STUPIDSID
Short Questions
1(a)
Sketch the waveform of the following signal:
x(t) = u(t + 1) ' 2u(t) + u(t ' 1)
x(t) = u(t + 1) ' 2u(t) + u(t ' 1)
1 M
1(b)
If the response of LTI continuous time system to unit step signal is \( \dfrac{1}{2}-\dfrac{1}{2}e^{-2t}, \) then find the impluse response of the system.
1 M
1(c)
Find inverse Laplace transform of \( \dfrac{1}{s}(1-e^{-2s}) \)
1 M
1(d)
Fill in the blanks: 'An energy signal has _______ average power, whereas a power signal has _______ energy.'
1 M
1(e)
Find the output of an LTI system with impulse response h(t) = δ(t-3) for the input x(t) = cos 4t + sin 7t.
1 M
1(f)
Evaluate the following integrals. \[(i)\int ^2_1(3t^2+1)\delta (t)dt\] \[(ii)\int ^{\infty}_0 t^2 \delta(t-6)dt\]
1 M
1(g)
Determine whether following signal is periodic or not. If it is periodic, find fundamental period. x[n] = (-1)n
1 M
1(h)
Determine z- transform and its ROC of x[n] = u[n]
1 M
1(i)
Write Fourier Transform of x(t) = sin(ω0t)
1 M
1(j)
If convolution is performed between two signals, x and h, with lengths Nx and Nh then what will be the length of resulting signal?
1 M
1(k)
Define Invertible system.
1 M
1(l)
Let \( x[n]=\left\{\begin{matrix}
1, & 0\leq n\leq 9\\
0, & Otherwise
\end{matrix}\right.\ \ \text{and}\ \ \ \ h[n]=\left\{\begin{matrix}
1, & 0\leq n\leq N\\
0, & Otherwise
\end{matrix}\right. \) Where N≤9 is an integer. Determine the value of N, given that y[n] = x[n] * h[n] and y[4] = 5, y[14] = 0.
1 M
1(m)
Integration of unit impulse function over (-∞, ∞
yields _______ signal and
differentiating a unit ramp function yields _______ signal.
1 M
1(n)
Find the even and odd components of following signal:
x(t) = 1 + t + 3t2 + 5t3 + 9t4
x(t) = 1 + t + 3t2 + 5t3 + 9t4
1 M
2(a)
Categorize the following signals as an energy or power signal and find energy or power of the signal.
(i) x(t) = 5cos(πt) + sin(5πt) ; -∞ < t < ∞
\( (ii) x[n]=\left\{\begin{matrix} \cos(\pi n) ;& n\geq 0\\ 0 ; & Otherwise \end{matrix}\right. \)
(i) x(t) = 5cos(πt) + sin(5πt) ; -∞ < t < ∞
\( (ii) x[n]=\left\{\begin{matrix} \cos(\pi n) ;& n\geq 0\\ 0 ; & Otherwise \end{matrix}\right. \)
3 M
2(b)
Consider a system S with input x[n] and output y[n] related by
y[n] = x[n]{g[n] + g[n-1]}
(i) If g[n] = 1 for all n, show that S is time invariant.
(ii) If g[n] = n, show that S is nor time invariant.
(iii) If g[n] = 1+(-1)n, show that S is time invariant.
y[n] = x[n]{g[n] + g[n-1]}
(i) If g[n] = 1 for all n, show that S is time invariant.
(ii) If g[n] = n, show that S is nor time invariant.
(iii) If g[n] = 1+(-1)n, show that S is time invariant.
4 M
Solve any one question from Q.2(c) & Q.2(d)
2(c)
A continuous time signal x(t) is shown in figure: Sketch and label carefully each of the following signals.
\( (i)x \left ( \dfrac{8-t}{2} \right ) \)
\( (ii)x(t) \left [ \delta \left ( t+\dfrac{3}{2} \right )-\delta \left ( t-\dfrac{3}{2} \right ) \right ] \)
\( (i)x \left ( \dfrac{8-t}{2} \right ) \)
\( (ii)x(t) \left [ \delta \left ( t+\dfrac{3}{2} \right )-\delta \left ( t-\dfrac{3}{2} \right ) \right ] \)
7 M
2(d)
Determine whether each of them is (i) memoryless (ii) stable (iii) causal and (iv) linear
(1) y[n] = 2x[2n]
(2) y(t) = x(t/2) Justify your answers.
(1) y[n] = 2x[2n]
(2) y(t) = x(t/2) Justify your answers.
7 M
Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)
3(a)
State and prove a condition for a discrete time LTI system to be invertible.
3 M
3(b)
The following are the impulse responses of LTI systems. Determine whether each system is causal and/or stable. Justify your answers.
(i) h[n] = (0.8)n u[n+2]
(ii) h(t) = e2t u(-1-t)
(i) h[n] = (0.8)n u[n+2]
(ii) h(t) = e2t u(-1-t)
4 M
3(c)
Find the convolution of two signals x(t) and y(t)
\( x(t)=\left\{\begin{matrix} 1 & |t|\leq 1\\ 0 & Otherwise \end{matrix}\right. \ \ \ \ \ \ \ \ y(t)=\left\{\begin{matrix} 1, & |t|\leq 1\\ \delta (t+2)+\delta (t-2) & \\ 0, & Otherwise \end{matrix}\right. \)
\( x(t)=\left\{\begin{matrix} 1 & |t|\leq 1\\ 0 & Otherwise \end{matrix}\right. \ \ \ \ \ \ \ \ y(t)=\left\{\begin{matrix} 1, & |t|\leq 1\\ \delta (t+2)+\delta (t-2) & \\ 0, & Otherwise \end{matrix}\right. \)
7 M
3(d)
State and prove a condition for a discrete time LTI system to be stable.
3 M
3(e)
Evaluate the step response for the LTI systems represented by the following impulse responses:
(i) h(t) = e-|t|
(ii) h[n] = (-1)n {u[n+2] ' u[n-3]}
(i) h(t) = e-|t|
(ii) h[n] = (-1)n {u[n+2] ' u[n-3]}
4 M
3(f)
Evaluate the discrete time convolution sum given below.
y[n] = βn u[n] * u[n-3], |β| < 1
y[n] = βn u[n] * u[n-3], |β| < 1
7 M
Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)
4(a)
Determine the output of the system described by the following differential equation with input and initial condition as specified.
\( \dfrac{d}{dt}y(t)+10y(t)=2x(t),\ \ \ y(0)=1,\ \ \ x(t)=u(t) \)
\( \dfrac{d}{dt}y(t)+10y(t)=2x(t),\ \ \ y(0)=1,\ \ \ x(t)=u(t) \)
3 M
4(b)
Write Differentiation in Time and Differentiation in Frequency property of Fourier Transform. Obtain Fourier Transform of \( \dfrac{d(e^{-at}u(t))}{dt}. \)
4 M
4(c)
Find the Fourier Series representation for the sawtooth wave depicted in below figure.
7 M
4(d)
Write modulation property of Fourier Transform. Use frequency differentiation property to find the Fourier Transform of x(t) = te-at u(t).
3 M
4(e)
Determine the Complex Exponential Fourier Series representation for the square wave depicted in below figure.
4 M
4(f)
Write Duality property of Fourier transform. Given \( e^{-|t|}\ ^{\underset{\longleftrightarrow }{FT}}\dfrac{2}{4\pi^2f^2+1}. \) Find the Fourier transforms of following:
\( (i)\dfrac{d}{dt}e^{-|t|}\ \ \ (ii)\dfrac{1}{2\pi (t^2+1)} \)
\( (i)\dfrac{d}{dt}e^{-|t|}\ \ \ (ii)\dfrac{1}{2\pi (t^2+1)} \)
7 M
Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)
5(a)
Compute DFT of the following sequence: x[n] = {0, 1, 2, 3}
3 M
5(b)
Determine z-transform of following sequences.
(i) x[n] = a-|n|, 0 < |a| < 1
(ii) x[n] = 2n u[n] + 3n u[-n-1].
(i) x[n] = a-|n|, 0 < |a| < 1
(ii) x[n] = 2n u[n] + 3n u[-n-1].
4 M
5(c)
Find the z-transform of the signal \[\displaystyle x[n]=\left \{ n \left ( \dfrac{-1}{2} \right )^n u[n]\right \}*\left \{ \left ( \dfrac{1}{4} \right )^{-n} u[-n]\right \}\]
7 M
5(d)
Find the impulse response h[n]
for each of the causal LTI discrete time systems satisfying the following difference equations.
(i) y[n] = x[n] ' 2x[n-2] + x[n-3
(ii) y[n] + 2y[n-1] = x[n] + x[n-1]
(i) y[n] = x[n] ' 2x[n-2] + x[n-3
(ii) y[n] + 2y[n-1] = x[n] + x[n-1]
3 M
5(e)
Write Differentiation in Z-domain property of z- transform. Obtain z-transform of x[n] = an cos(Ω0n)u[n], where a is real and positive.
4 M
5(f)
Find the inverse z-transform of \[X(z)=\dfrac{z^3-10z^2-4z+4}{2z^2-2z-4}\ \text{with ROC}\ |z|<1\]
7 M
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