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  • This right here is what we're going to build to, this video:

  • A certain animated approach to thinking about a super-important idea from math:

  • The Fourier transform.

  • For anyone unfamiliar with what that is,

  • my #1 goal here is just for the video to be an introduction to that topic.

  • But even for those of you who are already familiar with it,

  • I still think that there's something fun

  • and enriching about seeing what all of its components actually look like.

  • The central example, to start, is gonna be the classic one:

  • Decomposing frequencies from sound.

  • But after that, I also really wanna show a glimpse of how this idea extends well beyond sound and frequency,

  • and to many seemingly disparate areas of math, and even physics.

  • Really, it is crazy just how ubiquitous this idea is.

  • Let's dive in.

  • This sound right here is a pure A.

  • 440 beats per second.

  • Meaning, if you were to measure the air pressure

  • right next to your headphones, or your speaker, as a function of time, it would oscillate up and down

  • around its usual equilibrium, in this wave.

  • making 440 oscillations each second.

  • A lower-pitched note, like a D, has the same structure, just fewer beats per second.

  • And when both of them are played at once, what do you think the resulting pressure vs. time graph looks like?

  • Well, at any point in time, this pressure difference

  • is gonna be the sum of what it would be for each of those notes individually.

  • Which, let's face it, is kind of a complicated thing to think about.

  • At some points, the peaks match up with each other,

  • resulting in a really high pressure.

  • At other points, they tend to cancel out.

  • And all in all, what you get is a wave-ish pressure vs. time graph,

  • that is not a pure sine wave; it's something more complicated.

  • And as you add in other notes, the wave gets more and more complicated.

  • But right now, all it is is a combination of four pure frequencies.

  • So it seems...needlessly complicated, given the low amount of information put into it.

  • A microphone recording any sound

  • just picks up on the air pressure at many different points in time.

  • It only "sees" the final sum.

  • So our central question is gonna be how you can take

  • a signal like this,

  • and decompose it

  • into the pure frequencies that make it up.

  • Pretty interesting, right?

  • Adding up those signals really mixes them all together.

  • So pulling them back apart...feels

  • akin to unmixing multiple paint colors that have all been stirred up together.

  • The general strategy is gonna be to build for ourselves a mathematical machine

  • that treats signals with a given frequency...

  • ..differently from how it treats other signals.

  • To start,

  • consider simply taking a pure signal

  • say, with a lowly three beats per second, so that we can plot it easily.

  • And let's limit ourselves to looking at a finite portion of this graph.

  • In this case, the portion between zero seconds, and 4.5 seconds.

  • The key idea,

  • is gonna be to take this graph, and sort of wrap it up around a circle.

  • Concretely, here's what I mean by that.

  • Imagine a little rotating vector where each point in time

  • its length is equal to the height of our graph for that time.

  • So, high points of the graph correspond to a greater disance from the origin,

  • and low points end up closer to the origin.

  • And right now, I'm drawing it in such a way that moving forward two seconds in time

  • corresponds to a single rotation around the circle.

  • Our little vector drawing this wound up graph

  • is rotating at half a cycle per second.

  • So, this is important. There are two different frequencies at play here:

  • There's the frequency of our signal, which goes up and down, three times per second.

  • And then, separately, there's the frequency with which we're wrapping the graph around the circle.

  • Which, at the moment, is half of a rotation per second.

  • But we can adjust that second frequency however we want.

  • Maybe we want to wrap it around faster...

  • ..or maybe we go and wrap it around slower.

  • And that choice of winding frequency determines what the wound up graph looks like.

  • Some of the diagrams that come out of this can be pretty complicated; although, they are very pretty.

  • But it's important to keep in mind that all that's happening here

  • is that we're wrapping the signal around a circle.

  • The vertical lines that I'm drawing up top, by the way,

  • are just a way to keep track of the distance on the original graph

  • that corresponds to a full rotation around the circle.

  • So, lines spaced out by 1.5 seconds

  • would mean it takes 1.5 seconds to make one full revolution.

  • And at this point, we might have some sort of vague sense that something special will happen

  • when the winding frequency matches the frequency of our signal: three beats per second.

  • All the high points on the graph happen on the right side of the circle

  • And all of the low points happen on the left.

  • But how precisely can we take advantage of that in our attempt to build a frequency-unmixing machine?

  • Well, imagine this graph is having some kind of mass to it, like a metal wire.

  • This little dot is going to represent the center of mass of that wire.

  • As we change the frequency, and the graph winds up differently,

  • that center of mass kind of wobbles around a bit.

  • And for most of the winding frequencies,

  • the peaks and valleys are all spaced out around the circle in such a way that

  • the center of mass stays pretty close to the origin.

  • But!

  • When the winding frequency is the same as the frequency of our signal,

  • in this case, three cycles per second,

  • all of the peaks are on the right,

  • and all of the valleys are on the left..

  • ..so the center of mass is unusually far to the right.

  • Here, to capture this, let's draw some kind of plot

  • that keeps track of where that center of mass is for each winding frequency.

  • Of course, the center of mass is a two-dimensional thing, and requires two coordinates to fully keep track of,

  • but for the moment, let's only keep track of the x coordinate.

  • So, for a frequency of 0, when everything is bunched up on the right,

  • this x coordinate is relatively high.

  • And then, as you increase that winding frequency,

  • and the graph balances out around the circle,

  • the x coordinate of that center of mass goes closer to 0,

  • and it just kind of wobbles around a bit.

  • But then, at three beats per second, there's a spike as everything lines up to the right.

  • This right here is the central construct,

  • so let's sum up what we have so far:

  • We have that original intensity vs. time graph,

  • and then we have the wound up version of that in some two-dimensional plane,

  • and then, as a third thing, we have a plot

  • for how the winding frequency influences the center of mass of that graph.

  • And by the way, let's look back at those really low frequencies near 0.

  • This big spike around 0 in our new frequency plot

  • just corresponds to the fact that the whole cosine wave is shifted up.

  • If I had chosen a signal oscillates around 0,

  • dipping into negative values,

  • then, as we play around with various winding frequences,

  • this plot of the winding frequencies vs. center of mass

  • would only have a spike at the value of three.

  • But, negative values are a little bit weird and messy to think about

  • especially for a first example,

  • so let's just continue thinking in terms of the shifted-up graph.

  • I just want you to understand that that spike around 0 only corresponds to the shift.

  • Our main focus, as far as frequency decomposition is concerned, is that bump at three.

  • This whole plot is what I'll call

  • the "Almost Fourier Transform" of the original signal.

  • There's a couple small distinctions between this and the actual Fourier transform,

  • which I'll get to in a couple minutes,

  • but already, you might be able to see how this machine lets us pick out the frequency of a signal.

  • Just to play around with it a little bit more,

  • take a different pure signal, let's say with a lower frequency of two beats per second,

  • and do the same thing.

  • Wind it around a circle,

  • imagine different potential winding frequencies,

  • and as you do that keep track of where the center of mass of that graph is,

  • and then plot the x coordinate of that center of mass

  • as you adjust the winding frequency.

  • Just like before, we get a spike

  • when the winding frequency is the same as the signal frequency,

  • which in this case, is when it equals two cycles per second.

  • But the real key point, the thing that makes this machine so delightful,

  • is how it enables us to take a signal consisting of multiple frequencies,

  • and pick out what they are.

  • Imagine taking the two signals we just looked at:

  • The wave with three beats per second, and the wave with two beats per second,

  • and add them up.

  • Like I said earlier, what you get is no longer a nice, pure cosine wave;

  • it's something a little more complicated.

  • But imagine throwing this into our winding-frequency machine...

  • ..it is certainly the case that as you wrap this thing around, it looks a lot more complicated;

  • you have this

  • chaos (1) and

  • chaos (2) and chaos (3) and

  • chaos (4) and then

  • WOOP!

  • Things seem to line up really nicely at two cycles per second,

  • and as you continue on it's more chaos (5)

  • and more chaos (6)

  • more chaos (7)

  • chaos (8), chaos (9), chaos (10),

  • WOOP!

  • Things nicely align again at three cycles per second.

  • And, like I said before, the wound up graph can look kind of busy and complicated,

  • but all it is is the relatively simple idea of wrapping the graph around a circle.

  • It's just a more complicated graph, and a pretty quick winding frequency.

  • Now what's going on here with the two different spikes,