EC-402 Signal and System - Jun 2022 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 following signals with examples.

   1. i)Continuous Time and discrete time
   2. ii)Periodic and Aperiodic
   3. iii)Energy and Power
   How to answer

   • Define continuous-time vs discrete-time signals with mathematical form
   • Define periodic vs aperiodic with condition and example
   • Define energy vs power signals with formulas and examples
2. 7 Marks
   bCheck whether the following are stable, causal and memory less

   1. i)h(n) = 2 u(n) - 2 u(n-2)
   2. ii)h(n) = $\delta(n) + \sin (n\pi)$
   How to answer

   • Apply BIBO stability condition sum |h(n)| < infinity
   • Apply causality condition h(n)=0 for n<0
   • Apply memoryless condition h(n)=C delta(n)
   • Conclude for each h(n)
3. 14 Marks
   Given the impulse response of a discrete time LTI system $H(n) = [-2 (1/3)^n + 3 (1/2)^n] u[n]$

   1. i)Find the system function H(z) of the system.
   2. ii)Find the difference equation representation of the system.
   3. iii)Find the step response of the system.
   How to answer

   • Take Z-transform of $h(n)$ to get $H(z)$ with ROC
   • Cross-multiply to obtain difference equation
   • Find step response via $S(z)=H(z)U(z)$ and inverse Z-transform
4. 7 Marks
   aFind out whether the following signals are periodic or not. If periodic find the period $x(t) = 2 \cos(10t+1) - \sin (4t-1)$ $x(n) = \cos(0.1\pi n)$
   How to answer

   • Check periodicity condition for sum of sinusoids
   • Compute fundamental period of $x(t)$ as LCM of component periods
   • Check $\cos(0.1\pi n)$ for discrete periodicity with $N=20$
5. 7 Marks
   bObtain the parallel realization of the system given by $y(n) - 3 y(n - 1) + 2 y(n - 2) = x(n)$
   How to answer

   • Find transfer function $H(z)$ from difference equation
   • Perform partial fraction expansion
   • Draw two parallel first-order sections with adders and delays
6. 14 Marks
   Derive the following properties of Z Transform

   1. i)Time Shifting
   2. ii)Initial Value Theorem
   3. iii)Convolution
   How to answer

   • State Z-transform definition and derive time-shifting property
   • Derive initial value theorem using limit $z \to \infty$
   • Derive convolution property $x_1*x_2 \leftrightarrow X_1(z)X_2(z)$
7. 7 Marks
   aDefine the ROC and its Properties.
   How to answer

   • Define ROC as set of z for which Z-sum converges
   • List properties: ring/strip, poles, causality, stability
   • Give examples for finite, right, left, two-sided sequences
8. 7 Marks
   bIf $X(e^{j\omega})$ is the DTFT of $x[n]$. Find the DTFT of $(n-1)^2 x[n]$ in term of $X(e^{j\omega})$
   How to answer

   • Use frequency differentiation property of DTFT
   • Use time-shifting to handle $(n-1)^2$ expansion
   • Express result in terms of $X(e^{j\omega})$ and its derivatives
9. 7 Marks
   aState and prove any three properties of Fourier Transform
   How to answer

   • State linearity, time-shifting, frequency-shifting, scaling, duality etc.
   • Prove any three with Fourier integral
   • Mention conditions for existence
10. 7 Marks
    bUsing the properties of Fourier Transform find the $X(j\omega)$ and $G(j\omega)$

    1. i)x(t) = $e^{-\alpha|t|} u(t); \alpha>0$
    2. ii)x(t) = $2/(1+ t^2)$
    How to answer

    • Apply Fourier transform pair for exponential and $1/(1+t^2)$
    • Use duality and scaling properties
    • Obtain $X(j\omega)$ expressions
11. 7 Marks
    aState and explain sampling theorem with necessary equation and illustration.
    How to answer

    • State Nyquist sampling theorem $f_s \ge 2f_m$
    • Explain aliasing with spectrum diagram
    • Give reconstruction formula with sinc interpolation
12. 7 Marks
    bExplain various methods of evaluation of state transition matrix with suitable example.
    How to answer

    • Define state transition matrix $\phi(t)=e^{At}$
    • Explain methods: Laplace inverse, Cayley-Hamilton, diagonalization, series
    • Illustrate with 2x2 matrix example
13. 14 Marks
    Write a short notes on any Two

    1. i)Convolution
    2. ii)Unilateral Z transforms
    3. iii)Applications of DTFT
    4. iv)State space analysis
    How to answer

    • Select any two topics from list
    • Give definition, formula and significance for each
    • Add example/diagram where applicable