EC-601 Digital Signal Processing - May 2022 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 System. Determine whether the system $y[n] = 2x[n]u[n]$ is

   1. i)Memoryless
   2. ii)Stable
   3. iii)Time invariant
   4. iv)Causal
   How to answer

   • Definition of system as mapping of input to output
   • Check memoryless: output depends only on present input
   • Check BIBO stability with bounded input
   • Check time-invariance by shift test and causality by dependence on future
2. 7 Marks
   bDefine the terms: linearity, time invariance and causality for a discrete time system.
   How to answer

   • Define linearity via superposition: additivity and homogeneity
   • Define time invariance via shift preservation
   • Define causality via no dependence on future inputs
   • Give mathematical conditions for discrete-time system
3. 7 Marks
   aDescribe z-transform and its properties.
   How to answer

   • Definition of z-transform summation formula
   • Region of convergence concept
   • List properties: linearity, shifting, scaling, convolution
   • Relation to system analysis
4. 7 Marks
   bFind the transfer function of the given signal flow graph of the system.

   

   How to answer

   • Identify forward paths and loops from signal flow graph
   • Apply Mason's gain formula
   • Compute determinants Delta and path determinants
   • Simplify to get C(s)/R(s) transfer function
5. 7 Marks
   aExplain DFS (Discrete Fourier Series) and its properties.
   How to answer

   • Definition of DFS for periodic sequences
   • Analysis and synthesis equations
   • Key properties: periodicity, linearity, shift, convolution
   • Relation to DFT
6. 7 Marks
   bCompute 8-point DFT of the sequence $x(n) = \{1, 1, 1, 1, 1, 0, 0\}$.
   How to answer

   • Use 8-point DFT formula X(k)=sum x(n)W^nk
   • Compute twiddle factors for N=8
   • Evaluate summation for k=0 to 7
   • List 8 DFT coefficients
7. 7 Marks
   aCompute the FFT for the sequence $x(n) = n + 1$ where $N = 8$ using the DIT algorithm.
   How to answer

   • Form sequence x(n)=1 to 8 for N=8
   • Draw 3-stage DIT butterfly with bit-reversed input
   • Compute stage-wise outputs using twiddle factors
   • Obtain 8-point FFT values
8. 7 Marks
   bExplain about decimation in frequency FFT algorithm.
   How to answer

   • Principle of decimation in frequency decomposition
   • Derivation of even and odd frequency splits
   • Butterfly structure and bit-reversal of output
   • Comparison with DIT flow
9. 7 Marks
   aExplain briefly about different structures in FIR systems.
   How to answer

   • Direct form FIR structure
   • Cascade and linear-phase structures
   • Lattice structure
   • Diagrams for each realization
10. 7 Marks
    bDiscuss the characterization of IIR filters and compare with FIR filters.
    How to answer

    • Characterization: poles, feedback, infinite impulse response
    • Design from analog filters
    • Comparison table: stability, phase, complexity, memory
    • Applications of each
11. 7 Marks
    aExplain the classification of discrete time signals and systems.
    How to answer

    • Classification of signals: energy/power, periodic, even/odd, etc.
    • Classification of systems: linear, causal, stable, etc.
    • Examples for each category
    • Block representation
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 terms
    • Use standard pairs: 1/(z-a) and delay property
    • Apply inverse z-transform term-wise
    • Write final x(n) with unit step functions
13. 7 Marks
    aFind circular convolution for the following sequences $x_{1}(n) = \{1, 2, 3, 4\}$ and $x_{2}(n) = \{1, 2, 1, 2\}$.
    How to answer

    • Use circular convolution formula or matrix method
    • Compute for N=4 point
    • Perform modulo-4 index summation
    • Obtain 4-point result sequence
14. 7 Marks
    bCompute DFT of the sequence $x(n) = \{1, 2, 3, 4, 4, 3, 2, 1\}$ using the DIF FFT algorithm.
    How to answer

    • Arrange input in natural order for DIF
    • Draw 3-stage DIF butterfly with twiddle factors
    • Compute stage outputs
    • Obtain bit-reversed DFT output
15. 14 Marks
    Write short notes on:

    1. i)DFT
    2. ii)Bilinear transformation method
    3. iii)Rectangular window
    4. iv)Block diagram
    How to answer

    • Define DFT with formula
    • Explain bilinear transformation s to z mapping
    • Describe rectangular window function and effects
    • Explain general DSP block diagram