EX-302 Signals and Systems - Jun 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
   aDescribe periodic and aperiodic signals as well as energy and power signals with neat sketch.
   How to answer

   • Define periodic signal x(t+T)=x(t) with example and sketch
   • Define aperiodic signal with example and sketch
   • Define energy signal and energy formula with condition
   • Define power signal and power formula with condition
2. 7 Marks
   bAnswer i) and ii):

   1. i)Find whether the following signals are energy or power signal or neither: 1) $x(t)=t^2u(t-1)$ 2) $x(t)=2u(t)-u(t-3)$
   2. ii)Check whether the following systems are time invariant or not: 1) $y(t)=t^2x(t)$ 2) $y(n)=x(n)$
   How to answer

   • Use energy integral and power average formulas for continuous-time signals
   • Test $x(t)=t^2u(t-1)$ for finite energy/power - neither
   • Test $x(t)=2u(t)-u(t-3)$ for finite energy - energy signal
   • Test time-invariance by checking delayed input vs delayed output for $y(t)=t^2x(t)$ and $y(n)=x(n)$
3. 7 Marks
   aWrite down the properties of Fourier transform and Laplace transform.
   How to answer

   • Define Fourier transform and list linearity, shifting, scaling, differentiation properties
   • Define Laplace transform and list ROC, shifting, convolution properties
   • Give formulas for each property
   • Compare existence conditions
4. 7 Marks
   bCarry out the Fourier Series Analysis of a signal $x(t) = A \sin \omega t$.
   How to answer

   • Express $A\sin\omega t$ in exponential form
   • Identify fundamental frequency and period
   • Compute Fourier series coefficients $a_n, b_n$ or $C_n$
   • Write trigonometric/exponential Fourier series result
5. 7 Marks
   aExplain region of convergence with suitable example.
   How to answer

   • Define ROC as set of s/z where transform converges
   • Explain ROC for causal, anti-causal, two-sided signals
   • Give suitable example e.g. $e^{-at}u(t)$
   • Sketch s-plane/z-plane ROC
6. 7 Marks
   bCompare Wavelet transform with Fourier Transform and Laplace transform.
   How to answer

   • Define Wavelet, Fourier and Laplace transforms
   • Compare basis functions, time-frequency resolution
   • Compare applications and limitations
   • Summarize in table
7. 7 Marks
   aA LTI system is described by following differential equation. Find out its impulse response assuming all conditions to be zero $$\frac{d^3y}{dx^3}+\frac{4}{3}\frac{d^2y}{dx^2}+\frac{2}{3}\frac{dy}{dx}+\frac{1}{3}y(t) = x(t)$$
   How to answer

   • Take Laplace transform with zero initial conditions
   • Obtain transfer function $H(s)=1/(s^3+...)$
   • Perform partial fraction expansion
   • Inverse Laplace to get $h(t)$ impulse response
8. 7 Marks
   bDiscuss Analog filters with their types, neat diagrams and applications.
   How to answer

   • Define analog filter and its frequency selective function
   • Explain types: low-pass, high-pass, band-pass, band-stop with neat magnitude plots
   • Give applications in communications, audio, instrumentation
   • Draw block/diagram for each type
9. 7 Marks
   aA LTI system is characterized by the transfer function $$H(z) = \frac{3-4z^{-1}}{1-3.5z^{-1}+1.5z^{-2}}.$$ Determine the $h(n)$ for the following conditions:

   1. i)The system is stable
   2. ii)The system is causal
   How to answer

   • Factor denominator to find poles at z=0.5 and z=3
   • Write partial fraction for H(z)
   • Choose ROC for stable case including unit circle
   • Choose ROC for causal case outside outermost pole and write h(n)
10. 7 Marks
    bWhat is first order discrete time LTI system? Write down the step response and impulse response of first order Discrete time LTI system.
    How to answer

    • Define first-order discrete-time LTI system with difference equation
    • Derive impulse response $h(n)=a^n u(n)$ form
    • Derive step response using convolution/summation
    • State stability/causality condition
11. 7 Marks
    aExplain even and odd signals with example and also derive the relationship between impulse and unit function.
    How to answer

    • Define even $x(-t)=x(t)$ and odd $x(-t)=-x(t)$ with examples and sketches
    • Show decomposition into even and odd parts
    • Define impulse $\delta(t)$ and unit step $u(t)$
    • Derive $u(t)=\int_{-\infty}^{t}\delta(\tau)d\tau$ and $\delta(t)=du/dt$
12. 7 Marks
    bDetermine whether or not each of the following is periodic? If it is periodic, find its period :

    1. i)$x(t) = \sin \frac{2\Pi}{3}t$
    2. ii)$x(t) = \cos \frac{1}{3}\Pi t + \sin \frac{1}{4}\Pi t$
    How to answer

    • Apply periodicity condition $x(t+T)=x(t)$
    • Find period of $x(t)=\sin(2\pi/3 t)$ as $T=3$
    • Find periods of components $\cos(1/3\pi t)$ and $\sin(1/4\pi t)$ and LCM for sum
    • Conclude periodic/non-periodic with fundamental period
13. 7 Marks
    aDetermine Z-transform with ROC of the following sequence.

    1. i)$x(n) = (n+1)u(n)$
    2. ii)$x(n) = [3(4^n + 3^n) u(n)]$
    How to answer

    • Use standard Z pairs $Z\{u(n)\}$, $Z\{n u(n)\}$
    • Compute Z of $(n+1)u(n)$ with ROC $|z|>1$
    • Compute Z of $3(4^n+3^n)u(n)$ with ROC $|z|>4$
    • State ROC reasoning from poles
14. 7 Marks
    bProve the convolution of Laplace and write its statement.
    How to answer

    • State convolution theorem: $L\{f*g\}=F(s)G(s)$
    • Start from Laplace integral definition
    • Interchange order of integration to prove
    • Give example application
15. 14 Marks
    Write short notes on two of the following:

    1. i)Digital filters
    2. ii)Analog filters
    3. iii)Linear time invariant systems and properties
    How to answer

    • Select any two topics from the three given
    • Define each with key properties/formulas
    • Give diagram/example and applications
    • Conclude with significance