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形象展示傅里叶变换 – 译学馆
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形象展示傅里叶变换

But what is the Fourier Transform? A visual introduction.

上面展示的 就是我们要在这期视频中
This right here is what we’re going to build to,
探讨的内容
this video:
我们将用动画来辅助考虑
A certain animated approach to thinking
数学中一个极其重要的概念
about a super-important idea from math:
傅里叶变换
The Fourier transform.
对于不熟悉傅里叶变换的观众来说
For anyone unfamiliar with what that is,
我的首要目标是
my # 1 goal here is just
借这个视频介绍概念
for the video to be an introduction to that topic.
但是 即便你
But even for those
很熟悉傅里叶变换
of you who are already familiar with it,
我还是觉得 了解下
I still think that there’s something fun
每一部分长什么样 既有趣 又能加深理解
and enriching about seeing what all of its components actually look like.
首先 最核心的例子很经典:
The central example, to start, is gonna be the classic one:
就是分解声音中的频率
Decomposing frequencies from sound.
但是在这之后 我还很想稍作说明
But after that, I also really wan na show a glimpse
这个概念的适用范围远不止声音和频率
of how this idea extends well beyond sound and frequency,
它在许多看似无关的数学领域 甚至是物理中都有体现
and to many seemingly disparate areas of math, and even physics. Really,
真是无所不在 令人震惊
it is crazy just how ubiquitous this idea is.
让我们直接进入正题吧
Let’s dive in.
现在播放的是纯A音
This sound right here is a pure A.
它的频率是440Hz 意思是说
440 beats per second. Meaning,
如果你测试耳机或者扬声器
if you were to measure the air pressure
附近的气压
right next to your headphones,
把它当做一个关于时间的函数 这个函数
or your speaker, as a function of time, it would oscillate up and down
也会在它的平衡点附近上下振荡
around its usual equilibrium, in this wave.
每秒产生440次振荡
making 440 oscillations each second.
对于一个更低的音 比如D音
A lower-pitched note, like a D,
波形是相同的 只是每秒振荡次数变少了
has the same structure, just fewer beats per second.
当这两个音同时播放时
And when both of them are played at once,
你认为最终的压强-时间图像是什么样的?
what do you think the resulting pressure vs. time graph looks like? Well,
在任意时刻 压强的变化就是
at any point in time, this pressure difference
各个音
is gon na be the sum of what it would be
产生的压强的总和
for each of those notes individually. Which,
说实在的
let’s face it,
这个东西有点复杂 很难想象
is kind of a complicated thing to think about.
在某些时刻 两个峰值相互重合
At some points, the peaks match up with each other,
产生很高的气压
resulting in a really high pressure.
而在其它时刻 它们又相互抵消
At other points, they tend to cancel out.
总而言之
And all in all,
你得到的类波形压强-时间图像
what you get is a wave-ish pressure vs. time graph,
并不是纯粹的正弦波 而是更复杂的波形
that is not a pure sine wave; it’s something more complicated.
当你加入更多音调
And as you add in other notes,
波形也会越来越复杂
the wave gets more and more complicated.
但是就目前来说 它不过是四个纯音的组合
But right now, all it is is a combination of four pure frequencies.
这个波形
So it seems…
虽然已知信息这么少 已经是很复杂了
needlessly complicated, given the low amount of information put into it.
麦克风在记录声音时
A microphone recording any sound
只能获取不同时刻的气压
just picks up on the air pressure at many different points in time.
它只能读取最终的复合波形
It only”sees” the final sum.
所以核心问题是 如何把
So our central question is gonna be how you can take
一个这样的信号
a signal like this,
分解为其中纯音的频率呢?
and decompose it into the pure frequencies that make it up.
有意思吧
Pretty interesting, right?
信号叠加会让它们全混在一起
Adding up those signals really mixes them all together.
所以把它们再分开 感觉就像
So pulling them back apart…feels
把混合好的颜料还原一样
akin to unmixing multiple paint colors that have all been stirred up together.
我们的大致策略是这样的
The general strategy is gonna
建造这么一台数学机器
be to build for ourselves a mathematical machine
使得它能够
that treats signals with a given frequency…
区别对待各个不同的频率
..differently from how it treats other signals.
首先 考虑只有一个信号的波
To start, consider simply taking a pure signal say,
假设它的频率是3Hz
with a lowly three beats per second,
我们能轻松画出它的图像
so that we can plot it easily.
而且我们只关注这个图像的一部分
And let’s limit ourselves to looking at a finite portion of this graph.
在这里 我们关注的部分是0到4.5秒
In this case, the portion between zero seconds, and 4.5 seconds.
关键思想在于
The key idea,
我们要把这个图像
is gon na be to take this graph,
缠绕在一个圆上 更具体点
and sort of wrap it up around a circle. Concretely,
解释下我的意思
here’s what I mean by that.
想象一个转动的向量
Imagine a little rotating vector where each point in time
在任意时刻下它的长度等于这个时刻的图像高度 所以
its length is equal to the height of our graph for that time. So,
在图像中
high points
高处的点对应于离原点较远
of the graph correspond to a greater disance from the origin,
低处的点对应于离原点较近
and low points end up closer to the origin.
我现在使用的绘图方法
And right now, I’m drawing it
是这样的 每过两秒
in such a way that moving forward two seconds in time
这个向量就转过一整圈
corresponds to a single rotation around the circle.
在缠绕图像中 这个向量
Our little vector drawing this wound up graph
每秒转过半圈
is rotating at half a cycle per second. So,
这一点很重要
this is important.
现在有两个不同的频率在起作用
There are two different frequencies at play here:
一个是信号的频率
There’s the frequency of our signal,
每秒上下振荡三次
which goes up and down, three times per second.
除此之外
And then, separately,
另一个是图像缠绕中心圆的频率
there’s the frequency with which we’re wrapping the graph around the circle. Which,
目前是每秒旋转半圈
at the moment, is half of a rotation per second.
但是我们可以自由地改变第二个频率
But we can adjust that second frequency however we want.
比如说 我们让它转得快一些
Maybe we want to wrap it around faster…
或者让它转得慢一点
..or maybe we go and wrap it around slower.
而且缠绕频率决定了缠绕图像的样子
And that choice of winding frequency determines what the wound up graph looks like.
有些缠绕图像可能相当复杂 但是
Some of the diagrams that come out of this can be pretty complicated; although,
的确很漂亮
they are very pretty.
要记住重要的一点
But it’s important to keep
我们所做的就是
in mind that all that’s happening here
把信号缠绕在一个圆上
is that we’re wrapping the signal around a circle.
顺便提一句 我在最上面的图像中画了些竖线
The vertical lines that I’m drawing up top, by the way,
它们是为了标清楚
are just a way to keep track
绕着圆旋转了整周时
of the distance on the original graph
原始图像中对应的位置
that corresponds to a full rotation around the circle. So,
如果竖线间隔是1.5秒
lines spaced out by 1.5 seconds
这就说明旋转一周需要1.5秒
would mean it takes 1.5 seconds to make one full revolution.
到现在为止
And at this point,
你可能隐约猜到接下来要发生有趣的事了
we might have some sort of vague sense that something special will happen
当缠绕频率和信号频率相等时 会出现特别的情况
when the winding frequency matches the frequency of our signal: three beats per second.
所有高处的点恰好
All the high points on the graph happen
落在圆右侧
on the right side of the circle
而所有低处的点恰好落在圆左侧
And all of the low points happen on the left.
但是我们要如何利用这一点
But how precisely can we take advantage of that
来建造一台频率分离机呢?
in our attempt to build a frequency-unmixing machine? Well,
那就把这个图像看成
imagine this graph is having some kind
一个有质量的东西 比如金属丝
of mass to it, like a metal wire.
这个小红点代表的是金属丝的质心
This little dot is going to represent the center of mass of that wire.
我们改变缠绕频率的时候图像的缠绕方式会发生变化
As we change the frequency, and the graph winds up differently,
质心的位置也会有所摆动
that center of mass kind of wobbles around a bit.
对于大部分缠绕频率来说
And for most of the winding frequencies,
图像的峰和谷会
the peaks and valleys are all spaced out
均匀分布在圆上
around the circle in such a way that
导致质心一直待在原点附近
the center of mass stays pretty close to the origin.
但是
But!
当缠绕频率等于
When the winding frequency is the same
信号频率时
as the frequency of our signal,
也就是 等于3Hz时
in this case, three cycles per second,
所有的峰落在右边
all of the peaks are on the right,
所有的谷落在左边
and all of the valleys are on the left..
特别地 质心就会偏向右侧
..so the center of mass is unusually far to the right. Here,
为了刻画这个现象 我们来画一个圆
to capture this, let’s draw some kind of plot
记录每个缠绕频率对应的质心位置
that keeps track of where that center of mass is for each winding frequency.
当然了 质心的位置是二维的
Of course, the center of mass is a two-dimensional thing,
需要两个坐标来完整表述
and requires two coordinates to fully keep track of,
但就目前来说 我们只记录它的横坐标
but for the moment, let’s only keep track of the x coordinate. So,
频率为0时
for a frequency of 0,
所有的点都堆在右边
when everything is bunched up on the right,
质心的横坐标相对较大
this x coordinate is relatively high.
当你增加缠绕频率时
And then, as you increase that winding frequency,
图像就会平均分布在圆上
and the graph balances out around the circle,
质心的横坐标
the x coordinate of that center
也就趋于0
of mass goes closer to 0,
之后它也只是在0附近不停摆动
and it just kind of wobbles around a bit.
但是 但频率等于3Hz时
But then, at three beats per second,
会出现一个尖峰 因为图像全都绕在右边
there’s a spike as everything lines up to the right.
这就是我们的核心构造
This right here is the central construct, so let
让我们来总结下到目前为止的内容
‘s sum up what we have so far: We
一个是原始的强度-时间图像
have that original intensity vs. time graph,
一个是
and then we have the wound
二维平面中的缠绕图像
up version of that in some two-dimensional plane,
除此之外 还有一个图像
and then, as a third thing, we have a plot
记录了缠绕频率如何影响缠绕图像的质心
for how the winding frequency influences the center of mass of that graph.
插一句
And by the way,
我们回头看看0附近的低频
let’s look back at those really low frequencies near 0.
在新图像中 0附近有个很大的尖峰
This big spike around 0 in our new frequency plot
它只是因为余弦曲线整体上移
just corresponds to the fact that the whole cosine wave is shifted up.
如果我选择的信号在0附近振荡
If I had chosen a signal oscillates around 0,
允许原信号有负值 那么
dipping into negative values, then,
我们在改变缠绕频率时
as we play around with various winding frequences,
质心-缠绕频率图像
this plot of the winding frequencies vs. center of mass
就只会在3处出现一个尖峰 不过
would only have a spike at the value of three. But,
负值
negative values are a
考虑起来既奇怪又麻烦
little bit weird and messy to think about
何况这是第一个例子
especially for a first example,
所以还是考虑上移的图像
so let’s just continue thinking in terms of the shifted-up graph.
你只需要明白
I just want you to understand
在0附近的尖峰只对应于上移罢了
that that spike around 0 only corresponds to the shift.
想要分解频率 我们的主要关注点就是
Our main focus, as far as frequency decomposition is concerned,
那个在3处的凸起
is that bump at three.
我会将这张图称为
This whole plot is what I’ll call
原信号的“近傅里叶变换”
the”Almost Fourier Transform” of the original signal.
它和真正的傅里叶变换相比
There’s a couple small distinctions
还是有几点小小的不同
between this and the actual Fourier transform,
过几分钟我就会讲
which I’ll get to in a couple minutes,
但你可能已经看出
but already, you might be able to see how
这个操作是如何帮我们挑出信号频率来的了
this machine lets us pick out the frequency of a signal.
我们再
Just to play around with it a little bit more,
换一个信号
take a different pure signal,
就拿这个稍低的频率吧
let’s say with a lower frequency of two beats per second,
施以同样的操作
and do the same thing.
绕成一圈 想像几个不同并且可能的缠绕频率
Wind it around a circle, imagine different potential winding frequencies,
与此同时 注意观察质心
and as you do that keep track of where the center
在什么地方
of mass of that graph is,
然后一边调整缠绕频率
and then plot the x coordinate of that center of mass
一边画出质心的x坐标
as you adjust the winding frequency.
和之前一样
Just like before,
在缠绕频率和信号频率相等时 出现了一个尖峰
we get a spike when the winding frequency is the same as the signal frequency,
这时就是每秒转两圈的时候了
which in this case, is when it equals two cycles per second.
但是真正的关键点
But the real key point,
——这个机器之所以让人喜闻乐见——
the thing that makes this machine so delightful,
是因为它能
is how it enables us to
读取包含好几个频率的信号
take a signal consisting of multiple frequencies,
再把不同的频率分出来
and pick out what they are.
就想一下我们刚看到的这两个信号吧
Imagine taking the two signals we just looked at:
3Hz的波
The wave with three beats per second,
和2Hz的波
and the wave with two beats per second,
捣鼓在一起
and add them up.
如我之前所说
Like I said earlier,
这么得到的就不是个纯余弦波了
what you get is no longer a nice, pure cosine wave;
而是更复杂的信号
it’s something a little more complicated.
但是想象一下 把这东西扔进我们的缠绕频率机器里去的话
But imagine throwing this into our winding-frequency machine…
肯定是越绕
..it is certainly the case that as you wrap this thing around,
看上去越复杂的
it looks a lot more complicated;
你得到的只是
you have this
混沌
chaos (1) and
杂乱 无序
chaos (2) and chaos (3) and
以及乱七八糟的东西 然后就
chaos (4) and then
哦?!
WOOP!
在每秒2圈的时候
Things seem to line up really nicely
图像整齐排列起来了
at two cycles per second,
再继续 乱七八糟
and as you continue on it’s more chaos (5)
乱七八糟
and more chaos (6)
乱七八糟
more chaos (7)
乱七八糟X3
chaos (8), chaos (9), chaos (10),
哦?!
WOOP!
在每秒3圈时又排列得超整齐了
Things nicely align again at three cycles per second. And,
跟我之前说的一样
like I said before,
绕起来的这个图像可能看起来又乱又复杂
the wound up graph can look kind of busy and complicated,
但这只不过是
but all it is is the relatively simple idea
把图像绕着圆缠起来了罢了
of wrapping the graph around a circle.
只不过是图像更复杂 缠绕频率更快罢了
It’s just a more complicated graph, and a pretty quick winding frequency.
这有两个不同的尖峰是因为
Now what’s going on here with the two different spikes,
如果你拿来两个信号
is that if you were to take two signals,
再分别对它们使用“近傅里叶变换”
and then apply this Almost-Fourier transform to each of them individually,
再把结果汇总
and then add up the results,
得到的结果 和先把信号加起来
what you get is the same as if you first
再进行“近傅里叶变换”是一样的
added up the signals, and then applied this Almost-Fourier transorm.
细心的观众可以停下想一想
And the attentive viewers among you might wan na pause and ponder, and…
自己体会下我所言不假
..convince yourself that what I just said is actually true.
对你来说这也是一个不错的挑战
It’s a pretty good test to verify
来看看这个缠绕机器
for yourself that it’s clear what exactly is being measured
内部到底测量的是个啥
inside this winding machine.
这一性质现在对我们来说是超级有用了
Now this property makes things really useful to us,
因为对单纯频率的转换
because the transform of a pure frequency
除了在其频率附近会出现一个尖峰以外 其它地方几乎都是0
is close to 0 everywhere except for a spike around that frequency.
所以在将两个单纯频率加起来以后
So when you add together two pure frequencies,
转换后的图像就在
the transform graph just has these little peaks
输入进的频率处出现了小尖峰了
above the frequencies that went into it.
这个小数学机器实现的正是我们想要的功能
So this little mathematical machine does exactly what we wanted.
把原有频率从一团糟里分离出来
It pulls out the original frequencies from their jumbled up sums,
使混在一起的颜料相互分离
unmixing the mixed bucket of paint.
在继续讲解此操作的完整数学描述之前
And before continuing into the full math that describes this operation,
我们来快速看看
let’s just get a quick glimpse
一个傅里叶变换特有用的场景
of one context where this thing is useful:
音频编辑
Sound editing.
比如说你现在手上有段录音
Let’s say that you have some recording,
里面有个超烦的高音啸叫
and it’s got an annoying high pitch that
你想把它过滤出去
you’d like to filter out. Well,
首先
at first,
信号以函数的形式输入 横轴是时间 竖轴是强度
your signal is coming in as a function of various intensities over time.
通过你的麦 每毫秒输入不同的电压
Different voltages given to your speaker from one millisecond to the next.
但我们想要以频率的角度来看待这个问题
But we want to think of this in terms of frequencies, so,
所以 对这一信号做傅里叶变换
when you take the Fourier transform of that signal,
烦人的啸叫看上去就会是
the annoying high pitch is going to show up just
某一高频率上的尖峰
as a spike at some high frequency.
把这个尖峰敲下去
Filtering that out, by just smushing the spike down,
看到的就是你
what you’d be looking at is the Fourier transform
所原本录下来声音的傅里叶变换了
of a sound that’s just like your recording,
只不过没了高音啸叫罢了
only without that high frequency. Luckily,
很幸运 我们还有“傅里叶逆变换”的概念
there’s a notion of an inverse Fourier transform
也就是说能通过傅里叶变换来推出变换前的信号来
that tells you which signal would have produced this as its Fourier transform.
我会在下期视频中
I’ll be talking
更多地探讨逆变换
about inverse much more fully in the next video,
长话短说 对傅里叶变换
but long story short, applying the Fourier transform
再施以傅里叶变换 得到的就是和原函数差不多的东西
to the Fourier transform gives you back something close to the original function. Mm,
唔 差不多吧
kind of… this is…
这么说是有点骗人
..a little bit of a lie,
但大方向没错
but it’s in the direction of the truth.
之所以说有点骗人主要是因为
And most of the reason that it’s a
我到现在也没说
lie is that I still have yet to tell
真正的傅里叶变换是什么样的
you what the actual Fourier Transform is,
毕竟它比这个“质心的x坐标”的概念要复杂那么一点点
since it’s a little more complex than this x-coordinate-of-the-center-of-mass idea.
首先 把这个绕起来的图再拿出来
First off, bringing back this wound up graph,
观察它的质心
and looking at its center of mass,
x坐标只能反映一半的事实 对吧
the x coordinate is really only half the story, right?
我是说 这东西毕竟是个2维图形
I mean, this thing is in two dimensions,
还有y坐标呢
it’s got a y coordinate as well. And,
往往在数学中
as is typical in math,
无论什么时候 处理二维问题的时候
whenever you’re dealing with something two-dimensional,
把它看做是复平面都是很方便的
it’s elegant to think of it as the complex plane,
此时质心就是个复数
where this center of mass is gonna be a complex number,
有着实部和虚部
that has both a real and an imaginary part.
之所以以复数角度看待事物
And the reason for talking in terms of complex numbers,
而不是简单地说
rather than just saying,
有两个坐标
“It has two coordinates,”
是因为复数能很好地描述
is that complex numbers lend themselves to really nice descriptions
和缠绕
of things that have to do with winding,
旋转等相关的事物
and rotation.
比如
For example:
超有名的欧拉公式告诉我们
Euler’s formula famously tells us
取e的n乘i次方
that if you take e to some number times i,
你就会落在
you’re gonna land on the point that you get
从右边开始 沿着半径为1的单位圆
if you were to walk that number of units around a circle
逆时针走了n个单位弧长的点上
with radius 1, counter-clockwise starting on the right.
所以说
So,
比如说你想要描述一个每秒钟转一圈的旋转
imagine you wanted to describe rotating at a rate of one cycle per second.
那你就可以
One thing that you could do
用e^2πit来表示
is take the expression”e^2π*i*t,”
其中t表示经过的时间 因为
where t is the amount of time that has passed. Since,
对半径为1的单位圆来说
for a circle with radius 1,
2π就是一圈的长度 不过
2π describes the full length of its circumference. And…
这看起来有点眼晕
this is a little bit dizzying to look at,
那就换一个
so maybe you wan na describe a different frequency…
更低也更合理的频率
..something lower and more reasonable…
这时
..and for that,
只需要在指数的t前面
you would just multiply that time t in the exponent
乘上频率f就好
by the frequency, f.
比如 如果f是1/10
For example, if f was one tenth,
那么此向量就每十秒转一整圈
then this vector makes one full turn every ten seconds,
因为只有在t增长
since the time t has to increase all
到10的时候 整个指数部分才是2πi
the way to ten before the full exponent looks like 2πi.
如果你感兴趣的话
I have another video giving some intuition
我在另一期视频里讲了
on why this is the behavior of e^x for imaginary inputs,
e的虚数次方为什么长这样的一些直观解释
if you’re curious ,
但就现在而言
but for right now,
我们拿来用就好
we’re just gon na take it as a given.
你现在可能问 “讲这些干嘛啊” 是这样的
Now why am I telling you this you this, you might ask. Well,
它能给我们一种超棒的方法
it gives us a really nice way
来将“缠绕图像”的思想
to write down the idea of winding up the
写成简洁紧凑的公式
graph into a single, tight little formula.
首先 在傅里叶变换的语境下 通常认为
First off, the convention in the context of Fourier transforms
旋转是沿顺时针方向的
is to think about rotating in the clockwise direction,
我们就在指数项前面加个负号吧 现在
so let’s go ahead and throw a negative sign up into that exponent. Now,
拿一个描述信号强度和时间关系的函数出来
take some function describing a signal intensity vs. time,
就比如我们之前用的这个纯余弦波好了
like this pure cosine wave we had before,
命名为g(t)
and call it g(t).
将这个指数函数乘上g(t)
If you multiply this exponential expression times g(t),
意思就是这个旋转的复数
it means that the rotating complex
依照函数值大小
number is getting scaled up and down
被缩放了
according to the value of this function.
这样就能
So you can think
将这个长度不断变化的旋转向量
of this little rotating vector with its changing length
看作是在画出缠绕起来的图像了
as drawing the wound up graph.
仔细想想 岂不美哉
So think about it, this is awesome.
这么个漂亮的小公式就囊括了
This really small expression is a super-elegant way to encapsulate
整个 按照可变频率f
the whole idea of winding a graph around a circle
将图像绕圆缠起来的想法
with a variable frequency f.
别忘了
And remember,
我们要这个缠绕图像的目的
that thing we want to do with this wound up graph
是跟踪它的质心
is to track its center of mass.
想想用什么公式能捕捉这一特性
So think about what formula is going to capture that. Well,
先至少估计一下吧
to approximate it at least,
你可以试试在原信号上
you might sample a whole bunch
取一堆样本点
of times from the original signal,
看看这些点在绕好的图上处在什么位置
see where those points end up on the wound up graph,
然后取个平均
and then just take an average.
也就是说 把它们都作为复数加起来
That is, add them all together, as complex numbers,
然后再除以样本点总数
and then divide by the number of points that you’ve sampled.
当你取的点更多 它们也就挨得更近 结果也就更准确
This will become more accurate if you sample more points which are closer together.
取极限时
And in the limit,
不再认为是把一大堆点加起来
rather than looking at the sum of a whole bunch
再除以点数
of points divided by the number of points,
而是对函数做积分
you take an integral of this function,
再除以时间区间的长度
divided by the size of the time interval that we’re looking at.
对复函数做积分看起来可能怪怪的
Now the idea of integrating a complex-valued function might seem weird,
对那些看到微积分就发抖的同学来说 可能就更吓人了
and to anyone who’s shaky with calculus, maybe even intimidating,
但这背后的思想并不需要微积分的知识
but the underlying meaning here really doesn’t require any calculus knowledge.
整个表达式不过是绕好的图的质心罢了
The whole expression is just the center of mass of the wound up graph.
所以
So…
很棒!一步步地
Great! Step-by-step,
我们就建立起了这个
we have built up this
有点复杂 但是仔细一看 其实还挺小的表达式
kind of complicated, but, let’s face it, surprisingly small expression
来表达我刚才说的整个缠绕机器的思想
for the whole winding machine idea that I talked about.
现在 离真正的傅里叶变换
And now, there is only one final distinction to point out
就只剩下最后一处不同了
between this and the actual, honest-to-goodness Fourier transform. Namely,
即 不用除以时间间隔
just don’t divide out by the time interval.
傅里叶变换就只是其中的积分部分
The Fourier transform is just the integral part of this.
其含义
What that means is that instead
不再是质心
of looking at the center of mass,
而是把它倍增
you would scale it up by some amount.
如果说
If the portion
原图像持续了3秒
of the original graph you were using spanned three seconds,
那就把质心乘上3
you would multiply the center of mass by three.
持续了6秒
If it was spanning six seconds,
就乘上6
you would multiply the center of mass by six.
物理上的效果就是
Physically, this has the effect
如果某个频率持续了很长时间
that when a certain frequency persists for a long time,
这个频率的傅里叶变换的模长
then the magnitude of the Fourier transform
就被放得很大
at that frequency is scaled up more and more.
比如我们正在看的这个
For example, what we’re looking at right here
就是频率为
is how when you have a pure frequency
2Hz的信号
of two beats per second,
以每秒2圈的圈速
and you wind it around the graph
绕起来时
at two cycles per second,
质心始终待在同一点 对吧
the center of mass stays in the same spot, right?
一直是同一个形状
It’s just tracing out the same shape.
但这个信号存在得越久
But the longer that signal persists,
此频率的傅里叶变换的值就越大
the larger the value of the Fourier transform, at that frequency.
对其他频率来说
For other frequencies, though,
就算你稍微增大了一点
even if you just increase it by a bit,
也会被抵消掉 因为
this is cancelled out by the fact
时间越长
that for longer time intervals
缠绕图像就越很可能
you’re giving the wound up graph more
在圆上均匀布开
of a chance to balance itself around the circle.
这次讲的内容多了点
That is… a lot of different moving parts,
我们还是停下来 总结一下目前为止做了什么
so let’s step back and summarize what we have so far.
对强度-时间函数 比如g(t)的
The Fourier transform of an intensity vs. time function, like g(t),
傅里叶变换是个新函数 其因变量不是时间 而是频率
is a new function, which doesn’t have time as an input, but instead takes in a frequency,
也就是我之前所称的缠绕频率
what I’ve been calling”the winding frequency.”
提一句 在记号上
In terms of notation, by the way,
我们通常把这个新函数
the common convention is to call this new function
叫做“g帽” 头上戴一个“^”号
“g-hat,” with a little circumflex on top of it.
新函数的输出值是一个复数
Now the output of this function is a complex number,
也就是2维平面上的一个点
some point in the 2D plane,
它对应原信号中某一频率的强度
that corresponds to the strength of a given frequency in the original signal.
这里傅里叶变换的图像
The plot that I’ve been graphing for the Fourier transform,
只是这一实部 也就是x坐标
is just the real component of that output, the x-coordinate
如果你想更完整描述的话
But you could also graph the imaginary component separately,
也可以把虚部单独画出来
if you wanted a fuller description.
所有这些都囊括在了我们刚刚建立起的那个公式里了
And all of this is being encapsulated inside that formula that we built up.
不难想象
And out of context,
抽象地看 这一公式有多吓人
you can imagine how seeing this formula would seem sort of daunting.
但是只要你理解了指数项和旋转的联系
But if you understand how exponentials correspond to rotation…
以及把它和函数g(t)乘起来
..how multiplying that by the function g(t)
意味着画一张缠绕版的图像
means drawing a wound up version of the graph,
还有如何通过质心的思想
and how an integral of a complex-valued function
解释对复函数的积分
can be interpreted in terms of a center-of-mass idea,
就不难看出这些符号的背后
you can see how this whole thing carries
其实是非常直观的
with it a very rich, intuitive meaning. And,
别急
by the way,
还得加一个小注解 这个话题才能算真正地结束
one quick small note before we can call this wrapped up.
经管在实际操作中 比如说编辑音频
Even though in practice, with things like sound editing,
你是对有限的时间进行了积分
you’ll be integrating over a finite time interval,
但在描述傅里叶变换时
the theory of Fourier transforms is often phrased where the bounds
积分上下限通常为正负无穷 说到底
of this integral are -∞ and ∞. Concretely,
其含义是 考虑此表达式
what that means is that you consider this expression
在每个有限时间区间上的值
for all possible finite time intervals,
然后看
and you just ask,
当时间区间趋近无穷时 极限是什么
“What is its limit as that time interval grows to ∞?” And…man,
天呐 要说的东西真的太多了
oh man, there is so much more to say!
多到我不想让它在这结束
So much, I don’t wanna call it done here.
傅里叶变换涉及的数学领域
This transform extends to corners of math well
绝不仅限于提取信号频率 所以
beyond the idea of extracting frequencies from signal. So,
在下期视频中
the next video I put out
我会挑其中几个讲一讲
is gon na go through a couple of these,
那才是真正有意思的部分
and that’s really where things start getting interesting. So,
所以请关注这个频道
stay subscribed for when that comes out,
或者
or an alternate option is to
连刷几个3blue1brown的视频
just binge a couple 3blue1brown videos
新视频推出时 好让Youtube更愿意
so that the YouTube recommender is more inclined
给你自动推荐 当然
to show you new things that come out… ..really,
决定权在你手上
the choice is yours!
结束之前 我还有一道有趣的数学问题 这个问题来自本期视频的赞助商
And to close things off, I have something pretty fun: A mathematical puzzler from this video’s sponsor,
Jane Street 他们正在招募更多技术人才
Jane Street, who’s looking to recruit more technical talent. So,
假设三维空间中有一个有界
let’s say that you have a closed,
闭凸集C
bounded convex set C sitting in 3D space,
B是集合C的边界
and then let B be the boundary of that space,
也就是这个复杂图形的表面
the surface of your complex blob.
考虑平面上所有的二元点对
Now imagine taking every possible pair of points on that surface,
按照向量加法 把它们想加
and adding them up, doing a vector sum.
所有可能加和结果的集合叫做D
Let’s name this set of all possible sums D.
你的任务是 证明D也是凸集
Your task is to prove that D is also a convex set. So,
Jane Street是一家量化交易公司
Jane Street is a quantitative trading firm,
如果你是那种
and if you’re the kind
喜欢数学和解决难题的人
of person who enjoys math and solving puzzles like this,
由于他们的团队非常看重求知欲 所以
the team there really values intellectual curiosity. So,
他们或许会想要聘用你
they might be interested in hiring you.
他们正在招聘全职员工和实习生
And they’re looking both for full-time employees and interns.
就我而言
For my part,
我接触过这家公司的几个人 他们似乎
I can say that some people I’ve interacted with there just seem to
都热爱数学 也可以分享数学
love math, and sharing math,
他们招聘时并不过于看重金融背景
and when they’re hiring they look less at a background in finance
而更看重你的思考方式 学习方法
than they do at how you think, how you learn,
以及你解决问题的方法
and how you solve problems,
所以他们赞助了一期3Blue1Brown的视频
hence the sponsorship of a 3blue1brown video.
如果你想知道之前谜题的答案
If you want the answer to that puzzler,
或是了解Jane Street的工作
or to learn more about what they do,
抑或是应聘其岗位
or to apply for open positions,
就去访问janestreet.com/3b1b吧
go to janestreet.com/3b1b

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译制信息
视频概述

动态地认识傅里叶变换:把图像以一定频率盘绕在圆上。

听录译者

收集自网络

翻译译者

Aidenlazz

审核员

审核员 EM

视频来源

https://www.youtube.com/watch?v=spUNpyF58BY

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