The Laplace Transform: A Generalized Fourier Transform

[@Eigensteve]

Error: @10:20, should be e^{-st}

[@SRIMANTASANTRA]

But, this typo is understandable. Anyway, thank you, Prof. Steve.

[@gaelc13]

the H(t) definition should rather be 0 for t=0, isn't it ?

[@TURALOWEN]

Gael C. That is exactly how it is defined in the lecture.

[@gaelc13]

@@TURALOWEN Exact, my error : the space is so crowded that I missed the fact that the system as it is written at @7:30 refers to F(t)

[@nidhigoyal8893]

Sir is there any upcoming webinars or workshop of yours so that we could a bit more out of it ?

[@hashirroshinvaliyaparambil70]

Your 16 minutes video on Laplace transform gave me a deep understanding in this domain thane my 4 years bachelor's degree. You are priceless Mr Steve Brunton

[@Eigensteve]

Thanks!

[@윤진-n2y]

I'm Korean. I do a study of Laplace transform in high school. I also studied Fourier transform but couldn't find their common points, but your help is wonderful. Thank you for your detailed lecture!!

[@JHS-gu4lw]

캬 한쿡인 여기서 보네요

[@jonathanuis]

I'm doing my masters in control, I never really understood how Laplace works, Thanks a lot Steve, you make the concepts very understandable. regards from Germany

[@Eigensteve]

Happy to help!

[@vimostan269]

Agree! My ODE text book starts with the usage directly. I didn't even notice those badly behaved functions.

[@Physicsandmathswithpraveen]

In books and school they teach laplace before fourier and we never get a chance to sit back and relate them yes 🙂

[@Amine-gz7gq]

laplace transform scans for sinusoidals and exponentials in your transfer function so you can locate poles (region where you have resonance between your TF denominator and the e^-st function) and zeroes.

[@justin.booth.]

This is the best lighting I have EVER seen in a math lecture video. Sheer perfection!

[@paxdriver]

This is the best video on KZbin. On the entire internet, this is the best one made. Thank you and kudos for being such a rad teacher

[@Eigensteve]

Wow, thank you!

[@adityatandon2994]

This is probably the best explanation of the Laplace Transform that I've come across on the internet. 20 minutes did what 4 years of my bachelors degree failed to do - solidify my engineering math concepts.

[@volkerblock]

Excellent representation. Almost 60 years ago I learned the Laplace transformation, now I finally (hopefully) understand it. So, never give up, enlightenment will come at some point. ​

[@TKR911]

Dear professor, you do a really good job with these explanations ! Thank you

[@Eigensteve]

You are welcome!

[@jamen1993]

I thought that I grasped an intuitive understanding of the laplace transform once I recognised that it is essentially the correlation of a function with a decaying exponential oscillation, yet your presentation gave me additional insights.

[@plamenyankov2182]

I am a Data Science student and I thank KZbin's algorithm for suggesting your channel to me! For what I've seen because it's mind-blowing and I plan to watch all of your content and learn it by heart! Thank you Professor, you are doing amazing and very important job!

[@Eigensteve]

Cool, thanks!

[@dashjinn]

You having only 186K subscribers with so many really interesting and impactful videos just says about the direction of our society so much. I wish I had your videos during my bachelors... my love for math would have remained.. Thanks.

[@mrmister3507]

Im starting my master in Robotics in a few months and Im binging all of your videos. You're such a great teacher and you help me to get a true understanding of the theory. Thank you for posting all of these videos. Your students are extremely lucky to have a someone who understands the theory so thoroughly and is also excellent at teaching. That's a combination most professors can only dream of!

[@jurepustoslemsek7882]

holy sh*t! I've been trying to figure out what Laplace transform actually does and you've finally explained it in a way that I understand. thank you so much!

[@Eigensteve]

You're very welcome!

[@tsalVlog]

I don't know why, but I laughed really hard at "I think of it as a political Fourier transform".

[@Eigensteve]

Nice

[@Aziqfajar]

I can see why. Nice one

[@tractatusviii7465]

yeah, that's a great gimmick. useful too

[@jamesduff2647]

So did I..😂😂😂

[@douglasvalerio8880]

I`ve been first introduced to the Laplace Transform and only later to the Fourier Transform, and never before seen this approach, this generalization makes so much more sense Thanks for sharing this knowledge

[@krinkovakwarfare]

Not only did you broach the topic in a concise yet comprehensive way, you have written all this mirrored for our sake Impressive 💪

[@mikefredd3390]

I thought to myself, “self”, how can an Integral that looks the same as the FT but has a reduce integration range be a more general function? But lo and behold in the most straight forward and simplified presentation you explained it! Most productive use of my time in quite awhile. Thanks and I’ll watch some more videos.

[@danilomartins1901]

It’s just so hard o to find an intuitive video on what the Laplace transform actually is, other than just a random integral. You’re a genius! Key takeaway: Laplace is a weighted, one sided Fourier transform.

[@volkerblock]

Hervorragende Darstellung. Vor fast 60 Jahren lernte ich die Laplace Transformation, nun endlich habe ich sie (hoffentlich) verstanden. Also, nie aufgeben, irgendwann kommt die Erleuchtung.

[@pratapbhanusolanki6613]

Professor Burton, Thank you for the insightful video. I am wondering what happens to the heavy side function H(t) in the inverse Laplace derivation? Can we reconstruct the f(t) for negative t?

[@zwww_ee5235]

This is the video i came back to through my eng degree for laplace transform refresh, so concise and well explained, thank you Steve!

[@rene0]

Only after watching you write an i with a serif facing the 'wrong' way i was sure you were writing mirror script. Well done.

[@koninja1986]

This was randomly suggested to me by youtube. I don't know why, I never got past calc 2 and don't watch math vids much on youtube anymore. If I was still climbing the calc ladder I'd want Steve as a prof though. The enthusiasm is quite engaging.

[@philosoraptor3]

Pretty excellent overview, though it bugs me a bit to call the Laplace transform as a generalized Fourier, as it's more a restriction of the domain of the Fourier transform so that you can enlarge the space of allowed functions. But you were clear enough about this in your actual exposition!

[@Eigensteve]

Thanks, and I appreciate the note.

[@rajeshviky]

Steve Brunton has never failed me even once :) Yet, an another impressive video. Thank you!

[@lunakid12]

Very nice visuals and lovely structure, great performance, drawing skills, handwriting, even the colors! :) One minor advice, if I may: the act of chopping off of the < 0 half could be better communicated (before the "reveal" at ~10:45) by not talking (only; and perhaps a little too lovingly :) ) about the technicalities of H(t), but a) simply stating that we're just going to ignore everything < 0, and b) why that's both necessary and OK to do. Using H for that is trivial, use the time for explaining the rationale (of why the - half is treated differently from the +) instead, so that following it up in the math could feel natural and straightforward.

[@_notch]

This is a bit above my level, yet i managed to understand most of it! Great summaries of what just happened.

[@Eigensteve]

Thanks!

[@naveensd101]

I wish my math prof had this good handwriting.

[@alexanderbeliaev5244]

Finally, the misery resolved! now I see the logic behind s variable. Highly insightful channel, I wish I had these videos 10 years ago...

[@MaksymCzech]

Once again, thank you for your lectures!

[@Eigensteve]

Glad you like them!

[@peepeefrog_]

Amazing! A million times better than what I had in university in my days

[@mitchjust6688]

Really efficient way for video lecturing. Looks nice, I assume it's cheap(er) in time and processing power (for making them) and most importantly, does the job.

[@AlexAlex-bk5io]

When you made the inverse transform and multiplied by e^{\gamma t} to recover f(t) how can you got rid of H(t)? I mean f(t)H(t)=F(t)e^{\gamma t}.

[@hugod1276]

It's for t>0. When you define F(t), you lose the information for t

[@jamma246]

imo this isn't a sensible convention and should be ignored: you should consider your functions as only being defined for t bigger than or equal to 0. Indeed, all information about f(t) for t

[@ivanmazzalay7736]

This is great... I studied and always forget it, but you gave some elements of the definitions that are the keys to remember the process! Thank you so much!

[@MojoMonkeyMan67]

Brilliant, absolutely brilliant. Im speechless at how amazing this explanation is. Thank you Mr. Brunton

[@6Oko6Demona6]

Steve, you're left-handed, you write on the glass so it's readable from your side and then you mirror the whole video. Your handwriting character is unexplainable otherwise.

[@DanaWebb2017]

He knows his stuff backward and forwards.

[@Eigensteve]

I love it!

[@donotletthebeeswin]

I just noticed that too lol. You can confirm it by looking at his wedding ring

[@philippemichelvidori7248]

he writes well for a teacher ( left handed )

[@lawrencedoliveiro9104]

Leonardo da Vinci, I think it was, taught himself to write backwards and used that as a form of encryption for his diaries.

[@lucasbarroca8919]

Amazing, the Laplace transform was presented to me as magic wand, I've never been told how it works or why it works. This video clarified a lot for me. Thanks

[@felipegabriel9220]

That lecture was undoubtedly perfect, 100/10!

[@moustholmes]

Then why are you giving it 100/3628800? That's not a very high score

[@felipegabriel9220]

@@moustholmes i forgot some parenthesis

[@spitimalamati]

I made a T-shirt in the ‘70s with the Laplace Transform on it. In grad school, I loved using the Heaviside Theorem in digital process control. ChemE here.

[@bassboosted9708]

Now I finally understand solipsism with that formula. The math behind it opened my eyes.

[@emilywong4601]

Fourier and Laplace transforms are used in electronic music for converting sound to and from digital to analog signals.

[@emilywong4601]

Electronic music uses sin waves for sound.

[@hari8568]

Can you talk about why some functions have different Laplace and Fourier transform despite Laplace being a generalized version for example sinusoidal Laplace is different from sinusoidal Fourier.Similarly the step function has different transform in Fourier and Laplace.Also it would be helpful to know why we always use Fourier in communication subjects rather than Laplace which is way easier to handle

[@bjornfeuerbacher5514]

I think this is due to the use of the Heaviside function H(t), that changes the results quite dramatically...

[@AJ-et3vf]

Awesome teaching! Very insightful! I've watched tons of others videos about Laplace transform, but even in this I felt like I learned something new or gained a new perspective on Laplace. Thank you very much.

[@MrGeometres]

@3:30 But you can Fourier Transform all those functions... F(exp(λt)) = √(2π)δ(ω-iλ), F(H(t))= + √(π/2)δ(ω) + i/(2πω) etc.

[@Eigensteve]

Ok, that is fair enough. But it requires some new exotic functions, like the delta function.

[@1243576891]

Awesome videos! I followed this series from the first one to here. Glad to learn the connection between Fourier Transform, Wavelet Transform and Laplace Transform!

[@mortezakhoshbin]

you teach differently than others, and i learn new things about the subjects that im sure im so knowlegble on them! you say the basics so beauty

[@hupa1a]

Wow! This series is gold!

[@Eigensteve]

Awesome!

[@fhz3062]

I think it would also be interesting to briefly show why is not so simple to perform the inverse Laplace Transform. I mean, some Engineer courses don't have any complex Calculus lectures, so it is quite common to students try to perform the inverse Laplace integral without describing the path on the complex plane given by s = gamma + i*omega.

[@Eigensteve]

Great point. In my ME565 course (all videos in a playlist), I spend 6 lectures developing enough complex analysis to be able to take the inverse Laplace transform. Definitely not as simple as the forward transform.

[@bjornfeuerbacher5514]

So the price one has to pay for being able to transform more functions is that the inverse transform now becomes much more difficult?

[@Ajaykumaraita]

Dear professor you are such a great orator with visualisation.. Thank you. Please keep posting videos for this Laplace series.,

[@Eigensteve]

So nice of you!

[@preetymala3189]

Teaching way and writting technique both are outstanding. It help me a lot. Thank you 😊 SIR

[@branarch3878]

As a person who’s starting a control systems engineering / control theory course next semester - thank you so much!!!

[@Eigensteve]

Awesome, glad it helped!

[@IosueCanadensis]

2 questions: 1 - how do you 'undo' the heavyside function? is it not irreversible since you are losing information when you effectively truncate f(t)..? 2 - in the final formula, could we not just drop the gamma, since it will always be

[@gia-baoha5446]

Hello Prof. Brunton, I have seen in some control textbooks that the Fourier Transform and the Laplace Transform contain the same information about the characteristics of a system and thus either can be used to analyse the system. Here I have a question that I hope you could help me with: Why do they describe the same characteristics? I suspect it has something to do with Cauchy's Integral Formula that yields the same result when integrating the modulus of the transfer function over the Nyquist D-contour. Follow-up question: if my suspicion above is correct, then is the relationship only valid for RH-infinity systems (due to maximum modulus principle)? Many thanks!

[@calebgeballe2724]

Is there a named transform similar to the Laplace transform but instead of multiplying f(t) by H(t) and e^(-gamma*t), f(t) is multiplied by e^(-t^2)? This could satisfy the condition that the function f(t)*e^(-t^2) is "well behaved" at negative and positive infinity. I'm guessing the tricky part would be integrating the e^(-t^2) portion with non-special cases of f(t).

[@bjornfeuerbacher5514]

I think that's called "Gabor transform".

[@hakobarshakyan7177]

Hey Sir, Thank you so much for this useful material. 12:00 Why didn't you divide the equation with H(t)? Am I missing something??

[@bjornfeuerbacher5514]

I think he forgot to mention that the inverse transformation only works for t > 0. And in that region, we simply have H = 1, so we don't need to consider it explicitly.

[@theideamachine]

I heard at the beginning that the Laplace transform can be used to convert ODEs to algebraic equations, which can be more easily solved. But I've also heard that the Laplace transform is useful for identifying which exponentials and sinusoids exist in a particular signal. What is the connection between these two statements?

[@schkvty]

Hi Steve, I am a postdoc and have found your lectures useful when learning new concepts or brushing up old ones. I also find the mode of the lecture recording fascinating. Would it be possible to share an overview of the process of how your lectures are recorded? Thank you and keep up the good work.

[@ChaoS-pn3ic]

The trick is actually simple. The lecturer stands in front of a glass board and writes notes on the board normally as we have in class, and a camera records the process from the other side of the glass. Then, after the video is recorded, use editing software such as (opencv) to flip every images (left -> right) recorded in the video. That's it!

[@Eigensteve]

You're very welcome!

[@schkvty]

@@ChaoS-pn3ic Thank you.

[@motbus3]

ode ordinary differential equations must say, first video i watched in this channel. kept my attention trying to figure out how he writes mirrored

[@el_witcher]

He writes just like we do. There's a camera in front of him which does the reversal.

[@AntoineDennison]

@@el_witcher Really? He's righting from right to left... But he's writing with such ease, I guessed there must be some tech employed.

[@complex_variation]

@8:48 instead of "stable gaussian" is stable exponential, or "sufficiently stable"

[@SirajIssani]

Dear Prof. Brunton, I have a question. Isn't the inverse derivation process starting 11:46 missing out the Heaviside function? Or is it so that the inverse only valid for f(t) for t > 0. I tend to think I am missing something in this derivation. But I have to thank you a ton for the amazing way of teaching. Deriving the intent behind the transform is so much more interesting and insightful. I loved the Fourier transform series as well.

[@utkarshsrivastava4451]

I have the same question. Do tell us if you find the answer!

[@ivarangquist9184]

0:45 "I'm gonna walk you through how to derive the Fourier transform from the Fourier transform"

[@Eigensteve]

Whoops!

[@AshishPatel-yq4xc]

I realized it was just a slip but for a moment I was thinking , this is getting recursive :)

[@abhaykela]

Thank you for sharing this lecture video. I find it as one of the best explanations on Laplace and Fourier transformation.

[@noouch]

Love your minimalist setup, always nicer to have a teacher draw and gesticulate.

[@electricdreamer]

For those of you who wonder how he writes "backwards". He's not. The trick is, he writes normally onto a piece of glass in front of a mirror, if you point the camera from the same side towards the mirror through the glass, this is what you get.

[@DargiShameer]

Never seen such a great explanation for Laplace transform 🤩🤩🤩

[@DiggaDiggaDug]

So the Laplace transform is like doing a Fourier transform but instead of picking one specific window like Hanning or something, it is uses tunable exponential window, which when observing multiple inputs, gives us a corresponding Fourier transform at each point on the real axis in the s-plane. So the s-plane contour plot is like a waterfall plot of Fourier transforms using different windows. Is that right?

[@juanjosegiraldogutierrez8297]

Thanks a lot for the lecture I enjoyed it. I was just wondering why did you get rid of the H(t) at 12:05 when multiplying e^{\gamma t}F(t)? shouldn't it be f(t)H(t)=e^{\gamma t}F(t)? Thanks again!

[@bjornfeuerbacher5514]

I think he forgot to mention that the inverse transformation only works for t > 0. And in that region, we simply have H = 1, so we don't need to consider it explicitly.

[@a.b3203]

9:21 so you change the lower bound to 0 because the lowest value we can obtain from solving the integral is now 0, rather than negative infinity, since you've defined it to be that way using H(t)? Is that about right?

[@davidwilkie9551]

It's like mathematical Chocolate Cake, only the best ingredients. Very well done. And I once knew what it was about as a rool for Electronic Engineering, so the basic connection between Pi related sine waves, and e exponential "transformation", should now be the obvious QM-TIMESPACE Temporal vector coordination of e-Pi-i partial differentiates in Superspin Superposition-point interference of hyper-hypo modulating Conformal fields/interference positioning. (If you know what I mean)

[@trip_on_earth]

Thanks a lot for explaining this so clearly. Regards from India

[@chimetimepaprika]

Nice. I understood FT from this explanation in a way I never have previously.

[@SaeedAcronia]

But from the perspective of Fourier Transform , what does it really mean if a transfer function has a pole at s = 0 for example. I know that shows the general solution of it's ODE but, of all the frequencies (omegas), why is the sinusoid associated with 0 frequency so important in this case? What is the connection between roots of a characteristic equation and the Fourier Transform of it's corresponding ODE?

[@omarfarouk3848]

In 12.03 when we multiply F by e(omega t) we dnt really get f, we only get the right part of f Once we multiply f by heaviside, its left part is lost forever, i dnt really understand how the inverse works

[@ailtonteixeira4730]

I'm glad to find this lecture, now i saw the meaning and beauty.

[@bjornfeuerbacher5514]

Why don't we use e^(-gamma |t|) or e^(-gamma t²) instead of e^(-gamma t)? That would yield a function F(t) which goes to zeri for t ---> -infinity, too, so we would be able to transform even more functions. My guesses are: The first isn't used because this is not an analytic function, and the second is not used because the exponent is quadratic instead of linear, which makes calculations and general formula much more difficult...?

[@sridharc92]

"One-sided, Weighted Fourier transform, or a political Fourier transform". Pure Gold! :-D

[@Eigensteve]

:)

[@atlas2919]

What happened to the step function @12:07? f(t)=F(t)exp(yt) only for t>=0 according to the definition

[@mouryatejagn9592]

Around 12:05 - 12:08 how could we equate f(t) to e^(gamma*t)*F(t) what happened to H(t)🤔

[@nigelferrer557]

Is he writing backwards on a window?

[@BrightBlueJim]

... or is he writing forward on a window, and then mirroring it in editing?

[@王珂-k7d]

That would hard to be produced this smooth, since the direction he moves would be then the opposite direction of the sketch's.

[@BrightBlueJim]

@@王珂-k7d No: the whole frame is mirrored. There is a clear window between him and the camera. He writes normally on his side of the window, but that makes it reversed left-to-right from the camera's view. Mirroring the video applies to both him and the writing. This is much, much simpler than, say, deriving the Laplace transform. Or learning to write backwards. Not that he couldn't have learned to write backwards; I once learned to write backwards, but that was on a dare.

[@inisipisTV]

@@BrightBlueJim - It's been mirrored. He is known to be left-handed as you can see his ring finger.

[@anikethaldankar4642]

mirror effect

[@nishapawar3368]

there r so many videos about laplace transform but I loved this one.....#mustwatch

[@shivamagarwal6649]

This approach explains the unilateral laplace transform so well. Thank you very much. But the other day, I saw there is something called bilateral laplace transform that is calculated from negative infinity to positive infinity. I cannot find a way to use your approach to explain the bilateral laplace transform. Can you help me that @Steve Brunton ??

[@ElMalikHydaspes]

really a well done explanation of bringing the two concepts together ... 🎉

[@syllogismo]

What happened to the heaviside step function when you are deriving the inverse laplace? how is f(t) = e^(gamma*t)*F(t)?

[@alexanderuyttendaele1625]

Since the Laplace Transform only integrates over a positive domain, you can only expect it to return a valid inverse over that same positive domain, i.e. where the Heaviside is one and can be ignored in the equation.

[@guanchenpeng5180]

I think you didn't mention although the inverse Laplacian transform has definition from -inf to +inf for t or x, it only equal to the original function for t>0

[@robertbillette4671]

Wow Steve! Such a good teacher. Wish I had you in my undergrad as a teacher

[@shchowdh]

Great video! how would the proof change with a bilateral Laplace transform? Specifically, when would one use a bilateral versus a unilateral Laplace transform? Is there such a thing as a unilateral versus bilateral inverse Laplace transforms?

[@huijiewu8395]

Hi Professor Brunton, very nice lecture. I am wondering what do we need to pay attention before we take fft for the real data. I mean I am dealing with the fluid feild data, it is kind of imperfect. When I fft it, the results is not very good, especially to calculate its derivetives.

[@prandtlmayer]

TOP QUALITY and really enjoyable!

[@Eigensteve]

Glad you enjoyed it!

[@superuser8636]

I remember your older video which always stuck with me “one-sided Fourier transform”. I also love the history in your lessons! However, I do recall you saying that the bounds of integration started at 0- due to a mathematical technicality which I did not see here. Can you explain that? Thanks !!

[@Eigensteve]

Glad you liked it! Yes, the bounds of integration start at 0 because we multiplied our function by the Heaviside function, so everything is 0 for t

[@superuser8636]

@@Eigensteve I do see that, thanks so much for clarifying/affirming.

[@Amb3rjack]

A fascinating video which I found utterly compelling. I actually almost sort of understood a tiny part of some of it . . . . .

[@יעקבמישייב]

Hey steve i have a research and i need to learn hilberts transform..i will happy if you make a video on this

[@Eigensteve]

Thanks for the note. It is on the list!

[@agihtiassalam8628]

Very impressive lecture Prof, I enjoyed it ! Anyway, if you think numerical laplace inversion is useful in order to inverted back to real domain, please make a video about it. I use laplace transform to transfrom the fluid flow equation (Pde) then I use gaver-stefhest numerical laplace inversion to transform it back to time domain. I think there are couple of methods to numerically inverted the equation in laplace space but I only know one, the gaver stefhest. Thank you for the great lecture.

[@iabhisekdas]

could you give us a problem or tell us about its application, especially where to understand a specific part of nature we require this mathematical tool...as you said in filtering noise in data, so I think in climate studies it will have a huge impact on co2 data application...and thanks a lot for your demonstration.

[@shubhamdeshmukh1900]

Only if they could teach so articulately in college 🙌 I have become your fan!🙌🙌🙌

[@JoaoVitorBRgomes]

Regards from Brazil! Thx

[@UMESH-lj3ye]

12:03 why we are ignoring Heaviside function here

[@gagra1234]

Very nice lecture, professor, thank you! However I would like to notice, that it's not perfectly correct to say, that Fourier Transform is unapplicable to ugly functions like constants and sins (in general non rapidly decreasing functions). A Fourier Transform can be generalized to such functions by defining FT of a generalized function aka distribution. This generalization allows to handle ugly functions and work with things like deltas in a mathematically correct way. And I belive that it's more appropriate to call this type of generalization a "Generalized Fourier Transform". I'm not saying that you are wrong, it's true that classical Fourier Transform has problems with this ugly functions, but generalization through distributions solves this problem just as good as the Laplace Transform.

[@Eigensteve]

You are definitely right. I was glossing over some more technical details here, but you are right that with generalized functions it is possible to FT these "ugly" functions.

[@neurosock]

@@Eigensteve Thanks to both of you for clarifying. Immediately had this question.

[@neurosock]

Are distributions something like cut out or pieces of functions?

[@HerChip]

Really nice studio (video/lights etc) setup!

[@Eigensteve]

Thanks!

[@antonnilukshan935]

Can't thank you enough sir! The best 16 mins ever spent on KZbin for me. Thank you! Hilarious comment there on politics 😂