EC-402 Signal and System - 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
   aDistinguish between the following:

   1. i)Continuous time signal and discrete time signal
   2. ii)Unit step and Unit Ramp functions
   3. iii)Periodic and Aperiodic Signals
   4. iv)Deterministic and Random Signals
   How to answer

   • Define each pair with key distinguishing feature
   • Give mathematical definition / example for each
   • Tabular comparison of properties
   • Mention continuous vs discrete domain representation
2. 7 Marks
   bDetermine whether the following function is periodic or not. If so find the period. $$x(t) = 3\sin 200\pi t + 4\cos 100t$$
   How to answer

   • Find fundamental periods of $3\sin 200\pi t$ and $4\cos 100t$
   • Check ratio $T_1/T_2$ for rationality
   • Conclude periodic or non-periodic
   • Compute LCM for overall period if periodic
3. 7 Marks
   aCheck linearity and casualty of the following systems:

   1. i)$\frac{d^3y(t)}{dt^3} + 7\frac{d^2y(t)}{dt^2} + 9t\frac{dy(t)}{dt} + y(t) = x^2(t)$
   2. ii)$y(n) = x(n)x(n-2)$
   How to answer

   • Apply superposition test for linearity on each system
   • Check time-invariance / time-varying coefficient for (i)
   • Check causality: output depends only on present/past inputs
   • Conclude linear/nonlinear and causal/non-causal
4. 7 Marks
   bWhat is LTI system? How discrete time LTI system differs from continuous time LTI system. What is the importance of impulse response to LTI system? Explain in detail.
   How to answer

   • Define LTI system: linear and time-invariant
   • Contrast difference equation vs differential equation representation
   • Explain convolution with impulse response $h(t)$ / $h[n]$
   • State importance for analysis of LTI systems
5. 7 Marks
   aCalculate the output of the LTI system whose impulse response and input are given by $$h(t) = e^{-at}u(t) \text{ and } x(t) = e^{at}u(-t)$$
   How to answer

   • Write convolution integral $y(t)=x(t)*h(t)$
   • Substitute $h(t)=e^{-at}u(t)$ and $x(t)=e^{at}u(-t)$
   • Evaluate integral for $t$ ranges
   • Obtain closed-form output expression
6. 7 Marks
   bA LTI System is described by the following differential equation. Find out its impulse response by assuming all initial condition to be zero. $$3\frac{d^3y}{dt^3} + 4\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + y(t) = 3x(t)$$
   How to answer

   • Assume zero initial conditions and take Laplace transform
   • Find transfer function $H(s)=Y(s)/X(s)$
   • Find $H(s)$ for given third-order equation
   • Take inverse Laplace to get $h(t)$
7. 7 Marks
   aState and prove the time shifting, differentiation and integration properties of Z transform.
   How to answer

   • State definitions of Z-transform properties
   • Prove time-shifting using summation shift
   • Prove differentiation (multiplication by n) property
   • Prove summation/integration property
8. 7 Marks
   bFind the inverse Z-transform of $X(z) = 1/(1 + z)$ with ROC $|z| < 1$.
   How to answer

   • Rewrite $X(z)=1/(1+z)$ in terms of $z^{-1}$
   • Identify standard pair for $|z|<1$ (left-sided sequence)
   • Apply time-reversal / geometric series expansion
   • Write $x[n]$ with $u[-n-1]$
9. 7 Marks
   aProve that the sequence $x(n) = a^n u(n)$ and $x(n) = -a^n u(-n-1)$ have the same $X(z)$ and differ only in ROC. Also plot their ROC.
   How to answer

   • Derive $X(z)$ summation for $a^n u(n)$ with ROC $|z|>|a|$
   • Derive $X(z)$ for $-a^n u(-n-1)$ with ROC $|z|<|a|$
   • Show same algebraic expression $1/(1-az^{-1})$
   • Sketch z-plane ROCs inside/outside circle $|z|=|a|$
10. 7 Marks
    bFind the Fourier transform of $e^{-3t}\sin w_0 t\, u(t)$.
    How to answer

    • Write $\sin w_0 t$ in exponential form
    • Use Fourier transform of $e^{-3t}u(t)$ shifted in frequency
    • Apply linearity to combine $w_0$ shift terms
    • Obtain $X(jw)$ expression
11. 7 Marks
    aFind the Fourier series of the following discrete-time signal. $$x(n) = 1 + \sin\left(\frac{2\pi}{5}\right)n + 3\cos\left(\frac{2\pi}{5}\right)n + \cos\left(\frac{4\pi}{5}n + \frac{\pi}{2}\right)$$
    How to answer

    • Identify fundamental frequency from $2\pi/5$ terms
    • Expand sines/cosines into complex exponentials
    • Determine discrete Fourier series coefficients $a_k$
    • Write series in magnitude-phase form
12. 7 Marks
    bList and explain various properties of Fourier transform.
    How to answer

    • List linearity, time/frequency shift, scaling, duality
    • Explain convolution, multiplication, Parseval properties
    • Give mathematical statement for each property
    • Give brief proof or example
13. 7 Marks
    aState and prove sampling theorem for band-limited signals.
    How to answer

    • State Nyquist sampling condition $f_s \ge 2f_m$
    • Prove using impulse-train sampling and spectrum replication
    • Explain aliasing and reconstruction with low-pass filter
    • Draw spectra before and after sampling
14. 7 Marks
    bDescribe state-transition matrix and its role.
    How to answer

    • Define state-transition matrix $\phi(t)=e^{At}$
    • List properties: $\phi(0)=I$, $\phi^{-1}$, semigroup
    • Explain role in solving $\dot{x}=Ax+Bu$
    • Mention discrete-time counterpart
15. 14 Marks
    Write a short note on any two :

    1. a)State space analysis
    2. b)Convergence of discrete time Fourier Transform
    3. c)Unilateral Z-Transform
    4. d)Convolution
    How to answer

    • Select any two topics from (a)-(d)
    • Give definition and key formulas for each
    • Explain with example / diagram
    • Mention applications in signals and systems