1) find the z-transform, x(z), given x(n) = (0.75^n)u(n) - (-0.75^n-1)u(n-1) a)

Question

1) find the z-transform, x(z), given x(n) = (0.75^n)u(n) - (-0.75^n-1)u(n-1)

a) z/(z-0.75) -1/(z+0.75)

b) z/(z-0.75) +1/(z+0.75)

c) 1/(z-0.75) -z/(z+0.75)

d) 1/(z-0.75) -1/(z+0.75)

2) find the inverse z-transform of x(z) = (2z-1)/(z+0.5) and determine its numerical value for n=2

a) 1

b) -2

c) 0.5

d) 2

3) find the first seven seven value (i.e.,x(n) for n=0 to 6) of the function x(n) = [-3nu(n)] - [2(n-3)] +[4u(n-4)]

a) 0, -3, -6, -7, -9, -11, 15
b)  0, -3, -5, -9, -8, -11, 15
c)      0, -3, -6, -9, -12, -15, -20
  d)     0, -3, -6, -9, -10, -15, -20

4) A system has the transfer function given below. To such a system we apply a unit impulse. (A unit impulse has a value of 1 at n = 0 and is 0 everywhere else). Find the first seven outputs using the iteration method as outlined in the lecture. (Hint: First convert the transfer function into a difference equation.) Assume that all initial conditions are 0.

2z+3/z^2+z-0.5

a) 0.2. 1, 0, 0.5, 0.5, 0.75

b) 0.2. 1, 0, 0.5, -0.5, -0.75

c) 0.2. 1, 0, 0.5, -0.5, 0.75

d) 0.2. 1, 0, 0.5, -0.75, 0.95

5) A system has the following transfer function. Determine its poles and zeroes and whether it is a stable or unstable system.

H(z) = (z^2 + 2.89)/(z^2 + 0.91)

a) zerosatz=?0.6

Explanation / Answer

1) find the z-transform, x(z), given x(n) = (0.75^n)u(n) - (-0.75^n-1)u(n-1)

x(z) = z/(z-0.75) -z^-1 z/(z+0.75)
x(z) = z/(z-0.75) -1/(z+0.75)

a) z/(z-0.75) -1/(z+0.75)   is ANSWER

2) find the inverse z-transform of x(z) = (2z-1)/(z+0.5) and determine its numerical value for n=2

x(z) = 2z/(z+0.5)-1/(z+0.5)   TAKE INVERSE Z TRANSFROM

x(n) = 2 (-0.5)^n u(n) - (-0.5)^(n-1)u(n-1)
x(2) = 2(-0.5)^2 - (-0.5)^1 = 2 * 0.25 + 0.5 = 0.5 + 0.5 = 1

a) 1 IS ANSWER.

3) find the first seven seven value (i.e.,x(n) for n=0 to 6) of the function x(n) = [-3nu(n)] - [2(n-3)] +[4u(n-4)]

x(n) = [-3nu(n)] - [2*(n-3)u(n-3)] +[4u(n-4)]

x(0) = -3*0 - 0 + 0   = 0
x(1) = -3*1 - 0 + 0   = -3
x(2) = -3*2 - 0 + 0   = -6
x(3) = -3*3 - 2*0 + 0 = -9
x(4) = -3*4 - 2*1 + 4   = -12-2+4 = -14+4 = -10
x(5) = -3*5 - 2*2 + 4   = -15-4+4 = -19+4 = -15
x(6) = -3*6 - 2*3 + 4   = -18-6+4 = -24+4 = -20

d) 0, -3, -6, -9, -10, -15, -20 IS ANSWER

4) A system has the transfer function given below. To such a system we apply a unit impulse.
(A unit impulse has a value of 1 at n = 0 and is 0 everywhere else).
Find the first seven outputs using the iteration method as outlined in the lecture.
(Hint: First convert the transfer function into a difference equation.) Assume that all initial conditions are 0.

h(z) = 2z+3/z^2+z-0.5
y(z)/x(z) = 2z+3 /(z^2+z-0.5)

y(z) (z^2+z-0.5) = x(z) (2z+3)
y(n+2) +y(n+1)-0.5y(n) = 2x(n+1) + 3x(n)
when x(n) = del(n) then
x(n+2) +x(n+1)-0.5x(n) = 2del(n+1) + 3del(n)

c) 0.2. 1, 0, 0.5, -0.5, 0.75 IS ANSWER.

5) A system has the following transfer function. Determine its poles and zeroes and whether it is a stable or unstable system.
H(z) = (z^2 + 2.89)/(z^2 + 0.91)

poles are z^2+.91 = 0 {z=((-sqrt(91)*i)/10), z=((sqrt(91)*i)/10)}
ZEROS are z^2+2.89 = 0 {z=((-17*i)/10), z=((17*i)/10)}

since poles are inside unit circle.

c) zeros at z=0.0

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