EX-703-C Digital Signal Processing - Jun 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aThe impulse response of a linear time invariant system is $h(n) = \{1, \underset{\uparrow}{2}, 1, -1\}$. Determine the response of the system to the input signal $x(n) = \{\underset{\uparrow}{1}, 2, 3, 1\}$.
   How to answer

   • State linear convolution y(n)=x(n)*h(n)
   • Use tabular / matrix method for length 4 and 4 giving length 7
   • Compute samples y = {1,4,8,8,3,2,-1} with origin marking
   • Verify duration and origin position
2. 7 Marks
   bDiscuss the concept of sampling and reconstruction of signals.
   How to answer

   • Define sampling: continuous to discrete, sampling theorem
   • Explain Nyquist rate, aliasing, types of sampling
   • Explain reconstruction via ideal low-pass / sinc interpolation
   • Mention practical reconstruction with zero-order hold
3. 7 Marks
   aDetermine the impulse response $h(n)$ for the system described by the second order difference equation. $$y(n)-3y(n-1)-4y(n-2) = x(n)+2x(n-1)$$
   How to answer

   • Take z-transform / assume impulse input with zero initial conditions
   • Find transfer function H(z) from difference equation
   • Perform partial fraction / long division for h(n)
   • State causal impulse response form
4. 7 Marks
   bWhat do you understand by linear time invariant causal system?
   How to answer

   • Define linearity with superposition
   • Define time-invariance
   • Define causality h(n)=0 for n<0
   • Give example / condition for LTI causal system
5. 14 Marks
   Determine the z-transform of the following:

   1. i)$x(n) = \alpha^{n}u(n) + b^{n}u(-n-1)$
   2. ii)$x(n) = (\cos \omega_{0}n)u(n)$
   3. iii)$x(n) = \frac{1}{2}(n^{2}+n)\left(\frac{1}{3}\right)^{n-1}u(n-1)$
   4. iv)$x(n) = (-1)^{n}(\cos \frac{\pi}{3}n)u(n)$
   How to answer

   • Apply standard z-transform pairs and linearity
   • Use ROC rules for right-sided vs left-sided terms
   • Use differentiation / shifting properties for n-multiplied and delayed terms
   • Write final X(z) with ROC for each case
6. 7 Marks
   aDetermine the inverse z-transform of following system for the given three conditions $$X(z) = \frac{1}{1-1.5z^{-1}+0.5z^{-2}}$$ if

   1. i)ROC $|z| > 1$
   2. ii)ROC $|z| < 0.5$
   3. iii)ROC $0.5 < |z| < 1$
   How to answer

   • Factor denominator to poles at z=1 and z=0.5
   • Do partial fraction expansion of X(z)/z or X(z)
   • Map each term to right/left/two-sided sequence per ROC
   • Write x(n) for ROC |z|>1, |z|<0.5, 0.5<|z|<1
7. 7 Marks
   bDetermine the causal signal $x(n)$ whose z-transform is given below. $$X(z) = \frac{1+z^{-1}}{1-z^{-1}+0.5z^{-2}}$$
   How to answer

   • Assume causal ROC outside outermost pole
   • Perform partial fraction / long division
   • Invert to form with damped sinusoid / exponentials
   • Multiply by u(n) for causal signal
8. 7 Marks
   aExplain about multiplication of two DFT's and circular convolution.
   How to answer

   • State DFT multiplication property
   • Define N-point circular convolution formula
   • Explain IDFT of product equals circular convolution
   • Mention relation to linear convolution via zero-padding
9. 7 Marks
   bPerform the circular convolution of the following two sequences. $$x_{1}(n) = \{\underset{\uparrow}{1}, 2, 1, 2\}$$ $$x_{2}(n) = \{\underset{\uparrow}{2}, 1, 3, 4\}$$
   How to answer

   • Set N=4 and use circular convolution sum / matrix method
   • Compute each output sample y(m)
   • Obtain y = {14, 15, 13, 18} (to verify)
   • State result with origin
10. 7 Marks
    aExplain the designing of FIR filter using window method.
    How to answer

    • State ideal impulse response hd(n) for desired filter
    • Explain truncation by window w(n) and types of windows
    • Give design steps: specifications, hd(n), multiply, realize
    • Mention Gibbs phenomenon and window comparison
11. 7 Marks
    bExplain the designing of IIR filter using Butterworth method.
    How to answer

    • Review analog Butterworth prototype and specifications
    • Explain impulse-invariant / bilinear transformation steps
    • Give order and cutoff calculation formulas
    • Mention magnitude response characteristics
12. 7 Marks
    aWhat is parametric and nonparametric spectral estimation.
    How to answer

    • Define parametric model-based approach (AR, MA, ARMA)
    • Define nonparametric FFT/periodogram based approach
    • Compare resolution, assumptions, data length needs
    • Give applications of each
13. 7 Marks
    bExplain about Linear Mean Square Estimation.
    How to answer

    • Define MMSE criterion E[(d - estimate)^2]
    • Derive orthogonality principle / Wiener-Hopf equations
    • Explain linear estimator structure
    • Mention applications in filtering and prediction
14. 14 Marks
    Write short notes on any two of the following:

    1. a)Parseval Identity
    2. b)Optimal filtering using ARMA model
    3. c)Wiener filter
    How to answer

    • State Parseval identity for energy in time and frequency
    • Explain ARMA model for optimal filtering
    • Explain Wiener filter principle for MMSE
    • Give formulas and use-cases for any two chosen