STUPIDSID
1 (a)
Determine the fundamental period of the following signals:- \[ i) \ x(t) = \cos \dfrac {\pi}{3} t+ \sin \dfrac {\pi}{4} t \\ ii) \ x[n] = \cos ^ 2 \dfrac {\pi}{8}n \]
4 M
1 (b)
State and prove Time Shifting and Time Scaling property of continuous time Fourier Transform.
4 M
1 (c)
For the following system, determine whether it is. (i) memory less, (ii) causal, (iii) linear, (iv) time-invariant y[n]=x[n2]
4 M
1 (d)
Find out even and odd component of the following two signals:
\[ i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n) \]
\[ i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n) \]
4 M
1 (e)
Determine whether the signals are power of energy signals. Calculate energy/power accordingly:
i) x(t) = 0.9 e-3t u(t)
ii) x[n]=u[n]
i) x(t) = 0.9 e-3t u(t)
ii) x[n]=u[n]
4 M
2 (a)
Find the inverse Laplace Transform of \(\dfrac {s-2}{s(s+1)^3}\)
5 M
2 (b)
Let x(t)=1.......0 ≤ t≤ 2T and; h(t)=e-at.... 0≤ t≤ T. Compute y(t) using graphical convolution approach.
10 M
2 (c)
State and discuss the properties of the region of convergence for z-transform.
5 M
3 (a)
An LTI system is characterized by the system function: \[ h(z) = \dfrac {z} { \left ( z- \dfrac {1}{4} \right ) \left ( z+ \dfrac {1}{4} \right ) \left ( z- \dfrac {1}{2} \right )} \] write down possible ROCs. For different possible ROCs, determine causality and stability and impulse response of the system.
10 M
3 (b)
Calculate Z transform of the following signals: \[ i) \ x[n] =n \left ( - \dfrac {1}{4} \right )^n u[n]\times \left ( - \dfrac {1}{6} \right )^{-n} u [-n] \\ ii) \ x[n] = u[n-6] -u [n-10] \]
10 M
4 (a)
For the periodic signal x(t)=e-t with a fundamental period T0=1 second. Find the exponential form of Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).
10 M
4 (b)
Consider a continuous time LTI system described by \[ \dfrac {dy(t)}{dt} + 2y(t)=x(t) \] Using the transform, find out output to each of the following input signals.
i) x(t)=e-t u(t)
ii) x(t)=u(t)
i) x(t)=e-t u(t)
ii) x(t)=u(t)
10 M
5 (a)
Convolute \[ x[n] = \left ( \dfrac {1}{3} \right )^n u[n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n] \] using convolute sum formula and verify your answer using z transform.
10 M
5 (b)
Explain Gibb's phenomenon. Also explain conditions necessary for the convergence of Fourier Series.
5 M
5 (c)
A system is described by the following difference equation. Find out its transfer function H(z). \[ y[n]= \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y[n-2]+ x[n]+ \dfrac {1}{2} x [n-1] \]
5 M
6 (a)
For the signal x(t) depicted in the figure given below, sketch the signals:
i) x(-t)
ii) x(t+6)
iii) x(3t)
x(t/2)
i) x(-t)
ii) x(t+6)
iii) x(3t)
x(t/2)
10 M
6 (b)
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
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