A certain animated approach to thinking about a super-important idea from math:

以動畫的方式來思考一個數學中超級重要的概念

The Fourier transform.

傅里葉變換

For anyone unfamiliar with what that is,

對不熟悉的人來說

my #1 goal here is just for the video to be an introduction to that topic.

我的首要目標是來介紹“傅里葉變換”這個概念

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

但是 即使你已經熟悉了

I still think that there's something fun

我還是覺得你能在這裡可以獲得更加有趣和更深入的理解。

and enriching about seeing what all of its components actually look like.

首先，我們舉一個很經典的例子

The central example, to start, is gonna be the classic one:

從聲音中分解頻率

Decomposing frequencies from sound.

但是在此之後，我還想要說明一下，這個想法的適用範圍遠不止聲音和頻率，

But after that, I also really wanna show a glimpse of how this idea extends well beyond sound and frequency,

在許多看似無關的數學領域，甚至物理世界中都可以用到。

and to many seemingly disparate areas of math, and even physics.

真的，這個想法無處不在，令人興奮。

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

讓我們進入主題

Let's dive in.

現在播放的是純A音，

This sound right here is a pure A.

每秒440拍。

440 beats per second.

意思是，如果你要測量耳機或發聲器周圍的空氣壓力

Meaning, if you were to measure the air pressure

把他當成一個關於時間的函數，那這個函數也會在它的平衡點附近上下震蕩

right next to your headphones, or your speaker, as a function of time, it would oscillate up and down

每秒鐘產生440次振盪。

around its usual equilibrium, in this wave.

對於一個低音音符，比如D音，一樣具有相同的結構，只是每秒跳動的次數變少了。

making 440 oscillations each second.

當這兩個音同時播放時，你認為最終壓力與時間的關係圖是怎麼樣的呢？

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

那麼，在任意時刻，這個壓力差的變化就是

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

每個音調產生的壓力總和

Well, at any point in time, this pressure difference

好吧，這個東西其實有點複雜，而且很難想象

is gonna be the sum of what it would be for each of those notes individually.

在某些點上，兩個峰值會相互重合

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

產生了更高的氣壓

At some points, the peaks match up with each other,

而在其他時刻，他們又會相互抵消。

resulting in a really high pressure.

總而言之，你得到的是一個波浪型壓力與時間的關係圖，

At other points, they tend to cancel out.

那不是純粹的正弦波，而是更複雜的東西。

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

當你加入更多音調時，波浪也會變得越來越複雜。

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

但就目前來說，他就是個四個純音的組合

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

所以看起來...這個波形的信息量明明很少，看起來卻相當複雜。

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

麥克風在記錄任何聲音時

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

只是獲取了在不同時間點上的氣壓。

A microphone recording any sound

它只“看到”了氣壓最後的總和。

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

所以我們的核心問題就是：

It only "sees" the final sum.

要如何把一個這樣子的信號，

So our central question is gonna be how you can take

分解之後

a signal like this,

找到其中純音的頻率呢？

and decompose it

聽起來很有趣，對吧？

into the pure frequencies that make it up.

把這些信號加起來，它們就全都混合在一起了。

Pretty interesting, right?

所以把他們分開...感覺就像

Adding up those signals really mixes them all together.

把混合在一起的顏料分開。

So pulling them back apart...feels

我們的策略大概是：建造一台數學機器

akin to unmixing multiple paint colors that have all been stirred up together.

讓它能夠區別不同的頻率

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

首先，

that treats signals with a given frequency...

只考慮一個純粹的信號

..differently from how it treats other signals.

假設它每秒只有3拍，我們就可以很容輕鬆的畫出它的圖像。

To start,

讓我們只關注這個圖像的一部分。

consider simply taking a pure signal

在這種情況下，關注0秒到4.5秒

say, with a lowly three beats per second, so that we can plot it easily.

關鍵思想在於，

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

我們要把這個圖像纏繞在一個圓上

In this case, the portion between zero seconds, and 4.5 seconds.

具體來說，

The key idea,

想像一個旋轉的矢量，在任意時刻

is gonna be to take this graph, and sort of wrap it up around a circle.

它的長度等於這個時刻的圖像高度。

Concretely, here's what I mean by that.

所以，高處的點，對應圓上離圓心較遠的點

Imagine a little rotating vector where each point in time

低處的點，對應圓上離圓心較近的點

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

而現在，我作圖的方法是這樣的：每過2秒，

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

這個矢量就轉一整圈

and low points end up closer to the origin.

在纏繞圖像中

And right now, I'm drawing it in such a way that moving forward two seconds in time

這個矢量每秒轉半圈

corresponds to a single rotation around the circle.

所以這很重要， 我們有兩種不同的頻率：

Our little vector drawing this wound up graph

一個是信號的頻率，每秒上下震蕩3次。

is rotating at half a cycle per second.

另一個是圖像纏繞中心圓的頻率

So, this is important. There are two different frequencies at play here:

目前是每秒轉半圈。

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

但是我們可以隨意調整第二個頻率。

And then, separately, there's the frequency with which we're wrapping the graph around the circle.

比如，我們想讓它轉得更快

Which, at the moment, is half of a rotation per second.

或者，我們想讓他變慢。

But we can adjust that second frequency however we want.

而且，纏繞的頻率決定了纏繞圖像的樣子。

Maybe we want to wrap it around faster...

有些圖可能會非常複雜，雖然他們非常漂亮。

..or maybe we go and wrap it around slower.

但是要記住的是，我們所做的其實就是

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

把信號纏繞在一個圓上

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

順便說一下，我在最上面的圖中畫了一些豎線，

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

他們只是為了標明，繞著圓旋轉了整周時，

is that we're wrapping the signal around a circle.

原始圖像對應的位置

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

所以，如果線間隔1.5秒

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

意味著需要1.5秒才能完成一次完整的旋轉。

that corresponds to a full rotation around the circle.

到目前位置，你可能大概猜到了

So, lines spaced out by 1.5 seconds

纏繞頻率和信號頻率相等時（每秒3拍），會出現很特別的現象

would mean it takes 1.5 seconds to make one full revolution.

所有高點都剛好都在圓的右側

And at this point, we might have some sort of vague sense that something special will happen

所有的低點都發生在左側。

when the winding frequency matches the frequency of our signal: three beats per second.

但是，我們要如何充分利用這點，來建造一臺頻率分離器呢？

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

好吧，那就想像一下這個圖形有質量，比如金屬絲。

And all of the low points happen on the left.

這個小點代表該金屬絲的質量中心。

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

當我們改變頻率時，圖像的纏繞方式會發生變化，

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

質心的位置搖晃了一下。

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

而對於大部分的纏繞頻率，

As we change the frequency, and the graph winds up differently,

圖像的峰和谷都以這樣的方式圍繞在圓周上

that center of mass kind of wobbles around a bit.

質心與原點非常接近。

And for most of the winding frequencies,

但！

the peaks and valleys are all spaced out around the circle in such a way that

當纏繞頻率與我們信號的頻率相同時，

the center of mass stays pretty close to the origin.

在這種情況下，也就是每秒三個週期，

But!

所有的波峰都在右邊，

When the winding frequency is the same as the frequency of our signal,

所有的波谷都在左邊

in this case, three cycles per second,

所以，質心就會非常偏右。

all of the peaks are on the right,

在這裡，為了捕捉這個現象，讓我們畫一個圖

and all of the valleys are on the left..

跟踪每個纏繞頻率的對應的質心位置。

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

當然，質心是一個二維的東西，所以需要兩個坐標來表述，

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

但是我們暫時只跟踪x坐標。

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

當頻率為0時，所有點都聚集在右邊，

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

質心的x坐標比較大。

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

然後，當你增加纏繞頻率時，

So, for a frequency of 0, when everything is bunched up on the right,

圖像就會平均分佈在圓上

this x coordinate is relatively high.

該質心的x坐標也就趨近於0，

And then, as you increase that winding frequency,

並且在0附近擺動。

and the graph balances out around the circle,

但是，當頻率等於每秒三拍時，會出現一個尖峰，因為圖像全都繞在右邊

the x coordinate of that center of mass goes closer to 0,

這就是我們的核心構造，

and it just kind of wobbles around a bit.

讓我們總結一下到目前為止的內容

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

我們有原始的強度與時間的關係圖

This right here is the central construct,

一個二維平面上的纏繞圖像，

so let's sum up what we have so far:

除此之外，還有一個圖像

We have that original intensity vs. time graph,

記錄了纏繞頻率如何影響纏繞圖像的質心

and then we have the wound up version of that in some two-dimensional plane,

順便說一下，讓我們回顧一下0附近的低頻。

and then, as a third thing, we have a plot

在我們新的頻率圖中，這個在0附近的地方有一個很大的尖峰

for how the winding frequency influences the center of mass of that graph.

這只是因為餘弦曲線整體上移

And by the way, let's look back at those really low frequencies near 0.

如果我選擇一個信號在0附近振盪，

This big spike around 0 in our new frequency plot

允許出現負值，

just corresponds to the fact that the whole cosine wave is shifted up.

那麼，我們改變纏繞頻率時

If I had chosen a signal oscillates around 0,

質心與纏繞頻率的關係圖上

dipping into negative values,

只會在3這裡有一個高峰。

then, as we play around with various winding frequences,

但是，負值考慮起來又奇怪又麻煩

this plot of the winding frequencies vs. center of mass

何況這是第一個例子

would only have a spike at the value of three.

所以讓我們繼續考慮上移的圖像。

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

你只需要明白，0附近的尖峰只是對應於上移而已。

especially for a first example,

就頻率分解而言，我們的主要焦點就是在3那裡的凸起

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

我會把這張圖稱為

I just want you to understand that that spike around 0 only corresponds to the shift.

原始信號的“近似傅立葉變換”。

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

這與實際的傅里葉變換之間有一些小的區別，

This whole plot is what I'll call

我會在幾分鐘內提到，

the "Almost Fourier Transform" of the original signal.

但是你已經可以看到這台機器是如何幫我們挑出一個信號的頻率的。

There's a couple small distinctions between this and the actual Fourier transform,

讓我們多看兩眼

which I'll get to in a couple minutes,

換一個純信號，就假設每秒2拍的稍低頻率

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

以同樣的做法處理

Just to play around with it a little bit more,

繞一圈，

take a different pure signal, let's say with a lower frequency of two beats per second,

想象幾個不同可能的纏繞頻率

and do the same thing.

與此同時 注意盯著質心在哪裡

Wind it around a circle,

然後一邊調整纏繞頻率，

imagine different potential winding frequencies,

一邊畫出質心的x坐標

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

和之前一樣，在纏繞頻率和信號頻率相等時

and then plot the x coordinate of that center of mass

我們得到了一個高峰

as you adjust the winding frequency.

在這種情況下，它等於每秒兩個週期

Just like before, we get a spike

但真正的關鍵是，這台機器之所以那麼受歡迎

when the winding frequency is the same as the signal frequency,

是因為他能讀取好幾個頻率的信號

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

並把它們挑出來。

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

就想像一下我們剛才看到的兩個信號：

is how it enables us to take a signal consisting of multiple frequencies,

每秒三拍的波，以及每秒兩拍的波，

and pick out what they are.

全部加在一起

Imagine taking the two signals we just looked at:

正如我之前所說，你所得到的不再是一個很好的，純粹的餘弦波;

The wave with three beats per second, and the wave with two beats per second,

而是一個有點複雜的波

and add them up.

但是想像一下，把它扔到我們的捲繞機裡面

Like I said earlier, what you get is no longer a nice, pure cosine wave;

肯定是看上去越來越複雜

it's something a little more complicated.

混亂

But imagine throwing this into our winding-frequency machine...

很混亂

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

超級混亂

you have this

無敵的混亂

chaos (1) and

然後，哦？

chaos (2) and chaos (3) and

每秒兩圈的時候，圖像整齊的排列了起來，

chaos (4) and then

然後再繼續混亂

WOOP!

很混亂

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

非常混亂

and as you continue on it's more chaos (5)

亂到沒朋友之後

and more chaos (6)

哦！！

more chaos (7)

每秒三圈的時候，又排的超級整齊。

chaos (8), chaos (9), chaos (10),

就像我之前說過的那樣，曲線圖看起來可能很繁雜，

WOOP!

但這一切不過是把圖像繞著圓纏起來罷了

Things nicely align again at three cycles per second.

不過是圖像越複雜，纏繞頻率越快而已

And, like I said before, the wound up graph can look kind of busy and complicated,

現在這裡產生了兩個不同的尖峰，

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

如果你拿兩個信號，再分別對他們使用“近似傅里葉變換”，再把結果加在一起

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

你得到的結果

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

和先把信號加起來，再進行“近似傅里葉變換”是一樣的

is that if you were to take two signals,

細心的觀眾可以想停下來思考

and then apply this Almost-Fourier transform to each of them individually,

稍微體會一下我所說的都是正確的

and then add up the results,

這是一個很不錯的挑戰，來感受這個測量機

what you get is the same as if you first

到底測量的是個什麼東西

added up the signals, and then applied this Almost-Fourier transorm.

現在這個屬性對我們來說非常有用

And the attentive viewers among you might wanna pause and ponder, and...

因為純粹的頻率轉換

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

除了在其頻率附近的尖峰以外，

It's a pretty good test to verify for yourself that it's clear what exactly is being measured

其他地方幾乎都是0

inside this winding machine.

所以當你把兩個純頻率相加時

Now this property makes things really useful to us,

轉換后的圖像就是在輸入的頻率處出現小巔峰了

because the transform of a pure frequency

所以這個數學機器就是我們想要的。

is close to 0 everywhere

把原始頻率從一團糟中挑出來，

except for a spike around that frequency.

使混在一起的顏料分開

So when you add together two pure frequencies,

在繼續這個操作的數學描述之前，

the transform graph just has these little peaks above the frequencies that went into it.

讓我們快速看看這個東西有用的場景：

So this little mathematical machine does exactly what we wanted.

聲音編輯。

It pulls out the original frequencies from their jumbled up sums,

假設你有一段錄音，並且有一個煩人的高音，你想過濾掉。

unmixing the mixed bucket of paint.

那麼，首先，隨著時間的推移，信號的強度高低起伏。

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

通過麥克風，每毫秒輸入不同的電壓

let's just get a quick glimpse of one context where this thing is useful:

但是我們想從頻率的角度考慮這個問題，

Sound editing.

所以，

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

當你對信號進行傅里葉變換時，

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

令人討厭的高音將在高頻時出現。

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

（如果你可以的話）把這個高峰敲下去，

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

你會看到的就是原本錄音的傅里葉變換

so,

只有沒有了高音。

when you take the Fourier transform of that signal,

幸運的是，有一個反傅立葉變換的概念

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

就是說能透過傅里葉變換推出變換前的信號

Filtering that out, by just smushing the spike down,

我將在下一個視頻中更充分地討論逆變換，

what you'd be looking at is the Fourier transform of a sound that's just like your recording,

但長話短說，對傅里葉變換

only without that high frequency.

再用一次傅里葉變換，就能得到和原始函數差不多的圖形

Luckily, there's a notion of an inverse Fourier transform

嗯...差不多...就是這樣

that tells you which signal would have produced this as its Fourier transform.

這麼說有點唬人，但大方向沒錯

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

之所以說有點唬人是因為，我到現在也沒說真正的傅里葉變換是什麼

but long story short, applying the Fourier transform

因為它比“質心的x坐標”這個想法稍微複雜一些。

to the Fourier transform gives you back something close to the original function.

首先，把這個纏繞圖再拿出來，看看它的質量中心，

Mm, kind of... this is...

X坐標只能反應一半的情況，對吧？

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

我的意思是，這個東西是二維的，它還有y坐標。