EC-402 Signal and System - Nov 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
   aDefine signal. Classify the different types of signal in detail using time domain technique.
   How to answer

   • Definition of signal as function conveying information
   • Time-domain classification: continuous/discrete, analog/digital, periodic/aperiodic
   • Energy/power, causal/non-causal, even/odd, bounded/unbounded with examples
   • Diagrammatic illustration of types
2. 7 Marks
   bIdentify the following signals are energy signal or power signal or neither-

   1. i)$x(t) = e^{j\left(2t+\frac{\pi}{4}\right)}$
   2. ii)$x(t) = \left(\frac{1}{3}\right)^{n} u(n)$
   How to answer

   • Formula for energy $E=\int|x(t)|^2dt$ / $\sum|x[n]|^2$ and power $P$
   • Check i) complex exponential magnitude =1, infinite energy, finite power
   • Check ii) decaying exponential times $u(n)$, summable square => energy signal
   • Conclude energy / power / neither
3. 7 Marks
   aDefine system. Explain the different types of system with examples.
   How to answer

   • Definition of system as entity relating input to output
   • Classification: linear/non-linear, time-invariant/variant, causal/non-causal
   • Static/dynamic, stable/unstable, invertible/non-invertible with examples
   • Continuous/discrete examples
4. 7 Marks
   bDetermine whether the following systems-

   1. i)$y(t) = tx(t)$
   2. ii)$y(n) = a^{\{x(n)\}}$
   How to answer

   • Test properties: linearity, time-invariance, causality, stability, memory
   • Analyze i) $y(t)=tx(t)$ for linearity, time-variance
   • Analyze ii) $y(n)=a^{x(n)}$ for linearity, causality
   • Conclude classification
5. 7 Marks
   aWhat is Convolution? Find the Convolution integral of the signals $x(t) = e^{-2t} u(t)$ and $h(t) = e^{-4t} u(t)$.
   How to answer

   • Definition of convolution for CT signals
   • Convolution integral formula $y(t)=\int x(\tau)h(t-\tau)d\tau$
   • Substitute $e^{-2t}u(t)$ and $e^{-4t}u(t)$ and evaluate limits
   • Result $(e^{-2t}-e^{-4t})u(t)$
6. 7 Marks
   bComment on the causality and stability of the given system $h(t) = (2 + e^{-3t}) u(t)$ Also find the step response of the system.
   How to answer

   • Condition for causality: $h(t)=0$ for $t<0$
   • Condition for BIBO stability: $\int|h(t)|dt < \infty$
   • Test $h(t)=(2+e^{-3t})u(t)$ for both
   • Find step response $s(t)=\int_{-\infty}^{t}h(\tau)d\tau$
7. 7 Marks
   aShow a direct form-I realization of the transfer function- $$H(s) = \frac{s+5}{s^{2}+2s+4}$$
   How to answer

   • Rewrite $H(s)$ as ratio of polynomials in $s^{-1}$
   • Identify feedforward and feedback coefficients
   • Draw direct form-I with two cascaded delay chains
   • Label adders, multipliers $5,2,4$
8. 7 Marks
   bState and prove initial value theorem of z transform. Also find the initial value and final value of the given signal. $$X(z) = \frac{z(z+1)}{z^{2}+z-2}$$
   How to answer

   • Statement of initial value theorem: $x[0]=\lim_{z\to\infty}X(z)$
   • Proof from definition $X(z)=\sum x[n]z^{-n}$
   • Apply to $X(z)=z(z+1)/(z^2+z-2)$ for initial value
   • Apply final value theorem $\lim_{n\to\infty}x[n]=\lim_{z\to1}(z-1)X(z)/z$ with ROC check
9. 7 Marks
   aFind z-transform of the given signal- $$x[n] = \begin{cases} n, & 0 \le n \le N-1 \\ N, & N \le n \end{cases}$$
   How to answer

   • Definition $X(z)=\sum x[n]z^{-n}$ split into two sums
   • Evaluate finite sum $\sum_{n=0}^{N-1}nz^{-n}$ and infinite tail $N\sum_{n=N}^{\infty}z^{-n}$
   • Closed form using geometric series formulas
   • Mention ROC $|z|>1$
10. 7 Marks
    bState and prove any four z-transform properties.
    How to answer

    • List linearity, time-shifting, scaling, time-reversal etc.
    • Proof using $X(z)$ definition with substitution
    • Show ROC changes where applicable
    • Give example for each
11. 7 Marks
    aFind the exponential Fourier series and plot the magnitude and phase spectra of the following triangular wave form.

    How to answer

    • Determine period $T=2$ sec, $\omega_0=\pi$ from triangular plot
    • Compute exponential coefficients $C_n = (1/T)\int_T x(t)e^{-jn\omega_0t}dt$
    • Magnitude $|C_n|$ even and phase $\angle C_n$ spectra
    • Plot line spectra vs $n\omega_0$
12. 7 Marks
    bFind the Fourier Transform of Rectangular pulse. Sketch the signal and Fourier transform.
    How to answer

    • Define rectangular pulse $rect(t/\tau)$
    • Evaluate Fourier integral $X(\omega)=\int_{-\tau/2}^{\tau/2}Ae^{-j\omega t}dt$
    • Result $A\tau Sa(\omega\tau/2)$ sinc shape
    • Sketch time pulse and sinc spectrum
13. 7 Marks
    aDemonstrate time shifting property and time-scaling property of Fourier transform.
    How to answer

    • Fourier transform definition $X(\omega)=\int x(t)e^{-j\omega t}dt$
    • Proof of shifting: $x(t-t_0) \leftrightarrow X(\omega)e^{-j\omega t_0}$
    • Proof of scaling: $x(at) \leftrightarrow (1/|a|)X(\omega/a)$
    • Interpretation with examples
14. 7 Marks
    bObtain DTFT of following signals-

    1. i)$x[n] = a^{n} \sin(w_{o}n) u(n);\; a < 1$
    2. ii)$x[n] = n3^{-n} u(-n)$
    How to answer

    • Use DTFT pair for $a^n u(n)$ and modulation by sine
    • Evaluate i) $a^n\sin(\omega_0n)u(n)$ via Euler formula
    • Evaluate ii) $n3^{-n}u(-n)$ using time-reversal and differentiation
    • State convergence conditions
15. 7 Marks
    aState and prove sampling theorem and discuss the effect of under sampling.
    How to answer

    • Statement: $f_s \ge 2f_m$ for exact reconstruction
    • Proof via impulse sampling and spectral replication
    • Aliasing effect when $f_s < 2f_m$ with spectral overlap diagram
    • Nyquist rate and reconstruction filter
16. 7 Marks
    bWrite short notes on -

    1. i)State space analysis
    2. ii)State transition matrix
    How to answer

    • State-space model: $\dot{x}=Ax+Bu, y=Cx+Du$ meaning
    • State transition matrix $\phi(t)=e^{At}$ properties
    • Methods to compute: Laplace, Cayley-Hamilton
    • Applications in system analysis