Umal-qura university

Example: Triangular wave with N=3

clc; clear all; close all t=0:0.001:5;

x=(‐8/(pi*pi))*exp(i*(2*pi*0.5*t)); y=(‐ 8/(9*pi*pi))*exp(i*(2*pi*0.5*3*t)); s=x+y;

plot(t,real(s),'linewidth',3); title('Triangular Wave with N=3'); ylabel('Amplitude'); xlabel('Time');

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grid;

Example: Triangular wave with N=11 clc; clear all; close all t=0:0.01:0.25;
ff=25;


x1=(‐8/(pi^2))*exp(i*(2*pi*ff*t)); for k=3:2:21,
fh=ff*k; x=(‐8/(pi^2*k^2))*exp(i*(2*pi*fh*t)); y=x1+x;
end
plot(t,real(y),'linewidth',3); title('Triangular Wave with N=11'); ylabel('Amplitude');
xlabel('Time'); grid;
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Lab # 10

OBJECTIVES OF THE LAB

This lab aims at the understanding of:

• 
  Fourier Series Representation of Continuous Time Period Signals
  
• 
  Convergence of CT Fourier Series
  

1. 
   FOURIER SERIES REPRESENTATION OF CONTINUOUS TIME PERIOD SIGNALS
   

A signal expressed by the formula
is periodic with period T, as it is linear combination of harmonically related complex exponentials that are all periodic with T. Any well‐behaving periodic function can be expressed as a linear combination of harmonically related complex exponentials. The representation of periodic signal in this way is known as
^{Fourier series representation and the weight a}k^{’s are referred to as Fourier series coefficients. Given a} periodic signal x(t), it is possible to determine its Fourier series coefficients through the following integral.

This integral can be done over any time interval of length T, the period of the signal x(t).

1. Synthesis of a Simple Periodic Signal

Following example demonstrates that the linear combination of harmonically related complex exponentials leads to a periodic function. The signal used in example is