CE335 DIGITAL SIGNAL PROCESSING Undergraduate Examinations 2019

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CE335-6-AU

Undergraduate Examinations 2019

DIGITAL SIGNAL PROCESSING

Question 1

(a) If x1(n) and x2(n) are odd sequences, determine whether y(n) = x1(n)x2(n) is an odd or an even sequence. [6%]

(b) Compute the convolutional sum h[n] = h1[n] ⊗ h2[n] when h1[n] = {0.5, 0.5}, for 0 ≤ n ≤ 1 and h2[n] = {0.5, 0.5}, for 0 ≤ n ≤ 1. If h[n] is the impulse response of an FIR filter, what is the type of this filter?        [10%]

(c) One of the applications of Principal Component Analysis (PCA) is signal denoising. Describe the PCA algorithm.     [7%]

(d) The moving average filter input-output equation is given by y[n] = 4/1 3Pi=0x[n − i], for n ≥ 0. Prove that the filter is a linear time-invariant (LTI) system.      [10%]

Question 2

(a) The input-output relationship of a Finite Impulse Response (FIR) filter is given by         [16%]

y[n] = x[n] − 2x[n − 1] − 3x[n − 2] + 3x[n − 3] + 2x[n − 4] − x[n − 5]

Compute the impulse response of the filter and then find the magnitude and the phase response. What is the group delay of this filter?

(Hint: Use Euler’s formulae cos(ω) = e jω+ 2 e−jω and sin(ω) = e jω−e−jω2j)

(b) Assume a signal x[n] = δ[n], 0 ≤ n ≤ 49 that is generated by sampling an analog signal xa(t) with sampling period 0.01 sec.        [10%]

(i) Sketch the magnitude response of the signal. [6%]

(ii) The signal x[n] is filtered in the frequency domain with a low pass filter with cutoff frequency 20 Hz. Sketch the magnitude response of the filtered signal. [4%]

(Hint: Use the Discrete Fourier Transform (DFT) definition X(m) = N−1Pn=0x[n]WN mn with WN = e −j2π/N .)

(c) When we compute the DFT of a signal, we assume that the signal is periodic and that we observe a complete cycle of the signal. What happens if this assumption does not hold?        How can one overcome this problem?        [5%]

Question 3

A linear time invariant (LTI) system is realized by the flow graph shown in Figure 3.1

(a) Derive the relationship between the input x[n] and the output y[n] of the filter shown in the block diagram of Figure 3.1.   [6%]

(b) Find the transfer function H(z) of the filter depicted in Figure 3.1 and determine the magnitude response of the filter.        [10%]

(c) Find the zeros and the poles of the filter. Comment on the stability of the filter.

[Hint: az2 + bz + c = (z − z1)(z − z2), where z1 = −b + √ b2 − 4ac 2a and z2 = −b − √ b2 − 4ac 2a]              [8%]

(d) Find a causal and stable filter that has the same magnitude response as the original filter.

[Hint: Consider the use of an all pass filter.]        [6%]

(e) IIR filters do not have linear phase. Explain how this problem can be fixed offline and in real time.          [6%]