Applications of Fourier Series in Signal Processing and Engineering

IntroductionThe Fourier series is a mathematical tool widely used in engineering and signal processing to represent periodic signals in terms of a sum of sinusoids. 
It was discovered by Joseph Fourier in the early 19th century and is based on the idea that any periodic signal can be represented as an infinite sum of harmonically related sine and cosine functions. 
The Fourier series is an important tool for analyzing and processing signals in many engineering and physics applications, including filtering, audio and image processing, and communication systems.
ExplanationFourier series utilises the orthogonality (that being the 90-degree phase shaft) of sine and cosine functions. By using this principle one can effectively break down any repeating wave into a series of equations that can be solved individually and recombined to obtain an extremely close approximation of the original wave. 
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Fourier series waves explanation
As can be viewed in the above figure, these repeating functions are effectively broken down and reconstructed using a sum of sine and cosine functions. The accuracy of the approximation depends on the number of summations that have taken place. As per the graph, as the number of sine and cosine waves increase (denoted by n), the graph starts to approximate the desired function. It can also be thought that as n approaches infinity, the approximation will be exactly equal to the original function. 
Fourier Series Representation of a Periodic Signal A periodic signal f (t) with period T can be represented as a Fourier series as follows:
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f(t)=a02+∑n=1∞(ancos⁡(2πntT)+bnsin⁡(2πntT))f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{2\pi nt}{T} \right) + b_n \sin \left( \frac{2\pi nt}{T} \right) \right)f(t)=2a0​​+∑n=1∞​(an​cos(T2πnt​)+bn​sin(T2πnt​))
Where a0, an, bn and n are represented as: 
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a0=2T∫0Tf(t)dtan=2T∫0Tf(t)cos⁡(2πntT)dtbn=2T∫0Tf(t)sin⁡(2πntT)dtn=1,2,3...a_0 = \frac{2}{T} \int_{0}^{T} f(t) dt \\ a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(2\pi n \frac{t}{T}) dt \\ b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(2\pi n \frac{t}{T}) dt \\ n = 1,2,3...a0​=T2​∫0T​f(t)dtan​=T2​∫0T​f(t)cos(2πnTt​)dtbn​=T2​∫0T​f(t)sin(2πnTt​)dtn=1,2,3...
Interactive GraphTry increasing the number of samples and see how the graph changes! (Max: 10)
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Applications of Fourier Series in Engineering and Signal ProcessingOne of the most critical uses of Fourier series is in the representation of periodic signals. By representing a periodic signal as a sum of harmonically related sine and cosine functions, it is possible to analyse and manipulate the signal in various ways. For example, by only keeping the first few terms of the Fourier series, it is possible to simplify and approximate a complex signal, which can be useful in filtering and compression applications.
Another important application of the Fourier series is in the analysis of non-periodic signals. By using the Fourier transform, it is possible to transform a non-periodic signal into the frequency domain, where the signal can be analysed in terms of its frequency components. This is particularly useful in filtering applications, where it is possible to remove unwanted frequency components from a signal.
The Fourier series is also widely used in audio and image processing, where signals are often represented in terms of a sum of sinusoids. By using the Fourier series, it is possible to analyse and manipulate audio and image signals in terms of their frequency components, which can be useful in a variety of applications, such as noise reduction, compression, and filtering.
ConclusionThe Fourier series is a powerful mathematical tool that is widely used in engineering and signal processing. By representing a periodic signal as a sum of harmonically related sine and cosine functions, it is possible to analyze and manipulate signals in a variety of ways, including filtering, audio and image processing, and communication systems. With its wide range of applications and its simplicity and versatility. The Fourier series is an essential tool for any engineer or physicist working in the fields of signal processing and engineering.
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