Finally some dark theme animations that don't destroy your eyes at 3 AM

As an electrical engineering student who wants to understand these concepts, this video is gold!

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!

That's one of the best explanations of Fourier Transform I have ever seen !!!

As Leonard Euler, I confess that it’s too damn complicated.

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!

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!

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.

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 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!

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.

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.

3Major1Prep?

One day i'll understand this....

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

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.

This is incredibly helpful. Currently reviewing my understanding on Fourier Transforms and this really helps me visualize and intuitively understand it

14:01 can we just take a minute to appreciate how intuitively powerful this animation is?

thanks! the visuals are the best I've seen so far towards helping me understand these transforms!

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!

This is great. Please continue to make these videos. They are helping a lot of people!

Finally a great and simple explanation of Fourier Transform after days of searching and jumping from video to another. Thanks man, much appreciated.

I think this is the best explanation of Fourier transform so far. Thank you!

I must thank you this is by far the best visual explanation I've ever seen !

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.

I've watch some video about this topic - Fourier, but your video give me a clear understanding about Fourier. Thank you very much!

This was a great explanation of these transformations I encountered with after close to 35 years of dealing with them!

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.

Honestly, this is the video I always needed. Like for real, I thought I'd never understand it although 3 different professor tried to explain. It makes so much more sense to explain it with sin and cos than with an e-function. Thank you so much!

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!!

Zach, thanks for the video. I haven't found anyone else explain it the way you do!

Thank you! I've been studying this topic all day without much success and I think it finally clicked in my brain!

Bro. I love you. Genuinely, I love you. This video is a life saver. Best introductory video to Fourier and Laplace transforms out there.

This is magnificent! I never believed that I would understand Fourier and Laplace transform ever! but Your videos are miraculous! You are amazing!!! :)

This is some serious explanation. I wish all people would explain it like this. This needs to be preserved

Amazing. Thank you for sharing your intuition. I rarely feel compelled to leave a comment on a vid, but I'm just so thankful

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!

Fall in love with your explanation. It is excellent content with graphics that are very to understand.

Excellent explanation! Finally someone tried to explain beyond just calculating the Fourier transform!

Thank you so much sir. I had really expected you to make a video on this.

Man, I spend years trying to understand Fourier transform. You help me a lot. Thanks

This was really a brilliant, little lecture!! Thank you very much, Zach!!

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.

MajorPrep Converges to 3blue1brow

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.

The best explanation I have ever seen of the math involved in transforming to the frequency domain.

I've been enjoying your videos, and especially this one! As someone who is very interested in Fourier analysis, I have to thank you for giving me better intuition of what is actually going on :)

Hi Im self-quarantined @home. KZfaq is my school now. Thanks for the great lecture!

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.

Now that is a blow up !!! Great effort! Thanks a lot

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.

Best explanation i've ever gotten, thank you!

This is the best explanation i have ever seen. Thanks a lot!

He attac He protec But most importantly it doesn't want to go in my Head

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!

wow the intuitive explanation of the Laplace transform blew my mind!!

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

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.

You earned a subscriber. The only math video that has been on the same level as 3B1B. I love it.

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.

This is THE definitive and the most comphrehensive video ever made on laplace transform. To anyone reading my comment, I would like to say that this is the ONLY video you'll ever need to understand the intuition behind this ingenious mathematical tool.

Excellent presentation. Very very useful and simplified explanation. Thank you 👍

This is a fantastic video! Really great way to learn visually.

This was amzingly brilliant. Thank you so so so much.

Very good video! Great use of illustrations!

...... 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.

Very nice presentation! The animation from 5:20 to 5:30 in particular was worth a thousand words to me.

just what I needed. I just came across fourier in my quest for self studying physics. Thanks man

This is a great little introduction to the subject!

This was so beautiful. I love you 💕

Thanks for this nice approach to the topic!!!!

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.

My God. 3 university courses compressed in 15 minutes and still making it more understandable. My God. Amazing video man, it brought my hands to my head when you arrived to the zeroes and poles

this is real gold. thank you creators

AWESOME!Just what I needed.

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!

thank you so much for this explanation! really helped with the intuition behind all this

I kind of wish you had mentioned the "process" starting at 9:27 as "correlation" (sum of products in discrete terms), which would have been the perfect little bow wrapped around this great amazing gift you're giving to us!

Very good explanation. Really appreciate the material.

A spectrum analyzer can be thought of as a Fourier Transform machine. Oscilloscopes with Fast Fourier Transform (FFT) capabilities can do so as well.

Best explanation I've come across. Thanks a lot for the help.

Great video, makes much more sense than what I've been taught. If you ever have the time, I'd be greatly appreciative of something on impulse and the dirac functions.

Thanks for making simple to understand ❤️👍 giving us a clear explanation of this ...

Pure magic. Brilliant animations 👏🏻👏🏻

Fantastic explanation - I really struggled learning the Fourier transform / series during school.

this has helped me so much! THank you!!!

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

Fantastic just working on this subject in my research

OMG just in time !!! I love it!!!

So clear! Thanks!

You really helped me with this man, thank you. Something clicked while watching the pi hit infinity... :)

Love You're work man So helpful as a student 😇

Thank you for this AWESOME video 😊❤️

literally the first video making it clearer to me

Beautiful visualization!

I am grateful to you, this is really interesting!

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.

@09:45 I literally shouted genius. Really awesome and intuitive explaination Zach 👌

simply brilliant! thanx, a ton it was really helpful.