EC-601 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
   aDiscuss the direct form-I and direct form-II implementations of discrete-time systems. What are the main differences between these implementations?
   How to answer

   • Define Direct Form-I structure with separate feedforward and feedback sections
   • Define Direct Form-II with shared delay line / canonic structure
   • Compare number of delays, computational complexity and memory
   • Illustrate with block diagrams and difference-equation realization
2. 7 Marks
   bSolve the second-order difference equation $y[n]-0.5y[n-1]+0.25y[n-2] = x[n]$ for the input $x[n]=\delta[n]$, given the initial conditions $y[-1] = 1$ and $y[-2] = 0$.
   How to answer

   • Write difference equation with $x[n]=\delta[n]$ and given initial conditions
   • Solve iteratively for $n \ge 0$ or using Z-transform with initial conditions
   • Obtain homogeneous plus particular solution
   • Present final $y[n]$ for $n \ge 0$
3. 7 Marks
   aFor the system described by $y[n]=0.6y[n-1]+x[n]$, determine whether the system is stable and causal or not. Justify your answer.
   How to answer

   • Find transfer function / impulse response $h[n]=(0.6)^n u[n]$
   • Test causality from difference equation and $h[n]$
   • Test BIBO stability via absolute summability / pole at 0.6 inside unit circle
   • Conclude stable and causal with justification
4. 7 Marks
   bExplain the significance of pole-zero analysis in understanding the behavior of discrete-time systems. How do poles and zeros affect system stability and frequency response.
   How to answer

   • Define poles and zeros of $H(z)$
   • Explain relation of pole locations to stability and ROC
   • Explain effect on magnitude/phase frequency response
   • Give examples of pole-zero plots and interpretations
5. 7 Marks
   aGiven $X(z)= z/(z-0.5)$, use the properties of the z-transform to find the z-transform of $x[n] = (0.5)^n \cos(\pi n)u[n]$.
   How to answer

   • Recall $Z\{(0.5)^n u[n]\}=z/(z-0.5)$
   • Use modulation / frequency-shift property for multiplication by $\cos(\pi n)=(-1)^n$
   • Apply property to get $X$ shifted at $z \to -z$
   • Simplify to final Z-transform and state ROC
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

   • Identify state matrices A, B, C from given equations
   • Draw delay branches for state variables
   • Add gain branches 0.5, 1 and summing nodes for input/output
   • Show complete signal flow graph
7. 7 Marks
   aState and prove following properties of DFT

   1. i)Linearity
   2. ii)Circular time shifting
   3. iii)Circular frequency shifting
   4. iv)Time reversal
   How to answer

   • State DFT definition and each property mathematically
   • Prove linearity using summation linearity
   • Prove circular shift properties using modular index substitution
   • Prove time reversal property
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 on concentric circles
   • Perform circular folding, shifting and multiply-add for each n
   • Compute 4-point circular convolution values
   • Verify result
9. 7 Marks
   aCompute the 2D DFT of the following $2\times 2$ image matrix: $$\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$$
   How to answer

   • Apply 1D DFT along rows of 2x2 matrix
   • Apply 1D DFT along columns of intermediate result
   • Compute all four 2D DFT coefficients
   • Present final 2x2 DFT matrix
10. 7 Marks
    bDescribe the Decimation in Time (DIT) FFT algorithm. How does it decompose the computation of the DFT? Explain.
    How to answer

    • State principle of decimation in time by even/odd indices
    • Derive decomposition of N-point DFT into two N/2-point DFTs
    • Explain butterfly computation and twiddle factors
    • Discuss recursive stages and complexity reduction
11. 7 Marks
    aFor a sequence of length $N = 6$ (where $N = 2\times 3$), describe how the FFT can be applied using the Cooley-Tukey algorithm. Perform the computation for $x[n] = \{1, 2, 3, 4, 5, 6\}$.
    How to answer

    • Explain factorization N=2x3 into smaller DFTs
    • Describe Cooley-Tukey index mapping and stages
    • Compute 2-point and 3-point DFTs for x[n]={1,2,3,4,5,6}
    • Combine with twiddle factors to get 6-point DFT
12. 7 Marks
    bImplement the Radix-2 FFT algorithm for the sequence $x[n] = \{1, 1, 1, 1, 0, 0, 0, 0\}$. Provide the detailed steps and the final DFT result.
    How to answer

    • Arrange 8-point sequence for DIT/DIF stages
    • Compute butterflies stage by stage with twiddle factors
    • Show intermediate results after each stage
    • Present final 8-point DFT result
13. 7 Marks
    aCompare and contrast the impulse invariant transformation and bilinear transformation methods for IIR filter design.
    How to answer

    • Define impulse invariant transformation and mapping of s to z
    • Define bilinear transformation and frequency warping
    • Compare aliasing, stability preservation and design steps
    • Summarize advantages/disadvantages in table
14. 7 Marks
    bDesign a low-pass FIR filter with a cutoff frequency of $0.4\pi$ radians and a filter length of 21 using the rectangular window method. Provide the filter coefficients and plot the frequency response.
    How to answer

    • Compute ideal low-pass impulse response with cutoff 0.4π and M=21
    • Multiply by rectangular window to get filter coefficients
    • List symmetric coefficients h[n]
    • Sketch/plot magnitude frequency response
15. 14 Marks
    Write a short note on any two

    1. a)Rational Z-Transform
    2. b)Properties of DFS
    3. c)Discrete time signals
    4. d)Decomposition for 'N' Composite Number
    How to answer

    • Select any two topics from list
    • Give definition, mathematical form and key properties
    • Add example/diagram where relevant
    • Conclude with applications