EC-402 Signal and System - 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
   aExplain the classification of signals in detail.
   How to answer

   • Definition of signal and need for classification
   • Continuous vs discrete, analog vs digital, periodic vs aperiodic
   • Energy vs power, causal vs non-causal, deterministic vs random
   • Even vs odd, bounded vs unbounded with examples
2. 7 Marks
   bDetermine which of the following signals are energy signal, power signals, neither energy nor power signals.

   1. i)$x(t) = \cos t$
   2. ii)$x(n) = \left(1/3\right)^{n} u(n)$
   How to answer

   • Formula for energy $E$ and power $P$ for CT and DT signals
   • Check $x(t)=\cos t$: infinite energy, finite non-zero power -> power signal
   • Check $x(n)=(1/3)^n u(n)$: convergent geometric sum -> finite energy -> energy signal
   • Conclusion with calculated values
3. 7 Marks
   aCompare linear and non-linear system with examples.
   How to answer

   • Definition of linearity: superposition, homogeneity and additivity
   • Test for linear vs non-linear systems
   • Examples of linear systems e.g. $y=2x$, $y=x_1+x_2$
   • Examples of non-linear e.g. $y=x^2$, $y=\cos x$ with comparison table
4. 7 Marks
   bExplain causal, non-causal and anti-causal system with suitable diagram.
   How to answer

   • Definitions based on dependence on present, past, future values
   • Causal: $h(n)=0, n<0$; non-causal: depends on future; anti-causal: only future
   • Block / timing diagrams for each type
   • Examples of each system
5. 7 Marks
   aExplain the properties of impulse response representation of LTI system.
   How to answer

   • LTI representation by convolution with $h(t)$/$h(n)$
   • Properties: memory, causality, stability, invertibility via $h$
   • Step response $s = u*h$, interconnections of LTI systems
   • Eigenfunction property for complex exponentials
6. 7 Marks
   bDetermine the impulse response of given causal LTI system: $$y(n)-\left(\frac{1}{4}\right)y(n-1)-\left(\frac{3}{8}\right)y(n-2) = -x(n)+3x(n-1)$$
   How to answer

   • Take z-transform of difference equation assuming causality and zero initial conditions
   • Find $H(z)=Y(z)/X(z)$ and partial fraction expansion
   • Inverse z-transform to get $h(n)$ with ROC $|z|>$ max pole
   • Verify causality $h(n)=0, n<0$
7. 7 Marks
   aExpress the z-transform of given signal in terms of X(z). $$y(n) = \sum_{k=-\infty}^{n} x(k)$$
   How to answer

   • Use accumulation property: $y(n)-y(n-1)=x(n)$
   • Take z-transform: $Y(z)(1-z^{-1})=X(z)$
   • Obtain $Y(z)=X(z)/(1-z^{-1})$ with ROC
   • State summation / integration in z-domain interpretation
8. 7 Marks
   bWrite short note on:

   1. i)Properties of z-transform
   2. ii)Unilateral z-transform
   How to answer

   • Definition and list of z-transform properties: linearity, shifting, scaling, convolution
   • Definition of unilateral z-transform with summation $0$ to $\infty$
   • Differences from bilateral transform and use for initial conditions
   • Examples / ROC notes
9. 7 Marks
   aExplain the properties and application of Discrete time Fourier series.
   How to answer

   • Definition of DTFS for periodic $x(n)$ with period $N$
   • Properties: linearity, shifting, periodicity, Parseval, convolution
   • Applications in filtering, spectral analysis of periodic sequences
   • Relation to DFT
10. 7 Marks
    bExplain the Fourier transform and its properties.
    How to answer

    • Definition of CTFT/DTFT $X(\omega)$
    • Properties: linearity, time/frequency shift, scaling, duality, Parseval
    • Convergence and symmetry for real signals
    • Examples of use
11. 7 Marks
    aWhat is state transition matrix? Explain in detail.
    How to answer

    • Definition $\Phi(t)=e^{At}$, state equation $\dot X=AX+BU$
    • Properties: $\Phi(0)=I$, $\Phi^{-1}(t)=\Phi(-t)$, $\Phi(t_1+t_2)$
    • Methods: Laplace $ (sI-A)^{-1}$, Cayley-Hamilton, diagonalization
    • Use in zero-input and zero-state response
12. 7 Marks
    bState and prove sampling theorem in detail.
    How to answer

    • Statement: $f_s \ge 2f_m$ for exact reconstruction
    • Proof via spectrum of sampled signal and aliasing diagram
    • Nyquist rate and Nyquist interval definitions
    • Reconstruction by ideal low-pass interpolation formula
13. 7 Marks
    aDefine ROC and its properties. Explain Inverse z-transform.
    How to answer

    • Definition of ROC as set of $z$ for which $X(z)$ converges
    • Properties for right/left/two-sided, finite duration, causal/anti-causal
    • Methods of inverse z-transform: partial fraction, power series, contour
    • Example relating ROC to causality/stability
14. 7 Marks
    bFind the DTFT of the signal and draw its spectrum. $$x(n) = a^{n}u(n); |a| < 1$$
    How to answer

    • DTFT sum $X(e^{j\omega})=\sum a^n u(n)e^{-j\omega n}=1/(1-ae^{-j\omega})$
    • Magnitude $|X|=1/\sqrt{1+a^2-2a\cos\omega}$ and phase expression
    • Sketch magnitude low-pass shape and linear-like phase for $|a|<1$
    • Periodicity $2\pi$ and convergence condition
15. 14 Marks
    Write a short note on :

    1. i)Time Invariant and time varying system
    2. ii)Properties of LTI system
    3. iii)Periodic and non-periodic signals.
    How to answer

    • Time-invariant vs time-varying: delay test with examples
    • LTI properties: linearity, causality, stability, memory via $h$
    • Periodic vs non-periodic: condition $x(t)=x(t+T)$ with examples
    • Diagrams and mathematical tests for each