**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.

Lena Image

**Question 5.** Genrate and plot the samples using stem function for the following signals:

i) x_{1} [n] = 5δ[n+1]+n^{2} (u[n+5]-u[n-4] )+10(0.5)^{n} (u[n-4]-u[n-8] )

ii) x_{2} [n] = (0.8)^{n} cos?(0.2πn + π/4), 0 ≤ n ≤ 20.

iii) x_{3} [n] = ∑_{(m=0)}^{4}(m+1){δ[n-m]-δ[n-2m]} , 0 ≤ n ≤ 20

b) What is Imaging and Aliasing? How their Spectrum differ from each other? What is the need for Anti Aliasing filter prior to down sampling?

Note: Support your answers with appropriate explanation and figures. Provide references also.

**Question 6.** Let the signal to be filtered be the first 100 samples from MATLAB's "train" signal. To this signal add some Gaussian noise to be generated by randn, multiply it by 0.1, and add it to the 100 samples of the train signal. Design three discrete filters, each of order 20, and a half frequency (for Butterworth butter) and passband frequency (for the Chebyshev filters) of ω_{n} = 0.5. For the design with cheby1 let the maximum passband attenuation be 0.01 dB, and for the design with cheby2 let the minimum stopband attenuation be 60 dB. Obtain the three filters and use them to filter the noisy "train" signal.

Use MATLAB plot the following for each of the three filters:

a) Using fft function compute the DFT of the original signal, the noisy signal and the noise and plot its magnitude. Is the cut-iff frequency of the filters adequate to get rid of the noise? Explain.

b) Compute and plot the magnitude and poles & zeros for each of the three filters. Comment on the differences in the magnitude responses.

c) Use the filter function to obtain the output of each of the filters. Also plot the original noiseless signal and filtered signal. Compare them.

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