Question 1:

The transfer function for a digital system is given by:

H(z) = 0.33z²/(z - 0.09)(z - 0.55)

For the following. give all answers to 4 decimal places.

PART 1

Determine the factors in the following recursion for the system:

h[k]+ ____ x h[k - 1] + _____ x h[k - 2] = ___ x δ[k]

where δ[k - v] is the unit impulse function.

PART 2

If the input sequence to the system is x[n] and its output sequence is y[n]. derive the difference equation which specifies the current output in terms of previous outputs and current and previous inputs.

To answer this question, complete the following:
y[n] =  _____ x x[n] + _____ x y[n - 1] + ____ x y[n - 2]

PART 3

Perform partial fraction expansion with

A/(z-0.09) + B(z - 0.55) and complete the following:

PART 4

Determine the pulse response of the system in the form:

h[k] = x (0.09^k) + _____ x (0.55^k).

Question 2:

A digital fitter has a sampling frequency of f_s = 88.2kHz and a pulse response given by:

h[n] = 1 x (1/7)ⁿ

Answer the following, giving answers to four decimal places.

PART 1

Determine the z-domain transfer function and complete the following:

H(z) = Az(Bz - 1)

where A =  ___ and B = ____

PART 2

Determine the magnitude of the gain at f_s/2.
Gain= ____ dB.

PART 3

If the input sequence to the filter is denoted by x[n] and the output sequence is denoted by y[n], derive the difference equation that specifies y[n] as a function of past outputs and current and past inputs.

y[n] = _____ x x[n] + ___________ x x[n -1] + __________ x y[n - 1]+ _______ x y[n - 2].

PART 4

If the input to the filter is given by:

x[n]= (0.2 for n > 0; 0 for n < 0.)

Determine the form of Y(z) and complete the following.

Y(z) = C x z²/((z - 1)(7z - 1))

where C =  ____

PART 5

Determine the partial fraction expansion of Y(z) in the form:

D/(z - 1) + E/(z -1/7)

where D = ____ and E = _____

Part 6

Determine the time domain difference equation for the output sequence where:

y[n] = ______ x u[n] + ______ x (_____)ⁿ.