STUPIDSID
1 (a)
Prove convolution property of Fourier Transform.
5 M
1 (b)
State and prove final value Theorem of Laplace Transform.
5 M
1 (c)
Prove shifting property of Z transform.
5 M
1 (d)
Determine energy and / or power of following signals. \[ i) \ x(n) = \left ( \dfrac {3}{5} \right )^n u(n) - (4)^n u(-n-1) \\ ii) \ x(t) = 4e^{-2t}u (t) \]
5 M
2 (a)
Obtain output y(t)=x(t)*h(t) using graphical convolution.
\[ \begin {align*} x(t)& =1 + t \ for \ -1\le t \le 0 \\ & = 1-t \ for \ 0\le t \le 1 \end{align*} \] \[ \begin {align*} h(t)& =1 \ for \ 0\le t \le 2 \\ &=0 \ elsewhere \end {align*} \]
\[ \begin {align*} x(t)& =1 + t \ for \ -1\le t \le 0 \\ & = 1-t \ for \ 0\le t \le 1 \end{align*} \] \[ \begin {align*} h(t)& =1 \ for \ 0\le t \le 2 \\ &=0 \ elsewhere \end {align*} \]
10 M
2 (b)
Obtain h(n) for all possible ROC conditions. Also plot the ROC comments on causality and stability at the system. \[H(z)= \dfrac {4z (z^2 - 8z +9)}{ \left ( z - \frac {1}{3} \right ) (z-3)(z+4)} \]
10 M
3 (a)
A.C.T. LTI system has \[ \dfrac {d^2y (t)}{dt^2} + \dfrac {5dy(t)}{dt} + 6y (t) = \dfrac {7dx (t)}{dt} - 3x(t) \] i) Determine Transfer function.
ii) Obtain impulse response
iii) Obtain unit Ramp response.
ii) Obtain impulse response
iii) Obtain unit Ramp response.
8 M
3 (b)
Plot the magnitude and phase spectrum of the periodic signal. Show below.
8 M
3 (c)
Obtain initial and final value: \[ if \ X(z) = \dfrac {3z^2} {4z^2 + 5z +1} \]
4 M
4 (a)
If two subsystem are connected in cascade
h1(n)=(0.9)n u(n) ? 0.5(0.9)n-1 u(n-1)
h2= (0.5)n u(n) ? (0.5)n-1 u(n-1)
Determine overall impulse response of the interconnected system.
h1(n)=(0.9)n u(n) ? 0.5(0.9)n-1 u(n-1)
h2= (0.5)n u(n) ? (0.5)n-1 u(n-1)
Determine overall impulse response of the interconnected system.
8 M
4 (b)
Obtain z transform of the following signal using properties of z transform. \[ x(n) = \left ( \dfrac {3}{4} \right )^{n-1} \sin \left ( \dfrac {\pi}{6} n \right ) u(n) \]
6 M
4 (c)
Prove Parsevals theorem of Fourier series.
6 M
5 (a)
Obtain circular convolution of
x1(n)= [3 2 1 4]
x2 (n) = [ 5 7 -8 2]
x1(n)= [3 2 1 4]
x2 (n) = [ 5 7 -8 2]
5 M
5 (b)
Obtain Laplace Transform of following waveforms using its properties.
i)
i)
5 M
5 (c)
Obtain zero input response, zero state response and total response of a D.T.L.T.I. system.\[ \begin {align*} \ \ \ \ \ &y(n)+7y (n-1) +12 y (n-2)=4 \ x(n)-11 x(n-1)\\ if \ \ \ \ \ \ \ \ \ \ \ \ \ & y(-1)=1 \ y(-2) \ x(-1)=0 \\ if \ input \ \ \ & x(n) = u(n)= \ unit \ step \ signal \end{align*} \]
10 M
6 (a)
Obtain Fourier transform of the following signal.
6 M
6 (b)
Plot even and odd parts of following signals.
6 M
6 (c)
Obtain h(t) for causal and stable system If \[ H(s) = \dfrac { s^2 ? 3s +11} {(s-1) (s+2) (s+3)} \] Plot the ROC and pole's and zero's of the system.
8 M
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