The Laplace Transform - A Graphical Approach

Get the map of control theory: www.redbubble.com/shop/ap/550...
Download eBook on the fundamentals of control theory (in progress): engineeringmedia.com
A lot of books cover how to perform a Laplace Transform to solve differential equations. This video tries to show graphically what the Laplace Transform is doing and why figuring out the poles and zeros of a system help us to reconstruct the time domain impulse response (which is the solution to a diffy Q.)
Don't forget to subscribe! I'm on Twitter @BrianBDouglas!
If you have any questions on it leave them in the comment section below or on Twitter and I'll try my best to answer them.
I will be loading a new video each week and welcome suggestions for new topics. Please leave a comment or question below and I will do my best to address it. Thanks for watching!

[@fatihersoy7559]

You're a "teacher". My 'professors' at uni, they're "tellers". Nice lecture, from a nice lecturer. Thank you!

[@ad2181]

My Controls Teacher at U of Florida was Dr. Bullock a walking incompetent idiot. I hope you get this message! I'm relearning controls.

[@vaughnmonkey]

That is the best and most accurate way I have ever heard this explained. You are absolutely right and its amazing that Brain can be such an amazing teacher without even having feedback from us. while our "professors" can't when we are sitting in front of them begging them to teach us.

[@MikoPellas]

Exactly. IMO teacher holds a higher status than professor. Teachers actually "teach", while professors merely "profess"

[@georgeclooney6208]

​@@vaughnmonkeynot begging fakin paying for them to teach us

[@MugiwaraSuponji]

imma be real, this video blew my fuckin mind. the part where you went from the 3D s-plane plot to the poles and zeros? holy shit. it's like i just found the secrets to the universe.

[@fzigunov]

Man, after so many classes and so many videos I finally understood it! Thanks for the "real world" approach! I was struggling just with correlating with reality! Awesome work, keep up!

[@SeraphisQ]

It's hard to put into words how good these videos actually are. What an amazing piece of work. I'll make sure to watch and like every one of them.

[@marialey7658]

THANKS A LOT ! first time someone explains it in a way that I can actually grasp the idea behind the Laplace transform

[@pratibharacheljohn3814]

I have been following your lectures since 6 months now and I can't thank you enough. I wish I had seen these way earlier. Awesome way of explaining even the most confusing concepts!!

[@bboysil]

JUST PERFECT! I came back to this after many years and I have to say there are a LOT of insights this video. Perfect for remembering or if you're trying to understand the intuition of what the Laplace transform does.

[@ThatGuy-mf9ye]

Studying for my FE exam after I've taken all my signals classes and control electives; this really helps bring home some of the intuition that they miss. Thanks!

[@Ropsch]

Brian, I love the way your videos are built up and edited. You have really put a lot of thought in it. Brilliant!

[@DanT2990]

Finally an interesting, intuitive and colourful series on control systems! I'm in my final year in my aerospace engineering program and I'm using your videos as a refresher for control systems. I'm actually learning new perspectives I never thought about and they are helping me to understand topics I didn't quite get. My final year design project is purely based on control systems so this is going to help me immensely. Thank you!

[@shishirsks]

Awesome! THis video will help thousands to understand laplace transform.

[@poppyblop484]

The clarity and detail into each topic is amazing, it is so clear and easy to understand. Thank you so much!

[@speedbump0619]

I took differential equations in 1994, and never understood what the s-plane was (honestly, I don't think my professor understood it either). I cannot thank you enough for finally providing a sensible explanation of what in the world the Laplace transform is actually doing. Now I've got to go back and re-read every control theory book I've ever bought, since I can probably make sense of them now.

[@90ben09]

I just wanted to say thank you so much for this video it has really helped me to understand laplace transforms in a way that I never did before. Also thank you for making these available to us all, I really appreciate what you do.

[@Centuries_of_Nope]

In computer engineering. Started this class and is the hardest part of the whole degree. Watching this, it took until you drew the circuit until things started to click. Thank you.

[@apoorvvyas52]

understood the whole point of doing Laplace transforms and finding poles and zeroes for the first time. Great work. Thank you very much for posting this videos

[@rileystewart9165]

I must say you have excellent hand writing. Makes following much easier.

[@Chadwikj]

High quality visuals keeping pace with your lecture was fantastic. Excellent job with this.

[@MarkNewmanEducation]

Thanks for the visual approach. At last someone who will draw a few pictures and not just fill a blackboard with greek letters!! I wish people would explain things more this way.

[@alanly3780]

VERY well explained! Thank you, the contour map of the laplace transform plane was really helpful to visualize whats actually going on.

[@jamesheadrick7206]

As a controls 2 student, reviewing your videos from Fourier transforms too classical controls theory I am very impressed with your videos! Keep it up!

[@RexGalilae]

It's a great idea you came up with instead of simply writing while talking, wasting time in the process. Good work!

[@BrianBDouglas]

By LR did you mean Laplace Transform? The simple explanation is that FT breaks time signals into just sinusoids (or their frequency content). You can't use the FT to solve differential equations because it doesn't cover the exponential part. But you can use them FT for all sorts of frequency related problems like noise, sound, filters, and so on. LT breaks time signals into sinusoids and exponentials (just like the solution to Diffy Q's) so that's the motivation.

[@kamilbudagov9335]

is it possible to know exact value of magnitude and phase for arbitrary frequency from continuous frequency spectrum?

[@ricojia7322]

Your video is unique. It answers my questions perfectly.Thank you so much Brian, I regret so much that I pay a ton to university, hoping to learn things step by step, but the only things I get are complications.

[@allenkkwong]

Direct and clear in explanation! Great lecture.

[@rajatjadhav1061]

This was really good for actual understanding and imaginative approach. Now we can really get what the plot is.

[@ludwigrasmijn8218]

AMAZING! best part was the 3d part going to 2d to show the poles and zero, best explanation ever

[@Beudd]

Absolutely clear. Brilliant. I like this kind of video because it shows that we can explain some abstracts concepts with precise words and illustrations.

[@maksoff]

First video on youtube, where one "thumbs up" is not enough. Amazing video, after so many years it is not magic for me anymore!

[@nezv71]

At 2m40s, the claim is way too broad. Exponentials are the only solutions to *homogeneous linear constant-coefficient* differential equations, or in physical terms, they are the only possible *transient* responses of *linear time-invariant* systems. For example, the linear time-invariant system y'' + y' + y = x^2 has a non-exponential (particular) solution y = x^2 - 2*x just due to its inhomogeneity. It'd be bad for people to believe an (incorrect) statement like "the solution to every differential equation is an exponential." That'd be an extremely powerful game-changer if it were true.

[@zedlepplin9450]

Can you think of a function or a signal (other than exp or any sinusoidal func for that matter) that if you take it's derivatives (1st, 2nd, etc) and if you add them all up will get a zero? With the mathematics that we know now, there isn't any. I'm not sure if there is a proof for that but for now it's a (very) valid statement.

[@twilightknight123]

I think you misunderstood what the original comment was saying. The video states that the solution to ALL differential equations are exponentials, sinusoids, or combinations of the two. This is just not true. It may be true for most physical differential equations such as Laplace's equation or the heat equation, but it is not true for ALL differential equations. Hell, most physically described systems are described by Legendre polynomials while are neither exponentials nor sinusoids. You can put sinusoids as the argument for Legendre polynomials, and most of the time you want to because of symmetries, but they are not inherently exponentials NOR sinusoids.

[@eavids128]

Thank god, I thought I was the only one who got super confused by the statement the video made. The first differential equation we learned in ordinary differential equations were ones where you could use simple integration to find solutions. However, I see how it could be a valid statement that every solution to a differential equation is *comprised* of sinusoidals and exponentials, as this is true of all signals.

[@achimbuchweisel2736]

Great visualization of the Laplace Transformation! Made my day.

[@shekharyadav380]

The 3d plot explanation was amazing.....cleared a lot of things......thanks a loooottttt !!!

[@dericc8611]

Kinda blew my mind at the end :D Thanks so much for this video!

[@DDDelgado]

2:30 interesting, solutions to differential equations representing physical phenomena results in exponentials or sinusoids, nice, it clears a lot of things.

[@closingtheloop2593]

Always a good refresher. Thanks!

[@faifai4]

This video is insanely good.

[@inzepinz]

Finally I understand what the laplace transform is for, thanks.

[@squidcaps4308]

Thanks for doing this in reverse, made so much more sense this way.

[@samfisherXXI]

Thank you for your brilliant explanation, I always hate when teachers "parachute" methods and equations without explaining the Why, well you did just the opposite and thank you for that :D

[@priced80]

Wow. This is a really excellent explanation. Well paced too and clearly drawn. I like the fact I don't have to wait for you to write / draw things. That can get a bit tedious on Khan Academy

[@GonzaloBelascuen]

Amazing Video, thank you!!

[@rajeshkanna4124]

Man your tutorials are awesome. Its a lot better to watch your tutorial than going to college. Applause !!

[@horacechen5894]

Excellent introduction!!! Thanks a lot.

[@rajdeepchatterjee3549]

genuine and digestable. thankyou sir!

[@jupatj24]

Such knowledge, much appreciated, well done good sir.

[@doktoren99]

Ohh man this is great! I wish there were more videos of graphic understanding in mathmatics as well!

[@danielurdiales2856]

You are really good at explaining this material!

[@neilphilip2320]

These talks are stunning!!!

[@exmuslim3514]

awesome explanation you give answer of lot of questions brother..

[@HassanAli-os3py]

Such intuitive explanation!

[@hansi98]

this is helping me so much understand the motivation of what i have to do thank you

[@user-iz3rg3qq3z]

Great Video! Your explanation is very clear and intuitive. Thank you =D

[@SomeSortOfLandCow]

Around 10:50 when you move from A to B to C, I believe the probing signal amplitude should be increasing (decaying slower and then growing faster) rather than decreasing. As sigma decreases, e^-(sigma*t) increases faster due to the negative sign. From my understanding, at points B and B' the probing signal should be increasing at the "opposite" rate that the impulse response is decaying so that the product of the two signals is a constant amplitude sign wave. If I'm wrong please to correct me anyone.

[@sgtcojonez]

You just blew my mind.

[@averytieh]

Great video to rough understanding on Laplace Transform!!!

[@Amb3rjack]

Wow! Just exactly what kind of a mind does it take to be able to just trip this stuff off the tongue like Brian does? I was mesmerized by this video and understood practically none of it . . . .!!

[@andrerenault]

This is the closest I've come to understanding Laplace. I still don't fully get it, but I have glimmers of it. Thank you so much.

[@thetompham]

I am finally beginning to connect all the stuff Ive been learning as a electrical engineering student...wow.

[@funcionamaldito]

I thought that "solution to differential equations must be either ..." was misleading. He's specifically talking about linear differential equations with constant coefficients.

[@SuHAibLOL]

yeah exact differential equations wouldn't behave that way for example

[@SuHAibLOL]

Athul Prakash no you can find non sinusoidal and non exponential from simply some separable equations

[@grandlotus1]

You go, girl! (I'm at a loss to say anything probative.) Is math a conspiracy of smart people over the rest of us? I mean, i'm not dumb (stop sniggering), but this could be total baloney and I have no way to discern. For example the quote "...just below infinity..." I don't believe in shaming myself, but, huh?

[@TheDavidlloydjones]

Hugo, No, you're not shaming yourself. This guy is a wonderful example of David Hilbert's wise remark "You get all sorts of nonsenses when you bring in infinity." What he says about the declining case of a sinusoidal signal being "unfathomably large but not infinite," for instance, is a hoot. How be you try "limitless," baby?

[@technoguyx]

You can even get terms of type t*exp(at), t^2*exp(at), ..., t^k*exp(at) if the characteristic polynomial of the linear diff. eq. has a root of multiplicity larger than one. These terms arise from taking the exponential of the Jordan form of the associated linear system.

[@IsaMelCoding]

MY IB LIFE SAVER!! THANK U SO MUCH

[@Friemelkubus]

I only get half of this because I haven't gotten much of the mathematics yet (was just looking for Laplace transform because we vaguely saw it) but this is epic. I'll so dig into this after my exams.

[@Arobinek]

First, I was sceptic, but then!!! Great!

[@seinfan9]

The black magic of math

[@theman83744]

Thats a great overview. Thanks

[@deltaexplorer47]

WOW !! IMPRESSIVE .... Thank you very much. An INSPIRING video as well. GOD bless you always.

[@pp_01123]

Brilliant Video (Y). Great Work!

[@Rockstar1376]

Splendid video, thanks!

[@fatihsarikoc570]

Hi Brian, That's the best explanation of Laplace Transform I have seen, which is touching on originating conceptual ideas. All your videos have this charecteristics. Sure, you have extraordinary talent and expertise. But, I would like to learn where this culture of conceptual understanding of you comes from. Is it related to university you graduated from or is there conceptual formal textbooks/resources that you can strongly suggest in this manner?

[@tarickgayle3145]

awesome information. at first the maths class look boring but after know what i'm doing. it get pretty interesting. don't fully understand but i think i will get there

[@quantummath]

Thanks a lot bro, well done man.

[@Obyak]

I really like your videos. You know your stuff 99.9%. please keep adding more vids on ME Controls. Thanks

[@SafeAndEffectiveTheySaid]

Thank you Mr Douglas!

[@subramaniantr2091]

Hi Brian great video. I've found that explaining the transform as projection over Orthogonal basis vectors is also a good interpretation which high school students grasp very easily. Dollar example becomes a scalar example which may not put the true perspective. Something I would like to add your points is that dimension of the fourier transform is not really magnitude but rather magnitude per hertz(density). Surprising why no text mentions about this particular thing.

[@Mordaxe]

This video helped me a lot ! Thanks

[@boling5755]

I am reading your ebook. Thanks a lot for you kindly sharing.

[@VrushangPatel9121992]

great explanation, thank you sir.

[@TheScottttttt]

I think this might be quite a good idea! Having an image to quickly scan over to refresh my mind at the end of each of these videos would be quite useful. Thanks for the videos Brian.

[@marwabarznjy3606]

How I can get a good report about (Laplace transform and fourier series )

[@BrianBDouglas]

Hi Shouvik, great suggestion! I've just filled out the form to get my channel reviewed by KZbin to see if it meets the criteria for their education filter. I don't know how long it'll take but hopefully it'll be available soon. Thanks for the comment.

[@gulshan1767]

Excellent work !!

[@anoop5611]

Very neatly put!

[@JudyAbbott494]

This is an amazing video. I've taken this stuff throughout my university classes but still I found this a little advanced for me. What kind of courses should I take to understand this? Is it differential equation course, is it signals? idk HELP :D

[@JordanEdmundsEECS]

Wow. Well done. Very well done.

[@MrHashmi90]

Brian thanks for this informative video and recommending this book " Steven W. Smith - The Scientist and Engineer's Guide to Digital Signal Processing "..... awesome book .

[@abhimanyupatwari4025]

this is awasome lecture

[@Engineerchick247]

Brian, I think your videos are amazing. I've watched each one at least 5-6 times and learn something new each time. One question... I noticed e^-at has an ROC of s>-a. However, when we look at the s-plane after doing the transform, we consider everything before and after -a equally as valid. I've heard there is a proof that shows why the s-plane is valid everywhere even outside the ROC, but I havent been able to find it... Any ideas?

[@pefrenos]

THANK YOU very much for your time

[@AjinkyaDeogade]

Awesome explanation Brian. I am really enjoying control theory. Can you provide link to some exercises to solve and get hand over it?

[@marctison1039]

It's this video that made me finally click. Can't thank you enough, I'm buying your book

[@danbishop7177]

great, got a preview of what ill be using this for next year...one thing that confused me about LT was that i had no idea what it was doing graphically other than the simple bland exponential behavior example they always give at the beginning...This makes more sense for practical purposes.

[@jean-michelgonet9483]

I’ve got it. Wow. Very nice explanation.

[@1495978707]

2:40 Only true for linear differential equations with constant coefficients. A couple examples of exceptions are the Hermite Equation (defines solutions to the quantum harmonic oscillator), the Associated Legendre Equation (which defines spherical harmonics), and the Bessel Equation (unsurprisingly defines the Bessel functions, which are defined as a power series). In addition, sinusoids are also exponentials, just complex exponentials ( cos(x) = (exp(ix) + exp(-ix))/2, or exp(ix) = cos(x) + i*sin(x) ). So really the solutions to linear differential equations with constant coefficients are just various combinations of exponentials with complex valued coefficients in the exponent.

[@angelsscroll]

Amazing explanation!!! Giving insight for the unseen nature of looks-weird math...

[@mnada72]

8:43 What is happening and what is the meaning of probing the impulse response ? For example how this probing is achieved, is it by changing v(t) ?

[@ECOMMUSK]

this is very good. thank you!

[@mohammadbabaee9635]

Thanks Brian. I really like your drawings.

[@Nuke_Gunray]

Cool video, thank you very much

[@grpagobo]

Thanks Brian.

[@badarikarthikguddugurike598]

Great Vedio Thank you Bro Explanation was awesome ..

[@lectrix8]

This video was great!

[@willashland]

your videos are sweet bro, keep em comin