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INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Determine whether the following systems are:
i) Memoryless ii) Stable iii) Causal iv) Linear and v) Time-invariant.
• y(n)=nx(n)
• y(t)=ex(t)
10 M
1 (b) Distinguish between;
i) Determine and random signals and
ii) Energy and periodic signals
6 M
1 (c) For any arbitrary signal x(t) which is an even signal, how that $\int^{\infty}_{-\infty}x(t)dt=2 \ \int^{+\infty}_{0}x(t)dt$
4 M

2 (a) Find the convolution integral of x(t) and h(t), and sketch the convolved signal, x(t)=(t-1){u(t-1)-u(t-3)} and h(t)=[(t+1)-2u(t-2)].
12 M
2 (b) Determine the discrete time convolution sum of the given sequences.
x(n)={1, 2 , 3, 4} and h(n)={1, 5 , 1}
8 M

3 (a) Determine the conditions of the impulse response of the system if system is
i) Memoryless
ii) Stable
6 M
3 (b) Find the total response of the system given by, $\dfrac {d^2y(t)}{dt^2}+3 \dfrac {d}{dt}y(t)+2y(t)=2x(t) \ with \ y(0)=-1; \ \dfrac {\frac {d}{dt}y(t)}{t=0}=1 \ and \ x(t)=\cos (t)u(t)$
14 M

4 (a) One period of the DTFS coefficients of a signal is given by, x(k)=(1/2)k, on 0?K?9. Find the time domain signal x(n) assuming N=10
6 M
4 (b) Prove the following properties of DTFs; i) Convolution ii) Parseval relationship iii) Duality
iv) Symmetry
14 M

5 (a) Find the DTFT of the sequence x(n)=?n u(n) and determine magnitude and phase spectrum.
4 M
5 (b) Plot the magnitude and phase spectrum of x(t)=e-j4 u(t).
8 M
5 (c) Find the inverse Fourier transform of the spectra, $x(j\omega)= \left\{\begin{matrix}2\cos (\omega) ,&|\omega|<\pi \\0, &|\omega|>0 \end{matrix}\right.$
8 M

6 (a) Find the frequency response and impulse response of the system described by the differential equation. $\dfrac{d^2}{dt^2}y(t)+5 \dfrac {d}{dt} y(t) + 6y(t)= - \dfrac {d}{dt}x(t)$
8 M
6 (b) State sampling theorem. Explain sampling of continuous time signals with relevant expressions and features.
6 M
6 (c) Find the Nyquist rate for each of the following signals:
i) x1(t)=sin e(200t)
ii) x2 = sin e2 (500t)
6 M

7 (a) Prove the complex conjugation and time advance properties.
6 M
7 (b) Find the Z-transform of the signal along with ROC.
$x(n)=n sin \left ( \dfrac {\pi} {2} n \right ) u(n)$
6 M
7 (c) Determine the inverse z-transform of the following x(z) by partial fraction expansion method, $x(z)= \dfrac {z+2}{2z^2-7z+3} \\ if \ tht \ ROCs \ are \\ i) \ |z|>3 \\ ii) \ |z|<\dfrac {1}{2} \ and \\ iii) \ \dfrac {1}{2}<|z|<3$
8 M

8 (a) A system has impulse response $h(n)= \left ( \dfrac {1}{2} \right )^n u(n)$ determine the input to the system if the output is given by, $y(n)=\dfrac {1}{3}u(n)+ \dfrac {2}{3} \left ( - \dfrac {1}{2} \right )^n u(n)$
8 M
8 (b) Solve the following difference equation using unilateral z-transform. $y(n) - \dfrac {3}{2} y(n-1)+ \dfrac {1}{2}y(n-2)=x(n), \ for \ n\ge 0, \ with \ initial \ conditions \ y(-1)=4, y(-2)=10, \ and \ x(n)= \left ( \dfrac {1}{4} \right )^n u(n)$
12 M

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