STUPIDSID
1 (a)
Determine the fundamental period of the following signals. \[ x(t)=14 + 40 \cos (60 \pi t) \\ ii) x[n] = \cos^2 \left [\dfrac {\pi}{4}n \right ] \]
4 M
1 (b)
Compare the nature of ROC of Z transform and Laplace transform.
4 M
1 (c)
For the given system, determine whether it is,
i) memory less
ii) causal
linear
iv) time-invariant.
y[n] = x[-n].
i) memory less
ii) causal
linear
iv) time-invariant.
y[n] = x[-n].
4 M
1 (d)
"Find out even and odd component of the following two signals. \[ i) \ x(t) = \cos^2 \dfrac {\pi t}{2} \\ ii) x(t) = \left\{\begin{matrix}t\cdots \cdots &0 \le t \le 1 \\ 2-t \cdots \cdots
& 1< 2\le 2 \end{matrix}\right. \]"
4 M
1 (e)
Determine whether the signals are power or energy signals. Calculate energy / power accordingly.
i) x(t)=Ae-αtu(t)........... α>0.
ii) x[n]=u[n].
i) x(t)=Ae-αtu(t)........... α>0.
ii) x[n]=u[n].
4 M
2 (a)
Expand the periodic gate function as shown in the figure by the exponential Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).
10 M
2 (b)
Find the inverse Laplace Transform of the following: \[ i) \ X(S) = \dfrac {s-3}{s^2+4s+13} \\ ii) \ X(S)= \dfrac {5s^2 - 15 - 11}{(s+1)(s-2)^3} \]
10 M
3 (a)
Obtain inverse Laplace transform of the function \[ X(s) = \dfrac {3S+7}{s^2 -2s-3} \] Write down and sketch possible ROCs. Find out inverse Laplace for all the possible ROCs.
10 M
3 (b)
Using the z transform method, solve the difference equation
y[n]-4y[n-1]+4y[n-2]=x[n]-x[n-1]
When y(-1)=y(-2)=0.
y[n]-4y[n-1]+4y[n-2]=x[n]-x[n-1]
When y(-1)=y(-2)=0.
10 M
4 (a)
Explain Gibbs phenomenon. Also explain conditions necessary for the convergence of Fourier Series.
5 M
4 (b)
Find out Fourier Transform of f(t)=10 δ(t-2). Sketch its amplitude and phase spectrum.
5 M
4 (c)
Perform convolution of
i) 2u(t) with u(t)
ii) e-2t u(t) with e-5t u(t)
iii) tu(t) with e-5t u(t).
i) 2u(t) with u(t)
ii) e-2t u(t) with e-5t u(t)
iii) tu(t) with e-5t u(t).
10 M
5 (a)
Convolve \(x[n]= \left ( \dfrac {1}{3} \right )^n u [n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n] \) using Fourier transform.
10 M
5 (b)
A system is described by the following difference equation. \[ y[n] = \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y [n-2]+ x[n] \] Determine the following.
i) The system Transfer function H(2)
ii) Impulse response of the system h[n]
iii) Step response of the system s[n].
i) The system Transfer function H(2)
ii) Impulse response of the system h[n]
iii) Step response of the system s[n].
10 M
6 (a)
A discrete time signal is given by \( x[n] = \{ \underset {1}, 1, 1, 1, 2\} \) . Sketch the following signals.
i) x[n]
ii) x[n-2]
iii) x[n] ⋅ u[n-1]
iv) x[3-n]
v) x[n-1]⋅δ[n-1]
i) x[n]
ii) x[n-2]
iii) x[n] ⋅ u[n-1]
iv) x[3-n]
v) x[n-1]⋅δ[n-1]
10 M
6 (b)
For the periodic signal x[n] given below, find out Fourier series coefficient. \[ x[n] = 1 +\sin \left ( \dfrac {2\pi} { N} \right ) n+3 \cos \left ( \dfrac {2\pi}{N} \right ) n+\cos \left ( \dfrac {4\pi}{N} + \dfrac {\pi}{2} \right ). \]
10 M
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