A Musician's Intuition on the Fourier Transform (feat. the Inner Product) | 3b1b SoME1 

CORRECTIONS / CLARIFICATIONS: Here I describe "the inner product" as an operation between two functions. This is kind of an abuse of nomenclature. The term "inner product" should actually be used to describe any operation that takes in two objects and returns a scalar quantity and obeys a set of properties (linearity, conjugate symmetry, positive definiteness); for example, the inner product on vectors in R^n is the familiar dot product from linear algebra. Indeed, you can view a real-valued function as a vector with an uncountably infinite number of entries! You can see lots of cool relationships between linear algebra and fourier analysis like this (most notably, the sense in which the FT is a change of basis, where the basis is a set of complex exponentials), but that should be its own video. Just keep in mind that the operation here is more accurately described as "the inner product defined over real-valued functions" than just "the inner product." (thanks to Kevin Lu on the SoME1 discord for pointing this out!) Also I refer to the low A on a guitar interchangeably as "A1" and "A2." This is because I keep forgetting the fact that guitars are tuned one octave lower than written, despite having played guitar for, like, my whole life. The actual name of the note is A2 (but I was right about it being 110Hz, so the arguments still stand).

I've been trying to spread the gospel, but the sum of angles formula is in my opinion one of the WORST ways to prove the Harmonic Addition Theorem! I've come up with what I think is a very elegant way. Consider the vectors and . Then, one one hand their dot product is a*sin theta + b*cos theta, but on the other hand, it is the product of their magnitudes times the cosine of the angle of between them, which is sqrt(a^2+b^2)cos(theta-arctan(a/b)) = sqrt(a^2+b^2)sin(theta+arctan(b/a)), which is what we were looking for!

This is amazing, one of my favorite submissions thus far. it really resonated with me :D

7:00 - 7:27 the most universal saying for computer scientists in their math classes “you can see it just by eyeballing”

This is such a fantastic video! You are wonderfully lucid, and a very talented expositor. I thought all your visuals were well motivated, and you constantly grounded the abstract mathematics with the physical context of our relevant problem.

Exactly the way i would have explained it, both to musicians AND electrical engineers.

Absolutely brilliant video. Found this from the 3B1B summer exposition playlist and it was super illuminating, even with the handwaviness of the math at parts. :) As a scientist and (fairly decent) amateur pianist with an undergrad degree in math and hours of obsessive reading about stuff like this, I've never seen the Fourier transform explained or "derived" like this. It's amazing relearning things from a new basis, and I hope you continue making videos after defending.

very cool video! as someone who recently tried (and failed miserably) to learn about the fourier transform from my grandfather's old textbook, (I got lost pretty quickly once it started dealing with complex numbers), I love that this makes things easy to understand without any complex numbers, and the complex numbers are just letting you describe the output in the complex plane as a single number instead of as two separate values, and making it a bit easier to integrate (because e^x is the nicest possible thing to do calculus with).

this was really cool! we've never seen the fourier transform explained this way but it makes a lot of sense!!! also the text onscreen at 16:35 absolutely warmed our heart thank you

Excellent explanation. I've been interested in math since only recently but have had a life long passion for synthesizers. Never thought i would get into math in my 40s. This gives students of sound synthesis a good introduction the FT and related concepts. please make more videos like this. 😁

Well i wish teachers explained it like that in electrical engineering classes. It brings up a problem to be solved and derives the solution rather than starting with hard to understand and long maths eqations and then showing their use

That's my new favorite video on the fourier transform !

Simply THE best!

Thank you very much for procrastinating for us, it did it for me, you‘re the best.

12:56 OMG I cannot believe i guessed it perfectly 😱😱😱 just a half second before you disclosed it, that clicked in my mind 🤩🤩🤩

This is the best Fourier transform explanation video I've ever seen. A fantastic video and definitely my absolute favourite about all the SoME1 submissions I've seen (and I've seen quite a lot of them) Thank you so much for making it

Sweet, sweet fruit picked ripe from the tree of procrastination :) Thank you so much for sharing it! Hope your dissertation is going (went?) well. This explanation traces through my exact sequence of questions about how FT works. Like, at one point I was thinking “Yeah, but how the f does all this work when the two frequencies are equal but the signals are out of phase?” and bam, you went right into that on the very next beat.

When you introduced the harmonic addition (not that I'd seen it before), I knew that this was where the complex numbers where hiding. 😃. Very neat. And I do appreciate that you recurred to Fourier himself, explaining that he developed the method without complex numbers originally. I hope to see more content from you, once you have finished your dissertation.

As an electronics R&D engineer, programmer, and musician, this is the best explanation of the Fourier Transform I've ever seen.

I take a virtual bow for the ability to explain and the effort that went into this video.

I really enjoyed this more than I would have thought. To me, FT's were just some tool that I "learned" in uni to switch from time domain to frequency domain. But from the moment you said you were going to explain it with trig and the inner product, my eyes were glued to the screen. I wish I had taken the time earlier to really see these "tools" from different perspectives as it were, it's super entertaining. Thank you!

This is so delightful, I wish you will film more in the future!!

Thank you for this explanation. Especially getting past the Harmonic addition theorem which it is now clear was the point when my own naive adventures came crashing to a frustrating wall.

Thank you so much for this video! I work with other transforms such as PODs, and we were taught something similar for the Fourier transform, but your video really helped me put all the pieces together again in my head.

One of the best Fourier transform tutorial in the world !!

Awesome wideo, i have ever thought how this could work.

insanely helpful!! Thank you!

Amazing video thanks!

man this is just so cool

amazing video, thanks a lot!

I enjoyed following your explanation. The visuals on the outcome of the correlation integrals were honestly worth the effort. Seeing the maths is one thing , but understanding the meaning behind the math is the real challange. Thanks for sharing this insight. I will recommend whomever is interested in fourier transforms your video. Keep on the great work.

Excellent video!!

Wonderful video! I love that you took a complicated real-world example, and used it to work the abstract math to a useful conclusion. I look forward to more from you, once you’ve earned your doctorate.

Really interesting - I've never heard the inner product described as measuring the correlation between two functions. I must remember this for this future - thank you!

Thanks a lot for this perspective!

Just one thing... if the frequency is unrelated, even by just a little bit, it will give an inner product of zero. So, how do you _find_ the frequencies in your trials? You might try 400Hz, 401Hz, 402Hz, etc. but the real signal is 401.53987647... so you missed it. I expected a chapter where you go over this. My thought (I don't know if this is right): in a real-world application you are not summing two abstract functions but a finite amount of time sampling. If that window is smaller than the beat frequency, they are not running for long enough to cancel out, and will show some correlation. The alignment will drift a bit over the interval being integrated over, but will not drift enough to reverse the sum. Of course, that means that noise and other nearby frequencies don't cancel completely either.

I immediately thought of 3B1B's Abstract Vector Spaces video when you brought up the inner product. I can definitely see how this can extend beyond just waves to arbitrary vector spaces. In a sense, sin(theta) and cos(theta) are acting like basis vectors and taking the inner product with them is acting like a vector projection of the waveform onto those bases. It's just that in this case, sin(theta) represents the basis i and cos(theta) represents the basis i. e^itheta is just a more compact form of writing that particular pair of bases.

Yessss! As a programming competition enthusiast, this is how I think of Fourier transforms: It's really all about inner products! The intuitive reason that the convolution formula works is that there is a bunch of implicit inner product fuckery going on, nothing to do with complex numbers and trigonometry

Thank you I used Ernie Ball Super Slinky number 9s for my string theories. 👍

Thank you!

great job! the right dose of rigour and intuition, experimentation and demonstration :-)

thank you!

Cool explanation about phase randomness. Lots of sources skip that part

So far my top 1 SoME1 video :D

Side note on biophysics and musical ear, and their relationship with this math explanation: Mathematically: The Fourier transform definition includes an integral over time from -inf to +inf. This implies that one would need to listen two notes forever in order to be sure they don't eventually differ (7:47 "...eventually they are going to get out of phase with one another...") Biophysical approach: The inner ear has tiny filaments with different lengths and they vibrate by sympathy if the sound contains a wave with the same length as theirs. If the frequency is not exactly the same they will start vibrating but will eventually "get out of phase" and stop vibrating, and then again in phase and again start vibrating. This is why if two strings in the guitar are slightly out of tune with each other and you play both with the same note, you hear a pulsating tone.

Nice :) It's an alternative approach than bitwise multiplying the signal with a citation (with a sine wave bitwise multiplied by a normal curve). It's asking "how equal" rather than "how much" the two are. I came up with a third way as well, based on how the ear does this.

I just want a real time continuous wavelet transform 😭

So good

4:50 Strictly speaking, it's the covariance, not the correlation. But correlation is just covariance after normalization, so yeah, it's basically the correlation.

Shame, I wanted to do this video but couldn’t figure out how to use manim soon enough for the competition lol- good job tho

Love it

this video slaps

🙏 sir, Please make a video on How Fourier Transform is the special case of Laplace Transform?

There's a video from a very famous educational youtuber (at uni level) in the spanish speaking community that presents the Fourier Transform in this way (with the inner product). Though I can understand why you would not have noticed since it's in another language xd

nice bruh

Holy shit this is fire

to fix the phase problem can't we shift the guitar sample by π/4 or π/2 radians and check for the presence of the sine wave

Love your nails lmao. The video was also good.

Yes, I heard about the fourier transform through music, but theres not much content that really applies it to sound

Very noice

what does they do in the doctorate of music technology? I too love both math and music equally and have some basic dsp knowledge

Just play a synthesizer and shift between waves to go through this.

Dude damn you literally musically bottom-upped the DFT

The video was well done, but I wonder if all that effort to circumvent complex exponentials is really simplifying things and helping the understanding. The Fourier Transform (FT) is actually the best way to learn about correlation and phasors as test functions to represent idealistic oscillations. The correlation (cross-correlation) not only works for sine waves, but actually any two signals and gives you the best fitting in terms of linear least squares regression. It's important to understand, that the FT expects the signal to be periodic - otherwise you will find frequencies needed to represent the irregularities and transitions at the edges (probably including a DC term). Sine and cosine are merely projections of the complex exponential function and not the other way around. You also would need a complex signal to represent all possible signals, where amplitudes, phases and frequency might change over time. A real signal is just a special case, where positive and negative frequencies are symmetric and phase is often disregarded anyways as it's pretty ambiguous in most cases. For some types of analysis you can make use of the phase component though, like with the "analytic signal" or "reassignment method".

23:52 Thats a lot of noise 🙃

Wait

Fruityloops4life.

I love your nails!

Do folk actually believe this compilation of misunderstandings?

Great video!