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這支影片是我們會製造的一支影片

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有一個更新的表現

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想像過一個很重要的感覺

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例如Math,48的影片

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為其他人不太肯定

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我希望這支影片

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我的第一支影片

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就是為了這支影片

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有一個更新的影片

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但也有些人

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已經會否認識

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我還以為有些人有甚麼好

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在看見看見所有的音量

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其實會有些人

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這支影片的標準是要做的

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是比較高的

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D-composing frequencies from sound

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但 after that, I also really want to show a glimpse of how this idea extends

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well beyond sound and frequency

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into many seemingly disparate areas of math and even physics

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really, it is crazy just how ubiquitous this idea is

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Let's dive in

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This sound right here is a pure A

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440 beats per second

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meaning if you were to measure the air pressure

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right next to your headphones or your speaker

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as a function of time, it would oscillate up and down

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around its usual equilibrium in this wave

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making 440 oscillations each second

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a lower pitch note, like a D, has the same structure

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just fewer beats per second

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and when both of them are played at once,

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what do you think the resulting pressure versus time graph looks like?

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Well, at any point in time, this pressure difference

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is going to be the sum of what it would be for each of those notes

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individually which, let's face it, is kind of a complicated thing to think about

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At some points, the peaks match up with each other

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resulting in a really high pressure

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At other points, they tend to cancel out

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And all in all, what you get is a wave-ish pressure versus time graph

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that is not a pure sine wave, it's something more complicated

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

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

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So it seems needlessly complicated

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given the low amount of information put into it

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A microphone recording any sound

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

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it only sees the final sum

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So our central question is going to be how you can take a signal like this

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and decompose it into the pure frequencies that make it up

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Pretty interesting, right?

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Adding up those signals really mixes them all together

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So pulling them back apart feels akin to unmixing multiple paint colors

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that have all been stirred up together

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

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that treats signals with a given frequency

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differently from how it treats other signals

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To start, consider simply taking a pure signal

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say with a lowly 3 beats per second

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so that we can plot it easily

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

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in this case, the portion between 0 seconds and 4.5 seconds

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The key idea is going to be to take this graph

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and sort of wrap it up around a circle

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concretely, here's what I mean by that

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Imagine a little rotating vector

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where at each point in time

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

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So high points of the graph correspond to a greater distance from the origin

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and low points end up closer to the origin

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And right now I'm drawing it in such a way

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that moving forward two seconds in time

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corresponds to a single rotation around the circle

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Our little vector drawing this one up graph

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is rotating at half a cycle per second

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So this is important

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There are two different frequencies that play here

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There's the frequency of our signal

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which goes up and down three times per second

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And then, separately, there's the frequency

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with which we're wrapping the graph around the circle

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which at the moment is half of the rotation per second

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But we can adjust that second frequency however we want

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maybe we want to wrap it around faster

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or maybe we go and wrap it around slower

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And that choice of winding frequency

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determines what the wound up graph looks like

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Some of the diagrams that come out of this

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can be pretty complicated

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although they are very pretty

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

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is that we're wrapping the signal around a circle

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

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

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that corresponds to a full rotation around the circle

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So lines spaced out by 1.5 seconds

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would mean it takes 1.5 seconds to make one full revolution

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And at this point, we might have some sort of vague sense

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that something special will happen

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when the winding frequency matches the frequency of our signal

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three beats per second

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

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and all of the low points happen on the left

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But how precisely can we take advantage of that

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in our attempt to build a frequency on mixing machine

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Well, imagine this graph is having some kind of mass to it

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like a metal wire

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

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As we change the frequency

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and the graph winds up differently

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that center of mass kind of wobbles around a bit

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And for most of the winding frequencies

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the peaks and the valleys are all spaced out around the circle

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in such a way that the center of mass stays pretty close to the origin

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But when the winding frequency is the same

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as the frequency of our signal

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in this case three cycles per second

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All of the peaks are on the right

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and all of the valleys are on the left

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So the center of mass is unusually far to the right

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Here, to capture this

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let's draw some kind of plot

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that keeps track of where that center of mass is

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for each winding frequency

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Of course, the center of mass is a two-dimensional thing

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it requires two coordinates to fully keep track of

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But for the moment, let's only keep track of the x-coordinate

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So for a frequency of zero

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when everything is bunched up on the right

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this x-coordinate is relatively high

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and then as you increase that winding frequency

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and the graph balances out around the circle

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the x-coordinate of that center of mass

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goes closer to zero

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and it just kind of wobbles around a bit

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But then, at three beats per second

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there's a spike

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as everything lines up to the right

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This right here is the central construct

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so let's sum up what we have so far

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we have that original intensity versus time graph

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and then we have the wound up version of that

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in some two-dimensional plane

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and then as a third thing

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we have a plot

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for how the winding frequency

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influences the center of mass of that graph

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

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This big spike around zero in our new frequency plot

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just corresponds to the fact

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that the whole cosine wave is shifted up

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If I chose in a signal that oscillates around zero

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dipping into negative values

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then as we play around with various winding frequencies

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this plot of the winding frequency versus center of mass

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would only have a spike at the value of three

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

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especially for a first example

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

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I just want you to understand that that spike around zero

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only corresponds to the shift

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Our main focus as far as frequency decomposition is concerned

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is that bump at three

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This whole plot is what I'll call the almost Fourier transform

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of the original signal

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There's a couple small distinctions between this

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and the actual Fourier transform

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which I'll get to in a couple minutes

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but already you might be able to see

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how this machine lets us pick out the frequency of a signal

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Just to play around with it a little bit more

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take a different pure signal

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let's say with a lower frequency of two beats per second

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and do the same thing

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Wind it around a circle

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Imagine different potential winding frequencies

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

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and then plot the X coordinate of that center of mass

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as you adjust the winding frequency

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Just like before

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we get a spike when the winding frequency is the same as the signal frequency

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

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But the real key point

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the thing that makes this machine so delightful

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is how it enables us to take a signal

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consisting of multiple frequencies

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and pick out what they are

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Imagine taking the two signals we just looked at

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the wave with three beats per second

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and the wave with two beats per second

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and add them up

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Like I said earlier

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what you get is no longer a nice pure cosine wave

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it's something a little more complicated

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but imagine throwing this into our winding frequency machine

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it is certainly the case that as you wrap this thing around

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it looks a lot more complicated

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to have this chaos and chaos and chaos and chaos

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and then whoop

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things seem to line up really nicely at two cycles per second

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that as you continue on

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it's more chaos and more chaos

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more chaos

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things nicely align again at three cycles per second

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and like I said before

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the wound up graph can look kind of busy and complicated

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but all it is is the relatively simple idea

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of wrapping the graph around a circle

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it's just a more complicated graph

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and a pretty quick winding frequency

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

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is that if you were to take two signals

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and then apply this almost Fourier transform

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to each of them individually

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and then add up the results

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what you get is the same

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as if you first add it up the signals

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and then apply this almost Fourier transform

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and the attentive viewers among you might want to pause and ponder

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and convince yourself that what I just said is actually true

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it's a pretty good test to verify for yourself

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that it's clear what exactly is being measured

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inside this winding machine

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now this property makes things really useful to us

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because the transform of a pure frequency

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is close to zero everywhere

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except for a spike around that frequency

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so when you add together two pure frequencies

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the transform graph just has these little peaks

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above the frequencies that went into it

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so this little mathematical machine

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does exactly what we wanted

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it pulls out the original frequencies

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from their jumbled up sums

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unmixing the mixed bucket of paint

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and before continuing into the full math

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that describes this operation

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let's just get a quick glimpse of one context

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where this thing is useful

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sound editing

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let's say that you have some recording

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and it's got an annoying high pitch

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that you would like to filter out

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well at first your signal is coming in

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as a function of various intensities over time

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different voltages given to your speaker

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from one millisecond to the next

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but we want to think of this in terms of frequencies

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so when you take the Fourier transform of that signal

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the annoying high pitched is going to show up

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just as a spike at some high frequency

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filtering that out by just smushing this spike down

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what you'd be looking at is the Fourier transform

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of a sound that's just like your recording

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only without that high frequency

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luckily there's a notion of an inverse Fourier transform

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that tells you which signal would have produced this

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as its Fourier transform

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I'll be talking about that inverse

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much more fully in the next video

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but long story short

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applying the Fourier transform

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to the Fourier transform

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gives you back something close to the original function

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hmm kind of this is a little bit of a lie

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but it's in the direction of truth

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and most of the reason that it's a lie

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is that I still have yet to tell you

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what the actual Fourier transform is

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since it's a little more complex

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than this x-coordinate of the center of mass idea

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first off bringing back this wound up graph

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and looking at its center of mass

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the x-coordinate is really only half the story

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right I mean this thing is in two dimensions

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it's got a y-coordinate as well

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and as is typical in math

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whenever you're dealing with something

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too dimensional

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it's elegant to think of it as the complex plane

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where this center of mass is going to be a complex number

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it has both a real and an imaginary part

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and the reason for talking in terms of complex numbers

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rather than just saying it has two coordinates

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is the complex numbers lend themselves

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to really nice descriptions of things

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that have to do with winding and rotation

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for example

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Oilers formula famously tells us

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that if you take e to some number times i

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you're going to land on the point that you get

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if you were to walk that number of units

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around a circle with radius one

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counterclockwise starting on the right

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so imagine you wanted to describe rotating

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at a rate of one cycle per second

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one thing that you could do

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is take the expression

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e to the 2 pi times i times t

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where t is the amount of time that has passed

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since first circle with radius one

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2 pi describes the full length of its circumference

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and this is a little bit dizzying to look at

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so maybe you want to describe a different frequency

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something lower and more reasonable

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and for that you would just multiply that time t

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in the exponent by the frequency

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f

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for example if f was one tenth

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then this vector makes one full turn every ten seconds

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since the time t has to increase all the way to ten

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before the full exponent looks like two pi i

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i have another video giving some intuition

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why this is the behavior of e to the x for imaginary inputs

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if you're curious

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but for right now we're just going to take it as a given

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now why am i telling you this you might ask

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well it gives us a really nice way to write down the idea

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of winding up the graph into a single tight little formula

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first off the convention in the context of Fourier transforms

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is to think about rotating in the clockwise direction

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so let's go ahead and throw a negative sign

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up into that exponent

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now take some function describing a signal intensity versus time

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like this pure cosine wave we had before

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and call it g of t

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if you multiply this exponential expression

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times g of t

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it means that the rotating complex number is getting scaled up and down

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according to the value of this function

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so you can think of this little rotating vector with its changing length

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as drawing the wound up graph

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so think about it this is awesome

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this really small expression is a super elegant way to encapsulate

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the whole idea of winding a graph around a circle with a variable frequency

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f and remember the thing we want to do with this wound up graph

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is to track its center of mass

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so think about what formula is going to capture that

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well to approximate it at least

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you might sample a whole bunch of times from the original signal

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see where those points end up on the wound up graph

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and then just take an average

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that is add them all together as complex numbers

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and then divide by the number of points that you've sampled

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this will become more accurate if you sample more points which are closer together

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and in the limit rather than looking at the sum of a whole bunch of points

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divided by the number of points

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you take an integral of this function

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divided by the size of the time interval that we're looking at

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now the idea of integrating a complex valued function might seem weird

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and to anyone who's shaky with calculus maybe even intimidating

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but the underlying meaning here really doesn't require any calculus knowledge

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the whole expression is just the center of mass of the wound up graph

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so great step-by-step we have built up this kind of complicated but let's face it

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surprisingly small expression for the whole winding machine idea that i talked about

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and now there is only one final distinction to point out between this

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and the actual honest to goodness Fourier transform

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namely just don't divide out by the time interval

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the Fourier transform is just the integral part of this

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what that means is that instead of looking at the center of mass

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you would scale it up by some amount

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if the portion of the original graph you were using spanned three seconds

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you would multiply the center of mass by three

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if it was spanning six seconds you would multiply the center of mass by six

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physically this has the effect that when a certain frequency persists for a

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long time then the magnitude of the Fourier transform at that frequency

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is scaled up more and more for example what we're looking at right here

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is how when you have a pure frequency of two beats per second

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and you wind it around the graph at two cycles per second

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the center of mass stays in the same spot right it's just tracing out the same

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shape but the longer that signal persists the larger the value of the

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Fourier transform at that frequency

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for other frequencies though even if you just increase it by a bit

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this is cancelled out by the fact that for longer time intervals

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you're giving the wound up graph more of a chance to balance itself around

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the circle that is a lot of different moving parts

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00:17:11,460 --> 00:17:14,740
so let's step back and summarize what we have so far

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the Fourier transform of an intensity versus time function like g of t

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00:17:19,779 --> 00:17:22,980
is a new function which doesn't have time as an input

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00:17:22,980 --> 00:17:28,579
but instead takes in a frequency what i've been calling the winding frequency

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00:17:28,579 --> 00:17:32,099
in terms of notation by the way the common convention is to call this new

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00:17:32,099 --> 00:17:35,779
function g hat with a little circumflex on top of it

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now the output of this function is a complex number

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00:17:38,980 --> 00:17:43,220
some point in the 2d plane that corresponds to the strength of a given

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00:17:43,220 --> 00:17:47,220
frequency in the original signal the plot that i've been graphing for the

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00:17:47,220 --> 00:17:51,299
Fourier transform is just the real component of that output

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00:17:51,299 --> 00:17:54,660
the x coordinate but you could also graph the imaginary component

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00:17:54,660 --> 00:17:57,460
separately if you wanted a fuller description

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00:17:57,460 --> 00:18:01,859
and all of this is encapsulated inside that formula that we built up

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00:18:01,859 --> 00:18:05,059
and out of context you can imagine how seeing this formula would seem sort of

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00:18:05,059 --> 00:18:10,259
daunting but if you understand how exponentials correspond to rotation

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00:18:10,259 --> 00:18:13,619
how multiplying that by the function g of t means

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00:18:13,619 --> 00:18:17,940
drawing a wound up version of the graph and how an integral of a complex

394
00:18:17,940 --> 00:18:22,420
valued function can be interpreted in terms of a center of mass idea

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00:18:22,420 --> 00:18:27,460
you can see how this whole thing carries with it a very rich intuitive meaning

396
00:18:27,460 --> 00:18:30,980
and by the way one quick small note before we can call this wrapped up

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00:18:30,980 --> 00:18:34,019
even though in practice with things like sound editing you'll be

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00:18:34,019 --> 00:18:36,500
integrating over a finite time interval

399
00:18:36,500 --> 00:18:39,859
the theory of Fourier transforms is often phrased where the bounds of this

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00:18:39,859 --> 00:18:42,980
integral are negative infinity and infinity

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00:18:42,980 --> 00:18:46,259
concretely what that means is that you consider this expression

402
00:18:46,259 --> 00:18:51,140
for all possible finite time intervals and you just ask what is its limit

403
00:18:51,140 --> 00:18:54,660
has that time interval grows to infinity

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00:18:54,660 --> 00:18:58,420
and man oh man there is so much more to say so much i don't want to call it

405
00:18:58,420 --> 00:19:02,099
down here this transform extends to corners of math well beyond the idea of

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00:19:02,099 --> 00:19:06,500
extracting frequencies from signal so the next video i put out is going to go

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00:19:06,500 --> 00:19:08,740
through a couple of these and that's really where things start getting

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00:19:08,740 --> 00:19:12,259
interesting so stay subscribed for when that comes out

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00:19:12,259 --> 00:19:15,619
or an alternate option is to just binge on a couple three blue on brown

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00:19:15,619 --> 00:19:18,740
videos so that the youtube recommender is more inclined to show you new

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00:19:18,740 --> 00:19:22,660
things that come out really the choice is yours

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00:19:22,660 --> 00:19:26,180
and to close things off i have something pretty fun a mathematical puzzler from

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00:19:26,180 --> 00:19:31,140
this video sponsor Jane street who's looking to recruit more technical talent

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00:19:31,140 --> 00:19:37,059
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and then let b be the boundary of that space the surface of your complex

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blob now imagine taking every possible pair of points on that surface

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00:19:46,259 --> 00:19:52,019
and adding them up doing a vector sum let's name this set of all possible sums d

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