Write a suitable matlab program with comments to indicate

Question

Question 1. a. In a digital communication transceiver, signal processing is used to model the signal received from a channel. When a signal is passed through the channel, the channel output can be modeled as y(n)=0.6x(n)-0.1x(n-2)-0.2x(n-3). If the transmitted signal is given by x(n)=sin(2*pi/12*n)+cos(2*pi/6*n)+cos(2*pi/24*n) for n=0 to 24
= cos(2*pi/6*n) for n=25 to 47
= 0 otherwise
Write a suitable matlab program with comments to indicate the logic used and generate the discrete plot for x(n) and y(n) for n=0 to 60.

b. Calculate poles and zeros of the system. Sketch the pole zero plot and comment on the stability of the system based on the pole zero plot.

Question 2. In an image processing application a digital signal processing system is represented by y(n)=med{x(n-3),x(n-2),x(n-1),x(n),x(n+1),x(n+2),x(n+3)}; where the operation med{} is given by the algorithm: arrange values in the ascending order and the answer is the middle value. For eg: med(5,3,9}=5. Whether the system is
causal
linear
time invariant system
Justify your answers.

Question 3. a. It is claimed that a stable and causal ideal frequency-selective filter can be obtained. Do you agree or disagree? If you agree provide an example of such a filter. If you disagree provide a proof of your disagreement.

b. The following five facts about a discrete-time signal x(n) with z-transform X(z) is given below:
x(n) is a real and right-sided sequence.
X(z) has exactly two poles.
X(z) has two zeros at the origin.
X(z) has a pole at z=(1/2)e^j.π/3.
X(1)=8/3
Determine X(z) and specify its ROC.

Question 4. Design a low-pass filter of order N=63 with a cutoff frequency ω_p = 0.3π and a stop band cutoff frequency ω_s = 0.32π.

a. What is the approximate stopband attenuation that would obtained if this filter was designed using the window design method with a Kaiser window.

b. Repeat part (a) for a equiripple filter assuming that we want δ_p = δ_s.