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)=e

i) Memoryless ii) Stable iii) Causal iv) Linear and v) Time-invariant.

• y(n)=nx(n)

• y(t)=e

^{x(t)}
10 M

1 (b)
Distinguish between;

i) Determine and random signals and

ii) Energy and periodic signals

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,

x(n)={1,

**, 3, 4} and h(n)={1,**__2__**, 1}**__5__
8 M

3 (a)
Determine the conditions of the impulse response of the system if system is

i) Memoryless

ii) Stable

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

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) x

ii) x

i) x

_{1}(t)=sin e(200t)ii) x

_{2}= sin e^{2}(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.

\[ x(n)=n sin \left ( \dfrac {\pi} {2} n \right ) u(n) \]

\[ x(n)=n sin \left ( \dfrac {\pi} {2} n \right ) u(n) \]

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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