EX-302 Signals and Systems - Dec 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
   aExplain even and odd signals with example and also derive the relationship between impulse and unit function.
   How to answer

   • Define even signal x(-t)=x(t) and odd signal x(-t)=-x(t) with sketches
   • Give examples e.g. cos(t) even, sin(t) odd
   • State unit step u(t) and impulse delta(t) definitions
   • Derive relation delta(t)=du/dt and u(t)=integral delta
2. 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

   • Recall condition for periodicity and period formula T=2pi/|omega|
   • Check i) single sinusoid for period
   • Check ii) ratio of periods for common period via LCM
   • Conclude periodic / non-periodic with period value
3. 7 Marks
   aWhat is ROC? Explain ROC of Z-transform and write down the properties of Z-transform.
   How to answer

   • Define ROC as set of z where Z-transform sum converges
   • Explain ROC for right, left, two-sided sequences with z-plane sketches
   • List Z-transform properties: linearity, shifting, scaling, etc.
   • Mention ROC implications for causality and stability
4. 7 Marks
   bFind the inverse Fourier transform of the signal $$X(j\omega) = \frac{2j\omega+1}{(2j\omega+1)^2}$$
   How to answer

   • Write given X(jω) in partial fraction form
   • Use standard Fourier pairs e.g. e^{-at}u(t)
   • Apply frequency differentiation / duality property
   • Obtain x(t) for t>0 and t<0
5. 7 Marks
   aWrite down the properties of Fourier transform and prove them.
   How to answer

   • List properties: linearity, time/frequency shifting, scaling, duality, differentiation, convolution
   • State mathematical form of each property
   • Provide proof for key properties using integral definition
   • Give example application
6. 7 Marks
   bWhat is wavelet transform? Explain with an example along with mathematical support.
   How to answer

   • Define wavelet transform and mother wavelet concept
   • Write continuous wavelet transform integral formula
   • Explain scaling and translation with example
   • Discuss advantage for time-frequency analysis
7. 7 Marks
   aA causal and stable continuous time LTI system is described by following differential equation. Determine whether the system is over damped, under damped or critically damped. $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y(t) = x(t)$$
   How to answer

   • Take Laplace transform to get characteristic equation s^2+4s+4=0
   • Find roots / damping ratio zeta
   • Compare zeta with 1 for damping classification
   • Conclude critically damped and comment on causality-stability
8. 7 Marks
   bWhat is interconnection of discrete time LTI system? Explain in detail.
   How to answer

   • Define series, parallel and cascade interconnections
   • Give input-output relations using impulse responses / transfer functions
   • Draw block diagrams for each interconnection
   • Mention overall system function for each case
9. 7 Marks
   aDraw the block diagram representation in direct form, cascade form and parallel form for discrete-time LTI system described by transform. $$H(z) = \frac{1}{\left(1+\frac{z^{-1}}{3}\right)\left(1-\frac{z^{-1}}{6}\right)}$$
   How to answer

   • Express H(z) in terms of z^{-1} polynomials
   • Draw direct form using delays, adders, multipliers
   • Factor into first-order sections for cascade form
   • Partial fraction expansion for parallel form with diagrams
10. 7 Marks
    bExplain the linear convolution of a sequence with an example.
    How to answer

    • Define linear convolution sum formula
    • Explain steps: folding, shifting, multiplication, summation
    • Work out small numerical example
    • Mention properties and distinction from circular convolution
11. 7 Marks
    aFind whether the following signals are energy or power signal or neither:

    1. i)$x(t) = t^2u(t)$
    2. ii)$x(t) = 2u(t-1) - u(t-3)$
    How to answer

    • Recall energy vs power signal definitions via integrals
    • Evaluate energy integral for x(t)=t^2 u(t)
    • Evaluate energy/power for finite-duration pulse 2u(t-1)-u(t-3)
    • Classify as energy, power or neither
12. 7 Marks
    bWhat are analog and digital signals? Explain with mathematical support.
    How to answer

    • Define analog continuous-time/amplitude signal
    • Define digital discrete-time/quantized signal
    • Give mathematical representation and sketches
    • Compare with examples
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 pair u(n) <-> 1/(1-z^{-1})
    • Apply differentiation in z-domain for (n+1)u(n)
    • Use linearity for combination of 4^n and 3^n
    • State ROC |z|>max pole magnitude
14. 7 Marks
    bProve the convolution of Laplace and write its statement.
    How to answer

    • State Laplace convolution theorem L{f*g}=F(s)G(s)
    • Write convolution integral definition
    • Prove by interchanging order of integration
    • Mention ROC / existence condition
15. 14 Marks
    Write short notes on two of the following:

    1. i)Digital filters.
    2. ii)Non ideal frequency selective filter.
    3. iii)Invertible and inverse system.
    How to answer

    • Select any two topics from list
    • Give definition and principle with diagram/formula
    • Mention applications and advantages/limitations
    • Conclude with example