1. Find the z-transform x(z ) of x ( n ) =cos (0.45n+0.25)u(n) . 2. Find the sys

Question

1.       Find the z-transform x(z)  of  x(n) =cos (0.45n+0.25)u(n)  .

2.       Find the system transfer function of a causal LSI system whose impulse response is given by

h[n] = (-0.5)^n-1sin[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.

3.       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(pi/2 n)u[n-1]

4.       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)(z0.5)

5. The transfer function of a system is given by

H(z) = Z/(z^2-0.8z+0.15)

To such a system we apply an input of the type  x[n[= e^-0.4n for n>= 0.

Explanation / Answer

1.Find the z-transform x(z) of x(n) =cos (0.45n+0.25)u(n)   .
cos(A+B) = cosA cosB-sinA sinB thus

x(n) = (cos(0.45n) cos(0.25) - sin(0.45n) sin(0.25) ) u(n) EXPANDING WE GET
x(n) = cos(0.25) cos(0.45n) u(n) - sin(0.25) sin(0.45n) u(n)

now taking z transform we get.
x(z) = cos(0.25) z(z-cos(0.45))/(z^2-2zcos(0.45)+1) - sin(0.25) * (z sin(0.45)/(z^2-2zcos(0.45)+1)

x(z) = (0.968912421711 *z(z-0.900447102353) - 0.434965534111*0.247403959255*z )/ (z^2-2z*0.900447102353+1)
     = (0.968912421711 *z(z-*0.900447102353) - 0.434965534111*0.247403959255*z )/ (z^2-1.8z+1)
   = (0.9689z^2 - 0.87201z - 0.10761z) / (z^2-1.8z+1)
     = (0.9689z^2-0.97962z) / (z^2-1.8z+1) IS ANSWER

2.Find the system transfer function of a causal LSI system whose impulse response is given by
h[n] = (-0.5)^n-1sin[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-1sin[0.5(n-2)]u[n-2]
h[n] = (-0.5)(-0.5)^(n-2) sin[0.5(n-2)]u[n-2]

consider z transform of sin(an) u(n) = (sin(0.5))z^-1/(1-(2cos(0.5))z^-1+z^-2)
now z transform of r^n sin(an)u(n) = (-0.5*sin(0.5))z^-1 / (1-(2*-0.5*cos(0.5))z^-1+0.25z^-2)
now shift in time domain by m units is equal to mutiplication by z^-2 in frequency domain.
thus
now z transform of (-0.5)(-0.5)^(n-2) sin[0.5(n-2)]u[n-2] is -.05* z^-2 * (-0.5*sin(0.5))z^-1 / (1+(cos(0.5))z^-1+0.25z^-2)


3.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(pi/2 n)u[n-1]

x[n] = 0.25*(0.25)^n-1 cos(pi/2(n-1))u[n-1]

4.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 = A(z+0.5) + B(z-0.5)
at z = 0.5
1 = A(0.5+0.5) = A = 1
at z = -0.5
1 = B(-0.5-0.5) = -B => B = -1

H(z) = 1/(z-0.5) - 1/(z+0.5)
take inverse z transform
h(n) = 0.5^(n-1) u(n-1) - (-0.5)^(n-1) u(n-1) IS ANSWER.


5. The transfer function of a system is given by
H(z) = Z/(z^2-0.8z+0.15)
To such a system we apply an input of the type x[n[= e^-0.4n for n>= 0.
x[n] = e^-0.4n
taking z transform for above we get.
x[z] = z/(z-e^-0.4)

now y(z) = h(z)*x(z) = z/(z^2-0.8z+0.15) *z/(z-e^-0.4)

y(z) = z/(z^2-0.8z+0.15) *z/(z-0.67)

y(z)/z = z/(z-0.5)(z-0.3(z-0.67) = A/(z-0.5)+B/(z-0.3)+C/(z-0.67)

z = A(z-0.3)(z-0.67) + B(z-0.5)(z-0.67) + C(z-0.5)(z-0.3)
at z = 0.5
0.5 = A(0.5-0.3)(0.5-0.67) => A = -14.7
at z = 0.3
0.3 = B(0.3-0.5)(0.3-0.67) => B = 4.05
at z = 0.67
0.67 = C(0.67-0.5)(0.67-0.3) => C = 10.65

y(z) = -14.7/(z-0.5) + 4.05/(z-0.3) + 10.65/(z-0.67)
y(z) = -14.7 z/(z-0.5) + 4.05z/(z-0.3) + 10.65z/(z-0.67)

taking inverse z transform
y(n) = -14.7 (0.5)^n + 4.05 (0.3)^n + 10.65 (0.67)^n

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