Fourier Series

The concept of the Fourier Series has revolutionized the way we analyze and understand periodic functions in mathematics and engineering. From signal processing to electrical engineering, the ability to decompose complex functions into simpler trigonometric components has widespread applications. In this blog, we'll explore the Fourier Series, discuss different types, provide examples, and highlight their practical uses.

1.0What is a Fourier Series?

A Fourier Series is a mathematical representation of a periodic function as an infinite sum of sine and cosine functions (or equivalently, complex exponentials). It is named after Jean-Baptiste Joseph Fourier, who introduced the idea that any periodic function can be expressed as the sum of sinusoidal waves.

The key idea is that any periodic signal can be broken down into a series of sine and cosine waves, each with different frequencies and amplitudes. This breakdown allows easier analysis and manipulation of signals, especially in fields like signal processing, communications, and vibration analysis.

2.0Fourier Series Formula and Equation

The general Fourier Series formula for a periodic function f(x) with a period T is given by:

$f(x)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }\left[a_n\cos \left(\frac{2\pi nx}{T}\right)+b_n\sin \left(\frac{2\pi nx}{T}\right)\right]$

Where:

• a_0 is the average (DC) component.
• a_n and b_n are the Fourier coefficients, representing the amplitudes of the cosine and sine terms, respectively.

These coefficients are calculated using the following formulas:

$a_0=\frac{2}{T}{\int }_0^{T}f(x)dx$

$a_n=\frac{2}{T}{\int }_0^{T}f(x)\cos \left(\frac{2\pi nx}{T}\right)dx$

$b_n=\frac{2}{T}{\int }_0^{T}f(x)\sin \left(\frac{2\pi nx}{T}\right)dx$

3.0Euler’s Formula for Fourier Series

Euler's formula is a fundamental equation in complex analysis that relates complex exponentials to sine and cosine functions. It is written as:

$e^{ix}=\cos (x)+i\sin (x)$

Using Euler’s formula, we can represent the Fourier Series in a more compact form, especially useful for analyzing the frequency components of a signal in the complex domain. The exponential Fourier series is given by:

$f(x)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{\frac{2\pi nx}{T}}$

Where C_n are the Fourier coefficients and ii is the imaginary unit. The exponential form of the Fourier series is useful in signal processing and communications as it simplifies the computation of Fourier coefficients. 

4.0Types of Fourier Series: Even and Odd Functions

Fourier series can be classified based on the symmetry of the function being represented. These types are:

• Even Fourier Series: A function is even if it satisfies the condition f(x) = f(-x). The Fourier series of even functions only contains cosine terms (no sine terms). This is because cosine is an even function, and the sine terms, being odd functions, cancel out.
• Odd Fourier Series: A function is odd if it satisfies the condition f(x) = -f(-x). The Fourier series of odd functions only contains sine terms (no cosine terms). Sine is an odd function, which matches the symmetry of the odd function.

The presence of only sine or cosine terms greatly simplifies the calculations for certain types of functions. 

5.0Example of a Fourier Series

Let’s consider an example of a simple periodic square wave. A square wave can be expressed as a Fourier series. The function f(x)f(x) for a square wave is typically defined as:

$f(x)=\{\begin{array}{ll}1,& 0\le x<\frac{T}{2}\\ -1,& \frac{T}{2}\le x<T\end{array}$

The Fourier series for this function contains only odd harmonics (i.e., sine terms), which gives the series:

$f(x)=\frac{4}{\pi }\left(\sin \left(\frac{2\pi x}{T}\right)+\frac{1}{3}\sin \left(\frac{6\pi x}{T}\right)+\frac{1}{5}\sin \left(\frac{10\pi x}{T}\right)+\dots \right)$

This demonstrates how a complex waveform like a square wave can be represented as a sum of simpler sine waves.

6.0Exponential Fourier Series

The exponential Fourier series is an alternative way of writing the Fourier series using complex exponentials. This form is preferred in many signal-processing applications because it simplifies calculations involving both positive and negative frequencies.

The exponential Fourier series is written as:

$f(x)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{\frac{2\pi nx}{T}}$

Where C_n are the Fourier coefficients and represent the amplitude and phase information of the signal’s frequency components. This representation is particularly useful in digital signal processing and communications systems.

7.0Use of Fourier Series

Fourier Series have a wide range of applications across various fields:

1. Signal Processing: Fourier series are extensively used in analyzing and filtering signals, especially in audio, image, and speech processing. By transforming signals into the frequency domain, engineers can manipulate and enhance the signal more easily.
2. Electrical Engineering: In circuits and systems, Fourier analysis helps in analyzing and designing filters, oscillators, and communication systems.
3. Vibration Analysis: Fourier series are used to analyze and solve problems related to the vibrations of mechanical structures. By breaking down complex vibration patterns into simpler sinusoidal components, engineers can understand the behavior of the structure more thoroughly.
4. Acoustics: Fourier series helps in sound synthesis, music production, and understanding how different frequencies contribute to musical notes or sound waves.
5. Quantum Mechanics: Fourier series are used in the analysis of wave functions, especially in quantum physics, to study the behavior of particles.

8.0Fourier Series Representation

Fourier series is an infinite series representation of periodic function in terms of trigonometry function sine and cosine.

$f(x)=a_0+{\sum }_{n=1}^{\infty }\left(a_n\cos nx+b_n\sin nx\right),\alpha <x<\alpha +2\pi$

It is known as Fourier Euler formula, where

$a_0=\frac{1}{2\pi }{\int }_{\alpha }^{\alpha +2\pi }f(x)dx$

$a_n=\frac{1}{\pi }{\int }_{\alpha }^{\alpha +2\pi }f(x)\cos nxdx$

$b_n=\frac{1}{\pi }{\int }_{\alpha }^{\alpha +2\pi }f(x)\sin nxdx$

In general formula

$f(x)=a_0+{\sum }_{n=1}^{\infty }\left[a_n\cos \left(\frac{n\pi x}{l}\right)+b_n\sin \left(\frac{n\pi x}{l}\right)\right],-l<x<l$

where   

$a_0=\frac{1}{2l}{\int }_{-l}^{l}f(x)dx$

$a_n=\frac{1}{l}{\int }_{-l}^{l}f(x)\cos \left(\frac{n\pi x}{l}\right)dx$

$b_n=\frac{1}{l}{\int }_{-l}^{l}f(x)\sin \left(\frac{n\pi x}{l}\right)dx$

9.0Solved Example of Fourier Series

Example: Find Fourier series for   

$f(x)=e^{i\pi n(0,2\pi )}$

Solution: 

The Fourier series is given by

$f(x)=a_0+{\sum }_{n=1}^{\infty }\left(a_n\cos nx+b_n\sin nx\right)$ … (i)

$a_0=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)dx-\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{ax}dx$ … (ii)

$a_0=\frac{1}{2\pi }\left(\frac{e^{ax}}{a}\right){|}_0^{2\pi }=\frac{1}{2\pi }\left(\frac{e^{2a\pi }-1}{a}\right)$

Also 

$a_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x)\cos nxdx$ 

$a_n=\frac{1}{\pi }{\int }_0^{2\pi }{e}^{ax}\cos nxdx$

$\left\{\int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}\left[a\cos bx+b\sin bx\right]\right\}$

$a_n=\frac{1}{\pi }{\left[\frac{e^{ax}(a\cos nx+n\sin nx)}{a^2+n^2}\right]}_0^{2\pi }$ 

$a_n=\frac{1}{\pi (a^2+n^2)}\left[e^{2a\pi }(a\cos 2n\pi +n\sin 2n\pi )-e^0(a\cos 0+n\sin 0)\right]$

$a_n=\frac{1}{\pi (a^2+n^2)}\left[a{e}^{2a\pi }\cos 2n\pi -a{e}^{a0}\cos 0\right]$

$\square \sin 2n\pi =0,n=1,2,3,\dots$

$a_n=\frac{1}{\pi (a^2+n^2)}\left(a{e}^{2a\pi }\cos 2n\pi -a\right)\text{where}\cos 2n\pi =1,n=1,2,3,\dots$ … (iii)

$a_n=\frac{a}{\pi (a^2+n^2)}\left[e^{2a\pi }-1\right]$ 

$b_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x)\sin nxdx$

$b_n=\frac{1}{\pi }{\int }_0^{2\pi }{e}^{ax}\sin nxdx$

$\left\{\int e^{ax}\sin bxdx=\frac{e^{ax}}{a^2+b^2}\left[a\sin bx-b\cos bx\right]\right\}$

$b_n=\frac{1}{\pi }{\left[\frac{e^{ax}(a\sin nx-n\cos nx)}{a^2+n^2}\right]}_0^{2\pi }$

$b_n=\frac{1}{\pi }\left[\frac{e^{2a\pi }(a\sin 2n\pi -n\cos 2n\pi )-e^0(a\sin 0-n\cos 0)}{a^2+n^2}\right]$

$b_n=\frac{1}{\pi }\left[-n\frac{e^{2a\pi }\cos 2n\pi -e^0\cos 0}{a^2+n^2}\right]$ … (iv)

$b_n=\frac{n}{\pi (a^2+n^2)}(1-e^{2ax})$

Using equation (ii), (iii) and (iv) in equation (i), then the

Fourier series is given by

$f(x)=\frac{1}{2\pi }\left(\frac{e^{2a\pi }-1}{a}\right)+{\sum }_{n=1}^{\infty }\frac{a(e^{2a\pi }-1)}{\pi (a^2+n^2)}\cos nx+{\sum }_{n=1}^{\infty }\frac{n(1-e^{2a\pi })}{\pi (a^2+n^2)}\sin nx$

$\Rightarrow f(x)=\frac{1}{\pi }(e^{2a\pi }-1)\left[\frac{1}{2a}+{\sum }_{n=1}^{\infty }\frac{1}{(a^2+n^2)}(-n\sin nx)\right]+{\sum }_{n=1}^{\infty }\frac{a(e^{2a\pi }-1)}{\pi (a^2+n^2)}\cos nx$