The Laplace Transform: A Generalized Fourier Transform

welcome back I'm Steve Brenton and today I'm gonna tell you about the Laplace transform which is one of my absolute favorite transformations at all of mathematics the Laplace transform so many of you have heard about the Fourier transform I just did a whole lecture series on the Fourier transform the fast Fourier transform and I kind of think of the Laplace transform as the culmination of all of the the work on Fourier transform so Laplace in some sense generalized the Fourier transform to a much larger more important class of functions that you can now transform ok so I'm gonna tell you all about that right now I'm gonna walk you through how to derive the Fourier transform from the Fourier transform and then how to use it to do things like compute derivatives and solve equations and I'll point out you know in in math usually there's no there's no magic wand but Laplace transform is about as close as it gets you can take a system and subtract about 2 or 3 years of advanced math from how hard it is to solve that system just by applying the Laplace transform so for example if you have a partial differential equation a PDE under certain circumstances you can Laplace transform it and turn it from a PDE into an OD e which is much simpler similarly you can take an OD e and under some conditions you can transform it with the Laplace transform into an algebraic equation which again this goes from college to high school kind of solution techniques and the Laplace transform is also extremely useful in control theory so the Laplace transform is going to crop up all over the place and today I'm going to derive it for you and show you how it's not actually separate from the Fourier transform it is in some sense a generalized Fourier transform okay so many of you have seen the Laplace transform and Fourier transform for years and you've noticed similarities today I'm going to show you how they are exactly the same thing the Laplace transform comes from the Fourier transform so the plus is one of my absolute favorite French mathematicians he was the son of a peasant farmer and went on to have his name on the Eiffel Tower that's one of the things I love about the French is that they Revere their great thinkers and mathematicians and scientists and the plaza was truly one of the greatest fun fact about Laplace he was one of the first researchers ever to realize that when you're dealing with real-world data which has noise and isn't perfect you have to look at that data through a probabilistic lens through the lens of probability theory for us we take that for granted but that was a huge deal back when the plos lived in the second half of the 1700s and the early 1800s okay so let's jump in the big idea here is that we we know that we can Fourier transform nice well-behaved functions that decay to zero at plus and minus infinity so I'm just going to draw an example of that if I have this nice Gaussian that goes to zero as X or T goes to plus or minus infinity I can Fourier transforms I'm going to say Fourier transform check we can do the Fourier transform but less well-behaved functions so there are nastier functions out there and I'm gonna draw a couple of them right now like e to the lambda T this function you can't Fourier transform because it does not go to 0 as T goes to 2 plus infinity so you cannot Fourier transform this function another example of a function that is tricky or impossible to Fourier transform is the Heaviside function I really like this Heaviside function that is 0 for negative time and 1 for positive time so let's be explicit and call this time and this is the Heaviside function named after oliver heaviside and it's 0 for T less than 0 and it's 1 for T greater than or equal to 0 this function also you can't easily Fourier transform because it doesn't taper off to 0 at plus infinity ok now there are other examples where technically you can Fourier transform it's just a little bit of a pain so let's think about this trigonometric function now again this doesn't decay to zero at plus and minus infinity but there are tricks you can play so the most common trick is to multiply this by a window function W where basically W is 1 on some window and 0 everywhere else so now if I multiply W by my sine function or cosine function it does have this nice property and then I can take the limit as this window becomes infinitely large that's one way you can Fourier transform these signals but again it's kind of a pain so what I'm gonna show you is how the Laplace transform is basically a weighted one-sided Fourier transform for these nasty functions okay that's that's all I'm gonna show you today and then we're gonna use it later on ok good and you know this all is due to pierre-simon laplace one of the great great mathematicians so the solution and make sure I can actually write where you can see so the solution is let's call these our little functions f of T ok these little functions f of T so our solution is to multiply f of t by some very stable e to the minus gamma t e to the minus gamma t that's an exponential function e to the minus gamma t so that f of t e to the minus gamma T goes to 0 as T goes to positive infinity ok so only as T goes to positive infinity so the first thing we're gonna do the solution is we're going to take our function f it's badly behaved and we're gonna multiply it by a decaying exponential function so we're gonna multiply it by a decaying exponential so that when those multiplied together it will go to 0 for as T goes to positive infinity now you might be thinking well we solved this problem in the positive T infinity direction but now my function might blow up at negative T infinity T equals negative infinity so we don't just multiply by e to the minus gamma T we also multiply by our handy Heaviside function H of T ok so now what we have is we're going to define this big function big f of T is going to equal a little F times e to the minus gamma T times Heaviside of T all right that's that's what we're gonna do we're gonna take our badly behaved little F of T and we're going to multiply it by a sufficiently stable exponential e to the minus gamma T where if I multiply them it goes to 0 at t infinity and my Heaviside function so that what I get is something that is zero for T less than zero that handles this thing blowing up at negative infinity and it equals F of T e to the minus gamma T for T greater than or equal to zero good so this this is the whole the whole thing we take our badly behaved function we multiply it by a stable exponential and a Heaviside function and now we're gonna Fourier transform big F so the Laplace transform of little F is the Fourier transform of big F good and I'm gonna write this down over here I'm actually going to box it so my I'm gonna write down my Laplace transform pair just like we write down our Fourier transform pair so we're gonna have f of T it's going to equal some Laplace transform it's re inverse Laplace transform and F bar of s is going to equal my my Laplace transform ok good so I'm just going to save these and we're gonna fill these out later so the Fourier transform of big F is going to be we're gonna call that big f hat and it's going to be a function of Omega and that's going to equal the integral from minus infinity to infinity of big f of T e to the minus I Omega T DT this is what we always do when we Fourier transform that's the Fourier transform a big F and remember I can Fourier transform big F because I've multiplied it by the stable Gaussian and the Heaviside function so it zero at plus or minus infinity so I can for you transform big s okay and now what I'm going to do is I'm just gonna substitute in my my formula right here for this big F the first thing I'm going to notice is that this Heaviside function is 0 for all T less than 0 so I can change the bounds of integral of my integral from instead of minus infinity to infinity its 0 to infinity so now this is from 0 to infinity and I can drop my Heaviside function of little F of T e to the minus gamma t e to the minus I Omega T DT ok so I've changed my bounds of integration because of my heavy side function so instead of negative infinity to infinity I'm doing 0 to infinity and then what I'm gonna do is I'm going to group these Exponential's here and I'm gonna say that this equals integral from 0 to infinity little F of T e to the minus gamma plus I Omega T DT and the last thing I'm gonna do is I'm gonna say that gamma plus I Omega is my Laplace variable s that's a pretty bad bracket so I'm gonna say s equals gamma plus I Omega that's my Laplace variable and so this equals integral 0 to infinity f of t e to the S T DT and that is my Laplace transform the Fourier transform of big F is my Laplace transform of little F that's the definition of the Laplace transform it's the the Laplace transform the Laplace transform is the Fourier transform of a one-sided weighted function f it's a one-sided waited for a transform I think of it as a political Fourier transform ok good and so I'm just literally gonna say that is defined this is f bar of s that's how I'm defining f bar of s sorry it got cut off a little bit f bar of s equals the integral from 0 to infinity of little f of T e to the minus s T DT that is the definition of a Laplace transform and now the inverse Laplace transform is just the inverse Fourier transform of this big f hat of Omega ok so that's what I'm going to do now and maybe I'll do a different color here so now what we're gonna do is we're gonna say big f of T is the inverse Laplace transform of this and if I remember correctly that's 1 over 2 pi integral negative infinity to infinity of F hat Omega e to the now instead of minus I Omega T it's plus I Omega T and we're integrating with respect to D Omega ok good and this f of T was little F times e to the minus gamma T remember I want a little F of T out for my inverse Laplace transform so I'm going to take this and I'm going to multiply both sides by e to the plus gamma T so e to the plus gamma t e to the plus gamma T and that's going to give me the little f of T that I want out so little F of T equals this weighted inverse Fourier transform of big F Omega and now I'm just gonna start working through what the mass here looks like ok so this equals and remember that F hat Omega is the same as f bar of s these are the the Laplace transform little F bar is the same as big f hat and so this is going to equal 1 over 2 pi integral minus infinity to infinity I'm gonna just swap this out and call this f bar of s and now my e to the gamma t times e to the I Omega T is e to the gamma plus I Omega T D Omega ok good and you'll recognize this is our handy Laplace variable s so now all I have to do is I have to change this D Omega and the bounds of my integration to a d s and I'm just going to do that right now for you so if I look at D s remember gamma is just a constant it's a constant that's big enough that this goes to zero so gamma is a constant so des is I times D Omega okay good and that means that D Omega is just 1 over I DS and so I'm just going to swap out my D Omega for 1 over I D s ok so this is I'm gonna put my eye out here in my in my coefficient 1 over 2 pi I think I'm gonna run out of space here so I'm going to do it down here so this is 1 over 2 pi I integral of f bar of s e to the positive s TDS ok and here's the last last last thing is that if Omega went from minus infinity to plus infinity if Omega went from minus infinity to plus infinity then s goes from gamma minus I infinity gamma minus i infinity to gamma plus I infinity ok so that is so my bounds of integration changed and that is the inverse Laplace transform that's all that there is to it so now F of T is just 1 over 2 pi 2 pi integral gamma minus I infinity to gamma plus I infinity F bar of s e to the positive st D s and this should look very much like the Fourier transform parrot because it is a Fourier transform here the Laplace transform pair if I have some function of time F of T u it's badly behaved I can take its Laplace transform this way and if I have the Laplace transform F Bar of s I can inverse Laplace transform and recover my function f of T my original function and this works for nasty poorly behaved functions that you could not normally take the Fourier transform of ok so let's take a step back the Laplace transform is a generalized Fourier transform for badly behaved functions so instead of just directly Fourier transforming those badly behaved functions what we do is we multiply them by a stable exponential so that they decay to 0 and a Heaviside function so that they don't blow up at negative infinity and then we Fourier transform that product so this is a one-sided because of the Heaviside function weighted because of the e to the minus gamma t fourier transform so it's a it's a one-sided waited for a transform for badly behaved functions that's all the little applause transform is and it is extremely useful because lots of the solutions of PD ES and OD es and control theory look more like this and this and this then our nicely behaved functions where we can easily Fourier transform so in the next lecture what I'm going to do is I'm going to walk you through some of the properties of the Laplace transform it inherits most of the same properties as the Fourier transform for example how you transform derivatives or convolutions and we're going to use those properties of the Laplace transform to simplify our PDE s to Odie's ro d east algebraic equations and we're also going to use this a lot in the control theory bootcamp all right thank you

Channel: Steve Brunton

Views: 196,818

Rating: 4.962801 out of 5

Keywords: Applied mathematics, linear algebra, Laplace transform, Fourier transform, FFT, Data science, Machine learning

Id: 7UvtU75NXTg

Length: 16min 28sec (988 seconds)

Published: Fri Jul 24 2020

Reddit Comments

[u/thodcrs]

Oh yeah, this guy has a great channel too. Very informative

[u/pen_n_run]

Steve Brunton! The go to guy for controls!

[u/BittyTang]

I think he's skimming over an important detail. The "sufficiently stable" exponential decay function has to be chosen relative to f(t). Namely, gamma has to be large enough. And for some f(t), you can't even choose such a gamma, for example, f(t) = e^t2 .

But still, good video! Very clearly explained by relating to the Fourier transform.

[u/wintergreen_plaza]

Very nice! Thanks for introducing me to this guy’s channel.

[u/[deleted]]

He is great, but I feel like I need more visuals, animations and the like. Algebra and symbolic transforms are a bit difficult to understand sometimes.

[u/smtdgrt]

His control bootcamp playlist is awesome