EC-402 Signal and System - Jun 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. 7 Marks
   aDefine signal and classify different types of signals with proper representation.
   How to answer

   • Define signal as function conveying information
   • Classification by continuity, periodicity, causality, determinism, energy/power
   • Give representation/examples for each type
2. 7 Marks
   bDifferentiate following

   1. i)Periodic and aperiodic signals
   2. ii)Random and deterministic signals
   3. iii)Even and odd signals
   How to answer

   • Define periodic vs aperiodic, random vs deterministic, even vs odd
   • Give mathematical condition and example for each
   • Tabulate differences
3. 7 Marks
   aDescribe following in brief

   1. i)Causal and noncausal system
   2. ii)Additivity and homogeneity
   3. iii)Causality and realizability
   How to answer

   • Define causal vs noncausal system with example
   • Define additivity and homogeneity for linearity
   • Define causality vs realizability condition
4. 7 Marks
   bWrite properties of the impulse response representation for LTI systems.
   How to answer

   • Define LTI impulse response $h(t)$/$h[n]$
   • List properties: convolution, causality, stability, memory
   • Explain eigenfunction and system interconnections
5. 7 Marks
   aDiscuss impulse response representation for LTI system and describe LTI system by difference equations.
   How to answer

   • Derive convolution sum/integral impulse response form
   • Explain difference equation representation
   • Relate coefficients to system order and realization
6. 7 Marks
   bExplain block diagram representations for following:

   1. i)Direct form - I and Direct form - II
   2. ii)Cascade and parallel form
   How to answer

   • Sketch Direct Form-I and Direct Form-II structures
   • Explain cascade and parallel interconnection
   • Compare delay/ multiplier requirements
7. 7 Marks
   aUsing partial fraction expansion find the inverse z-transform of $$X(z)=\frac{1-\frac{1}{3}z^{-1}}{(1-z^{-1})(1+2z^{-1})}, |z| > 2$$
   How to answer

   • Perform partial fraction expansion of $X(z)/z$ or $X(z)$
   • Identify poles $z=1, z=-2$ with $|z|>2$
   • Invert term-by-term to get right-sided $x[n]$
8. 7 Marks
   bConsider the signal $x[n]=\left(\frac{1}{5}\right)^{n} u[n-3]$. Evaluate the z-transform of this signal and specify the corresponding region of convergence.
   How to answer

   • Apply time-shift and scaling property to $(1/5)^n u[n-3]$
   • Evaluate $X(z)$ as rational function in $z$
   • Specify ROC as exterior of pole $|z|>1/5$
9. 7 Marks
   aDescribe following in brief

   1. i)ROC of finite and infinite duration sequence
   2. ii)Properties of the ROC and Z-transform
   How to answer

   • Define ROC for finite vs infinite duration sequences
   • List general properties of ROC and Z-transform
   • Give examples of pole locations and ROC
10. 7 Marks
    bDiscuss properties and application of discrete time Fourier series.
    How to answer

    • Define discrete-time Fourier series and coefficients
    • List periodicity, linearity, shift properties
    • Mention applications in filtering and spectral analysis
11. 7 Marks
    aUse the Fourier series analysis to calculate the coefficients $a_{k}$ for the continuous time periodic signal. $$x(t)=\begin{cases} 1.5 & 0 \le t < 1 \\ -1.5 & 1 \le t < 2 \end{cases}$$
    How to answer

    • Use analysis equation $a_k = (1/T)\int_T x(t)e^{-jk\omega_0 t}dt$ with $T=2$
    • Integrate piecewise 1.5 and -1.5 intervals
    • Simplify to closed-form $a_k$ and $a_0=0$
12. 7 Marks
    bCompute the Fourier transform of the following signals.

    1. i)$x(t)=\begin{cases} 1+\cos \pi t & |t| \le 1 \\ 0 & |t| > 2 \end{cases}$
    2. ii)$x(t)=\left[t e^{-2t} \sin 4t\right]u(t)$
    How to answer

    • Express $1+\cos \pi t$ via exponentials for finite-duration pulse
    • Apply Fourier integral / frequency-shift and differentiation properties for $t e^{-2t}\sin 4t u(t)$
    • Write final $X(j\omega)$ expressions
13. 7 Marks
    aDiscuss convergence of discrete time Fourier transform and write applications of DTFT.
    How to answer

    • State existence condition: absolute summability / square summability
    • Discuss convergence in terms of ROC including unit circle
    • List applications: frequency analysis, filtering, system response
14. 7 Marks
    bHow multi-input, multi-output systems are represented in state space? Take 3 input 2 output system and represent it in state space form.
    How to answer

    • Write MIMO state equations $\dot X=AX+BU, Y=CX+DU$
    • Give dimensions for 3-input 2-output case
    • Construct example A,B,C,D matrices representation
15. 7 Marks
    aWrite properties and role of state transition matrix. Also describe any two methods to determine state transition matrix.
    How to answer

    • Define state transition matrix $\phi(t)=e^{At}$
    • List properties: $\phi(0)=I$, $\phi^{-1}$, semigroup
    • Describe Laplace and Cayley-Hamilton methods
16. 7 Marks
    bDiscuss following in detail:

    1. i)Sampling theorem and its implications
    2. ii)Reconstruction of a signal from its samples
    How to answer

    • State Nyquist sampling theorem and aliasing implications
    • Explain ideal interpolation / sinc reconstruction formula
    • Mention practical sampling and reconstruction steps