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  Introduction to Fourier Series.Pérez Ruiz Ismael Eliézer

 Biomedical, Electric & Engineering Electronic Department, Mérida Institute of Technology

Tecnológico Avenue,4th Km. Plan de Ayala, Mérida, Yucatán, México

i.s.m123@hotmail.com

 Abstract  —   This document gives a formal description of the

concept of Fourier Series, and its solution by finding the Fourier

coefficients. Finally, is presented the graphics obtained by the

implementation on Fourier series on MatLab.

Keywords  —  Fourier Series, Continuous signal, Fourier

Coefficients. 

I.  I NTRODUCTION 

Fourier analysis is the result of the investigation made by

French mathematician Jean-Baptiste Joseph Fourier finding thesolution to a practical problem: the heat conduction in an iron

ring. During the investigation, Fourier developed the idea that

a periodic signal can be discomposed in terms of basic periodic

signals (sine and cosine) which frequencies are multiples of the

original signal. Fourier created a series for a periodic functionwith a period of “T”, also known as signal, defined in an

interval “T.” The series is defined by: 

  + ∑ cosnωt+ sinnωt∞=  

To obtain the Fourier series and the description of the signal

is necessary to get the Fourier coefficients, given by:

2  

2 /−/  

2 cos /−/  

2 sin /

−/  

Finding this coefficients is possible to build the Fourier

series replacing the values obtained in the original formula.

With this is possible to obtain an approximation to the signal.

The higher value for the series, a signal will be found closest to

the original.

II.  PROCEDURE OF SOLUTION.

For the function defined as

  1, 2 < < 0  1, 0 < < /2 

The procedure of solution is to apply the formulas given. For

a0=

2 /−/ 2 0/

−/  

A similar procedure is made to find the other coefficients:

2 cos

− 

∫ cos −/ +

∫ cos /  

2 1 |−/

+ 1 |/

 

2 {1 [sin0sin] + 1 [sin0] } 

0 

For the coefficient bn is necessary to use the sine function

instead of cosine:

2 sin /−/  

2 sin

−/ + 2 sin /

 

2 1 |−/ + 1 |

/ 

2 {1 [1 c o s] 1 [cos 1] } 

2 1 c o s 

But cos 1, so bn is defined as:

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0, 4 ,  

Substituting in the formula:

  4

1

sin∞

=

 

4 (sin+ 13 sin3+ 15 sin5+⋯) 

III. RESULTS

It is possible to ascertain the results obtained by making

graphics on a software, such as MatLab. The procedure is reallysimple. First is necessary to define the period T, and create

vectors that allow graphing the function. It is also necessary to

determine the number of elements on the vectors, the frequency

and define a vector that will take the output values. After that a

cycle for is used to plot the original signal, as was shown in past practices. Also, cycles for are used to plot the different Fourier

components, by replacing the “n” values on the Fourier seriesformula obtained with the procedure previously shown, and

giving the correct values to the variable “i” of cycle for is

 possible to obtain highest values for the Fourier series, and, thus,

a graph closer to the original. Three exercises are proposed to

show the graphics for different Fourier elements.

1. 

Fourier Series is defined by

  4 1 sin∞=

 

4 (sin+ 13 sin3+ 15 sin5+⋯) 

Plotting the components on MatLab and comparing with the

original, is shown the effect of the Fourier series

2. 

Fourier series formula

  

     .........5

25

13

9

18)(

2  wt Coswt Coswt Cost  f  

  

 

3. 

2/0,

02/,0

)(

T t t wSen A

t T 

t  f   

Fourier series formula:

 

  

    .........6

35

14

15

12

3

12

2)(   wt Coswt Coswt Cos

 Awt Sen

 A At  f  

    

 

IV. 

CONCLUSIONS 

Fourier series is an interesting and efficient way to describe

a periodic signal in terms of sine and cosine functions. It is a

 powerful tool in processing and treatment of signals because

allows to rebuild signals like electrocardiogram, and many

signals used normally in Electronic Engineering.

R EFERENCES 

[1]  J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles,

 Algorithms and Applications, 3rd ed., Ed. Prentice Hall, United States: 1998.

 

)()(2/0,1

02/,1)(   t  f  T t  f   y

T t 

t T t  f    

The figures show the result of the Fourier series for n=1, 3, 9 y 101. It

can be observed how the function changes as the value of “n” getshigher. The command hold on allows to have both graphics: the original

function and the Fourier series to compare the results obtained.

2/0,4

1

02/,4

1

)(

T t T 

t 

t T T 

t 

t   f