CA 2 - Z-transforms and applications

Question 1

Show using the example of an impulse δ[n] that multiplication by Z^n0 has the effect of time delaying any sampled data signal by n0 sampling intervals. Then derive an expression for the phase transfer function Φ_δ(Ω) of such a delayed unit impulse. What is the slope of the plot of Φ_δ(Ω) vs Ω?

Question 2

Find y[n] and Y(Z) where y[n]=h[n]*x[n] and Y(Z)=H(Z).X(Z) for:
x[n]=1,2,3,1,-1,1
h[n]=1,1,1
using both time and Z-domain methods. Show that Y(z) is indeed the Z-transform of y[n]

Question 3

Find the poles and zeros of the transfer function:
H(Z)=(z^3 - z^2 +0.9z -0.9) / (z^3 +0.9z^2)
Plot the poles and zeros of H(Z) on the Argand plane.
Sketch the approximate frequency response over the interval 0<= Ω <= π.
Find the difference equation that describes the action of the corresponding processor

Question 4

Using the smallest possible number of z-plane poles and zeros, design a digital filter with the following parameters:
1. complete frequency rejection at Ω=0 radians,
2. Complete frequency rejection at Ω=π/4 radians,
3. A narrow frequency passband at Ω= 2π/3 radians with poles at r=0.9
4. No time delay in the output y[n]

find the Z-transform of the filter H(z) and also the difference equation describing this filter, y[n] =f(x[n])

Question 5

A filter is described by the block diagram below
Find the difference equation for this processor y[n]=f(x[n]).
If the input x[n] is an impulse δ[n], find the Z-transform of the output i.e. Y(Z) for the following initial conditions
1. y[-1]=y[-2]=0

Does this case represent the Z-transform of the true, natural impulse response of the processor Y(Z)=H(Z)? If so, why?