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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) = Cos⁡〖(0.45n+0.25)u(n)〗. 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.


〖x(n)=Cos〗⁡〖0.45n cos⁡0.25-sin⁡〖0.45n sin⁡0.25 〗 nu(n)〗

〖X(z)=0.9689((1-cos⁡〖0.45 z^(-1) 〗)/(1-[2 cos⁡0.45 ] z^(-1)+z^- ))〗⁡〖-0.247((1-[sin⁡0.45 ] z^(-1))/(1-[2 cos⁡〖0.45]z^(-1)+z^(-2) 〗 ))〗

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

x[n]=〖(0.25)〗^n sin⁡〖(π/2 n)u[n-1]〗
x[n]=(0.25)(0.25)^(n-1) sin⁡〖(π/2(n-1)) cos⁡〖pi/2+cos⁡[pi/2〗 [n-1]]sin⁡〖pi/2〗 〗
x[n]=(0.25)[(0.25)^(n-1) cos⁡〖(π/2 (n-1) ) 〗
H(z)=(0.25)[(1-0.25 〖cos(〗⁡〖pi/2)〗 z^(-1))/(1-2(0.25) cos⁡〖(pi/2) z^(-1)+〖0.25〗^(-2) z^(-2) 〗 )] z^(-1)
H(z)=(0.25)[(1-0.25 〖cos(〗⁡〖pi/2)〗 z^(-1))/(1-2(0.25) cos⁡〖(pi/2) z^(-1)+〖0.25〗^(-2) z^(-2) 〗 )] z^(-1)

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
(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) )]

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