The intuition behind Fourier and Laplace transforms I was never taught in school
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- Опубликовано: 26 сен 2024
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This video covers a purely geometric way to understand both Fourier and Laplace transforms (without worrying about imaginary numbers).
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![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
Finally some dark theme animations that don't destroy your eyes at 3 AM
Many people were requesting it lol
Personally, I don’t like dark themes. All the programs on my computer are white.
So, please, don’t speak for everyone, Antonio. Too bad we cannot choose the design. Hey, is there a mathematical way to find an optimal solution that won’t annoy both Antonio and me? Looks like it doesn’t. Never mind, I’m used to be in the minority.
@@Scorpionwacom go get a life
Sorry, Nabil, only normal people do have a life. I know that the majority likes black software.
Scorpy_ If you use Windows, pressing Windows Key - + followed by Ctrl - Alt - I will invert the colours on your screen for pretty much every application. (If there is default magnification, you can switch it to "100%" to turn it off). This will leave you with the blazing spectral sun you so desire to reside within your computer screen.
Wow, Laplace transform is like Fourier but it looks for exponentials as well as sinusoids? Holy crap, now it makes sense why the transfer function of a control system is the laplace transform of the system dynamics, you're looking at the exponential part for stability, and the sinusoidal part can tell you about performance. Fascinating!
I will need some time to recover from this video
@@inigomeniego4906 yes, it hurt a little, good hurt, but hurt
i was looking thru the comments right before I watched this video, and I thought wtf is this
20 minutes later everything is crystal clear
That's one of the best explanations of Fourier Transform I have ever seen !!!
sudip banerjee look at the 3blue1brown video it dose not show you how to compute it but the intuition it provides it’s beautiful. It shows how the Fourier transform is a mathematical machine that wraps a function around a circle and measures the x output of its center of mass (measure) and lets you pick out frequencies from a mixed sum of frequencies. It’s awesome I love 3blue1brown
@@jmof0464 I was just going to comment it but have found you. Haha. Thank you
It is weird to think that the equation was developed before the complex machines we use today. Yet, the second example is omega t = pi x is a binary output of 0 or infinity, quite remarkable.
really
I find this even more intuitive than 3b1b's, and that is saying something. Considering how good his explanations are, there is no over estimating how good this is! Thank you!
Plus Zack also extends into engineering
You should also look for Up and Atom video on this. She also describes it very intuitively and in a beautiful way
As Leonard Euler, I confess that it’s too damn complicated.
but we couldn't do it without you?
*Leonhard
James Bra Indeed my child, just like you wouldn't have these symbols: √, i, e, f(x), Σ
It's Leonhard Euler...
@Bob Trenwith And the original conceiver of Nuka Cola Park
as a physicist, i found this to be the best, most intuitive no bullshit explanation of the fourier transform. the orthogonality and completeness relations make perfect sense now. fucking awesome job dawg!
HOLY SHIT this is the best explanation i ever got, i've been looking for so long for videos that would give the intuition behind those notions and i couldn't find one that resonated with my logic but this one hit hard, thank you so much!!!!
This one and 3b1b's video on the Fourier Transform are brilliant
Yes we could say this video resonated
3Major1Prep?
Exactly my thought 😂
yea, i thought this was some collab too
you're a funny guy
Ahh you did something to my brain
yeah what is going on here?
I've always thought the simplest way to understand Fourier series is from a linear algebra perspective: First, define continuous function space, where the "inner product" of two functions is the integral of their product. Then the infinite set of Sin(nwt) and Cos(nwt) functions form an orthoganal basis that spans this space,. So any continuous function can be expressed as a linear combination of these "eigen functions". I haven't thought about this in years, but I think I still have the terminology right.
Also, the Laplace transform is a special integration technique for solving the convolution integral. I believe that is where it came from. You should explain Laplace by introducing the concept of linear systems and superposition, then show how this leads to the convolution integral, then show how Laplace came up with a brilliant shortcut for solving these without having to do integrals. Just saying that's how I learned it centuries ago.
except this is better
I absolutely agree; introducing it using this linear algebra approach makes it far more intuitive. It then becomes glaring why we can filter by amplitude and frequency
I can't agree with you more.
Hahaha let me go ahead and confuse people even more by bringing up the abstract field that is linear algebra, which 99% of viewers watching this video are not familiar with
@@salimdebit7638 Confuse people "even more"? So you admit this video is confusing?
As for bringing up an "abstract" field (which actually has countless real world applications), people watch math videos because they want to learn MATH, otherwise they would watch cat videos.
14:01 can we just take a minute to appreciate how intuitively powerful this animation is?
Yes exactly my thoughts at that moment
It's from Wikipedia. Yes that's animation is really powerful.
Been working with integral transforms since ~7years and this is the first video that actually gave me a graphical understanding of the transform itself. Awesome videos, mate!
For a second I thought it was 3blue1brown video😂 also the timing of the video couldn’t be any better, I have signals and systems exam in 3 days.
Best of luck to you and hopefully there wont be any altercations with the exam in regards to the corona situation.
Same situation, dude! 3 days! 😂
Failed a signals midterm a bit ago; this video really helps...
This is really what I was looking for - a visualization that is explained slowly enough to catch its underlying thought while teaching a concept like FT. My professor always reminds us of the importance of understanding the underlying working principle, but fails to explain it in a way that would allow it. So really thank you for your effort in creating these animations.
You literally blew my mind. I have studied control systems and we've continuously used the concept of poles but after listening only do I understand the intuition behind using Laplace transforms. This is an absolute genius and a work of art! Thank you so much for this video @zach star.
One day i'll understand this....
Ok... I've been using LaPlace for control systems for years and never truly understood what was happening behind the scenes, I came here to understand how the hell the Fourier Series work and I'm completely mind blowed, congratulations, I'll have a hard time sleeping tonight with all the concepts and ideas taking shape in my little brain.
I had to learn this many years ago, and I wrestled with the concept until I figured it out. My girlfriend at the time said that I would sometimes stare off into space with a blank look on my face, and then suddenly have a "Eureka" moment and smile. This was also in the early days of microcomputers, and it took me quite a long time to visualize all of this since we didn't have access to videos like this. If I had seen a video like this as a student, the light bulb in my head would have lit up much earlier. This is the best video I have ever seen illustrating the concepts and nuances of the Fourier Transform. By the way, a short continuation of this video would also be a great way to show what windowing functions do by arbitrarily limiting the number of samples and showing the resultant "Gibbs Phenomenon" that Electrical Engineers who deal with digital filters learn to hate.
The connection between using impulses to find frequencies of the signal and showing a continuous fourier transform as a magnitude plot at varying frequencies, THEN showing how laplace is a generalisation. I'm amazed :) looking forward to my signals class now
I'm a freshman in high school in algebra 2 and don't rly understand much of what was said but I'm so excited for the day I can do complex maths
Wooow, I used the FFT (DFT) for years now and know the math. But honestly, watching 3B1B and the Veritasium version of it rather got me confused about my previous assumption. They do an ok job but they try to squash too much detail into a short YT video. I think what you did right here is the perfect explanation of how the FFT works, on a level that one can actually really fit in a youtube video. Well done!
he's not talking about FFT.
Hi Im self-quarantined @home. RUclips is my school now. Thanks for the great lecture!
A really great video explaining the intuition, Fourier used for a sinusoidal scan, while laplace is used for both both sinusoidal and exponential scan. Summarizes the whole video, loved it.
This is one of those great pieces of content where I can come back to it after months at a time and get something new out of it each time. Great work, Zach.
12:14
This is actually really interesting, because in music, a square wave produces only odd-numbered harmonic overtones. So, in this regard, the "omega sweep" kinda thing here, when applying these waves to music, would reveal all the overtones of a given wave form. Thats super cool
That's LITTERALLY how we produce a square wave. If you use a spectroscope to find the overtones of a square wave (actually, of ANY wave), the spectroscope is literally performing (a variation of called Fast) Fourier transforms of the incoming signal. Pretty cool stuff tbh. No matter what you do, you can't escape maths.
-A physicist, and a guitarist.
idk if you guys can hear graphs but the demostration of FM at 7:48 and AM at 11:09 is pretty interesting.
MajorPrep Converges to 3blue1brow
Everything does eventually
Lol
Bakdi Abderrahmane yah I came here right after watching 3blue1brown’s videos on the Fourier transform but I must say their approaches to the problem are a bit different
I've always looked to understand the fourier transform and series..... all I can say is I landed on GOLD today. Thank you very much for the video. Very intuitive and I always love intution first before I dive into the calculations.
He attac
He protec
But most importantly it doesn't want to go in my Head
🤣
seen lots of similar videos always amazed with them but as electriacl enginner I never really undestand the formula or how the magic happens but your video is simply super educative this is the first time ever after 3 years of studing it I finally really understand it I am helpless of expressing how this video gets straight into the core and smash it out that easy .
U own my life.
A spectrum analyzer can be thought of as a Fourier Transform machine.
Oscilloscopes with Fast Fourier Transform (FFT) capabilities can do so as well.
This is perfect! We have learned a month ago about Laplace Transforms and now we're learning the Fourier Complex Series. Since I'm studying an ICT Systems Engineering Degree, this is really useful for wave analysis and many other things. Kudos to you!
After having spent a good amount of time understanding DFTs and specifically the FFT algorithm, this video talking about how to intuitively understand transforms is just brilliant! Everything falls into place!
I think this is the best explanation of Fourier transform so far. Thank you!
I applied fourier transform to best teacher waveform and i got Zach Star ⭐
a small correction:
at 13:00
the integral is not infinity not because the a1 is too small but because f(t) is zero outside the interval -0.5 < t < 0.5
thanks
Finally a great and simple explanation of Fourier Transform after days of searching and jumping from video to another.
Thanks man, much appreciated.
3blue1brown has an excellent one, too.
extremely intuitive explanation
This was a great explanation of these transformations I encountered with after close to 35 years of dealing with them!
So grateful to you for such an intuitive, mind blowing and brilliant explanation of such an important topic, which I guess most indian college professors themselves have no intuitive and graphical understanding about!
I would have never understood the big picture behind these transforms it not for videos like yours!!
Thank you! I've been studying this topic all day without much success and I think it finally clicked in my brain!
This is so cool, Zach. I wish I had you as my professor when I was a student in engineering class. I had a lot of professors who can't teach at all in layman term.
because they don't really understand these tools
Before watching this video, I'd never expected that it is so easy to understand Fourier transform and Laplace transform and the connection between them.
Dammmmnnnnnnn! Why the hell was this so hard back in engineering college! This has been the best 40mins spent on a channel! I watched one of your Laplace transforms video as well! Honestly @3Blue1Brown and @ZachStar you guys should collab for such amazing videos!
This videoclip saved lives
This video, along with 3blue1brown videos really help visualizer what's going on. Back when I watched 3b1b vid about Fourrier transform it gave me the intuition for why the integral of two cos or sin is only non zero when you are multiplying 2 cos with same fréquency or two sin with same frequency. You can show with trigonometry and a little bit of calculus that if you take an integral of two cos functions which have different frequency, and the boundary of the integral is any common period of those two trig functions (I know this is not a term, my English is not good, but I hope you all understand what I want to say, for instance the boundaries of the Integral for cos(4pix) cos(8pix) can be one half or 1 or 3/2 or any multiple of one half (because one half is the lowest common denominator of the period of those two functions). The common period of two cos or sin functions is any denominator of the periods of two functions, or any common divisor of frequencies of two functions). But honestly, doing the math gives no intuition whatsoever of what is going on, it is just an interesting result that appears out of nowhere. after watching videos like this and videos from 3blue1brown I was like aha, so now I would have expected this result even before I calculated it.
Bravo for the video it really helps understanding what's going on.
Here's the math which to me doesn't give much intuition. We need these trig formulas.
cos(a+b) = cosacosb - sinasinb
cos(a+b) + cos(a-b) = 2cosacosb
(1/2)(cos(a+b) + cos(a-b))= cosacosb
β = a + b, θ = a-b
β + θ = 2a
a = (1/2)(β+θ)
θ - β = - 2b
b = (β - θ)/2
(1/2)cosβ + cosθ = cos(1/2 (β+θ))cos(1/2(β-θ))
cos(a-b) - cos(a + b) = cos(a) cos(b) + sin(a) sin(b) - cos(a) cos(b) - (-sin(a) sin(b) ) = 2sin(a) sin(b)
sin(a) sin(b) = 1/2 cos(a-b) - cos(a + b)
cos(a+b) + cos(a-b) = 2cosacosb
(1/2)cos(a+b) + cos(a-b) = cosacosb
cos(mx) cos(nx) = 1/2 cos((m+n)x) + 1/2 cos((m-n)x)
if m != n
Integrating over a period T can be done by setting boundaries to - T/2 and T/2 or 0 and T.
\int_{x=0}{x=T} cos(mx) cos(nx)dx = 1/2 \int_{x=0}{x=T} cos((m+n)x)dx + 1/2 \int_{x=0}{x=T} cos((m-n)x)dx
int_{x=0}{x=T} cos((m+n)x)dx = 1/(m+n) sin((m+n)x)
x = T
1/(m+n) sin(mT+ nT) = 1/(m+n) sin(mT)cos(nT) + sin(nT)cos(mT) = 1/(m+n) (0+0) = 0
x = 0
1/(m+n) sin(0) = 0
int_{x=0}{x=T} cos((m+n)x)dx = 1/(m+n) sin((m+n)x) = 0 - 0 = 0
if m=n
_||_
int
int_{x=0}{x=T} cos((m+n)x)dx = 1/(m+n) sin((m+n)x) = 0 - 0 = 0
\int_{x=0}{x=T} cos((m-n)x)dx = 1/(m-n) sin((m-n)x)
_||_
\int_{x=0}{x=T} cos((m-n)x)dx = 1/(m-n) sin((m-n)x) = 0 - 0 = 0
If m = n
cos(mx) cos(nx) = 1/2 cos(2nx) + 1/2 cos(0x) = 1/2 cos(2nx) + 1/2
\int_{x=0}{x=T} cos(mx) cos(nx) dx =
1/2 \int_{x=0}{x=T} cos(2nx) dx + 1/2 \int_{x=0}{x=T} dx = 1/4 sin(2nx) + x/2
x = T
1/4 sin(2nx) + x/2 = 0 + T/2
x = 0
1/4 sin(2nx) + x/2 = 0
\int_{x=0}{x=T} cos(mx)cos(nx) dx = T/2 - 0 = T/2, if m=n
Sin
sin(a) sin(b) = 1/2 (cos(a-b) - cos(a + b))
sin(mx) sin(nx) = 1/2 (cos((m-n) x) - cos((m+n)x)
if m != n
\int_{x=0}{x=T} sin(mx) sin(nx) dx = 1/2 \int_{x=0}{x=T} cos((m-n) x) dx - int_{x=0}{x=T} cos((m+n) x) dx
int_{x=0}{x=T} cos((m-n) x) dx = 1/(m-n) sin((m-n) x)
x = T
1/(m-n) sin((m-n) x) = 1/(m-n) * 0 = 0
x = 0
1/(m-n) sin((m-n) x) = 1/(m-n) * 0 = 0
int_{x=0}{x=T} cos((m-n) x) dx = 1/(m-n) sin((m-n) x) = 0 - 0 = 0
int_{x=0}{x=T} cos((m+n) x) dx = 1/(m+n) sin((m+n) x)
_||_
int_{x=0}{x=T} cos((m+n) x) dx = 0
int_{x=0}{x=T} sin(mx) sin(nx) dx = 0, m !=n
If m = n
sin(mx) sin(nx) = 1/2 (cos((m-n)x) - cos((m+n)x) = 1/2 (cos(0) - cos(2nx)) = 1/2 (1 - cos(2nx)) = 1/2 - 1/2 cos(2nx)
\int_{x = 0}{x = T} sin(mx) sin(nx) dx = int_{x = 0}{x = T} 1/2 - 1/2cos(2nx) dx = int_{x = 0}{x = T} 1/2 dx - 1/2 \int_{x=0}{x=T} cos(2nx) dx = 1/2 x - 1/4 sin(2nx)
x = T
1/2 x - 1/4 sin(2nx) = T/2 - 0 = T/2
x = 0
1/2 x - 1/4 sin(2nx) =0 - 0 = 0
int_{x = 0}{x = T} sin(mx) sin(nx) dx = T/2 -0 = T/2n
This is magnificent! I never believed that I would understand Fourier and Laplace transform ever! but Your videos are miraculous! You are amazing!!! :)
...... Allright I just stumbled on this channel looking up Fourier transforms. This video alone was worth the subscribe. Interested to explore more with your content.
This is incredibly helpful. Currently reviewing my understanding on Fourier Transforms and this really helps me visualize and intuitively understand it
Man, I spend years trying to understand Fourier transform. You help me a lot. Thanks
wow the intuitive explanation of the Laplace transform blew my mind!!
You can live the rest of your life in peace from here on, for you've done a service to mankind in making this video. I am happy to be a representative of the same species. Great work mate
i solved a lot of pole zero plots exercises but i never understood what they represent until i watched this vedio; zach star, you are a star.
literally the first video making it clearer to me
The best explanation I have ever seen of the math involved in transforming to the frequency domain.
This is some serious explanation. I wish all people would explain it like this. This needs to be preserved
I've watch some video about this topic - Fourier, but your video give me a clear understanding about Fourier. Thank you very much!
finally after years I understand what is Fourier transform and why we use it.
Dang, its not even been 4.5 minutes and I'm already connecting this to several things in my Vibration Control course. Nice stuff my good sir!
Guess who has a test in a few hours on Laplace transforms and systems of linear equations? *Me.*
Good Luck, is not that complicated!
Lol Nice 😂
may god help you :D this is insane.
@@numankaraaslan Thanks, it's not hard but there's a lot of things to remember
I just finished it. I wasn't as bad as I though it would be. I was studying like crazy 15 minutes before the test (I studied before going to the school but I still reviewed my notes one last time before going to the testing center).
I had to give a talk on Fourier Analysis but it was some time ago now. PCs had been invented only a couple of years earlier! Consequently, only the "important" people at work had a PC and I had to do my (static) graphs on a mainframe. I wish I could have made animated charts like yours. To try and make things simple I did this:
1. Used f instead of ω. The relationship between t and 1/f become more obvious and the inverse transform formula is almost the same as the forwards transform.
2. Concentrate on even functions only. That way you can ignore the sine part and the transform works both ways. That is while a boxcar function transforms to a sinc function the inverse is also true so you learn two transforms in one.
3. Show how stretching a time function compresses a frequency function and vice versa. Now you learn whole families of functions in one go.
4. We were interested in sampled functions so I showed how a set of sample pulses in one domain looked just the same in the other albeit with the t versus 1/f thing going on.
5. Then I added Convolution. Multiplication in one domain is equivalent to convolution in the other. Now you can use that to combine waveforms and spectra in extra ways and see the answer without any more maths.
At that point you can now imagine basic transforms, transforms of sampled functions and even "windowed" functions or ( same thing really) transforms of finite length signals.
Anyway, great presentation - keep up the good work!
You earned a subscriber. The only math video that has been on the same level as 3B1B. I love it.
I studied Fourier Transform 11 years ago and I just understand what it means by watching this video. Brilliant
and do you know why there are sinus and cosines ? it's because f(x) is wrapped around different circles (2D objects) that's why f(x) is multiplied by cos and sine or e to the power of a complex imaginary number which represents a circle. each circle has a different sampling frequency (wt).
@09:45 I literally shouted genius. Really awesome and intuitive explaination Zach 👌
This is a great little introduction to the subject!
this is real gold. thank you creators
Understood more in the first 1:48 than in a whole semester class on Fourier transforms...
Please continue this series
I was confused initially as an electrical engineer what the current had to do with anything in the Time/ Frequency domain, till I remembered why in electrical engineering we use "j" not "i", as the Square root of -1, to avoid this exact ambiguity!
Amazing. Thank you for sharing your intuition. I rarely feel compelled to leave a comment on a vid, but I'm just so thankful
This is the best explanation i have ever seen. Thanks a lot!
How a spectrum analyzer works... Wonderful! Thank you!
Best Fourier video I've watched.
Awesome brother 🙏🙌🙌
Bro. I love you. Genuinely, I love you. This video is a life saver. Best introductory video to Fourier and Laplace transforms out there.
just what I needed. I just came across fourier in my quest for self studying physics. Thanks man
This just cleared the confusion I've had for 4 years
Excellent explanation! Finally someone tried to explain beyond just calculating the Fourier transform!
thanks! the visuals are the best I've seen so far towards helping me understand these transforms!
Really all you need to understand the Fourier is the concept of complicated (not complex!) frequencies consisting of added sinusoids. The integral in the expanded transform through Euler's identity should be seen as a summation of these simple frequencies: you are adding all of them from negative infinity to infinity and checking whether any are present in the original function (multiplication).
It's a remarkably easy to grasp concept once it clicks.
Extremely good explanation
I've been bugging my professors to explain something like to me for about a year now. Thanks boss
Now I see the light ... after decades of darkness. I always think that Mathematics is intuitive why can't we learn that in College ! yes I know , it's very hard to explain without brilliant animations like this which was never available. I also suspect that the Profs, don't understand it that simply.
This is great. Please continue to make these videos. They are helping a lot of people!
3:18 - sine is an ODD function (not even). Odd integrals symmetric about y axis are zero, not even ones. Integrals of even functions can be doubled and have one boundary replaced with zero. ;)
I noticed that as well. Good catch
By the continuation of his sentence he is referring to the function being transformed (the rectangular pulse here) as even - for every even function, the sin component of the Fourier transform is zero.
@@kovatembel thanks. Otherwise it was confusing to me.
If only our professors could do the same approach instead of jumping into solving, everything makes sense if you know the concept.
This was the best video on fourier transform for me. Explained a complicated idea in such a simple and a intuitive way. I would have loved if it would have explained more on the meaning of the phase in fourier transform as well.
Hi from the Turkey.
Thank you for this concise and intense video , for visual understanding of complex integral equations.
I took nearly 13 pages of handwritten notes , just about this video.
It contains the fundamental understanding ability for the control systems and signals-systems course.
If you are outsider of the topic and you don't understand quite well , I suggest to watch it 3-4 times with extra sources. ( what I were did )
Waiting new videos about the engineering mathematics , and various engineering applications.
The Fourier Transform 'transforms' (converts) amplitude - time representation to amplitude - frequency representation. The Transform of ideal white light will yield roughly 7 frequencies and their corresponding amplitudes. Similarly the FT of an ideal alternating current or voltage of say 50Hz freq. and 220 voltage produces a graph of 220V at 50Hz and nothing at other frequencies because the input signal is 'monochromatic'. The Transform also explains, at least in part, how we recognise familiar voices on land line phone almost automatically (the mind 'draws' freq. spectrum of the speech).
The video is excellent by the by.
I must thank you this is by far the best visual explanation I've ever seen !
3:21 For f(t) = sin at, the graph on the right is not always equal to zero. When w = a, the area of the right graph gives +inf.
Now that is a blow up !!!
Great effort! Thanks a lot
Beautiful visualization!
Btw the point in the Fourier analysis where the magnitude was zero and there were several points that spiked to infinite magnitude, we call these Dirac delta functions and they are important in signal and modulation systems for determining the band. The poles and zeroes in Laplace Analysis are important in Root Locus analysis and Nyquist Plot analysis, as well as bode plot (phase and magnitude graphs) but bode plot is basically a Laplace representation in the Fourier domain because s, the complex number has a real value of 0.
This Explanation is a silver plated
I love this video. Now i can follow the poorly explained functions that we use to describe functions in uni
Wow i have seen an other hidden explanation of fourier series. Thank u so much.
Very nice presentation! The animation from 5:20 to 5:30 in particular was worth a thousand words to me.
I can’t thank you enough for this
Great video! Just a minor correction at 3:22 -- you say that an "even function is symmetric about the y-axis, hence the negative and positive areas will cancel". However, sine is an odd function that is symmetric about the origin.
EDIT: I read the other comments which also pointed it out; I see what you mean now!
Zach, thanks for the video. I haven't found anyone else explain it the way you do!
Aaaha ! Thus we have a spectrum analyser ! Thanks a lot for this video ! I am so glad I tripped on this !