What does the Laplace Transform tell us? A visual explanation [video]
youtube.com
youtube.com
The Fourier Transform decomposes a signal into its sinusoidal frequency components. It is used in a bunch of everyday appliances like your guitar tuner, JPEG images (wavelet compression) and speech recognition on your smartphone or smart speaker.
The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors (decaying or rising) AND its sinusoidal components. So the FT is just one slice of the Laplace Transform where the input signal has no exponential rise or decay.
In the electrical and mechanical domains, spring mass damper systems are super common, even your car's suspension! And to analyse and control them, engineers apply the Laplace Transform.
This is the OMG moment from the video. I wish my teachers would have put it this way a few years back. It now feels like an obvious secret hidden in place sight but Laplace was a very abstract concept before
>The Laplace Transform, on the other hand, decomposes a signal into both its exponential factors
in what sense is this true? yes the s in e^(-st) is complex but that factors the kernel of the transform into e^(-at)e^(-iwt). individually the factors do what you say (decompose over a basis) but as a product they do not do that.
I think DCT is a good entry to understand these kind of transformations. The set of base functions for DCTs is displayed in the image on this site:
https://edoras.sdsu.edu/doc/matlab/toolbox/images/transfo6.h...
I think imaging makes understanding the motivation of frequency dissection plausible and you can imagine any image being a sum of these base images with a weight(R) and a phase/offset(IM). From that understanding LaPlace and FT is far easier. You can even kick one dimension in most applications like audio.
If you view the spiral from the side, you see a sine wave.
If you view the spiral from the top, you see a cosine wave.
If you look down the barrel of the signal, you will see an infinitely bright circle with a radius equal to the amplitude of the signal. The brightness of the circle relates to the energy in the signal, since by looking down the barrel, we have collapsed time and relinquished all phase information.
Pure sinusoids are infinite, so real-world signals will be windowed (note: windowing always introduces artifacts).
Now, what happens if the signal is a composition of rotating circles all spinning at different frequencies and directions? Well, you'll see a bit of a blurry distribution of where the wavefront spends its time. If you think about this a bit, you can ask yourself what you have to do to "see" the frequency components of the signal :).
I have to go now, but will try to post more later.
>the brightness of the circle relates to the energy in the signal
What does this “energy” mechanically correspond to in the complex unit circle and how do these mechanics appear in the time-domain?
The ideal low pass filter’s mathematical application - of the frequency domain effects on natural signals - astonishes me and I can’t quite fathom the transformation; ie is the frequency space physical or a purely mathematical abstraction?
Ultimately: is the complex exponential function a natural algorithm?
I want to clarify something a tiny bit misleading about this. In general complex exponentials are not orthogonal w.r.t. the relevant inner product. You can't really think of them as independent components that compose a function.
If you think of a function as the impulse response of a linear time invariant system, then the laplace transform of that function tells you the result of an experiment where you drive the system with an exponentially damped sinusoid. This is why the poles of the transformed impulse response tell you about the stability of the system: those are the inputs that cause the system to explode!
Basically the video ignores the idea that the X(w) integral may not exist. It doesn’t for many x(t). I don’t want to type a book, but basically the X(s) integral exists (where it does is called the region of convergence) for many more x(t), so there are classes of functions which can now be analyzed where with just Fourier, they could not be.
The other thing is delta functions. With Fourier they are a pain but with Laplace they are a breeze. This is especially important if you want to mathematically model how the control system will respond to a jolt or impulse. I always figured that is why control engineers use Laplace.
The flicker was due to oscillation, due to a pole too close to the right half plane. I was about to stick a bigger capacitor in the feedback compensation circuit, but a bit of maths told me that that would make the problem worse (and probably blowing up my LED's) - so instead I used a smaller resistor value, and yay - perfect flicker free light!
By chance, it was mostly stable when warm (presumably because some resistance or capacitance slightly changed to push the pole to the left half plane), but when cold it flashed on and off at about 2Hz.
The cool part is that at twilight, when the light turned on, the little bit of reflected light bounced back to the photovoltaics, adding just enough light to turn the light off, which brought the system full circle. Amazingly, this behavior only occured for about 1 minute each day.
This is probably not related to Laplace Transforms, but I thought it was an interesting phenomena...
I'd guess in your case it's because each LED has it's own constant current power supply, and when the battery voltage gets low, the power supply becomes unstable, but the exact flicker rate depends on small variations in the components it's made from.
Because of such people efforts, I personally consider this to be a golden age for mathematics and an untold opportunity for younger folks (i.e. students) to finally understand the maths, get involved in it and perhaps use it in ways previously only math wizz would have used.
I wish these were available back when I was taking countless Calculus and Algebra courses at University -- all we had were Professors who couldn't explain in a simple way and books which nobody had time to read during course of semester. The end result was simply treating mathematical phenomenons as black-boxes and/or perform rote memorization to clear the course.
Where the Laplace Transform comes from (Arthur Mattuck, MIT) https://www.youtube.com/watch?v=hqOboV2jgVo
From what I recall, the transformations were so well defined and the solution space so self contained that I could solve the assignments with little difficulty.
I remember in one of the other modules using laplace to solve a problem and the lecturer remarking that it was "an interesting approach". I had no idea what it meant, (though had some slight intuition about how it worked, based on previous exposure to Fourier transforms), but to this day I can't understand how I could take to it so fluently, while floundering at everything else.
Could just have been that in 4th year I gave up part time work and drinking to focus on my finals, and this improved my abilities.
There is a bit of hand holding in engineering courses with it however. A lot of looking at transform tables, and maybe a few problems where you need to exploit the properties of the transform or remember its definition to solve a problem. That can trip people up, and it happens in the real world more often than you'd like.
I'd be interested if people with a science/tech/math background, but no specific training on laplace transforms, managed to understand this video. If they did, it might be a good time to replace all those university classes with this video!
Quite a lot of these topics are the kind of thing where you needed to find the explanation that made sense _for you_, and despite universities having a lot of books chances were there were only a handful on any topic such as this.
I have a feeling I'll retain the ideas better than I would have with a traditional lecture. There is something about how the visuals are presented and transition into each other that really helps make things intuitive. I don't expect to use this directly anytime soon, but it's nice to keep the concepts in the back pocket.
Also, apparently the Laplacian Transform can be used to convert Calculus derivative equations into algebraic ones...
Again, I am no expert; what I've noted is based solely on the claims in this video...
But, some fascinating stuff...
This trick is - AFAIR from my classes - the basis of control theory math. You convert a control system to algebra, do your work there, and in the end, you convert back to differential equations.
Are you thinking of the Jacobian matrix? I'm not sure what you mean by equational conversion. The Laplacian matrix is from graph theory and doesn't involve derivatives. The Laplacian operator involves differentiation, but is not a matrix.
What does the comment mean then?
> "Also, apparently the Laplacian Transform can be used to convert Calculus derivative equations into algebraic ones..."
Neither. It's "Laplace Transform".
> The Laplacian is a matrix of (partial) derivitives and is used for the equational conversion.
Bar the name, this is called Operational Calculus[0].
It was amazing to be able to just do a little bit of odd math here or there, and suddenly have the solution on how to fix something completely unrelated, like how to tilt an airplane safely or make a robot balance a pen upright or something, by modeling it. Granted, we did very simple models, but the sheer scalability and power of these techniques is what made me feel on top of the world.
The laplacian is the dot product of the gradient operator with itself, the Laplace transform is an integral transform with the kernel e^(-s*t).
The relation of the Fourier transform to the Laplacian is that FT diagonalises the Laplacian.
I had a really good teacher on this topic, but without a formulary for time domain -> spectral domain I wouldn't come very far. And they are all pretty hard to memorize.