Solve any one question from Q1 and Q2

1 (a)
Find odd and even components of following signals.

i) x(t) = 3-2t

ii) x[n]=u[n].

i) x(t) = 3-2t

^{2}+6t^{3}+9t^{4}.ii) x[n]=u[n].

4 M

1 (b)
Determine whether the following signals are periodic or aperiodic. If periodic find the fundamental period.

i) x(t)=3 sin (4πt)+7 cos (3πt).

ii) x[n]=cos(2n).

i) x(t)=3 sin (4πt)+7 cos (3πt).

ii) x[n]=cos(2n).

4 M

1 (c)
Find the response of the LTI system for i/p \( x(t) = rest \left ( \dfrac {t}{2} \right ). \) If the system is described by impulse response. h(t)=δ(t+1)-2δ(t)+δ(t-1).

4 M

2 (a)
Sketch the following signals: \[ i) \ x(t)=u (t) + u(t-2)+ u(t-4)-3 u(t-6) \\
ii) \ x(t) = \sum^{10}_{k=-10} \delta (t-2k) \]

4 M

2 (b)
Classify whether the following system are:

a) Causal/Non-causal.

b) Stable/unstable.

i) h(t)=e

ii) h[n]=δ[n]+δ[n-2]-2δ[n-3].

a) Causal/Non-causal.

b) Stable/unstable.

i) h(t)=e

^{2t}u(-t)ii) h[n]=δ[n]+δ[n-2]-2δ[n-3].

4 M

2 (c)
Find the overall impulse response of the system given below:

i) h

h

h

i) h

_{1}(t)=δ(t),h

_{2}(t)=u(t).h

_{3}(t)=u(t-2).

4 M

Solve any one question from Q3 and Q4

3 (a)
Find the Fourier Transform of following signals.

i) x(t) = sgn (t).

ii) x(t) = rect(t).

i) x(t) = sgn (t).

ii) x(t) = rect(t).

6 M

3 (b)
Find the transfer function and impulse response of the system describe by following differential equation. \[ \dfrac {d^2}{dt^2}y(t)+5 \dfrac {d}{dt}y(t)+6y(t) = \dfrac {dx}{dt} (t)+ x(t). \]

6 M

4 (a)
Using differentiation in frequency domain property. Find Fourier transform of:

y(t)=tx(t)

where x(t) = e

y(t)=tx(t)

where x(t) = e

^{-at}u(t).
6 M

4 (b)
Find initial and final value of following: \[ i) \ x(S) = \dfrac {2}{s(s^2 +3s+5)} \\
ii) \ x(S) = \dfrac {1}{S^2} \]

6 M

Solve any one question from Q5 and Q6

5 (a)
Prove that auto-correlation and ES.D from a Fourier transform pair. Verify the same for x(t)=e

^{-at}u (t).
7 M

5 (b)
Compute cross correlation between given two sequences: \[ x_1 [n] = \{1, 1, \underset{\uparrow}{2} , -1 \} \\
x_2 [n] = \{ 1, \underset{\uparrow}{2}, 3, 4 \} \] Using analytical or graphical methods only sketch the output sequence.

6 M

6 (a)
Find auto-correlation, PSD and power of given signal.

x(t)=2 cos t +3 cos 3t+5 sin 4 t.

x(t)=2 cos t +3 cos 3t+5 sin 4 t.

7 M

6 (b)
State and describe the properties of energy spectral density (ESD).

6 M

Solve any one question from Q7 and Q8

7 (a)
PDF of a random variable 'X' is given as F

Find:

i) Mean E[x].

ii) Mean square E[x

iii) Variance

iv) Std. deviation.

_{x}(x)=e^{-x}for x≥0.Find:

i) Mean E[x].

ii) Mean square E[x

^{2}].iii) Variance

iv) Std. deviation.

7 M

7 (b)
Explain Gaussian probability model with respect to its density and distribution function.

6 M

8 (a)
A random variable X is defined by the CDF. \[ F_x (x)= \left\{\begin{matrix}
0 & x<0 \\\ \dfrac {1}{2}x
& 0 \le x \le 1 \\ k &x \ge 1
\end{matrix}\right. \] i) Find value of K.

ii) Find and sketch PDF.

iii) P(x>2).

ii) Find and sketch PDF.

iii) P(x>2).

7 M

8 (b)
State and explain properties of PDF.

6 M

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