Signals & Systems : Question Paper May 2014 - Electronics & Telecomm. (Semester 4)

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Signals & Systems - May 2014

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 power and energy for the following signals
i)x(t)=3cos 5$\Omega_0t.$
ii)$X[n]=(\dfrac{1}{4})^n u[n]$

(5 marks) 1(b) State and prove the following properties of Fourier transform:
Time shifting property
Convolution property.(5 marks) 1(c) Compare linear conersion and circular convolution.(5 marks) 1(d) Define and Explain
Auto correlation
Cross correlation
Circular convolution.(5 marks) 1(e) e[x]=u[n]-u[n-5]
Sketch even and odd parts of x[n](5 marks) 2(a)

Determine Fourier series representation of the following signals:

(10 marks) 2(b)

For a continuous time signal x(t)=8cos 200πt
Find (1)Minimum sampling rate.
(2)If f_s=400Hz,what is discrete time signal?
(3)If f_s=150Hz,what is the discrete time signal?
(4)Comment on result obtained in 2 and 3 proper justification.

(10 marks) 3(a)

Determine the inverse z transform of the function using Residue method:
$X(z) =\dfrac{3-2z^{-1}+z^{-2}}{1-3z^{-1}+2z^{-2}}.$

(10 marks) 3(b)

Two LTI system in cascade have impulse response h₁[n] and h₂[n]
$h_{1}[n]=(0.9)^{n}u[n]-0.5(0.9)^{n-1}u[n-1]$
$h_{2}[n]=(0.5)^{n}u[n]-0.5(0.5)^{n-1}u[n-1]$
Find the equivalent response h[n]of the system.

(10 marks) 4(a)

A casual LTI system is described$ y[n]=\dfrac{3}{4}y[n-1]-\dfrac{1}{8}y[n-2]+x[n]$
Where y[n]response of the system and x[n]is excitation to the system.

Determine impulse response of the system.
Determine step response of the system.
Plot pole zero pattern and state whether system is stable.

(10 marks) 4(b)(i) Determine the z transform and the ROC of the discrete time signal. X[n] ={2,10,1,2,5,7,2}(5 marks) 4(b)(ii)

Determine the inverse z-transform for the function:$X[Z]=\dfrac{z^{2}+z}{z^{2}-2z+1}\space\space ROC>|z|$

(5 marks) 5(a) The impulse response of an LTI system h[n]={1,2,1,-1}.Find the response y[n]of the system for the input x[n]={1,2,3,1}using Discrete time Fourier Transform.(10 marks) 5(b)

Find the response of a system with transfer function $H(s) =\dfrac{1}{s+5}R_{e}>-5$.

Input $ x(t)=e^{-t}u(t)+e^{-2t}u(t)$

(10 marks) 6(a)

For the given LTI system,described by the differential equation:
$\dfrac{dy^{2}(t)}{dt^{2}}+\dfrac{3dy(t)}{dt}+2y(t)=x(t)$
Calculate output y(t) if input$ x(t)=e^{-3t}u(t)$is applied to the system.

(10 marks) 6(b) Find the autocorrelation,power spectral density of the signal x(t) =3cost +4cos 3t.(10 marks)

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