1 (a)
Determine whether the discrete time signal \[ x(n)= \cos \left ( \dfrac {\pi n}{5} \right ) \sin \left ( \dfrac {\pi n}{3} \right ) \] is periodic. If periodic, find the fundamental period.

4 M

1 (b)

Determine whether the signal shown in Fig. Q1(b) is a power signal or energy signal. Justify your answer and further determine its energy/power.

6 M

1 (c)
Given the signal x(n)=(6-n){u(n)-u(n-6)} make a sketch of x(n), y

_{1}(n)=x(4-n) and y_{2}(n)=x(2n-3).
4 M

1 (d)
Find and sketch the following signals and their derivatives:

i) x(t)=u(t)-u(t-a); a>0

ii) y(t)=t[u(t)-u(t-a)];a?0

i) x(t)=u(t)-u(t-a); a>0

ii) y(t)=t[u(t)-u(t-a)];a?0

6 M

2 (a)
The impulse response of discrete LTI system is give by, h(n)=u(n+1)-u(n-4). The system is excited by the input signal x(n)=u(n)-2u(n-2)+u(u-4). Obtain the response of the system y(n)=x(n)×h(n) and plot the same.

7 M

2 (b)
Given x(t)=t 0

7 M

2 (c)
Show that:

i) x(t)× h(t) = h(t)× x(t)

ii) {x(n)× h

i) x(t)× h(t) = h(t)× x(t)

ii) {x(n)× h

_{1}(n)} × h_{2}(n)=x(n)×{h_{1}(n)×h_{2}(n)}.
6 M

3 (a)
Solve the difference equation, y(n)-3y(n-1)-4y(n-2)=x(n) with x(n)=4

^{n}u(n). Assume that the system is initially relaxed.
6 M

3 (b)
Draw the direct form I and direct form II implementation for \[ i) \ y(n) - \dfrac {1}{2} y(n-1)-y(n-3)= 3x(n-1)+ 2x(n-2) \\ ii) \ \dfrac {d^2 y(t)}{dt^2}+ 5 \dfrac {dy (t)}{dt}+ 4y(t)= \dfrac {dx (t)}{dt} \]

10 M

3 (c)
Define causality. Derive the necessary and sufficient conditions for a discrete LTI system to be causal in terms of the impulse response.

4 M

4 (a)
Determine the DTFS coefficients of. \[ x(n)= 1+\sin \left \{ \dfrac {1}{12} \pi n + \dfrac {3\pi}{8} \right \} \]

6 M

4 (b)

Find the exponential Fourier series of the waveform shown in Fig Q4(b).

8 M

4 (c)
Explain the Direchlet conditions for the existence of Fourier series.

6 M

5 (a)
Find the DTFT of the signal x(n) given by x(n)=u(n)-u(n-N); where N is any +ve integer. Determine the magnitude and phase components and draw the magnitude spectrum for N=5.

10 M

5 (b)
Determine the Fourier transform of the following signals: i) x(t)=e

ii) x(t)=e

^{-3t}u(t-1)ii) x(t)=e

^{-a|t}
10 M

6 (a)
Determine the frequency response and the impulse response for the system described by the differential equation, \[ \dfrac {d^2 y(t)}{dt^2} + \dfrac {dy(t)}{dt}+ 6y(t)= \dfrac {-d}{dt}x(t) \]

10 M

6 (b)
Determine the Nyquist sampling rate and Nyquist sampling interval for, \[ i) \ x(t)=1+\cos 2000 \pi t + \sin 4000 \pi t \\ ii) \ x(t)= \left [ \dfrac {\sin (4000 \pi t)}{\pi t} \right ]^2 \]

6 M

6 (c)
Explain briefly, the reconstruction of continuous time signals with zero order hold.

4 M

7 (a)
Find the z-transform of the following and indicate the region of convergence:

1) x(n)=a

ii) x(n)=n(n+1)a

1) x(n)=a

^{n}cos?_{0}(n-2)u(n-2)ii) x(n)=n(n+1)a

^{n}u(n)
10 M

7 (b)
Find the inverse z-transform of the following: \[ i) \ x(z)= \dfrac {z^4 +z^2}{z^2-\frac {3}{4}z+\frac {1}{8}}; |z|>\dfrac {1}{2} by \] Partial fraction expansion method. \[ ii) \ x(z)= \dfrac {1-az^{-1}}{z^{-1}-a}; z>\dfrac {1}{a} \] by long division method.

10 M

8 (a)
A discrete LTI system is characterized by the different equation, y(n)=y(n-1)+y(n-2)+x(n-1)

Find the system function H(z) and indicate the ROC if the system, i) Stable ii) Causal. Also determine the unit sample response of the stable system.

Find the system function H(z) and indicate the ROC if the system, i) Stable ii) Causal. Also determine the unit sample response of the stable system.

10 M

8 (b)
Solve the following difference equation using unilateral z-transform. \[ y(n) - \dfrac {7}{12} y(n-1)+ \dfrac {1}{12}y(n-2)=x(n), \ for \ n\ge 0, \ with \ initial \ conditions \ y(-1)=2, y(-2)=4, \ and \ x(n)= \left ( \dfrac {1}{5} \right )^n u(n) \]

10 M

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