
What does the Laplace Transform tell us? A visual explanation [video] - peter_d_sherman
https://www.youtube.com/watch?v=n2y7n6jw5d0
======
pgt
CONTROL THEORY NERD ALERT! This is hands-down the best video on control theory
I've ever seen, clearly depicting the relationship of the Laplace Transform to
the Fourier Transform.

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.

~~~
danharaj
> The Laplace Transform, on the other hand, decomposes a signal into both its
> exponential factors (decaying or rising) AND its sinusoidal components.

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!

~~~
pgt
Aren't complex exponentials signal rotations? If you multiply a sinusoidal
signal by e^(-iwt), you are essentially shifting its phase modulus, no?

~~~
danharaj
Those are purely imaginary exponentials. General complex exponentials also
have a real part.

~~~
dreamcompiler
And that's how the FT differs from the LT. LT is complex while FT is purely
imaginary, as the video explains.

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londons_explore
I actually _used_ this math at home! - one of my LED lamps was flickering in
cold weather, and upon opening it up I saw a primary side controlled flyback
transformer to act as a constant current power supply (for ~20 LED's in
series).

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!

~~~
drhodes
I noticed outdoor LED lamps flickering too, but only at dusk, and for what I
thought was a feedback issue. These lamps are solar powered and turn on at
night.

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.

~~~
londons_explore
The normal solution to this is hysteresis. In most cases, it doesn't cost any
extra to add hysteresis.

~~~
Enginerrrd
It does if you pay for a competent engineer in the first place who would know
to do things like debounce switches and add hysteresis.

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m23khan
I really appreciate people (amateurs and professionals) who take time to make
explanation of mathematics concepts simpler and easier to understand -- even
for layman just familiar with basic primary school maths.

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.

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targonca
I prefer this one:

Where the Laplace Transform comes from (Arthur Mattuck, MIT)
[https://www.youtube.com/watch?v=hqOboV2jgVo](https://www.youtube.com/watch?v=hqOboV2jgVo)

~~~
gtycomb
A piece of jewel. Reminds me of chalk board talks of Gilbert Strang and Robert
Gallager (also on youtube, mit video lectures) that celebrates this gift for
teaching us something.

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rusk
This is tangential, and veering very much into the off-topic .. but the
Laplace transform has a special place in my heart, because after banging my
head against the wall of advanced calculus and other engineering mathematics
topics for 4 years, with varying degrees of success, when presented with
laplace I could "just do it".

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.

~~~
zozbot234
The LaPlace transform is also isomorphic to the net present value problem in
finance. The Wiki article on net present value even links to a paper pointing
this out from a finance POV. (The problem of how the net present value of a
flow varies with changes in the discount rate is also of some theoretical
interest in finance, since it clarifies the conditions under which the so-
called 'roundaboutness' of e.g. an investment is well defined.)

~~~
pgt
Can you expound on this? I recently applied some classic signal processing
techniques to help reconcile my personal tax accounting by tracking coherence.
This sorta works if you consider double-entry accounting as reverb in a linear
time-invariant chamber (or echoes with unknown delay). I would love hear about
any non-traditional applications of DSP.

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londons_explore
I learned all this at university, over many hours of classes, exams and
practice questions.

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!

~~~
lordnacho
Well I also did Fourier and Laplace at uni, but I'm certain it would have been
easier to learn if the current explosion of videos on tech topics had happened
before I'd gotten there. That 3D of the sums makes a lot of sense, because you
need to motivate taking that integral somehow. In my mind I did that, but it's
nice to see the 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.

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peter_d_sherman
Apparently (and I am no expert), The Laplacian Transform is a superset of the
Fourier Transform, and the Fourier Transform is a subset of the Laplacian.

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...

~~~
TeMPOraL
> _convert Calculus derivative equations into algebraic ones..._

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.

~~~
mikorym
It should say Laplacian in OP's post, not Laplacian Transform (if I am not
mistaken). The Laplacian is a matrix of (partial) derivitives and is used for
the equational conversion.

~~~
mkl
It should be "Laplace transform". This video has nothing about Laplacians at
all.

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.

~~~
mikorym
Sorry, sorry. You are right, I am thinking of the Jacobian.

What does the comment mean then?

> "Also, apparently the Laplacian Transform can be used to convert Calculus
> derivative equations into algebraic ones..."

~~~
mkl
That's the primary reason to use the _Laplace_ transform, as seen in the
video. A derivative x'(t) gets transformed into a product (and an initial
condition), s X(s) - x(0), and similar for higher derivatives, so a
differential equation transforms into an algebraic equation, which can be
solved by rearranging. This video assumed the initial conditions like x(0) =
0, and its notation was quite sloppy/confusing in places, as it didn't clearly
distinguish the names of the two functions, x(t) and X(s).

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timonoko
Transform of nightmares. Computers were not yet invented in 1974 and we were
trained to design analog electronics with massive laplace matrixes and
sliderules and tables. Now I not remember anything about matrix calculations
and know exactly nilch about any bloody transforms.

~~~
raxxorrax
If you know how to turn a dial on a pid-controller, you are probably already
on par with 95% of the engineering work force.

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.

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dreamcompiler
I'd love to see a follow-on that described the z-transform. The z-transform is
the digital or discrete version of the Laplace transform. It's very useful for
designing digital filters.

~~~
bainsfather
Previous video from the same author is on the z-transform.

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dreamcompiler
This is a very good, quick video that describes both of the amazing properties
of the Laplace transform (how it's the complex version of the Fourier
transform and how it makes linear differential equations almost trivial to
solve).

------
XnoiVeX
Also interesting to note that stock market returns map better to a Laplacian
distribution. [https://sixfigureinvesting.com/2016/03/modeling-stock-
market...](https://sixfigureinvesting.com/2016/03/modeling-stock-market-
returns-with-laplace-distribution-instead-of-normal/)

