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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,

• 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.