## 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] = n^{2}; 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)^{n}u[n] + (0.3)^{n}u[-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]=2^{n}u[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^{-2t}u(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)