EC-601 Digital Signal Processing - May 2023 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. 8 Marks
   aExplain in detail about different representations of discrete time signals.
   How to answer

   • Define discrete-time signal and representation forms
   • Tabular / sequence representation
   • Functional / graphical representation and impulse decomposition
   • Example of each representation
2. 6 Marks
   bState stability, causality and invertible properties of discrete time LTI system.
   How to answer

   • State BIBO stability condition for LTI system
   • State causality condition in terms of h(n) and ROC
   • State invertibility condition and inverse system
   • Relate to impulse response and system function
3. 8 Marks
   aFind the impulse response of the system described by the following difference equation: Assume all initial conditions are zero. $$y(n)-2y(n-1)+4y(n-2)=x(n)+x(n-1)$$
   How to answer

   • Take Z-transform of difference equation with zero initial conditions
   • Find H(z) = Y(z)/X(z) and partial fraction / pole form
   • Find inverse Z-transform to get h(n)
   • State ROC and causality assumption
4. 6 Marks
   bProve the following properties of Z.T.

   1. i)Convolution in time
   2. ii)Scaling in time domain
   How to answer

   • State definition of Z.T. X(z)=sum x(n)z^{-n}
   • Prove convolution in time: Z{x1*x2}=X1(z)X2(z)
   • Prove scaling/time-scaling or multiplication property in time domain
   • State ROC considerations
5. 7 Marks
   aDetermine the Inverse Z.T. of the $X(z)=\frac{1+z^{-1}}{1+\frac{1}{3}z^{-1}}$; ROC: $Re\{z\}>1/3$.
   How to answer

   • Rewrite X(z) in terms of z^{-1} and identify pole at z=1/3
   • Use standard pair a^n u(n) <-> 1/(1-a z^{-1})
   • Apply linearity / time-shift for numerator terms
   • Write x(n) with ROC Re{z}>1/3 implying right-sided sequence
6. 7 Marks
   bDetermine the Inverse Z.T. of the $X(z)=\frac{z^{2}}{(z-a)^{2}}$; ROC: $Re\{z\}>a$.
   How to answer

   • Identify X(z)=z^2/(z-a)^2 with double pole at z=a
   • Use known pair n a^n u(n) <-> a z^{-1}/(1-a z^{-1})^2
   • Manipulate to standard form X(z)/z
   • Write x(n)=(n+1)a^n u(n) for ROC |z|>|a|
7. 10 Marks
   aDerive the following properties of DFS.

   1. i)Time shifting
   2. ii)Time reversal
   3. iii)Convolution
   How to answer

   • Write DFS definition for periodic sequence
   • Derive time-shifting: x(n-m) <-> W_N^{km} X(k)
   • Derive time-reversal: x(-n) <-> X(-k)
   • Derive periodic convolution property
8. 4 Marks
   bFind the DFT of a sequence $x(n) = (1/2)^{n}u(n)$.
   How to answer

   • Write DFT definition for N-point sequence
   • Substitute x(n)=(1/2)^n u(n) truncated / periodic form
   • Evaluate geometric sum formula
   • Simplify to closed-form X(k)
9. 6 Marks
   aDetermine the DFT of a sequence X(n) = {1, 1, 0, 0}.
   How to answer

   • Write 4-point DFT matrix with W_4
   • Substitute x(n)={1,1,0,0}
   • Compute X(k)=sum x(n)W_4^{kn}
   • List X(0) to X(3) values
10. 8 Marks
    bDetermine DFT of the sequence $x(n) = \{1, 0, 0, 0\}$ using FFT algorithm.
    How to answer

    • Draw 4-point DIT-FFT butterfly / stages
    • Apply bit-reversed input order
    • Compute stage-1 and stage-2 butterflies with W_4
    • Obtain X(k) output
11. 7 Marks
    aDetermine the 4 - point IDFT of the sequence $x[k] = \{1, -j, 0, j\}$ using DIT-FFT algorithm.
    How to answer

    • Write 4-point IDFT formula with 1/N factor
    • Apply DIT-FFT flow with conjugated twiddles / inverse stages
    • Compute butterflies for X(k)={1,-j,0,j}
    • Scale by 1/4 to get x(n)
12. 7 Marks
    bFind inverse z-transform of $$X(z)=\frac{z^{-4}}{z-1}+z^{-6}+\frac{z^{-3}}{z+0.5}$$
    How to answer

    • Split X(z) into three delayed terms
    • Use pairs 1/(z-1) <-> u(n-1) and 1/(z+0.5) forms
    • Apply time-delay property z^{-k}X(z)
    • Combine to get x(n) as sum of delayed sequences
13. 7 Marks
    aOutline the steps involved in the design of FIR filter using Hanning window.
    How to answer

    • State ideal desired frequency response Hd(w)
    • Obtain hd(n) by inverse DTFT and shift for causality
    • Multiply by Hanning window w(n)=0.5-0.5cos(2pi n/(N-1))
    • List design steps and truncation effects
14. 7 Marks
    bWrite the magnitude and phase functions of FIR filter when impulse response is symmetric and N is odd.
    How to answer

    • State symmetric condition h(n)=h(N-1-n) with N odd
    • Derive H(w)=e^{-j w (N-1)/2} Hr(w)
    • Write magnitude function Hr(w) sum form
    • Write phase function -w(N-1)/2
15. 14 Marks
    Write short notes on any two of the following:

    1. i)Comparison of Butterworth and Chebyshev filter
    2. ii)Properties of convolution sum
    3. iii)R.O.C of Z-Transform
    4. iv)Circular convolution
    How to answer

    • Select any two topics with definition and key points
    • Give relevant formulas / properties / diagrams
    • Compare / illustrate with examples where applicable
    • Conclude with applications / significance