A Fourier series (/ˈfʊrieɪ, -iər/) is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or period), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any well behaved periodic function (see Pathological and Dirichlet conditions). The components of a particular function are determined by analysis techniques described in this article. Sometimes the components are known first, and the unknown function is synthesizedby a Fourier series. Such is the case of a discrete-time Fourier transform. Convergence of Fourier series means that as more and more components from the series are summed, each successive partial Fourier series sum will better approximate the function, and will equal the function with a potentially infinite number of components. The mathematical proofs for this may be collectively referred to as the Fourier Theorem (see ). The figures below illustrate some partial Fourier series results for the components of a square wave. * A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum). * The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums converge to (become more and more like) the square wave. * Function (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform is a frequency-domain representation that reveals the amplitudes of the summed sine waves. Another analysis technique (not covered here), suitable for both periodic and non-periodic functions, is the Fourier transform, which provides a frequency-continuum of component information. But when applied to a periodic function all components have zero amplitude, except at the harmonic frequencies. The inverse Fourier transform is a synthesis process (like the Fourier series), which converts the component information (often known as the frequency domain representation) back into its time domain representation. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis. (Wikipedia).
Fourier Series Coefficients (where did they come from?)
Learn how to derive the Fourier series coefficients formulas. Remember, a Fourier series is a series representation of a function with sin(nx) and cos(nx) as its building blocks. Meanwhile, a Taylor series is a series representation of a function with x^n as its building blocks. These are
From playlist Fourier Series
Intro to Fourier series and how to calculate them
Download the free PDF from http://tinyurl.com/EngMathYT This is a basic introduction to Fourier series and how to calculate them. An example is presented that illustrates the computations involved. Such ideas are seen in university mathematics.
From playlist Fourier
How to compute a Fourier series: an example
Free ebook http://tinyurl.com/EngMathYT This video is a demonstration on how to compute a Fourier series of a simple given function. I discuss how to calculate the Fourier coefficients through integration and the simplifications involved. Fourier series are an important area of applied
From playlist Engineering Mathematics
Free ebook http://tinyurl.com/EngMathYT A review question involving Fourier series, including their calculation and related concepts.
From playlist Engineering Mathematics
Free ebook http://tinyurl.com/EngMathYT A tutorial showing how to to calculate Fourier series. Several examples are presented to illustrate the ideas.
From playlist Engineering Mathematics
Free ebook http://tinyurl.com/EngMathYT A tutorial on Fourier series and how to calculate them. Plenty of examples are discussed to illustrate the ideas.
From playlist Engineering Mathematics
Intro to Fourier series & how to calculate them
Download the free PDF http://tinyurl.com/EngMathYT This is a basic introduction to Fourier series and how to calculate them. An example is presented that illustrates the computations involved. Such ideas are seen in university mathematics.
From playlist Several Variable Calculus / Vector Calculus
The Fourier Transform and Derivatives
This video describes how the Fourier Transform can be used to accurately and efficiently compute derivatives, with implications for the numerical solution of differential equations. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow
From playlist Fourier
Fourier series + Fourier's theorem
Free ebook http://tinyurl.com/EngMathYT A basic lecture on how to calculate Fourier series and a discussion of Fourier's theorem, which gives conditions under which a Fourier series will converge to a given function.
From playlist Engineering Mathematics
Lec 8 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 8: The discrete Fourier series Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6-008 Digital Signal Processing, 1975
Lecture 8, Continuous-Time Fourier Transform | MIT RES.6.007 Signals and Systems, Spring 2011
Lecture 8, Continuous-Time Fourier Transform Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
Data Science - Part XVI - Fourier Analysis
For downloadable versions of these lectures, please go to the following link: http://www.slideshare.net/DerekKane/presentations https://github.com/DerekKane/YouTube-Tutorials This lecture provides an overview of the Fourier Analysis and the Fourier Transform as applied in Machine Learnin
From playlist Data Science
Lecture 10, Discrete-Time Fourier Series | MIT RES.6.007 Signals and Systems, Spring 2011
Lecture 10, Discrete-Time Fourier Series Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
Lec 9 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 9: The discrete Fourier transform Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6-008 Digital Signal Processing, 1975
Pure Fourier series animation montage
Because why not? Learn the math behind this: https://youtu.be/r6sGWTCMz2k If you're curious about the number of vectors used for each animation: - Eighth note: 100 - Sigma: 200 - Britain: 500 - Fourier drawing: 300 - Nail and Gear: 200 - Treble clef: 100 - Hilbert curve: 300 (relatively
From playlist Fourier
Lecture 5 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood finishes up on Fourier series, then he talks about the transformation Fourier series compared to the Fourier Transformations and how one gets to th
From playlist Fourier
19. Relations Among Fourier Representations
MIT MIT 6.003 Signals and Systems, Fall 2011 View the complete course: http://ocw.mit.edu/6-003F11 Instructor: Dennis Freeman License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.003 Signals and Systems, Fall 2011
Lecture 9.2 Fourier Series (part 1)
We introduce the theory of Fourier series, which allows us to analyse a periodic function in terms an infinite series of simple sines and cosines. In this lecture, we show how to calculate the coefficients of the series using integrals over the period of the function.
From playlist MATH2018 Engineering Mathematics 2D