Pratica di Shadowing: But what is the Fourier Transform? A visual introduction. - Impara a parlare inglese con i video
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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, the Fourier Transform.
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For anyone unfamiliar with what that is, my number one goal 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 components actually look like.
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The central example to start is going to be the classic one, decomposing frequencies from sounds.
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But after that, I also really want to show a glimpse of how this idea extends well beyond sound
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and frequency into many seemingly disparate areas of mass, 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, 440 beats per second, meaning
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if you were to measure the air pressure right next to your 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 way, 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 versus time graph looks like?
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Well, at any point in time, this pressure difference is going to 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 point, 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 versus 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 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 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 differently from how it treats other signals.
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To start, consider simply taking a pure signal, say with a lowly 3 beats per second, 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 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, 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, 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 corresponds to a single rotation around the circle.
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Our little vector drawing with Roundup's graph 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 at play here.
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There's the frequency of our signal, which goes up and down three times per second.
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And then, separately, there's the frequency with
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which we're wrapping the graph around the circle 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 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, although they are very pretty.
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But it's important to keep in mind that all that's happening here is that we're wrapping the signal around the circle.
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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
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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 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 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, and all of the low points happen on the left.
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But how precisely can we take advantage of that in 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, that center of mass kind of wobbles around a bit.
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And for most of the winding frequencies, the peaks and the valleys are all spaced out around the circle in such a way
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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, in this case three cycles per second,
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all of the peaks are on the right 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, let's draw some kind of plot
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that keeps 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, 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, when everything is bunched up on the right, this x coordinate is relatively high,
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and then as you increase that winding frequency, and the graph balances out around the circle,
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the x coordinate of 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.
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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, 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 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 corresponds to the fact
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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, then as we play around with various winding frequencies,
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this plot of the winding frequency versus the center of mass, would only have a spike at the value of 3.
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But negative values are a little bit weird and messy to think about, 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 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, which I'll get to in a couple minutes,
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but already you might be able to 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 pure signal, let's say with a lower frequency of 2 beats per second,
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and do the same thing.
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Wind it around a circle, imagine different potential winding frequencies, and as you do that,
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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 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, 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 what they are.
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Imagine taking the two signals we just looked at, the wave with 3 beats per second and the wave with 2 beats per second,
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and add them up.
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Like I said earlier, what you get is no longer a nice pure cosine wave, 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 lot more complicated.
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You have this chaos and chaos and chaos and chaos and then whoop!
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Things seem to line up really nicely at two cycles per second.
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Then as you continue on it's more chaos
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and more chaos more chaos chaos chaos Whoop things nicely align again at three cycles per second
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and like I said before the wound -up graph can look kind of busy
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and Complicated 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
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that if you were to take two signals
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and then apply this almost Fourier transform to each of them individually and then add up the results,
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what you get is the same as if you 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
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and ponder 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 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 for a spike around that frequency.
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So when you add together two pure frequencies, 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, 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 you have some recording, and it's got an 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, what you'd be looking at is the Fourier transform 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 that tells 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 transform gives you back something close to the original function.
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Kind of.
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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 that it's a lie is
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that I still have yet to 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, 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, 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 and an imaginary part.
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And the reason for talking in terms of complex numbers, rather than just saying it has two coordinates,
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is that complex numbers lend themselves to really nice descriptions of 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
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if you were to walk that number of 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 one cycle per second.
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One thing that 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, 2 pi describes the full length of its circumference.
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And this is a little bit dizzying to look at, so maybe you want to describe 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 pi i.
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I have another video giving some intuition on why this is the behavior of e to the x for imaginary inputs,
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if you're curious, 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
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the idea of winding up the graph into a single tight little formula.
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First off, But the convention in the context of Fourier transforms is to think about rotating in the clockwise direction,
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so let's go ahead and throw a negative sign up into that exponent.
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Now take some function describing a signal intensity versus time, like this pure cosine wave we had before, and call it g .
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If you multiply this exponential expression times g ,
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it means that the rotating complex number is getting scaled up and down according to the value of this function.
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So you can think of this little rotating vector with its changing length as drawing the wound up graph.
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So think about it.
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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, f.
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And remember, the thing we want to do with this wound up graph 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, 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, 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 points divided by the number of points,
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you take an integral of this function divided by the size of the time interval that we're looking at.
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The idea of integrating a complex valued function might seem weird, and to anyone who's shaky with calculus,
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maybe even intimidating, 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. So great!
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Step by step, we have built up this kind of complicated,
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but let's face it, 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 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, 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, 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. For example, what we're looking at right here is how
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when you have a pure frequency of 2 beats per second
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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, right?
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It's just tracing out the same shape.
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But the longer that signal persists, the larger the value of the Fourier transform at that frequency.
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For other frequencies though, even if you just increase it by a bit, 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 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, 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 call this new 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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some point in the 2D plane that corresponds to the strength of a given frequency in the original signal.
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The plot that I've been graphing for the Fourier transform is just the real component of that output, the x coordinate.
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But you could also graph the imaginary component separately if you wanted a fuller description.
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And all of this is encapsulated inside that formula that we built up.
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And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation,
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how multiplying that by the function g means drawing a wound up version of the graph,
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and how an integral of a complex valued function can be interpreted in terms of a center of mass idea,
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you can see how this whole thing carries with it a very rich 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, 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 integral are negative infinity and infinity.
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Concretely, what that means is that you consider this expression for all possible finite time intervals,
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and you just ask, 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 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, and that's really where things start getting interesting.
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So stay subscribed for when that comes out, or an alternate option is to just binge on a couple 3BlueOnBrown videos
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so that the YouTube recommender is more inclined to show you new things that come out.
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Really the choice is yours.
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Thank you.
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Vocabolario e note di pronuncia per questa lezione
Questa lezione di conversazione si basa sul video “But what is the Fourier Transform? A visual introduction”. Le parole che ritornano più spesso: frequency, graph, signal, transform, mass. Questo video contiene 140 frasi e 3508 parole da ripetere con lo shadowing. Il parlato dura 19:28. Chi parla va veloce, circa 180 parole al minuto: aspettati suoni legati e ridotti. Solo il 81% delle parole rientra nelle 3.000 più comuni dell’inglese, quindi il lessico è impegnativo.
Vocaboli chiave di questo video
15 parole del video che vale la pena imparare, con pronuncia e significato:
| Parola | Pronuncia | Significato |
|---|---|---|
| frequency sostantivo | /ˈfɹiːkwənsi/ | frequenza |
| graph sostantivo | /ɡɹæf/ | grafico |
| transform verbo | /tɹænsˈfɔɹm/ | trasformare, modificare |
| complicated aggettivo | /ˈkɑm.plɪˌkeɪ.tɪd/ | complicato, intricato |
| chaos sostantivo | /ˈkeɪˌɑs/ | caos |
| coordinate verbo | /koʊˈɔɹ.dəˌneɪt/ | coordinare, accordarsi |
| spike sostantivo | /spaɪk/ | chiodo, lancia |
| wrap verbo | /ɹæp/ | avvolgere, avviluppare |
| correspond verbo | /ˌkoɹəˈspɑnd/ | corrispondere |
| interval sostantivo | /ˈɪntɚvəl/ | intervallo |
| rotate verbo | /ˈɹoʊ.teɪt/ | ruotare |
| formula sostantivo | /ˈfɔː.mjʊ.lə/ | formula |
| mathematics sostantivo | /mæθ(.ə)ˈmæt.ɪks/ | matematica |
| rotation sostantivo | /ɹoʊˈteɪʃən/ | rotazione, giro completo |
| intensity sostantivo | /ɪnˈtɛnsɪti/ | intensità |
I phrasal verb che sentirai
| Parola | Pronuncia | Significato |
|---|---|---|
| add up verbo | sommarsi | |
| build up verbo | accumulare | |
| draw up verbo | stilare, stendere | |
| end up verbo | andare a finire, finire | |
| go through verbo | attraversare | |
| go up verbo | salire | |
| line up verbo | /laɪnˈʌp/ | allineare |
| point out verbo | additare |
La grammatica di questo video
Le strutture che chi parla usa di più, con le parole esatte del video:
| Struttura | Nel video |
|---|---|
| Present perfect continuous have/has been + -ing — un’azione iniziata prima e ancora in corso | I've been calling · I've been graphing |
| Forma passiva be + participio passato — conta ciò che accade, non chi lo fa | are played · been stirred · is bunched |
| Present perfect have/has + participio passato — un’azione passata che conta ancora adesso | has passed · you've sampled · have built |
Pronuncia a cui fare attenzione
Chi parla usa 24 contrazioni e forme ridotte, come we're, you're, I'll. Pronunciale nella forma breve, così come le senti.
- I suoni “th”: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, tenth /tɛnθ/
- I suoni “sh” e “zh”: rotation /ɹoʊˈteɪʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, capture /ˈkæp.(t)ʃɚ/, distinction /dɪˈstɪŋkʃən/, notion /ˈnoʊʃən/
- Parole lunghe — attenzione all’accento: complicated /ˈkɑm.plɪˌkeɪ.tɪd/, coordinate /koʊˈɔɹ.dəˌneɪt/, mathematics /mæθ(.ə)ˈmæt.ɪks/, intensity /ɪnˈtɛnsɪti/, dimensional /daɪˈmɛn.ʃə.nəl/
I suoni difficili per chi parla italiano:
- Consonante finale — senza aggiungere una vocale dopo: graph /ɡɹæf/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, chaos /ˈkeɪˌɑs/, coordinate /koʊˈɔɹ.dəˌneɪt/, spike /spaɪk/
- /æ/ — più aperta della “e”: graph /ɡɹæf/, transform /tɹænsˈfɔɹm/, wrap /ɹæp/, mathematics /mæθ(.ə)ˈmæt.ɪks/, imaginary /ɪˈmæd͡ʒɪˌn(ɛ)ɹi/
Come esercitarsi con questo video
- Ascolta tutto il video una volta senza parlare e annota le parole che non conosci.
- Inizia a velocità 0,75×, fai shadowing frase per frase e torna alla velocità normale quando diventa facile.
- Registrati e confronta con l’originale, facendo attenzione a parole come frequency, graph, transform.
Cos'è la tecnica dello Shadowing?
Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.