(1);for brojac=2:N hML(brojac)=x(brojac)-0.5*x(brojac-1)+0.8*hML(brojac-1);end;h=0.375*0.8.^n;h(1)=h(1)+0.625;stem(n,[hML;h]'),xlabel('n'),ylabel('h(n)'),legend('h_{ML}','h');
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n
h(n)
hML
h
Zadatak 2 - MATLAB
close all, clear;b=[1 -0.5];a=[1 -0.8];figure,zplane(b,a);[H,w]=freqz(b,a);figure,plot(w/pi,abs(H));zx=-1.2:0.001:1.2;zy=-1.2:0.002:1.2;[zxm,zym]=meshgrid(zx,zy);z=zxm+1i*zym;Hz=(1-0.5*z.^(-1))./(1-0.8*z.^(-1));figure,mesh(zxm,zym,20*log10(abs(Hz))),xlabel('x'),ylabel('y'),xlim([-1.2 1.2]),ylim([-1.2 1.2]);figure,mesh(zxm,zym,20*log10(abs(Hz))),xlabel('x'),ylabel('y'),xlim([-1.2 1.2]),ylim([-1.2 1.2]),view([0 90]);
Zadatak 2 - MATLAB
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Part
Imag
inar
y P
art
Zadatak 2 - MATLAB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Zadatak 2 - MATLAB
Zadatak 2 - MATLAB
Zadatak 2 - MATLAB
Zadatak 2 - MATLAB