1.a. Sketch the following signals :

1. $u[n+2] - u[n-3]$
2. $r(t) u(2-t)$.

1.b. Find the convolution of $x(t)$ and $h(t)$ :

$x(t) = u(t+1)$

$h(t) = u(t-2)$.

2.a. Check whether the following system is static/dynamic, linear/non-linear, casual/non-casual, time variant/time invariant :

$y(t) = 10x(t) +5$.

2.b. Check whether the following signal is periodic or non-periodic. If periodic, find the functional time period.

$x(t) = 2 cos (10t + 1) - sin (4t - 1)$.

2.c. Determine the convolution sum of two sequences graphically :

3.a. Find the trigonometric Fourier series for the periodic signal $x(t)$.

3.b. Obtain the Fourier transform of a rectangular pulse :

$x(t) = A rect (t/T)$.

4.a. Obtain the exponential Fourier series of the unit impulse train

Sketch the Fourier spectrum.

4.b. Find the Fourier transform of the following signals :

1. $x(t) = \delta(t)$
2. $x(t) = e^{-at} u(t)$.

5.a. Find the laplace transform of : $x(t) = e^{-5t}[u(t) - u(t-5)]$ and its ROC.

5.b. Find the initial and final values for the following function : $x(s) = \frac {s+5} {s^{2}+3s+2}$

6.a. Determine the inverse laplace transform of : $x(s) = \frac {2} {s(s+1)(s+2)}$

6.b. Find laplace transform of given periodic signal :

7.a. In a random experiment, a trial consists of four successive tosses of a coin. If we define a random variable x as the number of heads appearing in a trial, determine PDF and CDF.

7.b. State and prove any three properties of PDF.

8.a. A certain random variable has the CDF given by :

$\begin{aligned} &=0 \text { for } x \leq 0 \\ F_{x}(x) &=k x^{2} \text { for } 0 \leq x \leq 10 \\ &=100 k \text { for } x\gt10 \end{aligned}$

Find the values of :

1. k
2. $P(x \le 5)$
3. $P(5 \lt x \le 7)$
4. Plot the corresponding PDF.

8.b. State and explain the properties of auto-correlation function for energy signal.