EX-302 Signals and Systems - 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
   aWhat are the basic operations on signals? Explain.
   How to answer

   • Define signal operations: time shifting, time scaling, time reversal, amplitude scaling, addition, multiplication
   • Explain each operation with mathematical expression
   • Give illustrative example / sketch for each
2. 7 Marks
   bDefine basic properties of systems.
   How to answer

   • Define system properties: linearity, time-invariance, causality, stability, memory, invertibility
   • Give condition / test for each property
   • Give brief example of each
3. 7 Marks
   aThe output of the system is given as $y(t) = x(t) + x(t+2) + x(t-3)$, then check the given system is time variant or not.
   How to answer

   • Recall test for time-invariance: delayed input must give equally delayed output
   • Apply delay $t_0$ to input and compute $y(t-t_0)$ vs response to $x(t-t_0)$
   • Conclude system is time-invariant
4. 7 Marks
   bDefine basic standard signals with suitable diagrams.
   How to answer

   • Define unit impulse, unit step, ramp, exponential, sinusoidal, signum signals
   • Give mathematical definition for each
   • Sketch waveform for each with labelled axes
5. 7 Marks
   aFind the Fourier Transform of $x(t)=\cos wt$.
   How to answer

   • Express $\cos w t$ using Euler formula in terms of complex exponentials
   • Apply Fourier transform of complex exponential / impulse
   • Obtain $X(\omega)=\pi[\delta(\omega-w)+\delta(\omega+w)]$
6. 7 Marks
   bList out the properties of Laplace transform.
   How to answer

   • Define Laplace transform
   • List linearity, time-shifting, frequency-shifting, scaling, differentiation, integration, convolution properties
   • Give formula for each property with ROC remark
7. 7 Marks
   aExplain with suitable example what is state variable representation of an LTI CT system.
   How to answer

   • Define state, state variables, state vector, state equation and output equation
   • Write form $\dot{X}=AX+BU$, $Y=CX+DU$ for LTI CT system
   • Explain with suitable RLC / differential equation example
8. 7 Marks
   bDefine impulse response, step response and frequency response.
   How to answer

   • Define impulse response as output for delta input
   • Define step response as output for unit step input and its relation to impulse response
   • Define frequency response as steady-state response to sinusoid / Fourier transform of h(t)
9. 7 Marks
   aWrite a short note on properties of convolution.
   How to answer

   • Define convolution integral / sum
   • List commutative, distributive, associative, shift, impulse properties
   • Give formula for each property
10. 7 Marks
    bFind the transfer function of the system governed by the following impulse response. $h(t)=u(t)+3e^{-4t}u(t)$.
    How to answer

    • Recall transfer function $H(s)=L\{h(t)\}$
    • Take Laplace transform of $u(t)$ and $e^{-4t}u(t)$ terms
    • Add to get $H(s)=1/s+3/(s+4)$ with ROC
11. 7 Marks
    aDefine sampling and how to avoid aliasing effect in sampling.
    How to answer

    • Define sampling of continuous-time signal
    • Explain Nyquist criterion $f_s \ge 2f_m$
    • Explain anti-aliasing filter and how aliasing is avoided
12. 7 Marks
    bFind the inverse z-transform of the function $X(z)=z/(z+1)(z+2)$.
    How to answer

    • Express $X(z)/z$ and perform partial fraction expansion
    • Use standard inverse pairs for $z/(z+a)$
    • Obtain $x[n]$ sequence with ROC / causality assumption
13. 7 Marks
    aDiscuss difference equations of Linear Time Invariant Discrete Time Systems.
    How to answer

    • Define difference equation relating input and delayed outputs
    • Give general Nth-order LTI form with coefficients
    • Explain recursive / non-recursive forms and example
14. 7 Marks
    bDescribe analysis of DT LTI systems using DTFT.
    How to answer

    • Define DTFT and frequency response $H(e^{j\omega})$
    • Explain convolution property and solution of difference equation via DTFT
    • Mention applications to magnitude-phase analysis and stability
15. 14 Marks
    Write short notes on:

    1. a)Exponential signals
    2. b)Region of Convergence (ROC) of Laplace Transform
    3. c)Digital filters
    How to answer

    • Define exponential signals and give CT/DT expressions
    • Explain ROC properties with examples for Laplace transform
    • Define digital filters, types and brief characteristics