The Fourier series is used to represent a complex function as an infinite summation of sine and cosine functions. In this article, you will see how it is done with a simple example.
In the domain of science and engineering, there are several phenomenons that are periodic in nature. For instance, sound waves, vibrations in structures, as well as current and voltage in AC circuits are examples. To study these things, it is essential to break them down into functions that can be analyzed. Fourier analysis is the process used to understand the constituent components of fundamental and harmonic frequencies in these systems.
A Fourier series is used to represent a periodic function f(x) as an infinite sum of sine and cosine functions. The study of the Fourier series is known as harmonic analysis. It is a highly useful tool to analyze arbitrary periodic functions by breaking them up into simpler terms that can be solved individually and then combined to solve the original problem.
A function is a unique relationship between members of two different sets. A periodic function is a relationship that repeats itself over a set of values. Consider the plot of the periodic function. F(x) = f(x np) where p is the period and n is a set of whole numbers n = 1, 2, … and so on. Here the graph shows a sine function with the least period of 2π as well as the periods -2π, 4π, 6π, etc.
Consider the functions sin(x) and sin(2x). The figure below shows each function as a waveform plotted on a graph. When you add the two, you make a new wave.
Complicated waves in many systems can be analyzed by breaking them down into smaller functions that can be solved individually and added up later. Consider the square wave shown below, for example:
It is possible to use sine waves to make a square wave. To that, we start with a sin(x) function:
Now take a new function sin(3x)/3 :
Add the two functions sin(x) sin(3x)/3 to generate this wave:
Now consider the wave of the function sin(5x)/5:
Add the above functions sin(x) sin(3x)/3 sin(5x)/5 to get:
Similarly if you add 20 more sine waves sin(x) sin(3x)/3 sin(5x)/5 ... sin(39x)/39 we get a wave that’s close:
If this process continues infinitely, we will arrive at a square wave, which is the principle behind the Fourier series. The same process can be used to approximate a lot of different functions like the ones shown below:
The Fourier series formula for a function f(x) (such as the square wave in the above example) between the interval [-L, L] is given by:
Fourier series is a branch of harmonic analysis. In music, if a note has a frequency of (f) which is called the fundamental frequency, its integer multiples such as 2f, 3f, 4f so on are called harmonics. These harmonics are where the instrument playing the note has its peak frequencies. For example, if the fundamental frequency of the first harmonic is 50 Hz, the second harmonic will be 100 Hz, and the third harmonic will be 150 Hz.
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