STUPIDSID
1 (a)
Determine the fundamental period of the following signals. \[ i) \ x(t) =2\cos \dfrac {2 \pi t}{3} + 3 \cos \dfrac {2\pi t}{7} \\ ii) x[n] = \cos^2 \left [ \dfrac {\pi}{4}n \right] \]
5 M
1 (b)
Prove and explain time scaling and amplitude scaling property of Continuous time Fourier Transform.
5 M
1 (c)
For the given system, determine whether it is, i) memory less, ii) causal, iii) time-invariant y[n]=nx[n].
5 M
1 (d)
Find out even and odd component of the following signal. \[ x(t)=\cos^2 \left ( \dfrac {\pi t}{2} \right ) \]
5 M
2 (a)
Determine the trigonometric form of Fourier Series of the waveform shown below.
10 M
2 (b)
State duality property of Fourier Transform. If Fourier Transform of \( e^{-t} u(t) \text{ is } \dfrac {1}{1+j\Omega}, \) then find the Fourier Transform of \( \dfrac {1}{1+t} \) using duality property.
10 M
3 (a)
Obtain inverse Laplace transform of the function. Write down and sketch possible ROCs. \( x(s) = \dfrac {8} {(s+2)^3 (s+4)} \)
10 M
3 (b)
Using the z transform, solve the difference equation and find out impulse response. y[n]-2y[n-1]+y[n-2]=x[n]+3x[n-3]
10 M
4 (a)
State and explain different properties of ROC of Z transform.
5 M
4 (b)
Convolve the sequences shown in the following figure using circular convolution.
5 M
4 (c)
A continuous time signal is shown below. Sketch the following transformed versions of the signal. \[ i)\ x(t-3) \\ ii) \ -2x(t) \\ iii)\ x(t-3)-2x(t) \\ iv) \ \dfrac {dx(t)}{dt} \]
10 M
5 (a)
Convolve \( x[n] = \left ( \dfrac {1}{3} \right )^n \ u[n] \text { with }h[n]= \left ( \dfrac {1}{2} \right )^n \ u[n] \) using convolution integral.
10 M
5 (b)
A second order LTI system is described by \( \dfrac {d^2 y(t)}{dt^2}+ 5 \dfrac {dy(t)}{dt}+6y(t)=x(t). \) Determine the transfer function and poles and zeros of the systems. Evaluate zero-state response to x(t)=u(t).
10 M
6 (a)
For the periodic signal x[n] given below find out Fourier series coefficient. \[ x[n] =1 + \sin \left ( \dfrac {2\pi}{N} \right ) n +3 \cos \left ( \dfrac {2\pi}{N} \right )n+\cos \left ( \dfrac {4\pi}{N}n + \dfrac {\pi}{2} \right ) \]
10 M
6 (b)
The input and impulse response of continuous time system are given below. Find out output of the continuous time systems using appropriate method. x(t)=u(t) & h(t)=e-2tu(t).
10 M
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