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# Ecet 345 Week 6 Homework Essay

1000 words - 4 pages

ECET345 Signals and Systems
Homework #6
Name of Student: Odair Torres

Find the z-transform x(z) of x(n) = Cosâ¡ã€–(0.45n+0.25)u(n)ã€—. Hint: Follow the method used in the lecture for Week 6. Also, when evaluating the numerical value of a trig function, keep in mind that the arguments of trig functions are always in radians and not in degrees.

ã€–x(n)=Cosã€—â¡ã€–(0.45n+0.25)u(n)ã€—

ã€–x(n)=Cosã€—â¡ã€–0.45n cosâ¡0.25-sinâ¡ã€–0.45n sinâ¡0.25 ã€— nu(n)ã€—

ã€–X(z)=0.9689((1-cosâ¡ã€–0.45 z^(-1) ã€—)/(1-[2 cosâ¡0.45 ] z^(-1)+z^- ))ã€—â¡ã€–-0.247((1-[sinâ¡0.45 ] z^(-1))/(1-[2 cosâ¡ã€–0.45]z^(-1)+z^(-2) ã€— ))ã€—

Find the system ...view middle of the document...

x[n]=ã€–(0.25)ã€—^n sinâ¡ã€–(Ï€/2 n)u[n-1]ã€—
x[n]=(0.25)(0.25)^(n-1) sinâ¡ã€–(Ï€/2(n-1)) cosâ¡ã€–pi/2+cosâ¡[pi/2ã€— [n-1]]sinâ¡ã€–pi/2ã€— ã€—
x[n]=(0.25)[(0.25)^(n-1) cosâ¡ã€–(Ï€/2 (n-1) ) ã€—
H(z)=(0.25)[(1-0.25 ã€–cos(ã€—â¡ã€–pi/2)ã€— z^(-1))/(1-2(0.25) cosâ¡ã€–(pi/2) z^(-1)+ã€–0.25ã€—^(-2) z^(-2) ã€— )] z^(-1)
H(z)=(0.25)[(1-0.25 ã€–cos(ã€—â¡ã€–pi/2)ã€— z^(-1))/(1-2(0.25) cosâ¡ã€–(pi/2) z^(-1)+ã€–0.25ã€—^(-2) z^(-2) ã€— )] z^(-1)
H(z)=z/(z^2+16)

The transfer function of a system is given below. Find its impulse response in n-domain. Hint: First expand using partial fraction expansion and then perform its inversion using z-transform tables

H(z)= 1/((z-0.5)(Z+0.5))

A/(z-0.5) + B/(z+0.5) = 1/(z-0.5)(z+0.5)

(A(z-5)(z+5))/(z-0.5)+(B(z-5)(z+5))/(z+0.5) = 1
A(z+0.5)+B(z-0.5) = 1
z=-0.5
0+B(-0.5-0.5)=1
B=-1
z=0.5
A(0.5+0.5)+0=1
A=1
(1/(z-0.5) -1/(z+0.5)) z^(-1)/z^(-1)

z^(-1)/(z-0.5z^(-1) ) - z^(-1)/(z+0.5z^(-1) )
z^(-1) [1/(1-0.5z^(-1) ) -1/(1+0.5z^(-1) )]
z^(-1)...

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