Shadowing Practice: But what is the Fourier Transform? A visual introduction. - Learn English Speaking with Video
Choose a practice mode
Each mode opens a full-screen practice room.
228 sentences
If sentences are too short or too long, click Edit to adjust them.
1
This right here is what we're going to build to this video,
2
a certain animated approach to thinking about a super important idea from math, the Fourier Transform.
3
For anyone unfamiliar with what that is, my number one goal here is just for the video to be an introduction to that topic.
4
But even for those of you who are already familiar with it,
5
I still think that there's something fun and enriching about seeing what all of its components actually look like.
6
The central example to start is going to be the classic one, decomposing frequencies from sounds.
7
But after that, I also really want to show a glimpse of how this idea extends well beyond sound
8
and frequency into many seemingly disparate areas of mass, and even physics.
9
Really, it is crazy just how ubiquitous this idea is.
10
Let's dive in.
11
This sound right here is a pure A, 440 beats per second, meaning
12
if you were to measure the air pressure right next to your headphones or your speaker as a function of time,
13
it would oscillate up and down around its usual equilibrium in this way, making 440 oscillations each second.
14
A lower pitch note, like a D, has the same structure, just fewer beats per second.
15
And when both of them are played at once, what do you think the resulting pressure versus time graph looks like?
16
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,
17
which, let's face it, is kind of a complicated thing to think about.
18
At some point, the peaks match up with each other, resulting in a really high pressure.
19
At other points, they tend to cancel out.
20
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.
21
And as you add in other notes, the wave gets more and more complicated.
22
But right now, all it is, is a combination of four pure frequencies,
23
so it seems needlessly complicated given the low amount of information put into it.
24
A microphone recording any sound just picks up on the air pressure at many different points in time.
25
It only sees the final sum.
26
So our central question is going to be how you can take a signal like this
27
and decompose it into the pure frequencies that make it up.
28
Pretty interesting, right?
29
Adding up those signals really mixes them all together,
30
So pulling them back apart feels akin to unmixing multiple paint colors that have all been stirred up together.
31
The general strategy is going to be to build for ourselves a mathematical machine
32
that treats signals with a given frequency differently from how it treats other signals.
33
To start, consider simply taking a pure signal, say with a lowly 3 beats per second, so that we can plot it easily.
34
And let's limit ourselves to looking at a finite portion of this graph.
35
In this case, the portion between 0 seconds and 4 .5 seconds.
36
The key idea is going to be to take this graph and sort of wrap it up around a circle.
37
Concretely, here's what I mean by that.
38
Imagine a little rotating vector where at each point in time, its length is equal to the height of our graph for that time.
39
So high points of the graph correspond to a greater distance from the origin, and low points end up closer to the origin.
40
And right now, I'm drawing it in such a way
41
that moving forward two seconds in time corresponds to a single rotation around the circle.
42
Our little vector drawing with Roundup's graph is rotating at half a cycle per second.
43
So this is important.
44
There are two different frequencies at play here.
45
There's the frequency of our signal, which goes up and down three times per second.
46
And then, separately, there's the frequency with
47
which we're wrapping the graph around the circle which at the moment is half of the rotation per second.
48
But we can adjust that second frequency however we want.
49
Maybe we want to wrap it around faster.
50
Or maybe we go and wrap it around slower.
51
And that choice of winding frequency determines what the wound -up graph looks like.
52
Some of the diagrams that come out of this can be pretty complicated, although they are very pretty.
53
But it's important to keep in mind that all that's happening here is that we're wrapping the signal around the circle.
54
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
55
that corresponds to a full rotation around the circle.
56
So lines spaced out by 1 .5 seconds would mean it takes 1 .5 seconds to make one full revolution.
57
And at this point, we might have some sort of vague sense
58
that something special will happen when the winding frequency matches the frequency of our signal, 3 beats per second.
59
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.
60
But how precisely can we take advantage of that in our attempt to build a frequency unmixing machine?
61
Well, imagine this graph is having some kind of mass to it like a metal wire.
62
This little dot is going to represent the center of mass of that wire.
63
As we change the frequency and the graph winds up differently, that center of mass kind of wobbles around a bit.
64
And for most of the winding frequencies, the peaks and the valleys are all spaced out around the circle in such a way
65
that the center of mass stays pretty close to the origin.
66
But when the winding frequency is the same as the frequency of our signal, in this case three cycles per second,
67
all of the peaks are on the right and all of the valleys are on the left,
68
so the center of mass is unusually far to the right.
69
Here, to capture this, let's draw some kind of plot
70
that keeps track of where that center of mass is for each winding frequency.
71
Of course, the center of mass is a two -dimensional thing, it requires two coordinates to fully keep track of,
72
but for the moment let's only keep track of the x coordinate.
73
So for a frequency of zero, when everything is bunched up on the right, this x coordinate is relatively high,
74
and then as you increase that winding frequency, and the graph balances out around the circle,
75
the x coordinate of that center of mass goes closer to zero, and it just kind of wobbles around a bit.
76
But then at 3 beats per second there's a spike as everything lines up to the right.
77
This right here is the central construct.
78
So let's sum up what we have so far.
79
We have that original intensity versus time graph, and then we have the wound up version of that in some two -dimensional plane,
80
and then as a third thing, we have a plot for how the winding frequency influences the center of mass of that graph.
81
And by the way, let's look back at those really low frequencies near zero.
82
This big spike around zero in our new frequency plot just corresponds to the fact
83
that the whole cosine wave is shifted up.
84
If I had chosen a signal that oscillates around zero, dipping into negative values, then as we play around with various winding frequencies,
85
this plot of the winding frequency versus the center of mass, would only have a spike at the value of 3.
86
But negative values are a little bit weird and messy to think about, especially for a first example,
87
so let's just continue thinking in terms of the shifted up graph.
88
I just want you to understand that that spike around zero only corresponds to the shift.
89
Our main focus, as far as frequency decomposition is concerned, is that bump at 3.
90
This whole plot is what I'll call the almost Fourier transform of the original signal.
91
There's a couple small distinctions between this and the actual Fourier transform, which I'll get to in a couple minutes,
92
but already you might be able to see how this machine lets us pick out the frequency of a signal.
93
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,
94
and do the same thing.
95
Wind it around a circle, imagine different potential winding frequencies, and as you do that,
96
keep track of where the center of mass of that graph is,
97
and then plot the x -coordinate of that center of mass as you adjust the winding frequency.
98
Just like before, we get a spike when the winding frequency is the same as the signal frequency,
99
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.
Download App
Everything you need to speak fluently
AI PronunciationScore every sentence
IPA PracticeMaster every sound
VocabularyBuild your word bank
Vocab GameLearn while playing
Same Channel
C1Differential equations, a tourist's guide | DE1
Daily ConversationUniversity Students
C15
Transformers, the tech behind LLMs | Deep Learning Chapter 5
American AccentBusiness EnglishUniversity StudentsWorking Adults
C12
Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C15
Fourier series
American AccentUniversity StudentsWorking Adults
C14
Euler's Formula Poem
American AccentIELTS PrepTOEFL PrepUniversity Students
C12
Vectors | Chapter 1, Essence of linear algebra
American AccentBusiness EnglishUniversity StudentsWorking Adults
C13
How might LLMs store facts | Deep Learning Chapter 7
American AccentIELTS PrepTOEFL PrepUniversity Students
C113
Essence of linear algebra preview
American AccentBusiness EnglishUniversity StudentsWorking Adults
C15
The inscribed square problem
American AccentUniversity StudentsWorking Adults
C14
The Newton art puzzle
American AccentUniversity StudentsWorking Adults
C249
Large Language Models explained briefly
American AccentBusiness EnglishUniversity StudentsWorking Adults
C12
Oh, wait, actually the best Wordle opener is not “crane”…
American AccentBusiness EnglishUniversity StudentsWorking Adults
C22
Hilbert's Curve: Is infinite math useful?
American AccentBusiness EnglishUniversity StudentsWorking Adults
C26
But what is a neural network? | Deep learning chapter 1
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Divergence and curl: The language of Maxwell's equations, fluid flow, and more
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1But what is quantum computing? (Grover's Algorithm)
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Tie random ends: How many loops?
American AccentUniversity StudentsWorking Adults
FOR YOU
Recommend Video
C13
279 sentences
LET THERE BE... Voltage? | Maxwell's Equation #2 Explained for Beginners
Daily ConversationUniversity Students
C11
430 sentences
Intensity, Loudness, and Timbre
Daily ConversationUniversity Students
C11
433 sentences
Give Me 24 minutes, I'll Give You 1,000+ hours of Claude Knowledge
Daily ConversationUniversity StudentsWorking Adults
C12
520 sentences
An Animated Introduction to Vibration Analysis by Mobius Institute
Daily ConversationUniversity Students
C13
62 sentences
A mind-blowing explanation of symmetry | Frank Wilczek for Big Think
Daily ConversationUniversity StudentsWorking Adults
C11
747 sentences
MIT Introduction to Deep Learning | 6.S191
Daily ConversationUniversity Students
C11
426 sentences
Learn the IPA for English Consonants!
Daily ConversationUniversity Students
C16
104 sentences
How real are the real numbers, really?
Daily ConversationUniversity Students
C14
72 sentences
Video 2: Shape
Daily ConversationUniversity Students
C11
166 sentences
Introduction to Optoelectronics and Photonics
Daily ConversationUniversity Students
C16
51 sentences
1 6 What Illusions Teach Us
Daily ConversationUniversity Students
Vocabulary and speaking notes for this lesson
This speaking lesson is built on the video “But what is the Fourier Transform? A visual introduction”. The speaker keeps coming back to these words: frequency, graph, signal, transform, mass. This video has 140 sentences and 3508 words to shadow. The speech runs for 19:28. The speaker talks fast, about 180 words per minute, so expect linked and reduced sounds. Only 81% of the words are among the 3,000 most common in English, so the vocabulary is demanding.
Key vocabulary in this video
15 words from the video worth learning, with pronunciation and meaning:
| Word | Pronunciation | Meaning |
|---|---|---|
| frequency noun | /ˈfɹiːkwənsi/ | The rate of occurrence of anything; the relationship between incidence and time period. |
| graph noun | /ɡɹæf/ | A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or… |
| transform verb | /tɹænsˈfɔɹm/ | To change greatly the appearance or form of. |
| complicated adjective | /ˈkɑm.plɪˌkeɪ.tɪd/ | Difficult or convoluted. |
| chaos noun | /ˈkeɪˌɑs/ | The unordered state of matter in classical accounts of cosmogony. |
| coordinate verb | /koʊˈɔɹ.dəˌneɪt/ | To place in the same order or rank. |
| spike noun | /spaɪk/ | A sort of very large nail. |
| wrap verb | /ɹæp/ | To enclose (an object) completely in any flexible, thin material such as fabric or paper. |
| correspond verb | /ˌkoɹəˈspɑnd/ | To be equivalent or similar in character, quantity, quality, origin, structure, function etc. |
| interval noun | /ˈɪntɚvəl/ | A distance in space. |
| rotate verb | /ˈɹoʊ.teɪt/ | To spin, turn, or revolve. |
| formula noun | /ˈfɔː.mjʊ.lə/ | Any mathematical rule expressed symbolically. |
| mathematics noun | /mæθ(.ə)ˈmæt.ɪks/ | An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. |
| rotation noun | /ɹoʊˈteɪʃən/ | The act of turning around a centre or an axis. |
| intensity noun | /ɪnˈtɛnsɪti/ | The quality of being intense. |
Phrasal verbs you will hear
| Word | Pronunciation | Meaning |
|---|---|---|
| wind up verb | /waɪnd ˈʌp/ | To wind (rope, string, mainsprings, etc.) completely. |
| add up verb | To take the sum of; to total. | |
| build up verb | To erect; to construct. | |
| come out verb | /ˌkʌm ˈaʊt/ | To be discovered; to be revealed. |
| pick out verb | To remove by picking. | |
| bring back verb | /ˈbɹɪŋ bæk/ | To fetch something. |
| draw up verb | To compose (a document), especially following a standard form; prepare a plan. | |
| end up verb | To bring to a conclusion. |
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| Present perfect continuous have/has been + -ing — an action that started earlier and is still going on | I've been calling · I've been graphing |
| Passive voice be + past participle — the focus is on what happens, not who does it | are played · been stirred · is bunched |
| Present perfect have/has + past participle — a past action that still matters now | has passed · you've sampled · have built |
Pronunciation to watch
The speaker uses 24 contractions and reduced forms, such as we're, you're, I'll. Say them the short way, as you hear them.
- The “th” sounds: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, tenth /tɛnθ/
- The “sh” and “zh” sounds: rotation /ɹoʊˈteɪʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, capture /ˈkæp.(t)ʃɚ/, distinction /dɪˈstɪŋkʃən/, notion /ˈnoʊʃən/
- Long words — get the stress right: complicated /ˈkɑm.plɪˌkeɪ.tɪd/, coordinate /koʊˈɔɹ.dəˌneɪt/, mathematics /mæθ(.ə)ˈmæt.ɪks/, intensity /ɪnˈtɛnsɪti/, dimensional /daɪˈmɛn.ʃə.nəl/
How to practise with this video
- Listen to the whole video once without speaking and note the words you do not know.
- Start at 0.75× speed, shadow it sentence by sentence, then go back to normal speed once it feels easy.
- Record yourself and compare with the original, paying attention to words like frequency, graph, transform.
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.