Laplace and Fourier transformation

[Anonymous No.7030372]

It seems like the time has com for me to understand Laplace and Fourier transformation.
Do i have to be a high IQ super smart person to understand them?
I could pass the derivation and integration stuff which needed for my bachelors.

Can you recommend a source which makes this stuff easy to understand?
(i need it for my EE studies)

[Anonymous No.7030394]

>>7030372
The Laplace Transform is taught in an ode course.
khan academy and patrickjmt have courses. so does mit ocw at a higher level.

Fourier is taught as you go along, youtube has lots of vids on the subject.

[Anonymous No.7030399]

>>7030372

You need the first few calculus and linear algebra courses. To do them without understanding them is one thing but if you really want to understand you should do at least calculus in one complex variable and maybe some babby functional analysis.

You could treat them as "just integrals" and learn tables of transforms and learn to use them in practice. It depends on what degree you are pursuing and what you want to be able to do in the end. Derive your own transforms and understand other linear transforms or just "build cool stuff" with them?

[Anonymous No.7030413]

I'm interested in learning fourier aswell but is there any CS related applications for this?

[Anonymous No.7030421]

>>7030413
Laplace and Fourier have so many applications I can't think of any field that doesn't use them anywhere. FFT was developed (or "re-discovered") in the 1960s just in time for transistors and digital technology to be able to use them. They are literally Everywhere.

[Anonymous No.7030442]

>>7030399
I already passed Calc I,II,III (i don't know how this translates to american education but the first one was about derivation and vector algebra and complex numbers, the second one was about integrals and matrices and the third one was about probability and related stuff, the courses were called Analysis 1,2 and 3)
I need to understand them at a level to be able to pass a Signals and Systems course.

>>7030394
The MIT stuff looks cool with it's exams and solutions, i wonder if it would be too deep for me though. I am just an europoor, i am not sure if i should pursue MIT levels of knowledge. I am not an autist and i dont think i am that smart.

[Anonymous No.7030505]

>>7030442

Maybe the coolest newest most advanced courses at MIT are really awesome, but Fourier and Laplace should not be that different in the US from in Europe.

[Anonymous No.7030515]

I was a math major and a chemistry major (with a focus on atomic physics) and I never saw a Fourier Transform in any class once, except briefly mentioned in a chem lab about FTIR (but without the mathemaitcal details). I was kind of surprised to never see them.

I saw Laplace Transforms in my DIff Eq class.

[Anonymous No.7030530]

This is the Fourier transform.
It takes a signal in the time domain and represents it in the frequency domain. In other words, it takes a sounds and tells you how hard you'd have to hit a load of tuning forks of various frequencies to reproduce that sound.

[Anonymous No.7030534]

>>7030442
You have enough to do ode and laplace transform. fourier is usually taught in pde, but you can learn from youtube

[Anonymous No.7030586]

>>7030530
shiet thats the theory behind my music production shit

[Anonymous No.7030594]

watch this video

https://www.youtube.com/watch?v=sZ2qulI6GEk

[Anonymous No.7030616]

>>7030515
If you were doing xray crystallography you might have heard of Fourier transforms. But really when I went over them it was basically "So the computer does a Fourier transform and bam, we have our configuration"

[Anonymous No.7030651]

>>7030530
If we imagine that gif as a cube, 2 sides are f and Fourier of f, what would be the third side? marked with ?
Is it anything at all?

[Anonymous No.7030672]

>>7030651
I think it'd be a bunch of parallel lines

[Anonymous No.7030693]

>>7030672
Id thought so too, doesnt make any sense.

[Anonymous No.7030748]

>>7030594
1:00 - 12:30

laplace explained like nowhere else ever.
awesom

[Anonymous No.7030776]

>>7030748
i dont understand fuckall of that

[Anonymous No.7030788]

>>7030776
What part don't you understand?

Maybe try Khan Academies videos instead.

khanacademy.org/math/differential-equations/laplace-transform

[Anonymous No.7030791]

>>7030372

If you understand how a fourier sine-cosine series works intuitively, then:

>replace sine-cosine by e-power
>replace the interval by -inf to inf
>a longer interval gives shorter gaps between the frequencies; an infinite interval gives infinitesimal spacing of the frequencies
>replace the sum by an integral because reasons

You could also work from the vertical bars drawn here: >>7030651

When the interval for f becomes infinitely large, the spacing between the frequency peaks becomes infinitesimal, creating a smooth line.

[Anonymous No.7030794]

If I want to truly understand everything that goes on in an ODE course, should I get calc III and linear algebra under my belt first?

I'm currently enrolled in all 3 courses + a physics course trying to haul ass and make up time where I didn't know what I was doing for the past year and a half, and everyone tells me I should "stick with all 3 it won't be bad," but I really hate not understanding mathematical concepts and just "using" them.

[Anonymous No.7030837]

Fourier transforms are much easier to understand compared to Laplace transforms. The Fourier transform takes some signal in time (i.e. f[t]) and converts it to the frequency domain, so FT[f[t]]=f[k], the inverse Fourier transform does the opposite. So if you have a square wave signal like in >>7030651 the FT decomposes the signal into weighted sums of sines and cosines at different frequencies. If an infinite number of terms are used you'll get back the original signal exactly when all of the sines and cosines are added together. This is because sine and cosine are orthogonal at different frequencies and form a complete basis or some math words like that.

You should check out the PDE section of Boas, since it will get you used to doing these. Fourier came up with them to solve the heat equation, the general idea is to say "I have a slab of metal kept at 0 on three sides and Temp=f(x) on the remaining side", the heat equation is a PDE with d/dt u=c*laplacian u, to get the answer you'll invariably use a fourier series to represent your function of temperature.

If you've done any quantum the square well drinks from the same well as Fourier transforms, as the wavefunction is made of a series of sines and cosines added together- and if you take the fourier transform of the position wave function you get the momentum wave function!

[Anonymous No.7030910]

>>7030788
all of it
i suck at math

[Anonymous No.7031033]

>>7030794
you only need calc 1-2 for ode, that is it. besides you have three different ways to learn the shit online.

[Anonymous No.7031104]

>>7030748
mind=blown.

[Anonymous No.7031113]

>>7030837
I always thought Laplace was far easier than Fourier.

[Anonymous No.7031118]

>>7031033
Calc 1-2 as in differential/integral and vectorial?

[Anonymous No.7031124]

>>7030794
Are you the guy who posts this course list in virtually every thread? Just try it out man, if it's too much just drop diff. eq. or linear. But definitely finish physics and calc III

[Anonymous No.7031127]

MIT's stuff really is superior to other schools when it comes to STEM. All schools have great professors, but MIT across the board is excellent.

[Anonymous No.7031132]

>>7031113
Because it is.

>>7030399
>need linear algebra for Laplace
Kek yea fukin rite

[Anonymous No.7031143]

>>7031033
Okay cool thanks anon. I wasn't sure, since I know ODE uses Partial Derivatives but that's about the only concept I can think of

>>7031124
no, this is the first time I've posted it actually. I just swapped majors, so I'm sorta testing the waters on all of this. (Wanna go to grad school now too) But it probably isn't that bad anyways. Cheers, anon.

[Anonymous No.7031171]

Aren't Fourier series a calc III concept rather than an ODE concept? We did some basic fouerir series at the end of my calc II course, but I figured that is more of a calc III thing.

[Anonymous No.7031198]

>>7031113
Doing Laplace transforms is easier because you're just going to look them up in a table, but understanding what they mean is boggling. Fourier takes time domain to frequency domain, Laplace takes time domain to the complex frequency domain, with the basis being moments. Fourier transforms can even be considered as just a special Laplace transform, as the LT evaluated at iw.

[Anonymous No.7031769]

>>7030791
so the vertical bars become like my drawing?

[Anonymous No.7031941]

>>7030372
I got them the first year in my bachelors study. They're fucking easy. Just start proving the basic properties of the Fourier transformation like translation,modulation,etc by using partial integration and you'll get used to them fast enough. Also, don't forget to use <div class="math">e^{ix}=\cos(x)+i\sin(x)</div>.

Then do the same for Laplace transforms. Only difference vs Fourier is you'll need to know what exponential order is.
If you get stuck, here you go (although I advice to try yourself first):
https://www.youtube.com/watch?v=hfKycVR4kSw

[Anonymous No.7031988]

>>7030515
>https://www.youtube.com/watch?v=sZ2qulI6GEk
what sort of chemistry did you study which didn't teach NMR?!

[Anonymous No.7032058]

>>7031769
don't know what you guys are talking about, but pretty/10

[Anonymous No.7032332]

>>7030594
OP here, this shit made me understand Laplace, i will rewatch it once or twice. I realized at the beginning of the video that i need to brush up on power series.

Is there a video somewhere where this guy talks about Fourier transformation?

[ojom™ No.7032419]

Fourier transforms really aren't that hard.
I taught them to myself after not even a half year of MV calc.
The hardest part about them is putting fundamentals like complex exponent form, integrals, and sinusoidal characteristics together. All math is is just putting together a few puzzle pieces to make a bigger chunk of the puzzle.

[Anonymous No.7032483]

>>7031769
Yea, but it isn't necessarily going to look like what you drew. Like with your blue line you're saying at 0 frequency the amplitude is going to infinity, but the zero frequency is actually just a constant. The zero frequency coefficient is kind of special, because it should always end up being the average of your function, so if the square wave in red has peaks at +A and -A then the line blue line should be 0 at 0, because the average amplitude is 0.

You should also try thinking about what happens if your red line (the time domain) is something like cos(kx). Sines and cosines are orthogonal, so you cannot make cos(kx) out of any combination of sines and cosines that do not include frequency k, and so the resulting fourier transform is a single spike at frequency k.

Another cool thing is to take some periodic dataset like atmospheric CO2 data and do the fourier transform, and you'll see two/three peaks, one at 0 frequency which is like adding a constant to everything, another at a low frequency that roughly relates to a linear rise in time, and a third at a high frequency that represents the yearly frequency. If you then take your fourier transform and set a cutoff frequency, you can remove the yearly variations (cut off high freq) or the overall rise (cut off low freq). Then you do an inverse fourier transform and get your data back without that specific part.

[Anonymous No.7032487]

>>7032483
So here is that same data set, with the smoothed curve being the IFT with basically a low pass filter, if frequency index is below 50 it gets IFT'd and only the overall rise is seen.

[Anonymous No.7032491]

>>7032483
I also forgot to mention that there are sampling issues, the second big peak is called the Nyquist frequency and should be double the frequency of the first big peak.

[Anonymous No.7032507]

>>7032487
That looks cool. How would it look like without the 0 frequency and low frequency component? Like a horizontal triangle wave?

[Anonymous No.7032516]

>>7032507
If you put a high pass filter on (high frequency is left) then it would basically look like a sine or cosine going up and down about 0. Maybe a little triangular since they look kind of triangly in the original.

[Anonymous No.7032557]

Fourier series have a really rather subtle theory. Of course, most of the time engineers can just 'plug and chug', but the question of convergence (in an appropriate sense) of Fourier series was one which took a great deal of effort to solve. In fact, it was only in 1966 that it was shown that the Fourier series of a continuous function converges almost everywhere. If you were to tell this to an engineer I think they would probably cry. If you want the proper theory, learn about measure theory and functional analysis.

[Anonymous No.7032808]

>>7030594

excellent video, really helpful.

thank you, based anon

[Anonymous No.7033479]

>>7030616

That is exactly what I was talking about, the computer thing.

[Anonymous No.7033491]

>>7030372
Fourier transforms have really nice physical interpretations, like converting a sound wave into a frequency distribution.

all I know about Laplace transforms is that they're helpful when integrating disgusting expressions

[Anonymous No.7033497]

Personally, I only really understood Fourier transforms once I started using wavelets.

I could always do them and use them, but never understand them.

>>7032557
Not if you talk to a control systems or signal processing engineer.
When analyzing discrete systems, you have to use the Laplace transform's cousin. The Z-transform only has a finite region of convergence.
This becomes important if you want to determine if a system is stable and causal.

[Anonymous No.7033502]

>>7033491
stability of poles.

[Anonymous No.7033507]

>>7033491
They both have the physical interpretations if you can see them.
The Fourier transform is this:
<span class="math">F(\omega)=\int_{-\infty}^{\infty} f(t) e^{-j \omega t} dt<span class="math">
What you are basically doing is multiplying your signal with a bunch of complex exponentials which allows you to see how strong the signal is at each frequency.

Mathematically, you are projecting the function onto a different basis. You do the exact same thing when you take the dot product of two vectors. You see how strong the vector is in another direction.

Now to Laplace.
F(s)=\int_{-\infty}^{\infty} f(t) e^{-s t} dt

All we did was use a complex number s=a+j\omega instead of a plain \omega .

What this means is that instead of projecting onto an infinite sum of sinusoids, you project onto an infinte sum of decaying sinusoids since:
e^{(-a+\omega)t}=e^{-a t}*(cos(\omega t) +j sin(\omega t)) [/spoiler][/spoiler]

[Anonymous No.7033508]

>>7033507
FFS

[Anonymous No.7033516]

>>7033491
Laplace is a Fourier transform but with e^(iω+σ)t instead of e^iωt to deal with functions that blow up when σ=0.

[Anonymous No.7033523]

>>7030413
I know they're used in image analysis. Just google Fourier Analysis in CS, you'll find something.

[Anonymous No.7033526]

>>7030586
What a fucking pleb tier DAW you should feel bad.
>shiet thats the theory behind my music production shit
you aren't doing any real music production you are just nigging around with some pirated software.

[Anonymous No.7033557]

>>7033526
But i only use free software.

[Anonymous No.7033577]

>>7031132
doesn't hurt to be able to look at things as both algebraic structures and as infinite dimensional vector spaces.

[Anonymous No.7033592]

>>7033526
What is wrong with Reaper?

[Anonymous No.7034356]

>>7033526
>Reaper
>Pirating
This software is almost a freeware for you have to wait 5 seconds each time you launch it ... without real expiring restriction

>Muh pro tools
>Muh completly obsolete interface
Last time a dude played me this string I discovered he was using Audacity as a compilator and Ableton as a DAW.
Ladies and gentlemen that was the Bashing-Guy.

[Anonymous No.7034395]

I enjoyed Systems and Signals. It's more about applications, but the theory went pretty deep sometimes. Indeed they are also taught in ODE, but it was more fundamental in S&S.