# Z-transform and Inverse Z- transform Examples

Z-transform & Inverse Z-transform  Examples

Example :01

x(z)   =     _______18z^3_______

18z^3 + 3z^2 -4z – 1

CODE:

b = [18 0 0 0];

a = [18 3 -4 -1];

[r p k] = residuez(b,a);

u = ones(1,10);

n = 0:9;

f = (0.3600.*0.5000.^n).*u + (0.2400.*-0.3333.^n).*u + (0.4000.*-0.3333.^n).*u;

subplot(2,1,1)

stem(n,f);

subplot(2,1,2)

impz(b,a);

OUTPUT:

Example :02

x(z)   =     _______1 – 2z^-1_______

(1 – 0.2z^-1)( 1+0.6z^-1)

CODE:

b = [1 -2 0];

a = [1 0.4 -0.12];

[r p k] = residuez(b,a)

u = ones(1,18);

n = 0:17;

f = (3.2500*-0.6000.^n).*u + (–2.2500.*0.2000.^n).*u

subplot(2,1,1)

stem(n,f);

subplot(2,1,2)

impz(b,a);

OUTPUT:

Example :03

Determine the Inverse z-transform of the following sequences, using partial fraction expansion method.

x(z)   =     _______1 – z^-1 – 4z^-2 + 4z^-3_______

1 – 2.75z^-1 + 1.625z^- 2 – 0.25z^-3

CODE:

b = [1 -1 -4 4];

a = [1 -2.75 1.625 -0.25];

[r p k] = residuez(b,a)

u = ones(1,18);

n = 0:17;

f = (0*2.^n).*u + (-10.*0.5.^n).*u + (27.*0.25.^n).*u;

subplot(2,1,1)

stem(n,f);

subplot(2,1,2)

impz(b,a);

OUTPUT:

Example :04

A digital filter is described by the difference equation,

Y(n) = x(n) + x(n-1) + 0.9y(n-1) – 0.81y(n-2)

Using the freqz function, plot the magnitude and phase of the frequency response of the filter.

CODE:

b = [1,1];

a = [1, -1, -0.9, 0.81];

w = [0:1:500]*pi/500;

H = freqz(b,a,w);

magH = abs(H);

phaH = angle(H)*180/pi;

subplot(2,1,1)

stem(w/pi,magH);

title(‘Magnitude’);

subplot(2,1,2)

stem(w/pi,phaH);

title(‘Phase’);

OUTPUT:

Example :05

x(z)   =     _______z^-1_______

1 – 2z^-1 + 3z^ – 2

CODE:

b = [0 1 0];

a = [1 -2 3];

[r p k] = residuez(b,a)

u = ones(1,30);

n = 0:29;

f = (0-0.356j.*(1.000+1.4142j).^n).*u + (0+0.356j.*(1.000-1.4142j).^n).*u

subplot(2,1,1)

stem(n,f);

subplot(2,1,2)

impz(b,a);

OUTPUT:

Example :06

x(z)   =     _______2 – 3z^-1_______

1 – 5z^-1 + 6z^ – 2

CODE:

b = [2 -3 0];

a = [1 -5 6];

[r p k] = residuez(b,a)

u = ones(1,10);

n = 1:10;

f = (3.*3.^n).*u + (-1.*2.^n).*u;

subplot(2,1,1)

stem(n,f);

subplot(2,1,2)

impz(b,a);

OUTPUT:

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