1) Given the transfer function shown below, convert it into a difference equatio

Question

1) Given the transfer function shown below, convert it into a difference equation.

z^2-0.7071z/z^2-1.41z+1

2.       The impulse response of a discrete time system is given by

h(n) = [ 1 -1 2 ].

To such a system we apply an input of the type

x(n) = [ 2 1 2 3 ].

3.       A system is described by the transfer function below.

H(z) = (z^2+0.64)/(z^2+1.8z+1.28)

Find the location of poles and zeros of the system and write them here.

4.The transfer function of a discrete time system is given by

H     H(z) = z/(z-0.5)

         The system has a sampling rate of 100 Hz. Calculate the frequency response of the system at 25 Hz input frequency as a complex number in both rectangular and polar form.

Explanation / Answer

1) Given the transfer function shown below, convert it into a difference equation.

H(z) = z^2-0.7071z/z^2-1.41z+1
Y(z)/X(z) = z^2-0.7071z/z^2-1.41z+1
Y(z) (z^2-1.41z+1) = X(z) (z^2-0.7071z)
Y(z)z^2-1.41Y(z)z+ Y(z) = X(z)z^2 - 0.7071X(z)z
taking inverse z transform
y(n+2) -1.41y(n+1) +y(n) = x(n+2)- 0.7071x(n+1)

2.       The impulse response of a discrete time system is given by
h(n) = [ 1 -1 2 ].
To such a system we apply an input of the type
x(n) = [ 2 1 2 3 ].

take following approach and mutliply by postion
0 0 0 1 -1 2
3 2 1 2
y(0) = 1*2 = 2
0 0 1 -1 2
3 2 1 2
y(1) = 1*1-1*2 = 1-2 = -1
0 1 -1 2
3 2 1 2
y(2) = 1*2-1*1+2*2 = 2-1+4 = 4+1 = 5
1 -1 2
3 2 1 2
y(3) = 1*3-1*2 + 2*1 = 3-2+2 = 3
1 -1 2
0 3 2 1 2
y(4) = -1*3+2*2 = 4-3 = 1
1 -1 2
0 0 3 2 1 2
y(5) = 2*3 = 6
linear convolution gives [2, -1, 5, 3, 1, 6]

3.A system is described by the transfer function below.
H(z) = (z^2+0.64)/(z^2+1.8z+1.28)

poles are z^2+1.8z+1.28 = 0 {z=((-(sqrt(47)*i+9))/10), z=((sqrt(47)*i-9)/10)}
zeros are z^2+0.64 = 0 {z=((-4*i)/5), z=((4*i)/5)}

since poles z = -0.9+0.685565 i and z = -0.9-0.685565 i are inside unit circle. SYSTEM IS STABLE

4.The transfer function of a discrete time system is given by
H(z) = z/(z-0.5)
The system has a sampling rate of 100 Hz.
Calculate the frequency response of the system at 25 Hz input frequency as a complex number in both rectangular and polar form.

H(z) = z/(z-0.5)
H(st) = e^(sT)/(e^(sT-0.5)
samping time = 1/f = 1/100 = 0.01
H(jw) = e^(i*2*pi*0.01*25) / (e^(i*2*pi*0.01*25)-0.5)

SIMPLFYING WE GET RECTANGULAR FORM 0.8-0.4 i
SIMPLFYING WE GET POLAR FORM r = 0.894427 (radius), theta = -26.5651

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