Signals & Systems - Dec 2012

Electronics & Telecomm. (Semester 4)

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 signals are power or energy signals.
(i) x(t)=0.9e^-3t u(t)
(ii) x[n]=u[n](4 marks) 1 (b) Convolve h[n] = n+1; 0 ≤ n ≤ 3 with
x[n] = n²; 0 ≤ n ≤ 2.(4 marks) 1 (c) Given Equation: y(t)=2r(t)-2r(t-1)-2u(t-3)
Sketch y(t) and odd part of y(t). (4 marks) 1 (d) Determine whether each of the signals is periodic. If so find its fundamental period-
(4 marks) 1 (e) Check Dynamicity, Linearity, Time variance and Causality of
y[n] = x[n] + x[n+2](4 marks) 2 (a) Determine the exponential Fourier Series of the signal x(t).
(10 marks) 2 (b) Perform convolution of :
(i) 2u(t) with u(t) (2 marks)
(ii) e^2t u(t) with e^-5t u(t) (4 marks)
(iii) tu(t) with e^-5t u(t) (4 marks)(10 marks) 3 (a) Sketch
x(t) = t ; 0<t<1<br>= 1 ; 1<t<2<br>= 3-t ; 2<t<3<br>Then sketch
(i) x(2-t)
(ii) x(t-3)
(iii) x(2t)
(iv) 0.5x(-t)</t<3<br></t<2<br></t<1<br>(10 marks) 3 (b) Consider an analog signal:
x(t) = 5cos(50πt) + 2sin(200πt) - 2cos(100πt)
(i) Determine Nyquist Sampling Rate. (2 marks)
(ii) If the given x(t) is sampled at the rate of 200Hz, what is the discrete time signal obtained after sampling? (3 marks)(5 marks) 3 (c) Find the DTFT of x[n] = {2,1,2} and compute its magnitude at ω = 0 and ω = π/2. (5 marks) 4 (a) Find the Z-Transform
(i) x[n] = (0.1)ⁿu[n] + (0.3)ⁿu[-n-1]
(ii) x[n]=(0.5)ⁿ[u[n] - u[n-2]].(5 marks) 4 (b) Prove convolution property of Z-Transform.(5 marks) 4 (c) Determine the response of an LTI discrete time system governed by the difference equation
y[n] - 2y[n-1] - 3y[n-2] = x[n] + 4x[n-1] for the input
x[n]=2ⁿu[n] with initial conditions
y[-2] = 0, y[-1] = 5. (10 marks) 5 (a) (i) Using Laplace Transform, determine the total response of the system described by the equation
y''(t) + 5y'(t) + 4y(t) = x'(t).
The initial conditions are y(0)=0 and y'(0)=1. The input to the system is x(t)=e^-2tu(t). (6 marks)
(ii) Also find the Impulse Response of the above system assuming initial conditions = 0. (4 marks)(10 marks) 5 (b) Realize Direct Form I, Direct Form II first order cascade and first order parallel structures
(10 marks) 6 (a) Find x[n] if

(i) ROC: |z|> 1/3
(ii) ROC: 1/4 < |z| < 1/3
(iii) ROC: |z|< 1/4(10 marks) 6 (b) Prove time shifting property of Fourier Transform.(5 marks) 6 (c) Determine the unit step response of the system whose impulse response is given as h(t) = 3u(t). (5 marks) 7 (a) The state space representation of a discrete time system is given as-

Derive the transfer function H(z) of the system. (10 marks) 7 (b) Using suitable method obtain the state transition matrix φ(t) for the system matrix.(10 marks)