STUPIDSID
1(a)
Check for periodicity of the following signals. Also find the new period.
\( (i)\ x(t)=3\cos(15\pi t)+4\cos\left ( 35\pi t-\dfrac{\pi}{4} \right )+8\sin(55\pi t) \)
\( (ii)\ x(n)=3\cos^2\left ( \dfrac{\pi}{6}n \right )+2\cos^2\left ( \dfrac{\pi}{4}n \right ) \)
\( (i)\ x(t)=3\cos(15\pi t)+4\cos\left ( 35\pi t-\dfrac{\pi}{4} \right )+8\sin(55\pi t) \)
\( (ii)\ x(n)=3\cos^2\left ( \dfrac{\pi}{6}n \right )+2\cos^2\left ( \dfrac{\pi}{4}n \right ) \)
4 M
1(b)
Determine whether the given signal is energy or power signal. Hence obtain its energy power accordingly.
\( (i)\ x(t)=4\sin t-\infty
\( (ii)\ x(n)=\left ( \dfrac{3}{7} \right )^n u(n) \)
\( (i)\ x(t)=4\sin t-\infty
\( (ii)\ x(n)=\left ( \dfrac{3}{7} \right )^n u(n) \)
4 M
1(c)
Plot x(t) = u(t) ' r(t) + r(t-1). Hence plot its even and odd parts also.
4 M
1(d)
Prove time shifting of Z-transform.
4 M
1(e)
Check for Dynamicity, Linearity, Time variance, causality of the system.
(i) y(t) = t x(t) + x (t-1)
(ii) y(n) = 3x (-n) + 4.
(i) y(t) = t x(t) + x (t-1)
(ii) y(n) = 3x (-n) + 4.
4 M
2(a)
Obtain inverse Z-transform for all possible ROC's. Also comment on Causality and Stability in each case. \[H(z)=\dfrac{z(3z-7)}{\left ( z-\dfrac{1}{4} \right )(z+2)}\]
10 M
2(b)
State and prove Time Shifting and Convolution property of Continous Time Fourier Transform.
10 M
3(a)
Obtain graphical concolution of following two signals.
10 M
3(b)
Obtain exponential Fourier series of the following signal.
10 M
4(a)
Determine h(t) for all possible ROC's.
If \( T.F.=H(s)=\dfrac{2s+7}{(s+2)(s-3)} \)
Also comment on Causality and Stability of the system for each case.
If \( T.F.=H(s)=\dfrac{2s+7}{(s+2)(s-3)} \)
Also comment on Causality and Stability of the system for each case.
10 M
4(b)
A causal DT LTI system is described as
y(n) = 3y(n-2) + 4y(n-1) +x(n)
Obtain:
(1) T.F. of system
(2) Obtain step response
(3) Obtain response if input \( x(n)=\left ( \dfrac{1}{2} \right )^n u(n) \)
(4) Also plot pole's and zeros of the T.F. and comment on causality and stability.
y(n) = 3y(n-2) + 4y(n-1) +x(n)
Obtain:
(1) T.F. of system
(2) Obtain step response
(3) Obtain response if input \( x(n)=\left ( \dfrac{1}{2} \right )^n u(n) \)
(4) Also plot pole's and zeros of the T.F. and comment on causality and stability.
10 M
5(a)
Determine Impluse response and step response of a CT LTI system. \[\dfrac{d^2y(t)}{dt^2}+\dfrac{7dy(t)}{dt}+12y(t)=x(t)\]
10 M
5(b)
Obtain auto-correlation of following signals
\( (i)\ \ x(t)=3e^{-2t}u(t) \)
\( (ii)\ \ x(n)=\left ( \dfrac{3}{4} \right )^n u(n) \)
\( (i)\ \ x(t)=3e^{-2t}u(t) \)
\( (ii)\ \ x(n)=\left ( \dfrac{3}{4} \right )^n u(n) \)
10 M
6(a)
Obtain DT Fourier Transform of following signal h(n) = [2 1 2] plot its magnitude and phase spectrum.
10 M
6(b)
Obtain :
(i) Z- transform of \[x(n)=n\left ( \dfrac{3}{4} \right )^n u(n)+u(n-1)\]
(ii) Laplace transform of x(t) = t . e-3t u(t) + t u (t-1)
Use properties of transform only.
(i) Z- transform of \[x(n)=n\left ( \dfrac{3}{4} \right )^n u(n)+u(n-1)\]
(ii) Laplace transform of x(t) = t . e-3t u(t) + t u (t-1)
Use properties of transform only.
10 M
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