Fourier Analysis: Overview
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This video presents an overview of the Fourier Transform, which is one of the most important transformations in all of mathematical physics and engineering. This series will introduce the analytic theory of the Fourier Transform, along with the Fast Fourier Transform (FFT) algorithm for efficient computations. We will explore lots of applications in image compression, audio analysis, and solving partial differential equations. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdfThese lectures follow Chapter 2 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz Amazon: https://www.amazon.com/Data-Driven-Sc...Brunton Website: eigensteve.com
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[Music]
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hi everyone I'm Steve Brennan from the
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University of Washington and this is a
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new video lecture series on the Fourier
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transform and wavelets this is gonna
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follow chapter 2 of our new book
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data-driven science and engineering by
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myself and Nathan Koontz and you can
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find all of the code and lecture
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material online at data book UW comm and
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you can also find it in the the comments
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and the links below ok so I'm really
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excited to talk about the Fourier
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transform because this is one of the
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most important and ubiquitous
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transformations in all of mathematical
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physics and engineering ok and if you go
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back through the history of science most
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of our major breakthroughs started with
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a coordinate system or a coordinate
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transformation that made our world make
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more sense that made physics simpler in
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some way and so the Fourier transform is
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another coordinate transformation that
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is particularly useful for for
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representing data and images for
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representing the solution of partial
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differential equations and I'm gonna
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walk you through all of the different
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well not all of lots of the different
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uses of this Fourier transform kind of a
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mathematical theory of the Fourier
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transform how to compute it in MATLAB
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and in Python with the fast Fourier
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transform the FFT and we're also going
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to talk about the history and kind of
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how this was derived so Fourier in the
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early 1800s derived the Fourier
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transform and the Fourier series in
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particular as a way of approximating
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solutions of partial differential
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equations so he was particularly
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interested in the heat equation so
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imagine that I have some rectangular
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piece of metal and this has some
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temperature distribution maybe U as a
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function of x and y and time in two
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dimensions U is the temperature
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everywhere in space and time then this
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would be governed by the heat equation
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which is a partial differential equation
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u sub t the partial derivative of U with
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respect to time is a positive constant
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times the laplacian operator of u so
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this is the
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second partial derivative of U with
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respect to X plus the second partial
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derivative of U with respect to Y in two
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dimensions and so what Fourier did he
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didn't just write down the Fourier
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transform for no reason he discovered
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that the Fourier transform is a
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coordinate transformation that
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essentially diagonalizes this laplacian
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operator in the heat equation so what he
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showed was that this this operator this
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laplacian has eigenvalues and
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eigenfunctions just like other linear
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operators do and the eigen functions are
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sines and cosines with a particular
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frequency determined by the boundary
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conditions and the geometry of this this
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object and the corresponding eigenvalues
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are those spatial frequencies so for you
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derived the Fourier transform
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specifically to make the heat equation
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simpler by transforming it into an
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eigenvector eigenvalue coordinate system
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okay and since then it's been used for
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all kinds of applications in image
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compression and solving other partial
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differential equations in fact we've
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actually seen that the singular value
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decomposition from chapter 1 of our book
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can be thought of as a data-driven
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fourier so SVD is kind of like
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data-driven extension of the fast
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Fourier transform or the FFT ok so we've
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already seen coordinate transformations
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before in Chapter one with the SVD this
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is kind of a generalization of the FFT
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and now we're gonna go back and talk
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about the more fundamental and older and
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more ubiquitous Fourier transform ok
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good so the Fourier transform is also I
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think very important because it changed
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our view of mathematics in general this
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idea that you can take an arbitrary
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function I'll just draw kind of a
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function here so you can take an
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arbitrary function let's just draw this
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kind of pointy hat function and you can
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approximate this by a sum of sines and
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cosines of increasing frequencies so if
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this function is defined on some domain
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I can approximate it by a sum of sines
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and cosines of increase
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singly high frequencies so that's that's
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one maybe now I say I'm gonna have a
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higher frequency sign something like
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that and so on and so forth and you can
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keep adding these up it's gonna get a
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little harder for me to draw this but
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I'll try and so on and so forth okay so
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this function you can approximate as a
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sum of increasingly high frequency sines
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and cosines okay and so this idea of
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function approximation is one of the
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really really important aspects of the
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Fourier transform and of wavelets okay
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this laid the foundation for our
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understanding of function spaces like
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the hilbert space which was pivotal in
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our understanding of quantum mechanics
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this idea that these or these these
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sines and cosines in fact form an
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orthogonal basis for the space of
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possible functions so all of these kind
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of functions that I could draw can be
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represented in this basis of sines and
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cosines so just like in a vector space
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you might have your X and your Y
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coordinates which form a basis for
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two-dimensional vector space your sines
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and cosines form a basis for function
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space okay so that was really important
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in this derivation and understanding of
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Hilbert spaces which have been really
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really foundational for our modern
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understanding of mathematical physics
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okay
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so fast forward 200 years now we have
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the fast Fourier transform okay so now
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we have the fast fourier transform this
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was developed relatively recently
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okay the fast fourier transform or the
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FFT this is how you compute a fourier
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series
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efficiently on a computer so this has
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been absolutely central in how we
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analyze and process data that we collect
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so for example audio sequences images
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videos these are all compressed and
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represented efficiently using fast
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Fourier transforms so pretty much all of
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our modern digital communication is
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built on the fast Fourier transform you
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wouldn't be able to send pictures or
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audio clips or really anything
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efficiently we wouldn't be able to have
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satellite communicate
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if it wasn't for the fast fourier
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transform so this really is one of the
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absolute most important algorithms ever
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developed and it's still important today
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and I'm going to tell you all about it
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so we're gonna walk through the fast
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Fourier transform the Fourier series the
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analytical Fourier transform and
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wavelets we're going to code up examples
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of how to solve partial differential
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equations in matlab in Python we're also
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going to use it for function
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approximation for image compression for
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audio compression for denoising so I'm
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really excited to tell you about this
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this is going to be one of the most
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powerful tools in your arsenal
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especially when you're working with data
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alright thank you