STUPIDSID
1(a)
Define the following with examples i) Signals and systems ii) Power and Energy Signals
5 M
1(b)
A continuous time signal is described by \[\begin{matrix}
X(t)=t; & 0 \le1 \le1\\
2-t; & 1\le t \le2
\end{matrix}\]
Sketch even and odd component of the signal.
Sketch even and odd component of the signal.
5 M
1(c)
A continuous time signal x(t) is shown in Fig Q1(c). Plot the following signals \[i)x[-2(t+1)]\ \ ii)x\left ( \frac{t}{2}+1 \right )\ \ ii)x(-2t-1)\]
:!mage
:!mage
6 M
1(d)
Check whether sequence y(t) = log x(n) is Linear, Time invariant, Memory, causal and stable?
4 M
2(a)
Given input x(n) = u(n) - u(n-3) and impulse response h(n) = [1, 3, 2, -1, 1]. Determine the response y(n) using convolution sum.
6 M
2(b)
Using convolution integral, determine the output of an LTI system for an input x(t) = e-at; 0≤t≤T and impulse response h(t) = 1; 0≤t≤2T.
8 M
2(c)
Determine the range of 'a' and 'b' for which the LTI system with impulse response h(n) = an; n≥0 is stable, bn; n<0 is stable
2 M
2(d)
Check whether the system whose impulse response is h(t) = et u(t-1) i) Stable, Memory less and causal.
4 M
3(a)
Determine the complete response of system whose difference equation is \( y(n)-\dfrac{1}{4}y(n-1)-\dfrac{1}{8}y(n-2)=x(n)+x(n-1)\) with input x(n) = 2n u(n) and initial conditions y(-1) = 2 and y(-2) = -1.
8 M
3(b)
Determine the natural response of the system whose differential equation is \(\dfrac{d^2y(t)}{dt^2}+4y(t)=3\dfrac{dx(t)}{dt} \) with initial conditions \( y(0)=1,\dfrac{d}{dt}y(0)=1\)
6 M
3(c)
Draw the direct form - I and direct form - II implementation of the following differential equation \( \dfrac{2d^3y(t)}{dt^3}+\dfrac{dy(t)}{dt}+3y(t)=x(t).\)
6 M
4(a)
State and explain following Fourier series properties.
i) Frequency shift
ii) Convolution
i) Frequency shift
ii) Convolution
10 M
4(b)
For the signal x(t) = sin ω0t, find the Fourier series and draw its spectrum.
5 M
4(c)
Find the time domain signal corresponding to the DTFs coefficient \[x(k)=\cos\left ( \dfrac{16\pi}{17}k \right )\]
5 M
5(a)
State and explain Parasavel's theorem.
6 M
5(b)
Obtain the Fourier transform of the following signals
i) x(t) = e-at u(t); a>0
ii) x(t) = δ(t)
i) x(t) = e-at u(t); a>0
ii) x(t) = δ(t)
8 M
5(c)
The impulse response of a continuous time signal is given by \( h(t)=\dfrac{1}{R_C}e^{-t/RC}u(t).\) Find the frequency response and plot the magnitude and phase response.
6 M
6(a)
State and explain following DTFT properties i) Time shift ii) Linearity.
6 M
6(b)
Determine the DTFT of the following signals
i) x(n) = u(n) ii) x(n) = 2n u(-n).
i) x(n) = u(n) ii) x(n) = 2n u(-n).
7 M
6(c)
Obtain frequency response and impulse response of the system described by the difference equation \( y(n)-\dfrac{1}{4}y(n-1)-\dfrac{1}{8}y(n-2)=3x(n)-\dfrac{3}{4}x(n-1)\)
7 M
7(a)
What is z-transform? Mention properties of Region of convergence (ROC).
5 M
7(b)
Determine z transformation and its ROC of the following signals
i) x(n) = u(n)
ii) x(n) = cos Ω0 nu(n).
i) x(n) = u(n)
ii) x(n) = cos Ω0 nu(n).
7 M
7(c)
Determine inverse z-transformation of following function x(z) \[x(z)=\dfrac{1}{1-\dfrac{3}{2}z^{-1}+\dfrac{1}{2}z^{-2}}\ \text{for}\ \ i) |z|>1\ \ ii) |z|<\dfrac{1}{2}\ \ iii)\dfrac{1}{2}<|z|<1.\]
8 M
8(a)
State and prove final value theorem of z transformation
6 M
8(b)
Determine natural, forced and complete response of the system described by \( y(n)-\dfrac{1}{2}\ y(n-1)=2\ x(n)\) with initial conditions y(-1) = 3 and input x(n) = 2(-½)n.
8 M
8(c)
A DT - LTI system is given by \[H(z)=\dfrac{3-4z^{-1}}{1-\dfrac{7}{2}z^{-1}+\dfrac{3}{2}z^{-2}}\]
Specify the ROC of H(z) and determine h(n) for
i) Stable system
ii) Causal system
iii) non causal system.
Specify the ROC of H(z) and determine h(n) for
i) Stable system
ii) Causal system
iii) non causal system.
6 M
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