1.a. Perform the following operations on the given signal $x(t)$ which is defined as : $x(t) = u(t+4)$

1. Sketch $z(t) = x(-t-1)$
2. Sketch $y(t) = x(t) +z(t)$.

1.b. Write the expression for energy and power of the signal. Also determine whether the following signals is energy or Power, and find energy or time averaged power of the signal : $x(t) = 5cos(10\pi t)+sin(20\pi t); -\infty \le t \le \infty$.

1.c. Determine whether the following system is Ststic/Dynamic, Casual/Non-casual and Stable/Unstable and justify : $h(t) = e^{-10t}u(t)$.

2.a. Compute the convolution integral by graphical method and sketch the output for the following signals :

1. $x(t) = u(t)$
2. $h(t) = e^{-2t}u(t)$.

2.b. Check whether the following signal is even or odd and determine the even and odd part of the signal : $x(t) = u(t)$.

2.c. Compute the convolution integral for the following signal :

$x(t)=u(t), h(t) =\delta (t+1) + \delta (t) + \delta (t-1)$.

3.a. Find the trigonometric Fourier series for the periodic signal $x(t)$ shown in the following figure :

3.b. State any six properties of Fourier transform.

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

1. $x(t)=sng(t)$
2. $x(t)=cos(\omega_{0} t) u(t)$.

4.b. Write expression for trigonometric Fourier series and exponential Fourier series.

4.c. Define amplitude and phase spectra of the signal.

5.a. Find the inverse laplace transform of $X(s)=\frac{2}{(s+4)(s-1)}$

If the region of convergence is :

1. $-4 \le Re(s) \lt1$
2. $Re(s) \gt 1$
3. $Re(s) \lt -4$.

5.b. A signal $x(t)$ has laplace transform : $X(s) = \frac{s+2}{s^{2}+4s+5}$

Find the laplace transform of the following signals :

1. $y_{1}(t) = tx(t)$.
2. $y_{2}(t) = e^{-t}x(t)$.

6.a. Find the laplace transform of the following signal and sketch ROC :

$x(t) = e^{-3t}u(t) + e^{-5t} u(t)$.

6.b. Find the initial and final value of the following signal :$X(s) = \frac {2s+3}{s^{2}+5s-7}$.

6.c. State the relationship between Fourier transform and laplace transform.

7.a. Find the following for the given signal $x(t)$ :

1. Auto-correlation
2. Energy from auto-correlation

$x(t)=e^{-10t}u(t)$.

7.b. Define probability and state the properties of PDF. Also state the relationship between CDF and PDF.

8.a. Suppose a certain random variable has CDF :

$F_{x}(x) = 0 , x \le 0$

$F_{x}(x) = k x^{2} , 0 \lt x \le 10$

$F_{x}(x) = 100k , x \gt 10$

Calculate $K$. Find the values of $P(X \le 5)$ and $P(5 < X \le 7)$.

8.b. A coin is tossed three times. Write the sample space which gives all possible outcomes. A random variable X, which represents the number of heads obtained on any tripple toss. Also find the probabilities of X and plot the C.D.F.