EC-402 Signal and System - Jun 2025 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 criteria
   • Give mathematical expression / example for each
   • Tabulate differences in tabular form
   • Mention continuous vs discrete domain and examples
2. 7 Marks
   bDetermine whether the systems described by the equations are linear, time invariant, dynamic and stable:

   1. i)$y_1(t) = x(t-3) + (3-t)$
   2. ii)$y_1(t) = \frac{dx(t)}{dt}$
   3. iii)$y(n) = nx(n) + bx^2(n)$
   How to answer

   • Test linearity via superposition / homogeneity
   • Test time-invariance via delayed input vs delayed output
   • Check memory / dynamic vs static and causality
   • Check BIBO stability via bounded input condition
3. 7 Marks
   aDetermine whether the following signals are energy or power signals.

   1. i)$x(t) = tu(t)$
   2. ii)$x(t) = u(t)e^{-at}$
   How to answer

   • Recall energy $E=\int|x|^2$ and power $P=\lim$ definitions
   • Evaluate integral for $x(t)=tu(t)$ for infinite energy
   • Evaluate integral for $x(t)=u(t)e^{-at}$ with parameter a
   • Classify as energy, power or neither
4. 7 Marks
   bGive block diagram representation of linear time invariant discrete time systems and also state their properties.
   How to answer

   • Draw direct-form block diagram with delays, adders, multipliers
   • Show convolution-sum representation $y(n)=\sum h(k)x(n-k)$
   • List properties: linearity, time-invariance, causality, stability, LTI convolution
5. 7 Marks
   aThe input and output of a causal LTI system are related by the differential equation: $$d^2y(t)/dt^2 + 6dy(t)/dt + 8y(t) = 2x(t)$$ Find the impulse response of the system
   How to answer

   • Take Laplace transform with zero initial conditions for causal system
   • Find transfer function $H(s)=2/(s^2+6s+8)$
   • Partial fraction expansion into first-order terms
   • Inverse Laplace to get $h(t)$ multiplied by $u(t)$
6. 7 Marks
   bFind the convolution of two signals $x(t) = u(t-1) - u(t+1)$ and $h(t) = e^{-at}u(t),\ a > 0.$
   How to answer

   • Write convolution integral $y(t)=\int x(\tau)h(t-\tau)d\tau$
   • Express $x(t)$ window from given step difference
   • Split integration limits based on $u(t)$ causality
   • Evaluate exponential integral to get closed form
7. 7 Marks
   aState and prove the time shifting, differentiation and integration properties of Z transform.
   How to answer

   • Define Z-transform $X(z)=\sum x(n)z^{-n}$
   • Prove time-shifting: $x(n-k) \leftrightarrow z^{-k}X(z)$
   • Prove differentiation / multiplication by n property
   • Prove summation / integration property with ROC change
8. 7 Marks
   bUsing properties of z-transform, find z-transform and ROC of signal. $$\frac{1}{(1-z^{-1})^2}$$
   How to answer

   • Recall standard pair $u(n) \leftrightarrow 1/(1-z^{-1})$
   • Apply differentiation-in-z-domain / multiplication by n property
   • Obtain transform of $n u(n)$ as $z^{-1}/(1-z^{-1})^2$
   • State ROC $|z|>1$ for right-sided sequence
9. 7 Marks
   aExplain the concept of ROC and discuss about different properties of ROC.
   How to answer

   • Define ROC as set of z where Z-sum converges
   • Explain ROC does not contain poles
   • Discuss ROC shapes for right, left, two-sided sequences
   • Relate ROC to causality and stability
10. 7 Marks
    bDiscuss about the DTFT and its properties.
    How to answer

    • Define DTFT $X(e^{j\omega})=\sum x(n)e^{-j\omega n}$
    • State periodicity, linearity, time/frequency shifting
    • Explain convolution and Parseval properties
    • Give existence / convergence condition
11. 7 Marks
    aFind the exponential Fourier series representation of the signal $x(t) = \cos^2 t$
    How to answer

    • Use identity $\cos^2 t = (1+\cos 2t)/2$
    • Identify fundamental frequency $\omega_0=2$
    • Compute exponential coefficients $C_k$ via Euler formula
    • Write series $x(t)=\sum C_k e^{jk\omega_0 t}$
12. 7 Marks
    bCompute the Fourier transform of the signal x(t) applying differentiation-in-time property of the Fourier transform.

    

    How to answer

    • Sketch triangular $x(t)$ and find its second derivative as impulses
    • Apply differentiation-in-time: $d^2x/dt^2 \leftrightarrow (j\omega)^2 X(\omega)$
    • Find Fourier transform of impulse train
    • Divide by $(j\omega)^2$ to obtain $X(\omega)$ as sinc-squared
13. 7 Marks
    aDiscuss state variable representation and matrix representation of systems.
    How to answer

    • Define state vector, state equation $\dot X=AX+BU$
    • Define output equation $Y=CX+DU$
    • Explain matrix representation with A,B,C,D matrices
    • Give example of obtaining state model from differential equation
14. 7 Marks
    bThe spectral range of a signal extends from 5.6 MHz to 6.8 MHz. Find the minimum sampling rate and maximum sampling time.
    How to answer

    • Identify highest frequency $f_H=6.8$ MHz and bandwidth $B=1.2$ MHz
    • Apply bandpass sampling criterion $f_s \ge 2f_H/k$
    • Compute minimum sampling rate
    • Compute maximum sampling interval $T_s=1/f_s$
15. 14 Marks
    Write a short note on any two:

    1. a)Sampling Theorem and its applications.
    2. b)Convergence of discrete time Fourier Transform
    3. c)Unilateral Z-Transform
    4. d)Basic operation on Signals for DT Signal
    How to answer

    • Select any two topics from a-d
    • Give definition, statement / mathematical form
    • Explain key conditions and diagram if needed
    • Mention applications / significance