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This right here is what we're going to build to this video,
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a certain animated approach to thinking about a super important idea from math,
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the Fourier transform.
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For anyone unfamiliar with what that is, my number one goal
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here is just for the video to be an introduction to that topic.
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But even for those of you who are already familiar with it,
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I still think that there's something fun and enriching about seeing what all of its
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components actually look like.
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The central example to start is going to be the classic one,
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decomposing frequencies from sound.
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But after that I also want to show a glimpse of how this idea extends well beyond
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sound and frequency into many seemingly disparate areas of math, and even physics.
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Really, it's 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, 440 beats per second,
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meaning if you were to measure the air pressure right next to your
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headphones or your speaker as a function of time,
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it would oscillate up and down 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, just fewer beats per second.
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And when both of them are played at once, what do you think the resulting pressure vs.
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time graph looks like?
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Well, at any point in time, this pressure difference is going to
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be the sum of what it would be for each of those notes individually,
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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, 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 vs.
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time graph 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 given the low amount of information put into it.
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A microphone recording any sound just picks up on the air pressure
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at many different points in time, it only sees the final sum.
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So our central question is going to be how you can take a signal
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like this 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 that have
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all been stirred up together.
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The general strategy is going to be to build for ourselves a mathematical machine that
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treats signals with a given frequency 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, so 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 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 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 that moving forward 2
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seconds in time corresponds to a single rotation around the circle.
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Our little vector drawing this wound up graph is rotating at half a cycle per second.
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So this is important, there are two different frequencies at play here.
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There's the frequency of our signal, which goes up and down 3 times per second,
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and then separately there's the frequency with which we're wrapping the graph
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around the circle, which at the moment is half of a 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 determines what the wound up graph looks like.
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Some of the diagrams that come out of this can be pretty complicated,
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although they are very pretty, but it's important to keep in mind that
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all that's happening here 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 that corresponds to
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a full rotation around the circle.
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So lines spaced out by 1.5 seconds would mean
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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 that something special will
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happen when the winding frequency matches the frequency of our signal, 3 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 in
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our attempt to build a frequency unmixing machine?
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Well, imagine this graph is having some kind of mass to it, 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 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, the peaks and valleys are all spaced out
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around the circle 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 as the frequency of our signal,
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in this case 3 cycles per second, all of the peaks are on the right,
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and all of the valleys are on the left, so the center of mass is unusually far
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to the right.
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Here, to capture this, let's draw some kind of plot that keeps
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track of where that center of mass is 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, but for the moment,
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let's only keep track of the x-coordinate.
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So for a frequency of zero, 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, the x-coordinate of
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that center of mass goes closer to zero, and it just kind of wobbles around a bit.
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But then, at 3 beats per second, there's a spike, as everything lines up to the right.
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This right here is the central construct, so let's sum up what we have so far.
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We have that original intensity vs time graph,
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and then we have the wound up version of that in some two-dimensional plane,
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and then as a third thing, we have a plot for how the winding frequency influences
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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 just
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corresponds to the fact that the whole cosine wave is shifted up.
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If I had chosen a signal that oscillates around zero, 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 vs center of mass would only have a spike
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at the value of 3.
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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, so let's just continue thinking in terms of the
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shifted up graph.
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I just want you to understand that that spike around zero only corresponds to the shift.
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Our main focus, as far as frequency decomposition is concerned, is that bump at 3.
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This whole plot is what I'll call the almost Fourier transform of the original signal.
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There's a couple small distinctions between this and the actual Fourier transform,
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which I'll get to in a couple minutes, but already you might be able to
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see 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, take a different Fourier signal,
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let's say with a lower frequency of 2 beats per second, and do the same thing.
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Wind it around a circle, 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 as you adjust the winding frequency.
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Just like before, we get a spike when the winding frequency is the same as
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the signal frequency, which in this case is when it equals 2 cycles per second.
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But the real key point, the thing that makes this machine so delightful,
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is how it enables us to take a signal consisting of multiple frequencies and pick out
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what they are.
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Imagine taking the two signals we just looked at,
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the wave with 3 beats per second and the wave with 2 beats per second, and add them up.
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Like I said earlier, 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 it looks a
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lot more complicated, you have this chaos and chaos and chaos and chaos,
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and then whoop, things seem to line up really nicely at 2 cycles per second.
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Then as you continue on it's more chaos and more chaos and more chaos and chaos
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and chaos and chaos, whoop, things nicely align again at 3 cycles per second.
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And like I said before, the wound up graph can look kind of busy and complicated,
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but all it is is the relatively simple idea of wrapping the graph around a circle.
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It's just a more complicated graph and a pretty quick winding frequency.
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Now what's going on here with the two different spikes is that if you were to
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take two signals and then apply this almost Fourier transform to each of them
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individually, and then add up the results, what you get is the same as if you
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first added up the signals and then applied 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 that it's clear
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what exactly is being measured 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 is close to zero everywhere except
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for a spike around that frequency, so when you add together two pure frequencies,
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the transform graph just has these little peaks above the frequencies that went into it.
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So this little mathematical machine does exactly what we wanted.
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It pulls out the original frequencies 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 that describes this operation,
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let's just get a quick glimpse of one context where this thing is useful, sound editing.
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Let's say that you have some recording and it's got an
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annoying high pitch that you would like to filter out.
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Well at first your signal is coming in as a function of various intensities over time,
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different voltages given to your speaker 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 pitch is going to show up just as a spike at some high frequency.
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Filtering that out by just smushing the spike down,
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what you'd be looking at is the Fourier transform of a sound that's just like your
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recording, only without that high frequency.
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Luckily there's a notion of an inverse Fourier transform that tells
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you which signal would have produced this as its Fourier transform.
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I'll be talking about that inverse much more fully in the next video,
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but long story short, applying the Fourier transform to the Fourier
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transform gives you back something close to the original function.
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Kind of, this is a little bit of a lie, but it's in the direction of truth.
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And most of the reason it's a lie is that I still have yet to
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tell you what the actual Fourier transform is,
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since it's a little more complex than this x-coordinate of the center of mass idea.
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First off, bringing back this wound up graph and looking at its center of mass,
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the x-coordinate is really only half the story, right?
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I mean, this thing is in two dimensions, it's got a y-coordinate as well.
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And as is typical in math, whenever you're dealing with something two-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 that has both a real
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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 that complex numbers lend themselves to really nice descriptions of
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things that have to do with winding and rotation.
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For example, Euler's formula famously tells us 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 if you were to walk that number of
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units around a circle with radius 1 counterclockwise starting on the right.
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So imagine you wanted to describe rotating at a rate of 1 cycle per second.
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One thing you could do is take the expression e to the 2 pi times i times t,
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where t is the amount of time that has passed, since for a circle with radius 1,
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2 pi describes the full length of its circumference.
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And this is a little dizzying to look at, so maybe you want to describe
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a different frequency, something lower and more reasonable,
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and for that you would just multiply that time t in the exponent by the frequency f.
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For example, if f was 1 tenth, then this vector makes one full turn every 10 seconds,
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since the time t has to increase all the way to 10 before the full exponent looks like 2
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pi i.
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I have another video giving some intuition on why this is the
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behavior of e to the x for imaginary inputs, 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 is to think about
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rotating in the clockwise direction, so let's throw a negative sign 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, and call it g of t.
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If you multiply this exponential expression times g of t,
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it means that the rotating complex number is getting scaled up and down according to
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the value of this function.
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So you can think of this little rotating vector with
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its changing length as drawing the wound up graph.
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So think about it, this is awesome, this really small expression
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is a super elegant way to encapsulate the whole idea of
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winding a graph around a circle with a variable frequency, f.
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And remember, the thing we want to do with this wound up graph is to track
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its center of mass, so think about what formula is going to capture that.
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Well, to approximate it at least, you might sample a whole bunch of times
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from the original signal, see where those points end up on the wound up graph,
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and then just take an average, that is, add them all together as complex numbers,
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and then divide by the number of points 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
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points divided by the number of points, you take an integral of this
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function divided by the size of the time interval we're looking at.
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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 I talked about,
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and now there is only one final distinction to point out between this and the actual
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honest-to-goodness Fourier transform, 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 3 seconds,
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you would multiply the center of mass by 3.
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If it was spanning 6 seconds, you would multiply the center of mass by 6.
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Physically, this has the effect that when a certain frequency persists for a long time,
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then the magnitude of the Fourier transform at that frequency is scaled up more and more.
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For example, what we're looking at here is how when you have a pure frequency of 2
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beats per second and you wind it around the graph at 2 cycles per second,
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the center of mass stays in the same spot, just tracing out the same shape.
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But the longer that signal persists, the larger the value of the Fourier transform
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at that frequency.
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For other frequencies, 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 the circle.
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That is a lot of different moving parts, so let's
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step back and summarize what we have so far.
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The Fourier transform of an intensity vs.
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time function, like g of t, is a new function, which doesn't have time as an input,
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but instead takes in a frequency, what I've been calling the winding frequency.
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In terms of notation, by the way, the common convention is to
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call this new function g-hat with a little circumflex on top of it.
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The output of this function is a complex number,
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some point in the 2d plane that corresponds to the strength of a given
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frequency in the original signal.
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The plot I've been graphing for the Fourier transform is just the real
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component of that output, the x-coordinate, but you could also graph
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the imaginary component separately if you wanted a fuller description.
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And all of this is encapsulated inside that formula we built up.
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And out of context, you can imagine how seeing this formula would seem sort of daunting,
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but if you understand how exponentials correspond to rotation,
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how multiplying that by the function g of t means drawing a wound up version of the
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graph, and how an integral of a complex valued function can be interpreted in terms
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of a center of mass idea, you can see how this whole thing carries with it a very rich
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intuitive meaning.
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And by the way, one quick small note before we can call this wrapped up.
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Even though in practice, with things like sound editing,
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you'll be integrating over a finite time interval,
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the theory of Fourier transforms is often phrased where the bounds of this
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integral are negative infinity and infinity.
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Concretely, what that means is that you consider this expression
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for all possible finite time intervals, and you just ask,
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what is its limit as that time interval grows to infinity?
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And man oh man, there is so much more to say.
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So much, I don't want to call it done here.
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This transform extends to corners of math well
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beyond the idea of extracting frequencies from signal.
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So the next video I put out is going to go through a couple of these,
00:19:06
and that's really where things start getting interesting.
00:19:10
So stay subscribed for when that comes out, or an alternate option
00:19:13
is to just binge on a couple 3Blue and Brown videos so that the
00:19:16
YouTube recommender is more inclined to show you new things that come out.
00:19:19
Really the choice is yours.
00:19:22
And to close things off, I have something pretty fun,
00:19:25
a mathematical puzzler from this video's sponsor, Jane Street,
00:19:28
who's looking to recruit more technical talent.
00:19:31
So let's say that you have a closed bounded convex set C sitting in 3D space,
00:19:36
and then let B be the boundary of that space, the surface of your complex blob.
00:19:42
Now imagine taking every possible pair of points on that surface and adding them up,
00:19:47
doing a vector sum.
00:19:48
Let's name this set of all possible sums D.
00:19:52
Your task is to prove that D is also a convex set.
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Thank you.
