Talking Head Teaches Laplace Transform

Most people who deal with electronics have heard of the Fourier transform. That mathematical process makes it possible for computers to analyze sound, video, and it also offers critical math insights for tasks ranging from pattern matching to frequency synthesis. The Laplace transform is less familiar, even though it is a generalization of the Fourier transform. [Steve Bruntun] has a good explanation of the math behind the Laplace transform in a recent video that you can see below.

There are many applications for the Laplace transform, including transforming types of differential equations. This comes up often in electronics where you have time-varying components like inductors and capacitors. Instead of having to solve a differential equation, you can perform a Laplace, solve using common algebra, and then do a reverse transform to get the right answer. This is similar to how logarithms can take a harder problem — multiplication — and change it into a simpler addition problem, but on a much larger scale.

We assume it was a choice, but [Steve] presents wearing all black on a black background, so we found ourselves imagining him as a floating head of math knowledge. Kidding aside, what a great explanation of the topic! You’ll probably need some calculus to get the most out of the video, but it is interesting to just watch him cut through the equations.

Once you tackle the transform, you’ll find it interesting to learn about the S-plane which is the graphical representation of the Laplace transform. If you need a brush up on calculus we have mentioned before that calculus isn’t really hard. We also like a very old book on the subject.

23 thoughts on “Talking Head Teaches Laplace Transform

  1. that old calculus book is very good, I gave it to my son a couple of years ago to use. Because it is so old it doesn’t assume knowledge you don’t have, doesn’t assume that you are doing everything on a computer, but also doesn’t assume you are complete idiot either. It also helps :-) to tell your teenager that these were fundamental skills that people over 100 years ago could do with a pen and paper, so there is no reason he can’t.. :-)

  2. > and then do a reverse transform to get the right answer.

    Once you get familiar with the LT, it’s sometimes possible to work completely in that domain. The “right answer” isn’t necessarily in the time domain.

    This is equivalent to the way it’s possible to analyse or design systems of amplifiers (etc.) completely in the logarithmic “dB” domain.

  3. “The Laplace transform is less familiar, even though it is a generalization of the Fourier transform.”
    Someone knowing Fourier probably also knows LaPlace… I have a hard time seeing how else to learn (Discrete) Fourier properly in the first place…

    1. Coming from a physics background, only Fourier was covered in the curriculum (and it’s everywhere in physics). Laplace never got so much as a mention. I’ve been very slowly picking up the use of Laplace transforms since getting into electronics, and this is a pretty good presentation of it for background like mine.

      1. Interesting. I always assumed Laplace Transform was covered in every background that touches differential equations, I guess because I was first introduced to them in a DE class before actually putting it to use in control theory,

        1. Yeah, I feel pretty shortchanged by that. Our version of a differential equations course went as far as Fourier series. Not even the continuous-frequency transform, just a sum of sinusoids. All the rest — real Fourier transforms, connecting to vector spaces, generalizing that to other orthogonal function expansions — was mostly left to be implicitly picked up from physics, where the professors and textbook authors assumed that math classes had already taught you this math.

          It may have something to do with the math department’s general attitude that anyone who wasn’t a math major was a moron incapable of being taught anything, so all of the math courses for non-majors were severely dumbed down.

  4. I once had it explained like this:

    So you’ve got this very complex math problem, and you don’t know how to solve it. You’ve been scratching your head down to the bone and it really hurts big time.
    Then you discover you’ve got a neighbour who is the biggest maths head in town.
    The problem is, he only speaks Greek.

    So now, you just have to translate your math problem into Greek, then he solves your problem quickly and easily, and you (optionally) translate the answer back from Greek.

    So my hat’s off for Pierre-Simon Laplace ( 1749 – 1827) who probably invented this before Charles Babbage (1791 – 1871) was playing with his cogwheels.

    https://en.wikipedia.org/wiki/Pierre-Simon_Laplace
    https://en.wikipedia.org/wiki/Charles_Babbage
    https://en.wikipedia.org/wiki/Laplace_transform#History

  5. fourier is about steady state solutions to diff equations.
    Laplace is about full discovery through the initial conditions of a system to output after a time. ( given initial cond and stimulus)

    That’s what I think is the difference. What u think?

    1. There are two differences.
      One is that the limit of integration is 0 to infinity (“one sided”) for the LT and -infinity to +infinity (“two sided”) for the FT.
      The other is that the ‘s’ variable is complex in the LT, whereas the ‘j omega’ FT frequency is equivalent to the Y axis in the s-plane.

      BTW, there is such a thing as a two sided Laplace transform. I’ve never used one.

    1. Yeah, Da Vinci could also write backward!

      I wonder if this professor decided long ago, that too much information is lost in the classroom by the instructor blocking much of what they write on the [chalk,white]board with their body.

  6. “…The Laplace transform is less familiar, even though it is a generalization of the Fourier transform…”

    The Laplace transform is no more related to the Fourier Transform than the Fourier Transform is related to the ‘logarithm transform’–or, for that matter, as “lightning bug” is related to “lightning”.

    You’re in serious need of a good Technical Editor, Hackaday.

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