Question 1. Consider a LTI system whose input output relation is

y[n] = ∑_(k=0)⁷ kx[n-k]

a) Calculate the impulse response h[n] of the system and verify the output with matlab program.

b) Is this a causal and stable system> Explain.

c) Determine the uint step response s[n] of the system and verify in matlab.

d) If the input x[n] for this system is bounded, i.e, |x[n] |<3, What would be a minimum bound 'M^' for the output (i.e,|y[n]|≤M)?

e) Use the MATLAB function filter to compute the ipulse response h[n] and the unit step response s[n] for the given system and plot them.

Question 2. The difference equation describing the input-output relationship of a discrete LTI system is

y[n] - 0.5y[n-1] + 0.125y[n-2] = x[n] + x[n-1]

Compute the following:

a) Transfer function of the system, H(z).

b) Impulse response (h[n] ) of the discrete time. Also implement in matlab.

c) Output response of the system when the input is unit step u[n]. Also compute the unit step output using matlab Simulink.

Question 3. Design a fifth order band pass linear phase filter for the following specifications.

Lower cut off frequency=0.4 π rad/sec

Upper cut off frequency=0.6 π rad/sec

Window type is Hamming

Also draw the filter structure.

Question 4. In this problem use Lena image to compute the output.

a) Load the Lena image in MATLAB and display using imshow function.

b) Consider the 1D (single dimensional) impulse response h[n] = 1/5 {(1,1,1_↑,1,1)}. Using it perform 1D convolution along the each row of the Lena image and display the resulting blurred image. Comment on the result.

c) Using the image obtained in part (b) perform 1D convolution along the each column of the Lena image and display the resulting blurred image. Compare this image with the image in part (b) and comment on the results.