EC-601 Digital Signal Processing - May 2024 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. 6 Marks
   aWrite about discrete time system with the help of examples.
   How to answer

   • Definition of discrete-time system as transformation of input sequence to output sequence
   • Block representation y[n] = T{x[n]} with examples e.g. accumulator, moving average, FIR filter
   • Examples illustrating input-output relation
2. 8 Marks
   bDescribe the classification of discrete time systems.
   How to answer

   • List classification criteria: linear/nonlinear, time-invariant/variant, causal/noncausal, stable/unstable, static/dynamic
   • Define each class with condition e.g. superposition, shift invariance
   • Give one example for each class
3. 7 Marks
   aDetermine the impulse response of the given sequence $y[n] = \alpha_1 x[n] + \alpha_2 x[n-1] + \alpha_3 x[n-2] + \alpha_4 x[n-3]$
   How to answer

   • Use impulse input x[n]=delta[n] to find h[n]
   • Substitute delayed impulses to get weighted impulses alpha1 delta[n]+...
   • Write final impulse response expression
4. 7 Marks
   bDescribe the typical sequences and show how they are represented?
   How to answer

   • List typical sequences: unit impulse, unit step, exponential, sinusoidal, etc.
   • Give mathematical definition for each
   • Show graphical / stem-plot representation
5. 7 Marks
   aDetermine the z-transform of an anticausal exponential sequence given as $x[n] = -\alpha^n \mu[-n-1]$
   How to answer

   • Write z-transform summation for anticausal x[n] = -alpha^n mu[-n-1]
   • Evaluate geometric series for |z| < |alpha|
   • State X(z)=1/(1-alpha z^-1) with ROC |z|<|alpha|
6. 7 Marks
   bDescribe the concept of region of convergence of a rational z-transform.
   How to answer

   • Define ROC as set of z for which z-transform sum converges
   • Properties for rational X(z): rings bounded by poles, no poles inside ROC
   • Different ROCs for causal, anticausal and two-sided sequences
7. 7 Marks
   aDetermine the inverse z-transform via partial fraction expansion of the given function : $$H(z) = \frac{z(z+2.0)}{(z-0.2)(z+0.6)}$$
   How to answer

   • Write H(z)/z and expand into partial fractions with poles 0.2 and -0.6
   • Find coefficients A and B
   • Invert each term to get h[n] for assumed ROC
8. 7 Marks
   bWrite about the convolution property of z-transform.
   How to answer

   • State convolution in time maps to multiplication in z-domain
   • Give formula Z{x1*x2}=X1(z)X2(z) with ROC intersection
   • Mention application to LTI system response
9. 7 Marks
   aWhat is concept discrete Fourier series? Drive an expression for it.
   How to answer

   • Define discrete Fourier series for periodic sequence with period N
   • Derive analysis and synthesis equations using orthogonality of complex exponentials
   • State expression for DFS coefficients
10. 7 Marks
    bIllustrate the properties of discrete Fourier series.
    How to answer

    • List properties: linearity, shift, duality, symmetry, convolution
    • Give mathematical statement for each property
    • Brief illustration / implication
11. 8 Marks
    aDetermine the 4-point circular convolution of the two length-4 sequences $g[n]$ and $h[n]$ given by $g[n] = \{1\ 2\ 0\ 1\}, h[n] = \{2\ 2\ 1\ 1\}, 0 \le n \le 3$.
    How to answer

    • Write 4-point circular convolution formula
    • Compute using concentric-circle / matrix method for given g[n], h[n]
    • State resultant 4-point sequence
12. 6 Marks
    bDefine DFT and write about some important properties of it.
    How to answer

    • Define N-point DFT analysis and synthesis equations
    • List properties: linearity, circular shift, symmetry, Parseval
    • Give formula for each property
13. 7 Marks
    aWhat is FFT? Derive an expression for it.
    How to answer

    • Define FFT as fast computation of DFT
    • Derive decimation-in-time/frequency decomposition reducing complexity to (N/2)log2 N
    • Mention butterfly structure
14. 7 Marks
    bGiven $x[n] = \{1, 2, 3, 4, 4, 3, 2, 1\}$, find $X(k)$ using DIF FFT algorithm.
    How to answer

    • Arrange 8-point sequence for DIF stages
    • Apply DIF butterfly with twiddle factors stage by stage
    • Obtain 8-point DFT X(k) in natural order
15. 7 Marks
    aWrite about the design techniques for the FIR filters.
    How to answer

    • List FIR design methods: windowing, frequency sampling, optimal equiripple
    • Explain steps of window method and design equations
    • Compare linear-phase advantage
16. 7 Marks
    bConvert the analog filter into digital filter whose system function is $H(s) = \frac{s+0.2}{(s+0.2)^2+9}$. Use the impulse invariant technique. Assume $T = 1s$.
    How to answer

    • Express H(s) in form for inverse Laplace to get h(t)
    • Sample h(t) with T=1s to get h[n] and take z-transform
    • Simplify to H(z) rational form