The 200-year-old mathematics behind half the internet

[@ImprobableMatter]

I’ve reopened the channel’s Discord server here: discord.gg/cTjcKqPCEk I’ve got time this month to do a quick livestream; let me know if there’s something you would like me to discuss.

[@rexmann1984]

This is the long way of saying vinyl is superior . 😜 Excellent presentation, I learned a lot.

[@4thjulybd809]

How is it you have 34.060 likes in this video and the new one? Is this some youtube bs or something?

[@ducksies]

Fourier Transforms are extremely powerful. They singlehandedly transformed my GPA from a 4 to a 3.5

[@tmd4951]

😂😂

[@jumbledfox2098]

hahahahaha

[@hcn6708]

How the hell

[@steevf]

I can relate. :)

[@tanakablack5271]

😂😂😂😂😂😂 the funniest comment i read this week...

[@Rosa-lv8yw]

This video was absolutely incredible and informative. Even during the bits I already knew; especially the maths at the beginning and the fourier transform bit, I was still hooked. You've made a video that is educational and accessible to people of most skill levels, and you stuck on topic without any silly distractions. I love this.

[@artysanmobile]

Spectral editing has utterly transformed live music production for me. In live situations, bad stuff happens, thank you, audience. Being able to decompose the sound into crowd and musicians has saved many a live mix.

[@Emu0181]

Thanks for explaining this. I had heard of Fourier transforms with regards to compression, but never heard it explained as to what it actually did. Seems to be a running theme in your videos, explaining the middle steps in a process that most explanations gloss over. I really appreciate that, thanks again.

[@user-sb3wh3dd4v]

FABULOUS presentation ! I have been studying audio engineering since 1978, at the cusp of the digital revolution. This is the most concise and cogent treatise on this subject I have ever seen. BRAVO!

[@ConradPino]

I see a red door and I want it painted black.

[@middleclassthrash]

They should hire this guy to instantly improve the world's education system. All the context and visuals he provides. Amazing work.

[@m.streicher8286]

I did poorly in high school math, the first 90 seconds of this video would've been huge for me..

[@redlinerer]

It's almost like the school's don't teach you to learn, but teach you to obey instead...

[@PunzL]

@@redlinerer somebody forgot to wear their tinfoil hat

[@LetMeSoloYKS]

​@@PunzLHere's your favorite boots

[@byronwilliams7977]

This isn't high school math at all. Did Applied Mathematics at an R1 institute in the US. Don't feel down.

[@morgengabe1]

​​@@PunzLe's not wrong. If grammar is innate (the linguists who say it's not usually don't understand recursion/nesting), then while syntax/language is learned, teach is just sharing research strategies by telling others the orders you like to use letters/words in. If all you do is memorize other people's strategies, it's basically brainwashing/plagiarism. I doubt this is done for reasons other than laziness/hegemonic-capitalism. We just need a better education system. A kid who struggles with maths simply doesn't know how to form an agreement with whatever near ancient mathematician whose work they lean by using the semantics the teacher suggests. Assuming that "the old way is the best/only way" is precisely why it took so long to get over Euclid's 5th postulate.

[@Zibonnn]

I must say, this video deserves more recognition than it currently has. I thoroughly enjoyed watching it and gaining a clear understanding of how things work. If my teacher had been as skilled as you and if I had been exposed to such visual-aided education earlier, I might have pursued a career in STEM.

[@slightlytwistedagain]

As someone who has written music and played with synths in digital audio workstations, this is an excellent video showing the science behind the magic. You didn't outright say it, but you pretty much described interpolation and do a much better job explaining the magic sauce (getting the computer to calculate with frequency and amplitude) than any audio site or wikipedia I've read.

[@RyanMcGuinness]

This is the clearest explanation of Fourier transforms I've ever seen. I've done 11 years of high school and college mathematics and it wasn't until this video that it just clicked!

[@t-housetv7580]

This video is very well done. Full of information, compact, and straight to the point. Subscribed.

[@phillipneal8194]

Excellent ! Thank you. Loved the opening riff from "Paint it Black"

[@Rhino_Aus]

A perfectly simplified explanation of the Fourier transform and its applications!

[@MrRyyi]

As a beginner music producer, I am so happy to have viewed this video. I did not expect it from the title and thumbnail, so it was like finding gold unexpectedly. Thank you for this hugely informative video that taught me several things I hadn't thought of before. Have a good day :)

[@lelsewherelelsewhere9435]

Very well explained! Direct, illustrated well, great examples, fast enough but not too fast, builds on itself, great video indeed! Going to try to teach my family with this!

[@louco2]

Mind Blowing, learned so much! Thank you so much for making these videos. This is one of the best channels on KZbin!

Very, Very Nice video ! I have my firsts contacts with all this stuff maths in near time. And your video helps me a lot. Join maths with compressing algorithms was very mind clear to explain how this "gear" behind modern computers works. Congratulations and thank you very much !

[@OnionKnight541]

i saw another video talking about machine learning and Fourier Transforms and i've become in love with this information. that machines take a bunch of data, look at it like a wave function. amazing.

[@andreaskouv522]

I wish this video be released few years before when I was in university. I was so confused and helpless. Thank you even now 🙏

[@joeprince7509]

Fantastic video and visuals, thank you!

[@sigmundwong2489]

Wonderful video. I thought it would also be worth mentioning that in the case of images, unlike music, there is an additional technique that is absolutely crucial: chrominance subsampling. It turns out that the human eye has much more resolution in brightness than in color. While RGB consists of three orthogonal color components, it turns out we can store any image as a black and white "brightness only" component, together with two color components. By storing the color components at lower resolution, we can save huge amounts of data without losing too much noticeable quality: a loss of resolution in any of the R,G, or B components would be much more noticeable to us, as changing any of R,G, or B would affect the brightness. Historically, this was also advantageous as it allowed color information to be simply "tacked on" to existing black and white TV signals, making color broadcasts backwards-compatible with black-and-white TV sets.

[@ImprobableMatter]

Yes, you're totally right.

[@saganandroid4175]

Funny that NTSC and JPEG have that much in common @sidmundwong2489 but I found this line confusing: "a loss of resolution in any of the R,G, or B components would be much more noticeable to us, as changing any of R,G, or B would affect the brightness."

[@n1matsu]

I've seen a great amount of videos about the matter, but this one was my personal best. Thank you very much!

[@honkhonk8009]

I love your channel lol. Its everything I want from a documentary series

[@amazing7633]

This was my high-school science fair project, but I didn't have a computer or the Cooley-Tukey FFT. They rediscovered it about 1965! Anyway, I decomposed sounds into sets of sines by using pins pushed onto graph paper through a photograph of an oscilloscope trace. Calculated the FT by hand, using an adding machine and a slide rule. The original work on the FFT was done in 1805 by Carl Friedrich Gauss.

[@ImprobableMatter]

Gauss was an amazing mathematician.

[@dopplereffect3077]

Thanks a lot! Very simple and clear. For me video came in as summary of each of what I knew already. Staying tuned!

[@oopsiesh]

Your videos are a work of art.

[@FavoritoHJS]

OBJECTION! Most of the file formats you said use Fourier Transforms actually use Discrete Cosine Transforms! I _think_ the reason for this is that a FT also gives phase information, while a DCT doesn't so storage is easier. EDIT: you mentioned this at the very end... :facepalm:

[@ImprobableMatter]

Yes, sorry for not going into more detail, but I figured it would be too confusing if you were seeing this material for the first time. If you are more experienced, I figured you would be able to extend the FT formalism as required, whether to the DCT, a complex FT of a real valued function (i.e. I used the rfft function to make the spectrograms you saw) and so on.

[@skilz8098]

Yeah but in truth the sine and cosine for all tense and purposes are basically the same exact except that they are physically a 90 degree or PI/2 radian horizontal translation of each other. They both have the same shape or composition as each other, and they both have the same range and domain. The only difference between them are the initial values from the I/O when starting at 0. In other words, sin(0) = 0 where cos(0) = 1 and sin(90) = 1 where cos (90) = 0 and they are equal at 45 degrees or PI/4 radians. They are mutually the same function but are completely orthogonal or perpendicular to each other. So if one is using discrete sine or discrete cosine is more of a matter of convention as to who's right and who's wrong. Either procedure will suffice to provide good enough results, what's more important is that when you choose a specific convention or format to work with is to stick width it: don't change in midstream! You can use either sine or cosine as it wouldn't matter which. Why? Go back to the Trigonometric Pythagorean Identities: sin^2 + cos^2 = 1. Also you can use the substitution identity as well sin(t)/cos(t) = tan(t). Now as for the the file formats in what they used: yes you're not wrong in that because those are the conventions that were used. I suspect that they prefered to use the cosine over the sine for a specific reason. There are a couple of properties that the the cosine function possess over the sine function are the following: Within a given linear equation in slope-intercept form y = mx+b the slope of the line m being defined as rise/run which can be found by the following formula of any two given points on that line is by m = (y2-y1)/(x2-x1) or dy/dx which is simply the ratio of the rate of change in height over the rate of change in width or depth. In contrast to the slope m in any linear equation is a direct relationship of this line's slope with the trigonometric functions. Within the context of the dy/dx and using the circle definitions of the trigonometric functions, dy = sin(t) and dx = cos(t). So in truth the slope of the line m is tan(t) where t, theta is the angle between the given line of y=mx+b and the +x-axis. So with this understanding the first property that the cosine function possesses over the sine function is that the cosine function is a change in x where the sine function is the change in y. The second property and probably the more prominent property of why the cosine function is chosen over the sine function is the direct relationship that the cosine function has with the dot product of two vectors. Yes, you could do a dot product with the sine function but it's much more complex to represent and when it comes to computer algorithms and computations we are always looking to provide the simplest terms with the least amount of calculations, we are always looking to optimize things. So when it comes to the cost of computations just by the nature of the two functions and how they relate to each other; by default the cosine is computationally cheaper when working with the algebra of vectors and matrices due to its direct relationship of the dot product. This wasn't to take away anything you said, just more of an observation and food for thought!

[@jb7650]

​@@skilz8098this is what being in an oral exam feels like. You hear the words sine and cosine and just puke everything you know about it on the table

[@skilz8098]

@@jb7650 Well there is a bit more to it than that too. If you look at the composition of the unit circle within the Complex Plane with a given coordinate (cos(t), sin(t))... The cosine function yields a Real Number where the sine function yields A Complex or Imaginary number. This could also be another reason they chose to use the cosine over the sine function as their convention. Just another observation.

[@ianthehunter3532]

Visualizing images with sine waves look cool.

[@wabumbo]

The thumbnail to this video is the most visually understandable graphic of Fourier transform I've ever seen so thank you

[@Stigmata195]

Thank you for this great video ! I love how you explain and visualize everything so clearly ! It shows you understand what you're talking about very well.

[@jensphiliphohmann1876]

The title image is a perfect way to depict what FOURIER transform is all about.

[@aradmnk1269]

Very high quality video for this amount of subs, very informative and easy to understand. Great job dude

[@KevinBalch-dt8ot]

I used FFT to process neutron detector signals to look at the vibration of components inside a nuclear reactor. You could see fuel assemblies moving slightly under flow induced vibration, the effects of the reactor coolant pump blades and many other interesting effects.

[@WungoBungo]

WOW! That is so fascinating. I never would have thought of using FFT to decompose the vibrations of a mechanical system. I mean it sounds simple when described that way. Are there any resources I could read up on these sorts of techniques? Thanks for sharing!

[@KevinBalch-dt8ot]

@@WungoBungo - I used the technical manual for the system to understand it and interpret the data which was recorded on strip chart recorders (this was the early 1980s). The system was used when the plant was commissioned and used to verify design information and then pretty much forgotten other than taking weekly data samples. We also used accelerometers mounted on the reactor and other structures that could be translated into sound and this system had indicated there was a loose part somewhere. When the reactor was shutdown, we found the loose part. It was a pin that held a cylindrical thermal shield in place. I was asked late on Friday afternoon to see if I could use the FFT system to deduce the chronology of what happened. I spent the weekend reading the tech manual and then spent a week recovering the data from the archives. You could definitely see discrete changes in the vibration spectra in the data as the shield failed. Also, the non-symmetry of the peaks was also obvious. I wrote up what I thought happened. The company also contracted with outside experts who concluded the same things I did - but they got a HELL of a lot more money! Anyways, the shield was removed (it was put in as a precaution and was not deemed necessary). I don’t know your level of Mathematical background, but as a start I would look for youtube videos such as this one. Another channel Two Black and One Brown also has excellent videos with graphics that touch on mathematical theory and at least one discusses FFT. From there, a textbook on vibration analysis or vibration monitoring would be useful. There’s probably a textbook just on FFTs as well - check Amazon. See if any are published by Dover publishing as they have numerous books on math, science and engineering that are very inexpensive. Many are classics but were initially published a few decades ago. Good luck!

[@Belgarathe]

You did excellent describing how analog signals(real world) is stored digitally (virtual /computer). I wish I saw your video when I was in my master program that had course in signal processing

[@usvalve]

Brilliant! I'm going to watch this again.

[@skauffman74]

What a beautiful lecture. Thank you vey much.

[@ARandomTroll]

I love how you explain very advanced subjects in a way that is understandable to a layman without losing sight of the technical complexities that matter. Also, audio spectrum apps are really useful. want to know how fast a fan is spinning? frequency/ number of blades. What's that weird noise keeping me awake at night? 50Hz... probably something electrical. etc.

[@widnyj5561]

Pretty neat explanation overall, nicely laid background

[@unknown-ql1fk]

I have know about this transform for years but NEVER had it explaned to me well, i jusr accepted that it was real and worked...BUT NOW, 15yrs on from my schooling you did it, and for free. THANK YOU, THANK,YOU, THANK YOU

[@micosstar]

nice video as someone who got your video from youtube recommend! (and thus I subbed!)

[@skilz8098]

Although being familiar with FFTs and how they relate to the transformations of wave functions and how they correlate with audio and graphics, your video does an excellent job at explaining or describing how they are used in the simplest of terms. Well done Sir! Great video and very informative as well as right to the point.

[@panjak323]

I think this is the most down to the Earth Fourier transform video on the KZbin.

[@MattNolanCustom]

Nice clear explanation of functions and the Fourier Transform however, I fear you may have over-simplified mp3 encoding. While it is true that jpegs store image data as DCT coefficients, I don't think mp3s store audio data as FFT coefficients. Unless I am grossly mistaken, the audio is divided into sub-bands and each sub-band is independently sampled and then FFTd (for analysis). The quantisation bit-depth of each sub-band (in the time domain) is then decided on a "least you can get away with" basis determined by whether the resultant quantisation noise is sufficiently well masked by loud signals in adjacent sub-bands. Some algorithms also employ temporal masking, IIRC (i.e. you can't hear small details immediately after a loud in-band signal). Both spectral and temporal masking effects (at least forward temporal masking) are limitations of our bio-mechanical "Fourier" transform in our cochlea. This is what mp3 takes advantage of. Some sub-bands can get quantised away to nothing, but this is not the same as FFTing the whole thing and not recording the near-zero coefficients. A future video could address how voices are encoded and sent over modern phone networks. That's really freaky and interesting.

[@WungoBungo]

Wow that bit about the cochlear limitation being leveraged by MP3s is fascinating. Technology Connections has a great video that goes over that but it’s like an hour long and you nailed in a paragraph! Thanks for sharing!

[@WungoBungo]

also if you have any resources/papers you could link about the voice encoding for phone calls you mentioned that would be hugely appreciated! If you say it’s freaky then I’ve gotta know what it is!

[@MattNolanCustom]

@@WungoBungo start here - en.wikipedia.org/wiki/Linear_predictive_coding

[@inDefEE]

I went to school for Electrical Engineering and remember very specifically learning Fourier transforms and having absolutely no idea what the point of them was. Wish I had this video 20 years ago!

[@jimbovitikan4848]

bro your videos have been incredible from like the get go tf

[@nujabraska]

Very well made video and good explanation of topics that I didn’t fully understand.

[@dentonzh]

I've watched a lot of videos try to explain FFT, but none are so thorough yet still accessible who hasn't studied this. I think I'm going to make a Quizbi on this!

[@kuba2ve]

Thank you!!! Finally I understood what´s behind all these mysterious algorithms.

[@Samgurney88]

Your exposition of the prerequisites was admirably clear and concise. I sometimes wonder if trig functions are introduced in an unhelpful way. I primarily think of sin and cos as scaling coefficients of the hypotenuse which give its y and x "projections", respectively; in fact, it is perfectly fine just to define them in this way with h•sinθ = y, and h•cosθ = x. Of course, this is equivalent to thinking of them as ratios - but later on it turns out to be much more natural to have the ‘hypotenuse scaling coefficients’ view of them ingrained. But perhaps this would be too confusing?

[@NomenNescio99]

When I saw the thumbnail I thought you were going to talk about Wavelength-division multiplexing in fibercom. Also, I just did a sudo apt install audacity, just to play with the spectogram in audacity.

[@ImprobableMatter]

I don't know if you're referencing the fact that I also use Audacity on Linux, but yes.

[@ЖанибекБекхожин]

Amazing, as always! One thing that is not obvious is that FT is something common so it gets applied all the time. There are fast algorithms to perform it and it does good job clustering information. But it is probably not the most optimal thing for images at least. Maybe there are better functions that capture the image patterns, requiring less information to be stored for the same quality (I imagine maybe a bunch of linear/quadratic interpolations between different regions. If you have big enough region that you can describe as a plane of varying intensity then it should save a bunch of space. Of course for that some sophisticated edge detectors are needed to efficiently outline the boundaries)

[@WungoBungo]

Yeah image compression is always a combination of different techniques, the things you described seem likely, although I don’t know any more than the fact that other lossy image formats use combinations of algorithms, often as you said to reduce the amount of FFT’s that need to be computed

[@yoshi4980]

going into some of the quirks of audio, image, and video compression specifically in 3 separate videos would be cool. albeit them being too large to truly cover in one video, i imagine you could do something similar to what you did in this video: introduction using something many viewers probably know about, and then expanding from there, covering the core techniques used in those specific types of compression

[@kwazar6725]

Nice work!

[@CatFish107]

Are you telling me that my mp3 player is a tiny additive synthesizer? Whoa!😮

[@alex_7547]

i love the way you explain things

[@hoots187]

Thanks for taking the time to make this sort of content, the care you take really comes through in your explanations.

[@Profesor449]

damn, I knew Fouriers from school (used it in dynamics and sound), and I knew that it should be everywhere, but it never crossed my mind that it is used for images and video. Thanks for the presentation. By the way, what sofware do you use to create your charts and graphics?

[@ImprobableMatter]

For graphics in general I use Inkscape to make vector graphics (.svg). For numerical stuff like the waves in the thumbnail, the spectrograms and so on, I use Python's matplotlib.

[@f_USAF-Lt.G]

🤔you thought it should be used everywhere... And found out _it is. 😌

[@minkone007]

I feel a lot smarter after watching this video

[@matiastripaldi406]

Also the basis to NMR and Xray crystallography among many other things

[@user-wc7em8kf9d]

Thanks for teaching in a such complete and clear way! Amazing video and professor. Are you an engineer?

[@pianoman47]

Excellent video!

[@J-dash-M]

Liam Neeson giving you a lecture…I feel like jedi.

[@angelaarizona3375]

Imagine how different society would be if this video had millions of views.

[@tarasst6887]

great video!!!

[@Rudenich911]

Excellent 🎉

[@v0ldy54]

I'd be interested to hear your take about thorium fission reactors!

[@TeamDman]

Great video!

[@graeme011]

Very well explained! Now, where can I watch this video again in a less "lossy" format?

[@shreyayadav4061]

Thank you it helped me understand FT

[@JoyDavidson]

It's still hard to imagine how frequencies are the foundation of everything.

[@ExEBoss]

*@**14:35* Actually, what you described is the *SVG* format. *PNG* achieves lossless compression by using the *DEFLATE* algorithm, which does something else.

[@ImprobableMatter]

The SVG is vector graphics, which of course doesn't care about rasterization. Under the hood, DEFLATE will do the same thing for pixels - use correlations in pixel values versus positions to compress. I use SVGs for the graphics in my videos, convert to PNG and then animate; funnily enough, the blue rectangle has almost identical file sizes for the SVG and PNG files.

[@HMPerson2]

​@@ImprobableMatter DEFLATE is a generic data compression algorithm; it has no concept of pixels or positions. PNG does have a "spatial DPCM" pre-compression step which is intended to make the image data more compressible, but it's a stretch to describe that as storing instructions to set pixels between these horizontal and vertical limits to this color.

[@jussiheino]

In otherwise excellent presentation, it is very very unfortunate that the lossless/PNG-topic is so deeply and horribly wrong. That part should be deleted anyway as this video is not about image compression in general. That part could be replaced by a mention about wavelets.

[@ImprobableMatter]

@@jussiheino 🙄

[@StarfilmerOne]

I don't understand this video but exactly something like this is what I wished to have, not just about bits n bytes, kinda explanation to physics frequency's and stuff, he even provided math for speaker into data bits/bytes from PC with frequency's and stuff. This Men should get a Teacher, he has big knwledge

[@anon_y_mousse]

Seeing the difference between the original Fourier transform and the modern "fast" variants is interesting. Definitely get an upvote for the music. Although I suspect a certain show may have influenced the decision of using that particular piece, it's a good one.

[@vanhetgoor]

Who would have thought that the complete contents of a CD or a DVD can be put in one equation.

[@izaakgray8521]

I haven’t studied mathematics since age 15, but I take an interest in Fourier transforms, and this is amazing. Particularly the visual way you’re showing the effects of transform, and the effect of modulating either side of F(x). I’m also learning to code so these concepts all have a lot of synergy!

[@PedroFigueiredo-q9x]

More sinus waves do not remove the Gibbs effect which then becomes a very thin spike of constant height at each edge of the step function. This is avoidable by substituting sinus by the cumbersome Gebauer functions . Is there any simple technique to remove the blurring Gibbs peaks at the transition of light to dark in the image?

[@ImprobableMatter]

Usually, a window function is used to avoid the Gibbs effect, or similar. It's fairly straightforward: en.wikipedia.org/wiki/Window_function

[@PedroFigueiredo-q9x]

Agree, a window function to cut the Gibbs overshoot is easily conceivable. However this cuts information, narrows the band. Needed were a function which keeps and solely normalises, smoothens the overshoots .

[@martinmauko]

This guy is very smart and explains well

[@satiremuch2643]

Well done

[@polypaulpylen]

I see a red graph and I want it painted black

[@emilmckellar4932]

Yes I know I am being a Dickensien, yet any pulse of a frequency, 500Hz as in the video, is not pure 500Hz. It is a multitude of waves to form it, e.g. The start and end of the wave. Ony a pure sine wave that start at the beginning of time and never ends come close to a real pure sine wave. My 2 cents, take me down if you feel like it😊

[@ImprobableMatter]

Not if you cut your domain into finite chunks (of length N, say), as in a DFT.

[@emilmckellar4932]

@@ImprobableMatter well taken down a peg or two. Yet... Pure theory allows me to mathematically to prove a multiverse, and in reality to date there is no means to do so.

[@storytimewithunclekumaran5004]

great video...

[@5Stars49]

Something new 😮🎉

[@bunnyben5607]

Don't think I didn't miss that Painted Black melody

[@JustinArmstrong-u5w]

A perfectly simplified explanation of the Fourier transform and its applications!. £10.00Superb. .

[@Glocky131]

Damn, I always wondered about this.

[@sunni1174]

Shannon limit, Nyquist...please share

[@oysa21]

amazing

[@f_USAF-Lt.G]

Hey !! A colorful spectragraph for the song "Paint It Black"( - Rolling Stones)😊

[@Richardj410]

Thanks

[@Steve85-GTX]

Thanks for using binary megabytes 👍

[@phillipneal8194]

Would it be possible for you to discuss dropping a HF radio frequency down to an intermediate frequency ?

[@rationalthinker9612]

When I saw the title to this video, I was wondering if it was going to be based on Fourier, I guess I was right

[@Bjowolf2]

It's just like projections of a vector onto a set of orthogonal basis vectors - just in an infinite number of dimensions. 😉

[@ImprobableMatter]

You mean you don't always use infinite-dimensional vectors? 🤯

[@Bjowolf2]

@@ImprobableMatter Not when I am working in 1, 2 or 3D 😂

[@pawarranger]

very clear and concise gg

[@w__a__l__e]

he really transformed things, also so we can technically blame him for "social" media :P

[@nemofunf9862]

14:27 so it Paints It, Black?

[@error220_33]

7:27 amplitude amplitude modulated A.M. radio frecuency frecuency modulated F.M. radio. interesting 2 data create a graphic for sound... wow amazing... my teachers just show boring text, not draws...and explaining.

[@jirioto6089]

Sample, but don't re-sample, except using oscilator-comparator(DA-AD) conversion. Issue is with steepness, overall steepness, steepness as secret relationships between samples inside waveform. Yes, there is interpolation(partial differentia equation) as the best solution, but its wrong due to steepness. Listen carefully to get knowledge. Any practical math over the signal curving steepnes, resampling is most destructive same as pitch bending(any bending), equalizer less destructive, softvolume very little but still hearable. Understanding real (not virtual, not mathematical) dynamics is awesome. Why i say this? There is two most used timebases around the world - 44100Hz at recording side and 48000Hz at replaying side. Everything what computers/filters doing with this is wrong. "Do it different" for multimedia devices manufacturers. (DA-ASP-AD) Fourier analysis is very handy, but Fourier transform not.