ECET 345 week 6 homework

ECET 345 week 6 homework

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 transfer function of a causal LSI system whose impulse response is given by
h[n]=〖(-0.5)〗^(n-1) sin⁡〖[0.5(n-2) ]u[n-2]〗 and express the result in positive powers of z. Hint: The transfer function is just the z-transform of impulse response. However, we must first convert the power of -0.5 from (n - 1) to (n - 2) by suitable algebraic manipulation.

h[n]=〖(-0.5)〗^(n-1) sin⁡〖[0.5(n-2) ]u[n-2]〗

〖〖(-0.5)〗^(-1) (-0.5)〗^(n-1) 〖(-0.5)〗^1=(-0.5)^(n-2) (-0.5)

(-0.5)(-0.5)^(n-2) sin⁡[0.5(n-2)]
(-0.5)[(-0.5)^(n-2) sin⁡[0.5(n-2) ]]
-0.5((-0.5 sin⁡〖(0.5) z^(-1) 〗)/(1-2*0.5 cos⁡(0.5) z^(-1)+(-0.5) z^(-2) )) z^(-2)
-0.5((-0.5 sin⁡〖(0.5) z^(-1) 〗)/(1-2*0.5 cos⁡(0.5) z^(-1)+(-0.5) z^(-2) )) z^(-2)

Express the following signal, x(n), in a form such that z-transform tables can be applied directly. In other words, write it in a form such that the power of 0.25 is (n-1) and the argument of sin is also expressed with a (n-1) multiplier.
x[n]=〖(0.25)〗^n sin⁡〖(π/2 n)u[n-1]〗
Hint: Express sin(π/2 n) as sin (π/2 (n-1+1)) = sin (π/2 (n-1) + π/2 ) and then expand using use the trig identity for Sin(A+B).
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)〗...

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