Signal and Systems - May 2012

Electronics Engineering (Semester 5)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Determine whether the following signals are energy signals or power signals > calculate their energy or power
(i) x(t)=Acos(2?f₀t+?)
(ii) x(n)=(1/4)ⁿ u(n)(5 marks) 1 (b) Let x(n)=u(n+1)-u(n-5). Find and sketch even and odd parts of x(n).(5 marks) 1 (c) Mention and explain the conditions for the system to be called as IIR.(5 marks) 1 (d) State and explain Gibb's phenomenon.(5 marks) 2 (a) (i) Plot the signals with respect to time.
x(t)=u(t)-r(t-1)+2r(t-2)-r(t-3)+u(t-4)-2u(t-5)
(ii) Find the even odd parts of the signal.(10 marks) 2 (b) State and prove the following properties of the Fourier transform
(i) Frequency Different and time integration.(10 marks) 3 (a) An analog signal x(t) is given by x(t)=2cos(2000?t)+3 sin(6000?t)+8 cos(1200?t)
(i) calulate nyquist sampling rate.
(ii) If x(t) is sampled at the rate F(s)=5 KHz. What is the discrete time signal obtained after sampling.
(iii) What is the analog signal y(t) we can reconstruct from the samples if the ideal interpolation is used(10 marks) 3 (b) Find the Laplace Transform of the signal show below (10 marks) 4 (a) Obtain the transfer function of the system defined by the following state space equations
$$\left[\begin{array}{ccc}x1\left(t\right) \\x2\left(t\right) \\x3\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}-1 & 1 & -1 \\0 & -2 & 1 \\0 & 0 & -3\end{array}\right]\left[\begin{array}{ccc}x1\left(t\right) \\x2\left(t\right) \\x3\left(t\right)\end{array}\right]+\left[\begin{array}{ccc}1 & 0 \\0 & 1 \\1 & 0\end{array}\right]\left[\begin{array}{cc}u1\left(t\right) \\u2\left(t\right)\end{array}\right]$$
$$\left[\begin{array}{cc}y1\left(t\right) \\y2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}1 & 1 & 1 \\0 & 1 & 1\end{array}\right]\left[\begin{array}{ccc}x1\left(t\right) \\x2\left(t\right) \\x3\left(t\right)\end{array}\right]$$(10 marks) 4 (b) Find the state equations and output equtions for the system given by
$$G\left(s\right)=\frac{1}{s^3+4s^2+3s+3}$$(10 marks) 5 (a) Find the Fourier series for the function x(t) defined by
$$f\left(x\right)=\left\{\begin{array}{cc}0 & \frac{-T}{2}<1<0 \\A\sin{{\bar{\omega{}}}_0t} & 0<1<\frac{T}{2}\end{array}\right\}$$
$$And\ x\left(t+T\right)=x\left(t\right),\ {\bar{\omega{}}}_0=\frac{2\pi{}}{T}$$(10 marks) 5 (b) Obtain the Fourier transform of rectangle pulse of duration 2 seconds and having a mangnitude of 10 volts.(10 marks) 6 (a) Develop cascade and parallel realization structures for
$$H\left(z\right)=\frac{\frac{z}{6}+\frac{5}{24}z^{-1}+\frac{1}{24}z^{-2}}{1-\frac{1}{2}z^{-1}+\frac{1}{4}z^{-2}}$$(10 marks) 6 (b) Determine the system function, unit sample response and pole zero plot of the system decribe by the different equations y(n)- 1/2 y(n-1)=2x(n) and also comment on types of the system.(10 marks) 7 (a) Explain the relationship between tha laplace transform and Fourier transform.(7 marks) 7 (b) State properties of state transition matrix.(6 marks) 7 (c) State and discuss the properites of the region of convergence for Z-transform.(7 marks)