シャドーイング練習: But what is the Fourier Transform? A visual introduction. - 動画で英語スピーキングを学ぶ

練習モードを選ぶ

各モードは全画面の練習ルームで開きます。

228 文

文が短すぎたり長すぎる場合は、Editをタップして調整してください。

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

a certain animated approach to thinking about a super important idea from math, the Fourier Transform.

For anyone unfamiliar with what that is, my number one goal here is just for the video to be an introduction to that topic.

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

I still think that there's something fun and enriching about seeing what all of its components actually look like.

The central example to start is going to be the classic one, decomposing frequencies from sounds.

But after that, I also really want to show a glimpse of how this idea extends well beyond sound

and frequency into many seemingly disparate areas of mass, and even physics.

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

Let's dive in.

This sound right here is a pure A, 440 beats per second, meaning

if you were to measure the air pressure right next to your headphones or your speaker as a function of time,

it would oscillate up and down around its usual equilibrium in this way, making 440 oscillations each second.

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

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

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,

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

At some point, the peaks match up with each other, resulting in a really high pressure.

At other points, they tend to cancel out.

And all in all, what you get is a wave -ish pressure versus time graph that is not a pure sine wave, it's something more complicated.

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

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

so it seems needlessly complicated given the low amount of information put into it.

A microphone recording any sound just picks up on the air pressure at many different points in time.

It only sees the final sum.

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

and decompose it into the pure frequencies that make it up.

Pretty interesting, right?

Adding up those signals really mixes them all together,

So pulling them back apart feels akin to unmixing multiple paint colors that have all been stirred up together.

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

that treats signals with a given frequency differently from how it treats other signals.

To start, consider simply taking a pure signal, say with a lowly 3 beats per second, so that we can plot it easily.

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

In this case, the portion between 0 seconds and 4 .5 seconds.

The key idea is going to be to take this graph and sort of wrap it up around a circle.

Concretely, here's what I mean by that.

Imagine a little rotating vector where at each point in time, its length is equal to the height of our graph for that time.

So high points of the graph correspond to a greater distance from the origin, and low points end up closer to the origin.

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

that moving forward two seconds in time corresponds to a single rotation around the circle.

Our little vector drawing with Roundup's graph is rotating at half a cycle per second.

So this is important.

There are two different frequencies at play here.

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

And then, separately, there's the frequency with

which we're wrapping the graph around the circle which at the moment is half of the rotation per second.

But we can adjust that second frequency however we want.

Maybe we want to wrap it around faster.

Or maybe we go and wrap it around slower.

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

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

But it's important to keep in mind that all that's happening here is that we're wrapping the signal around the circle.

The vertical lines that I'm drawing up top, by the way, are just a way to keep track of the distance on the original graph

that corresponds to a full rotation around the circle.

So lines spaced out by 1 .5 seconds would mean it takes 1 .5 seconds to make one full revolution.

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

that something special will happen when the winding frequency matches the frequency of our signal, 3 beats per second.

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.

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

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

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

As we change the frequency and the graph winds up differently, that center of mass kind of wobbles around a bit.

And for most of the winding frequencies, the peaks and the valleys are all spaced out around the circle in such a way

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

But when the winding frequency is the same as the frequency of our signal, in this case three cycles per second,

all of the peaks are on the right and all of the valleys are on the left,

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

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

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

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

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

So for a frequency of zero, when everything is bunched up on the right, this x coordinate is relatively high,

and then as you increase that winding frequency, and the graph balances out around the circle,

the x coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.

But then at 3 beats per second there's a spike as everything lines up to the right.

This right here is the central construct.

So let's sum up what we have so far.

We have that original intensity versus time graph, and then we have the wound up version of that in some two -dimensional plane,

and then as a third thing, we have a plot for how the winding frequency influences the center of mass of that graph.

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

This big spike around zero in our new frequency plot just corresponds to the fact

that the whole cosine wave is shifted up.

If I had chosen a signal that oscillates around zero, dipping into negative values, then as we play around with various winding frequencies,

this plot of the winding frequency versus the center of mass, would only have a spike at the value of 3.

But negative values are a little bit weird and messy to think about, especially for a first example,

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

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

Our main focus, as far as frequency decomposition is concerned, is that bump at 3.

This whole plot is what I'll call the almost Fourier transform of the original signal.

There's a couple small distinctions between this and the actual Fourier transform, which I'll get to in a couple minutes,

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

Just to play around with it a little bit more, take a different pure signal, let's say with a lower frequency of 2 beats per second,

and do the same thing.

Wind it around a circle, imagine different potential winding frequencies, and as you do that,

keep track of where the center of mass of that graph is,

and then plot the x -coordinate of that center of mass as you adjust the winding frequency.

Just like before, we get a spike when the winding frequency is the same as the signal frequency,

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

100

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

101

is how it enables us to take a signal consisting of multiple frequencies and pick out what they are.

102

Imagine taking the two signals we just looked at, the wave with 3 beats per second and the wave with 2 beats per second,

103

and add them up.

104

Like I said earlier, what you get is no longer a nice pure cosine wave, it's something a little more complicated.

105

But imagine throwing this into our winding frequency machine.

106

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

107

You have this chaos and chaos and chaos and chaos and then whoop!

108

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

109

Then as you continue on it's more chaos

110

and more chaos more chaos chaos chaos Whoop things nicely align again at three cycles per second

111

and like I said before the wound -up graph can look kind of busy

112

and Complicated but all it is is the relatively simple idea of wrapping the graph around a circle

113

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

114

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

115

that if you were to take two signals

116

and then apply this almost Fourier transform to each of them individually and then add up the results,

117

what you get is the same as if you first added up the signals and then applied this almost Fourier transform.

118

And the attentive viewers among you might want to pause

119

and ponder and convince yourself that what I just said is actually true.

120

It's a pretty good test to verify for yourself that it's clear what exactly is being measured inside this winding machine.

121

Now this property makes things really useful to us

122

because the transform of a pure frequency is close to zero everywhere except for a spike around that frequency.

123

So when you add together two pure frequencies, the transform graph just has these little peaks above the frequencies that went into it.

124

So this little mathematical machine does exactly what we wanted.

125

It pulls out the original frequencies from their jumbled up sums, unmixing the mixed bucket of paint.

126

And before continuing into the full math that describes this operation,

127

let's just get a quick glimpse of one context where this thing is useful, sound editing.

128

Let's say you have some recording, and it's got an annoying high pitch that you would like to filter out.

129

Well, at first, your signal is coming in as a function of various intensities over time,

130

different voltages given to your speaker from one millisecond to the next.

131

But we want to think of this in terms of frequencies.

132

So when you take the Fourier transform of that signal,

133

the annoying high pitch is going to show up just as a spike at some high frequency.

134

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,

135

only without that high frequency.

136

Luckily, there's a notion of an inverse Fourier transform that tells you which signal would have produced this as its Fourier transform.

137

I'll be talking about that inverse much more fully in the next video,

138

but long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function.

139

Kind of.

140

This is a little bit of a lie, but it's in the direction of truth.

141

And most of the reason that it's a lie is

142

that I still have yet to tell you what the actual Fourier transform is,

143

since it's a little more complex than this x -coordinate of the center of mass idea.

144

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?

145

I mean, this thing is in two dimensions, it's got a y coordinate as well.

146

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,

147

where this center of mass is going to be a complex number that has both a real and an imaginary part.

148

And the reason for talking in terms of complex numbers, rather than just saying it has two coordinates,

149

is that complex numbers lend themselves to really nice descriptions of things that have to do with winding and rotation.

150

For example, Euler's formula famously tells us that if you take e to some number times i,

151

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

152

if you were to walk that number of units around a circle with radius 1 counterclockwise starting on the right.

153

So imagine you wanted to describe rotating at a rate of one cycle per second.

154

One thing that you could do is take the expression e to the 2 pi times i times t,

155

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.

156

And this is a little bit dizzying to look at, so maybe you want to describe a different frequency, something lower and more reasonable.

157

And for that you would just multiply that time t in the exponent by the frequency f.

158

For example, if f was 1 tenth, then this vector makes one full turn every 10 seconds,

159

since the time t has to increase all the way to 10 before the full exponent looks like 2 pi i.

160

I have another video giving some intuition on why this is the behavior of e to the x for imaginary inputs,

161

if you're curious, but for right now we're just going to take it as a given.

162

Now why am I telling you this, you might ask.

163

Well, it gives us a really nice way to write down

164

the idea of winding up the graph into a single tight little formula.

165

First off, But the convention in the context of Fourier transforms is to think about rotating in the clockwise direction,

166

so let's go ahead and throw a negative sign up into that exponent.

167

Now take some function describing a signal intensity versus time, like this pure cosine wave we had before, and call it g .

168

If you multiply this exponential expression times g ,

169

it means that the rotating complex number is getting scaled up and down according to the value of this function.

170

So you can think of this little rotating vector with its changing length as drawing the wound up graph.

171

So think about it.

172

This is awesome.

173

This really small expression is a super elegant way to encapsulate

174

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

175

And remember, the thing we want to do with this wound up graph is to track its center of mass.

176

So think about what formula is going to capture that.

177

Well, to approximate it at least, you might sample a whole bunch of times from the original signal,

178

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,

179

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

180

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

181

And in the limit, rather than looking at the sum of a whole bunch of points divided by the number of points,

182

you take an integral of this function divided by the size of the time interval that we're looking at.

183

The idea of integrating a complex valued function might seem weird, and to anyone who's shaky with calculus,

184

maybe even intimidating, but the underlying meaning here really doesn't require any calculus knowledge.

185

The whole expression is just the center of mass of the wound up graph. So great!

186

Step by step, we have built up this kind of complicated,

187

but let's face it, surprisingly small expression for the whole winding machine idea that I talked about.

188

And now there is only one final distinction to point out between this and the actual honest -to -goodness Fourier transform.

189

Namely, just don't divide out by the time interval.

190

The Fourier transform is just the integral part of this.

191

What that means is that instead of looking at the center of mass, you would scale it up by some amount.

192

If the portion of the original graph you were using spanned 3 seconds, you would multiply the center of mass by 3.

193

If it was spanning 6 seconds, you would multiply the center of mass by 6.

194

Physically, this has the effect that when a certain frequency persists for a long time,

195

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

196

when you have a pure frequency of 2 beats per second

197

and you wind it around the graph at 2 cycles per second.

198

The center of mass stays in the same spot, right?

199

It's just tracing out the same shape.

200

But the longer that signal persists, the larger the value of the Fourier transform at that frequency.

201

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,

202

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

203

That is a lot of different moving parts, So let's step back and summarize what we have so far.

204

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,

205

but instead takes in a frequency, what I've been calling the winding frequency.

206

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.

207

Now the output of this function is a complex number,

208

some point in the 2D plane that corresponds to the strength of a given frequency in the original signal.

209

The plot that I've been graphing for the Fourier transform is just the real component of that output, the x coordinate.

210

But you could also graph the imaginary component separately if you wanted a fuller description.

211

And all of this is encapsulated inside that formula that we built up.

212

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,

213

how multiplying that by the function g means drawing a wound up version of the graph,

214

and how an integral of a complex valued function can be interpreted in terms of a center of mass idea,

215

you can see how this whole thing carries with it a very rich intuitive meaning.

216

And by the way, one quick small note before we can call this wrapped up.

217

Even though in practice, with things like sound editing, you'll be integrating over a finite time interval,

218

the theory of Fourier transforms is often phrased where the bounds of this integral are negative infinity and infinity.

219

Concretely, what that means is that you consider this expression for all possible finite time intervals,

220

and you just ask, what is its limit as that time interval grows to infinity?

221

And man oh man, there is so much more to say.

222

So much, I don't want to call it done here.

223

This transform extends to corners of math well beyond the idea of extracting frequencies from signal.

224

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.

225

So stay subscribed for when that comes out, or an alternate option is to just binge on a couple 3BlueOnBrown videos

226

so that the YouTube recommender is more inclined to show you new things that come out.

227

Really the choice is yours.

228

Thank you.

アプリをダウンロード

流暢に話すために必要なすべて

AI発音採点文ごとに採点

IPA練習すべての音をマスター

単語学習自分の単語帳を作る

単語ゲーム遊びながら覚える

QRをスキャン

Download on theApp Store Get it onGoogle Play

同じチャンネル

Differential equations, a tourist's guide | DE1

Daily ConversationUniversity Students

Transformers, the tech behind LLMs | Deep Learning Chapter 5

Business EnglishUniversity StudentsWorking Adults

Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Cambridge Exam PrepUniversity StudentsWorking Adults

FOR YOU

このレッスンの語彙とスピーキングのポイント

このスピーキングレッスンは、動画「But what is the Fourier Transform? A visual introduction」を教材にしています。繰り返し出てくる語は次のとおりです：frequency, graph, signal, transform, mass。この動画には、シャドーイング用の文が140文、単語が3508語あります。音声の長さは19:28です。話す速さは速く、1分あたり約180語です。音のつながりや弱く発音される音が多くなります。英語の頻出3,000語に含まれる単語は81%だけなので、語彙は難しめです。

この動画の重要語彙

動画に出てくる覚えておきたい単語15語を、発音と意味つきで紹介します。

単語	発音	意味
frequency 名詞	/ˈfɹiːkwənsi/	頻度
graph 名詞	/ɡɹæf/	グラフ, 図表
transform 動詞	/tɹænsˈfɔɹm/	変形する, 変換する
complicated 形容詞	/ˈkɑm.plɪˌkeɪ.tɪd/	複雑な
chaos 名詞	/ˈkeɪˌɑs/	混沌
spike 名詞	/spaɪk/	針
wrap 動詞	/ɹæp/	包む, 包装する
correspond 動詞	/ˌkoɹəˈspɑnd/	文通する
interval 名詞	/ˈɪntɚvəl/	間隔
rotate 動詞	/ˈɹoʊ.teɪt/	回る
formula 名詞	/ˈfɔː.mjʊ.lə/	公式, 定式
mathematics 名詞	/mæθ(.ə)ˈmæt.ɪks/	数学
rotation 名詞	/ɹoʊˈteɪʃən/	回転, 自転
intensity 名詞	/ɪnˈtɛnsɪti/	強度
divide 動詞	/dɪˈvaɪd/	分割する, 分ける

動画に出てくる句動詞

単語	発音	意味
come out 動詞	/ˌkʌm ˈaʊt/	出る
draw up 動詞		練る
go up 動詞		上がる, 登る
point out 動詞		指す, 指し示す
show up 動詞		現れる
write down 動詞		書き留める, 記す

この動画の文法

話し手がよく使っている文型を、動画の実際の表現とともに紹介します。

文型	動画での表現
現在完了進行形 have/has been + -ing — 以前に始まり今も続いている動作	I've been calling · I've been graphing
受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる	are played · been stirred · is bunched
現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している	has passed · you've sampled · have built

注意したい発音

話し手はwe're, you're, I'llなど、短縮形や弱形を24回使っています。聞こえたとおりの短い形で発音しましょう。

「th」の音: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, tenth /tɛnθ/
「sh」と「zh」の音: rotation /ɹoʊˈteɪʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, capture /ˈkæp.(t)ʃɚ/, distinction /dɪˈstɪŋkʃən/, notion /ˈnoʊʃən/
長い単語（アクセントの位置に注意）: complicated /ˈkɑm.plɪˌkeɪ.tɪd/, coordinate /koʊˈɔɹ.dəˌneɪt/, mathematics /mæθ(.ə)ˈmæt.ɪks/, intensity /ɪnˈtɛnsɪti/, dimensional /daɪˈmɛn.ʃə.nəl/

日本語話者が苦手な音：

/r/ と /l/ の区別 — /l/ は舌先を歯茎につけ、/r/ はどこにもつけない: integral /ˈɪntəɡɹəl/, differently /ˈdɪf.ə.ɹənt.li/, separately /ˈsɛp.ɹət.li/, precisely /pɹɪˈsaɪs.li/, variable [ˈvæɹ.i.ə.bl̩]
/v/ — /b/ にならないように、上の歯を下唇に当てる: interval /ˈɪntɚvəl/, divide /dɪˈvaɪd/, vector /ˈvɛktɚ/, individually /ˌɪndɪˈvɪd͡ʒ(u)əli/, versus /ˈvɝsəs/
/f/ — 「フ」ではなく、上の歯と下唇で出す: frequency /ˈfɹiːkwənsi/, graph /ɡɹæf/, transform /tɹænsˈfɔɹm/, formula /ˈfɔː.mjʊ.lə/, infinity /ɪnˈfɪnɪti/

この動画での練習方法

まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
まず0.75倍速で一文ずつシャドーイングし、慣れてきたら通常の速度に戻します。
自分の声を録音して元の音声と比べます。frequency, graph, transformなどの単語に特に注意しましょう。

シャドーイングとは？英語上達に効果的な理由

シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。

シャドーイングのやり方: ステップ別の完全ガイドを読む →

シャドーイング練習: But what is the Fourier Transform? A visual introduction. - 動画で英語スピーキングを学ぶ

練習モードを選ぶ

アプリをダウンロード

同じチャンネル

Differential equations, a tourist's guide | DE1

Transformers, the tech behind LLMs | Deep Learning Chapter 5

Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra

Fourier series

Euler's Formula Poem

Vectors | Chapter 1, Essence of linear algebra

How might LLMs store facts | Deep Learning Chapter 7

Essence of linear algebra preview

The inscribed square problem

The Newton art puzzle

Large Language Models explained briefly

Oh, wait, actually the best Wordle opener is not “crane”…

Hilbert's Curve: Is infinite math useful?

But what is a neural network? | Deep learning chapter 1

Divergence and curl: The language of Maxwell's equations, fluid flow, and more

But what is quantum computing? (Grover's Algorithm)

Tie random ends: How many loops?

おすすめ動画

LET THERE BE... Voltage? | Maxwell's Equation #2 Explained for Beginners

Intensity, Loudness, and Timbre

Give Me 24 minutes, I'll Give You 1,000+ hours of Claude Knowledge

An Animated Introduction to Vibration Analysis by Mobius Institute

A mind-blowing explanation of symmetry | Frank Wilczek for Big Think

MIT Introduction to Deep Learning | 6.S191

Learn the IPA for English Consonants!

How real are the real numbers, really?

Video 2: Shape

Introduction to Optoelectronics and Photonics

1 6 What Illusions Teach Us