Question Paper: Signals & Systems : Question Paper May 2014 - Electronics & Telecomm. (Semester 4) | Mumbai University (MU)
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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 fs=400Hz,what is discrete time signal?
(3)If fs=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)