STUPIDSID
Answer the following
1(a)
Determine if the following system is memoryless, causal, linear, time invariant
y(t) = x2 (t-t0) + 2
y(t) = x2 (t-t0) + 2
4 M
1(b)
Explain in brief ROC (Region of Convergence) conditions of laplace transform.
4 M
1(c)
Consider two LTI systems connected in series. Their impulse response are h1[n] and h2[n] respectively. Find the output of the systems if x[n] is the Input being to one of the systems.
x[n]={→ 1, 2} h1[n]={1,0,-1 ←} h2[n]={→ 2, 1,-1}
x[n]={→ 1, 2} h1[n]={1,0,-1 ←} h2[n]={→ 2, 1,-1}
4 M
1(d)
State and prove time reversal property of Continuous time Fourier Series.
4 M
1(e)
Find energy of a causal exponential pulse x(t) = e-at u(t)
4 M
2(a)
A DT sifnal is given by the following expression. Find its Z transform \[x[n]=n(-\dfrac{1}{2})^n u[n]*(\dfrac{1}{4})^{-n}u[-n]\]
10 M
2(b)
A CT signal x(t) is applied to the input of a CT LTI systems with unit impluse response h(t). Find out y(t) using Convolution integral.
\[\begin {align*} x(t)&=e^{-at}u(t)\ \ a>0 \ \\\ h(t)&=u(t) \end{align*}\]
\[\begin {align*} x(t)&=e^{-at}u(t)\ \ a>0 \ \\\ h(t)&=u(t) \end{align*}\]
10 M
3(a)
Consider a causal LTI system with \( H(j \omega )=\dfrac{1}{j\omega+2}. \) For a particular input x(t), this system produces output y(t) = e-2t u(t) - e-3t u(t). Find out x(t) using Fourier Transform.
10 M
3(b)
Obtain Inverse Laplace Transform of the function \( X(s)=\dfrac{3S+7}{S^2-S-12} \) for following ROCs. Also comment on the stability and causality of the system for each of the ROC conditions. Support your answer with appropriate sketches of ROCs.
i) Rs(S)>4
ii) Re(S)<-3
iii) -3
i) Rs(S)>4
ii) Re(S)<-3
iii) -3
10 M
4(a)
A DT signal has been shown. Sketch the following signals.
i) x[n-4] ii) x[4-n iii) x[-2n+2] iv) x[n]u[3-n]
8 M
Find out DEFT of the following
4(b)(i)
\( x[n]=\left \{ 1,-1,2,2 \right \} \)
3 M
4(b)(ii)
\( x[n]=\sin \left [ \dfrac{\pi n}{2} \right ]u[n] \)
3 M
4(c)
Determine inverse Z Transform of \[X(Z)=\dfrac{3}{(1-Z^{-1})(1+Z^{-1})(1-0.5Z^{-1})(1-0.2Z^{-1})}\]
6 M
5(a)
Find the trigonometric Fourier Series for the waveform shown in the following figure.
10 M
5(b)
Determine impluse response of h[n] for the system described by the second order difference equation.
y[n] - 4y[n-2] + 4y[n-2] = x[r] - x[n-1] when y[-1] = y[-2] = 0
y[n] - 4y[n-2] + 4y[n-2] = x[r] - x[n-1] when y[-1] = y[-2] = 0
10 M
6(a)
A Lti system has the following transfer function \[H(Z)=\dfrac{Z}{(Z-\dfrac{1}{4})(Z+\dfrac{1}{4})(Z-\dfrac{1}{2})}\]
10 M
6(b)(i)
Give app possible ROC conditions
5 M
6(b)(ii)
Show pole-zero diagram of a system
5 M
6(b)(iii)
Find impluse response of system
5 M
6(b)(iv)
Comment on the system stability and causality for all possible RoCs
5 M
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