Fourier Series

What is Fourier Series?

Let’s know what is fourier series.Most of the phenomena studied in the domain of Engineering and Science are periodic in nature. For instance, current and voltage in an alternating current circuit. These periodic functions could be analyzed into their constituent components (fundamentals and harmonics) by a process known as Fourier analysis. Let’s go through the Fourier series notes and a few fourier series examples..


Periodic functions occur frequently in the problems studied through engineering education. Their representation in terms of simple periodic functions such as sine function and cosine function, which leads to Fourier series (FS). The Fourier series is known to be a very powerful tool in connection with various problems involving partial differential equations. In this article,we will discuss the Fourier analysis with fourier series examples and fourier series notes.


A graph of periodic function f(x) that has period equal to L exhibits the same pattern every L units along the x-axis, so that f(x + L) is equal to f(x) for each value of x. If we know what the function looks like over one complete period, therefore we can sketch a graph of the function over a wider interval of x (that may contain many periods) 


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We can use this property of repetition that defines a fundamental spatial frequency k = 2π L to give a first approximation to the periodic pattern f(x): f(x)' c₁ sin(kx +a₁) equals to a₁ cos(kx) + b₁ sin(kx), where symbols with subscript equal to 1 are constants that determine the amplitude and phase of this first approximation.


A much better approximation of the periodic pattern that is function f(x) can be built up by adding an appropriate combination of harmonics to the fundamental (sine-wave) pattern. For example, adding c₂sin(2kx + c₂) = a₂ cos(2kx) + b₂ sin(2kx) ( which is the 2nd harmonic) c₃ sin(3kx + a₃) = a₃ cos(3kx) + b₃ sin(3kx) ( that is the 3rd harmonic) Here, symbols with subscripts are the constants that generally determine the amplitude and phase of each harmonic contribution .


We can even approximate a square-wave pattern with a suitable sum which involves a fundamental sine-wave and  a combination of harmonics of this fundamental frequency. This sum is known as the Fourier series.


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What is Fourier Series?

  • A Fourier series can be defined as an expansion of a periodic function f(x) in terms of an infinite sum of sine functions and cosine functions.

  • The fourier Series makes use of the orthogonality relationships of the sine functions and cosine functions.


Laurent Series Yield Fourier Series (Fourier Theorem)

It’s very difficult to understand and/or motivate the fact that arbitrary periodic functions have Fourier series representations. In this section, we are going to prove that periodic analytic functions have such a representation using the Laurent expansions.


Fourier Analysis for Periodic Functions

The Fourier series representation of analytic functions can be  derived from Laurent expansions. The elementary complex analysis can generally be used to derive additional fundamental results in the harmonic analysis including the representation of C∞ periodic functions by the Fourier series,Shannon’s sampling theorem ,the representation of rapidly decreasing functions by Fourier integrals and the ideas are classical and of transcendent beauty.


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A function is known as periodic of period L if f(x+L) is equal to f(x) for all the values of x in the domain of function f. The smallest positive value of L can be known as the fundamental period.


The two main trigonometric functions sin x and cos x are basically examples of periodic functions with fundamental period equal to 2π and tan x is periodic with fundamental period equal to π. A constant function is a periodic function with an arbitrary period equal to L.


It is easy to verify that if we have the functions f₁, f₂,... fₙ which are periodic of period L, then any linear combination here,

c₁f₁(x) + …. + cₙfₙ(x) is also known as periodic. Furthermore, if the infinite series


½ a₀ + bₙ sin consists of 2L - periodic functions that converges for all the values of x, then the function to which it converges will be periodic of period 2L. There are two symmetry properties of functions which will be useful in the study of the Fourier series.


Even and Odd Functions

A function f(x) is known to be an even function if f(−x) is equal to f(x).


The function f(x) is known  to be an odd function if f(−x) = −f(x).


Graphically, even functions have symmetry about the y-axis, whereas odd functions have symmetry around the origin.


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Fourier Series Examples:

Sums of odd powers of x are odd: \[x^{3} - 4\]


Sums of even powers of x are even: \[ -x^{6} + 4x^{4} + x^{2} - 3\]


Since x is odd, and  the value of cos x is even


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The product of any two odd functions is even: x sin x is even


The product of  any two even functions is even: \[x^{2}\] cos x is even


The product of an even function and an odd function is odd is :  sin x cos x is odd.


Note:

To find the Fourier series, we know from the fourier series definition it is sufficient to calculate the integrals that will give the coefficients a₀, aₙ and bₙ and plug these values into the big series formula as we know from the fourier theorem.


Typically, the function f(x) will be piecewise - defined.


The big advantage that Fourier series have over Taylor series is that the function f(x) can have discontinuities.


Use of Fourier Series

We know from the fourier series definition that Fourier series can be defined as a way of representing a periodic function as a (possibly infinite) sum of sine functions and cosine functions. It is analogous to a Taylor series, that represents functions as possibly infinite sums of  the monomial terms. 


What is the Fourier Series and its Applications?

A Fourier ( that can be pronounced foor-YAY) series is a specific type of infinite mathematical series that involves trigonometric functions.Fourier series are the ones which are used in applied mathematics, and especially in the field of physics and electronics, to express periodic functions such as those that comprise communications signal waveforms.

FAQ (Frequently Asked Questions)

Question 1)What are the Two Types of Fourier Series and what is a Complex Fourier Series?

Answer) According to the fourier series definition, we can say that there are two types of Fourier series are-

 

1)Trigonometric 

2) Exponential.


The complex Fourier series is generally presented first with periods equal to 2π, then with the general period. The connection with the real-valued Fourier series is explained and  the formulae are given for converting between the two types of representation.

Question 2)How do you use the Fourier Series and What is the Application of Fourier Series in Engineering?

Answer) So this is how we use Fourier Series:

  1. We will take our target function and multiply it by sine function (or cosine function) and integrate (find the area)

  2. Do that for the value of n=0, n=1, etc to calculate the value of each coefficient.

  3. And after we calculate all the coefficients, we put them into the series formula above.

The Fourier series has various applications in electrical engineering, vibration analysis, acoustics, optics,image processing,signal processing, quantum mechanics, econometrics, thin-walled shell theory, etc.