Energy calculation of signal using Parseval's Relation

1. Lab Task : 02
Submitted By :
Syed Abuzar Hussain Shah
Reg #
SP15-BEE-096
Submitted To:
Sir Ateeq-ul-Anam
Class:
BEE-5A
Dated: 03/04/2017

2. Problem Statement :
Find the band width of a rectangular pulse, having pulse width ‘ ’, using
Parsaval’s theorem. Also derive a formula for calculating the band width of a
pulse in terms of pulse width in hertz.
Literature Background :
Parseval's theorem usually refers to the total energy of the signal which is
equal to the integral of the square of a function.
Mathematically:
∫ ( ) ∫ ( ) (1)
Or
∫ ( ) ∫ ( ) (2)
These two equations represents the area under the curve of the signal which
defines the energy of the signal.
Pulse Width ( ) is the time interval between the leading edge and trailing edge
of a pulse at a point.
Band Width is the range of frequencies within a given band, in particular that
used for transmitting a signal. Its contain 90% of energy.
We can find pulse width using Parsevals’s theorem. We will also see a
relationship between pulse width and band width at the end of this lab task.
Procedure :
Step 1:
Take a Fourier transform of a rectangular pulse and derive frequency domain
function F(f) (Sinc function), where ‘f’ is frequency in hertz.
Figure No 1: Rectangular Pulse (Frequency Response of Sinc function )

3. Step 2: Fourier Transform
Now taking Fourier Transform of sinc function and then analyse it using
MatLab.
X(jw) = ∫ ( )
( ) { (3)
⇒ F(w) = ∫
F(w) =
F(w) = [ ]
F(w) = [ ]
F(w) = [ ]
F(w) = [ ]
F(w) = ( )
F(w) = ( )
F(w) = ( )
F(w) = ( )
Step 3: Matlab Coding:
clc
clear all
close all
Ta=input('Enter pulsewidth (from 0 to 1) = ')

4. df=0.001;
f=0;
Ef=0;
while ((Ef/Ta)*100)<45
F=Ta*sinc(f*Ta);
y=sum((F^2)*df);
Ef=Ef+y;
f=f+df;
end
B=f*2;
disp(['B = ' num2str(B) 'Hz'])
Step 4: Input and Output:
Enter pulsewidth (from 0 to 1) = 0.1
Ta =
0.1000
B = 16.94Hz
Enter pulsewidth (from 0 to 1) = 0.4
Ta =
0.4000
B = 4.212Hz
Enter pulsewidth (from 0 to 1) = 0.5
Ta =
0.5000
B = 3.364Hz
Enter pulsewidth (from 0 to 1) = 0.9

5. Ta =
0.9000
B = 1.856Hz
Analysis and Observation:
In this lab I observed the changing behaviour of the bandwidth of a signal with
changing pulse width. I find out the relation between bandwidth and pulse width
which is :
B (4)
Table No 1:
S.NO Pulse Width ‘ ’ in second Band Width in Hertz
1 0.1 16.94
2 0.4 4.212
3 0.5 3.364
4 0.9 1.856

6. Question & Answers :
Q1) Based on the above exercise, what can be possible definition of
bandwidth ?
Ans. Bandwidth can be defined as the range of the frequencies within a given band
containing 90% of energy that is used for transmission of a signal.
Q2) What is the relationship of proportionality between pulse width and
band width ?
Ans. Pulse width and Bandwidth are inversely proportional to each other. As
one quantity(pulse width) increases, second one(band width) decreases.
As mentioned in eq (1) and Table no 1.
Conclusion:
 From this lab I concluded that Bandwidth having 90% energy of the
signal.
 Also pulse width and band width have inverse relation. Increase in one
cause other to decrease.
 We can calculate the energy of the signal by Parseval’s Theorem.
 We can find Bandwidth from pulse width by formula:
B = Hz