EC-601 Digital Signal Processing - Dec 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. 7 Marks
   aDefine stability and causality for discrete-time systems. How can the stability of a system be determined from its impulse response?
   How to answer

   • Define stability: bounded input produces bounded output, summable impulse response condition
   • Define causality: output depends only on present and past inputs, h[n]=0 for n<0
   • Stability test from impulse response: sum |h[n]| < infinity
   • Example / interpretation for LTI systems
2. 7 Marks
   bConsider an LTI system with impulse response $h[n]=(0.5)^n u[n]$. Determine its frequency response $H(e^{j\omega})$.
   How to answer

   • Given h[n]=(0.5)^n u[n]
   • DTFT formula H(e^{j\omega}) = sum h[n]e^{-j\omega n}
   • Evaluate geometric series to get 1/(1-0.5e^{-j\omega})
   • Magnitude/phase interpretation
3. 7 Marks
   aImplement the system described by the difference equation $y[n] = 0.5y[n-1] + 0.25y[n-2] + x[n]$ using Direct Form-I and Direct Form-II structures. Provide the block diagrams for both implementations.
   How to answer

   • Rewrite difference equation in z-domain / delay form
   • Draw Direct Form-I with separate feedforward and feedback delays
   • Draw Direct Form-II with shared delay line
   • Label coefficients 0.5, 0.25 and adders, delays
4. 7 Marks
   bDiscuss the advantages of analyzing discrete-time LTI systems in the z-domain compared to the time-domain. Provide examples to support your discussion.
   How to answer

   • State time-domain analysis: convolution, difference equations
   • Advantages of z-domain: algebraic convolution, pole-zero stability, frequency response
   • Examples: transfer function H(z), stability from ROC
   • Conclusion with comparison
5. 7 Marks
   aDetermine the Z-transform of the following signal.

   1. i)$X(n) = -\alpha^n u(-n-1)$
   2. ii)$X(n) = \alpha^n u(n) + b^n u(-n-1)$
   3. iii)$X(n) = (1/2)^n u(n)$
   How to answer

   • Apply standard Z-transform pairs for left/right sided exponentials
   • Determine expression and ROC for each case i, ii, iii
   • Handle linear combination a^n u(n)+b^n u(-n-1)
   • Specialize for (1/2)^n u(n)
6. 7 Marks
   bRepresent the system described by the following state-space equation using a signal flow graph $$\begin{cases} x[n+1]=\begin{bmatrix}0.5 & 1 \\ 0 & 0.5\end{bmatrix}x[n]+\begin{bmatrix}1 \\ 0\end{bmatrix}u[n] \\ y[n]=\begin{bmatrix}1 & 1\end{bmatrix}x[n] \end{cases}$$
   How to answer

   • Write state equations for x1[n+1], x2[n+1] and output y[n]
   • Identify gains 0.5, 1 and input branches
   • Draw nodes, forward paths and feedback loops as signal flow graph
   • Label input u[n] and output y[n]
7. 7 Marks
   aState and prove any four properties of DFT.
   How to answer

   • State four DFT properties e.g. linearity, circular shift, duality, symmetry
   • Give mathematical statement for each
   • Provide proof using DFT definition
   • Mention conditions / N-point
8. 7 Marks
   bDetermine the circular convolution for the two sequences $x_1(n) = \{1, 2, 3, 4\}, x_2(n) = \{1, 5, 1, 3\}$ using concentric circles method.
   How to answer

   • Arrange sequences x1(n)={1,2,3,4}, x2(n)={1,5,1,3} on concentric circles
   • Perform circular shifts and multiply-add for each n
   • Obtain 4-point circular convolution values
   • Verify with matrix / DFT method
9. 7 Marks
   aCompute the DFS coefficients for the periodic signal $x[n] = \{1, 2, 3, 4\}$ with period 4.
   How to answer

   • Use DFS analysis formula c_k = (1/N) sum x[n]W^{-kn}
   • Substitute x[n]={1,2,3,4}, N=4
   • Compute coefficients for k=0,1,2,3
   • Simplify using twiddle factors
10. 7 Marks
    bWhat is the Fast Fourier Transform (FFT)? How does it improve the efficiency of computing the DFT? Discuss with suitable example.
    How to answer

    • Define FFT as fast algorithm for DFT
    • Explain decimation and complexity O(N log N) vs O(N^2)
    • Give example e.g. 8-point comparison
    • Discuss butterfly computation
11. 7 Marks
    aCompute the 8-point DFT of the sequence $x[n] = \{1, -1, 1, -1, 0, 0, 0, 0\}$ using the DIF FFT algorithm. Show all intermediate steps and results.
    How to answer

    • Arrange input x[n]={1,-1,1,-1,0,0,0,0} for 8-point DIF
    • Draw 3-stage butterfly with bit-reversed output
    • Compute intermediate stage results
    • Obtain final DFT values
12. 7 Marks
    bGiven the sequence $x[n] = \{1, 2, 3, 4, 5, 6, 7, 8\}$, perform a single stage of the DIT FFT algorithm. Illustrate the decomposition and computation process.
    How to answer

    • Split x[n]={1,2,3,4,5,6,7,8} into even and odd subsequences
    • Show first-stage butterfly with twiddle factors W8
    • Compute one stage outputs
    • Explain further decomposition principle
13. 7 Marks
    aDiscuss the advantages and disadvantages of IIR filters compared to FIR filters in digital signal processing.
    How to answer

    • Compare efficiency, phase linearity, stability, order required
    • Advantages of IIR: lower order, sharp cutoff
    • Disadvantages: nonlinear phase, feedback stability issues
    • Summary table FIR vs IIR
14. 7 Marks
    bDesign a high-pass IIR filter with a cutoff frequency of $0.5\pi$ radians using the bilinear transformation method. Assume the analog prototype is a Butterworth filter of order 2 with a cutoff frequency of 1 radian/second.
    How to answer

    • Prewarp cutoff 0.5c0 and apply bilinear transformation
    • Transform order-2 Butterworth lowpass prototype to highpass
    • Obtain analog prototype with cutoff 1 rad/sec
    • Derive digital transfer function H(z)
15. 14 Marks
    Write a short note on any two.

    1. a)Frequency domain representation of discrete time signals
    2. b)Properties of Z-transform
    3. c)Design of FIR digital Filter
    4. d)Rational Z-Transform
    How to answer

    • Select any two topics from a-d
    • Give definition, key formulas/properties and diagram
    • Add example or design steps as applicable
    • Conclude with applications