## 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)$X[n]=(\dfrac{1}{4})^n u[n]$

**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:

**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.

**3(a)**

Determine the inverse z transform of the function using Residue method:

$X(z) =\dfrac{3-2z^{-1}+z^{-2}}{1-3z^{-1}+2z^{-2}}.$

**3(b)**

Two LTI system in cascade have impulse response h_{1}[n] and h_{2}[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.

**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.

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