EX-302 Signals and Systems - Dec 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
   aExplain energy and power signal. Also determine the following signals are energy or power signals.

   1. i)$x(t) = 5u(t) - u(t-6)$
   2. ii)$x(t) = 4t^2u(t)$
   How to answer

   • Define energy signal (finite energy, zero power) and power signal (finite non-zero power) with formulas for E and P
   • Give conditions for continuous-time signals
   • Evaluate i) x(t)=5u(t)-u(t-6) - time-limited rectangular pulse - finite energy
   • Evaluate ii) x(t)=4t^2u(t) - unbounded growing - neither energy nor power (infinite both)
2. 7 Marks
   bList the standard signals. Also explain the properties of systems.
   How to answer

   • List standard signals: impulse, step, ramp, exponential, sinusoidal, sinc, signum
   • Define each briefly in continuous and discrete time
   • Explain system properties: linearity, time-invariance, causality, stability, memory, invertibility
   • Give condition/example for each property
3. 7 Marks
   aCompare the Fourier series and Fourier transform.
   How to answer

   • Define Fourier series for periodic signals vs Fourier transform for aperiodic signals
   • Compare on basis: type of signal, time domain, frequency spectrum (discrete vs continuous)
   • Compare formulas, Dirichlet conditions, applications
4. 7 Marks
   bWrite the properties of Fourier transform and Laplace transform.
   How to answer

   • List Fourier transform properties: linearity, shifting, scaling, convolution, duality
   • List Laplace transform properties: linearity, time shift, s-shift, differentiation, integration
   • Give mathematical statement for each property
   • Mention ROC implication for Laplace
5. 7 Marks
   aDescribe the differential equations representation of LTI continuous time systems.
   How to answer

   • Write general Nth-order linear constant-coefficient differential equation
   • Explain input x(t), output y(t) and coefficients a_k, b_k
   • Discuss order, initial conditions and system response
   • Relate to LTI continuous-time system characterization
6. 7 Marks
   bExplain the impulse and step response of LTI system.
   How to answer

   • Define impulse response h(t) as output for delta(t) input
   • Define step response s(t) as output for u(t) input
   • Show relation s(t) = integral h(tau) d tau
   • Explain significance for LTI characterization via convolution
7. 7 Marks
   aDerive the relation between complex exponential and sinusoidal signals.
   How to answer

   • State Euler relation exp(jw0t)=cos w0t + j sin w0t
   • Express cos and sin in terms of complex exponentials
   • Show sinusoidal as combination of two complex exponentials
   • Mention magnitude, frequency and phase interpretation
8. 7 Marks
   bWrite short note on :

   1. i)Analog filters
   2. ii)ROC
   How to answer

   • Define analog filter: types, purpose, frequency response
   • Explain low-pass, high-pass, band-pass, band-stop concepts
   • Define ROC: region of convergence for Laplace/z-transform
   • Explain significance and properties of ROC
9. 7 Marks
   aDetermine the z-transform of the signal. $$x(n) = (-1)^n \left[ \cos \frac{\pi}{3} n \right] u(n)$$
   How to answer

   • Write x(n)=(-1)^n cos(pi n/3) u(n) as exponential form
   • Apply standard z-transform pairs for cos and a^n
   • Combine using linearity / modulation property
   • State final X(z) with ROC |z| > 1
10. 7 Marks
    bExplain the properties of z-transform.
    How to answer

    • List linearity, time-shifting, scaling, time-reversal properties
    • Explain differentiation, convolution, initial/final value
    • Give mathematical statements with ROC effects
    • Mention examples of use
11. 7 Marks
    aState and prove the Fourier transform based convolution theorem.
    How to answer

    • State convolution theorem: convolution in time equals multiplication in frequency
    • Write F{x1*x2}=X1(w)X2(w) statement
    • Prove using Fourier integral definition and change of variables
    • Mention dual multiplication-convolution property
12. 7 Marks
    bExplain signal reconstruction from its sample signal.
    How to answer

    • State sampling theorem and Nyquist rate condition
    • Explain ideal low-pass interpolation formula
    • Describe sinc reconstruction from samples
    • Mention practical zero-order hold reconstruction
13. 7 Marks
    aDescribe the matrix representation and state variable equations of system.
    How to answer

    • Define state vector, state equation x_dot=Ax+Bu
    • Define output equation y=Cx+Du
    • Explain matrix representation A,B,C,D meaning
    • Mention advantages for MIMO analysis
14. 7 Marks
    bExplain block diagram representation of LTI discrete time signals.
    How to answer

    • Show basic elements: adder, multiplier, unit delay
    • Draw direct form I/II block diagram for difference equation
    • Explain signal flow for LTI discrete-time system
    • Mention cascade and parallel realizations
15. 14 Marks
    Write short notes on:

    1. i)Digital filters
    2. ii)DTFT and its properties
    3. iii)Even and odd signals
    4. iv)Aliasing
    How to answer

    • Define digital filters: FIR vs IIR, purpose
    • Define DTFT and list its key properties
    • Define even and odd signals with examples
    • Define aliasing, cause and prevention via Nyquist