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Fourier transform (Fourier Transform, sometimes written as "Fourier Transform") is a particularly common mathematical tools, you've probably learned in college, but I want to specifically talk about. Fourier transform method is to build a foundation of modern technology, it can be said to be everywhere - and I feel there is wisdom behind this operation, it is worth everyone pondering.
Even if you have not formally studied, you've probably heard, "Fourier transform" the word. Sound and image signal on a computer, any fluctuation information on the project, solving differential equations in mathematics, astronomy observations of distant stars, everywhere should be used Fourier transform. You use the phone to play MP3 music, look at pictures, voice recognition, these are the daily application of the Fourier transform. Wu recently been speaking opened a class of information theory, which is also referred to the Fourier transform [1].
What is the Fourier transform it? Wikipedia to say that "is a linear integral transformation for signal conversion between the time domain (or spatial) and frequency domain" ...... This sentence is probably more difficult to understand, and people understand this sentence may not understand the nature of the Fourier transform. This lesson we ignore all the mathematical details, do not have a formula, straight thinking.
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In my view, in essence, Fourier transform is a complex thing to disassembled into a bunch of simple things standardized.
Let's take an example voice [2]. Note Sonic just one example of the application of the Fourier transform, the Fourier transform is they do not have the sound on, do not have to be about volatility.
Let's start with what is the "simple things." In fact, the sound is vibration of air. You flip a string, ears heard a pure, but in a short time is a continuous sound. Like an A note, about 440 times per second to shake, so unless a bass, you can usually not be able to feel the vibration, but you can feel the volume and tone - that is, the magnitude of the volume of vibration, the vibration frequency of the tone is.
The picture below show a simple sound. The abscissa is time and the ordinate is the amplitude of vibration. This presents the perfect sound of cyclical changes, indicating its frequency is fixed, it has a pure tone. The shape of the curve is the "sine wave" that is the way through high school sine curve.
This is a simple thing. The vast majority of real-world sounds are not simple, such as our voice to speak clearly not a pure tone. Enlarge the look, the sound is so messy complex following the shock -
Well, now comes the key insights: complex vibration, can be seen as a series of simple superposition of vibration.
Curve such as the one above may seem complicated, in fact, obtained by adding three simple fluctuating -
You can figure the bottom of complex curves as you feel the temperature change during the day. On the surface, you feel the temperature change is very complicated, but in fact, you know that you are experiencing three things at the same time. The red curve in the figure is equivalent to natural variations in temperature of nature, green curve is equivalent to you or outdoors, the blue curve represents you put on or take off a jacket indoors.
Put a complex matter broken down into three simple things that you completely Nengkanmingbai it in the end is how is it a child.
The so-called Fourier transform is to say, if we first ordained a series of simple fluctuation, any fluctuation in a complicated, you can use these simple dismantling fluctuations.
For example, we see the following waveform -
This shape looks a bit strange, but seems to have a law neat, that in the end is what law? Fourier transform is a mathematical operation, any shape can curve disassembled into a series of simple superposition waveform. Above this waveform, in fact, is superimposed below these types of waves -
The blue, is a series of simple fluctuations. Fourier transform tell us figure the contribution of each simple fluctuations on the red curve how much, for example,
The red curve is the frequency of blue curve = 100 × 0.5 + blue frequency of curve 200 × 0.2 + blue frequency of curve 300 × 0.1 + blue curve is the frequency of 400 × 0.08 + ......
Now imagine that, if the blue curve of various frequencies that are above everyone convention of "simple things standardized", then you want to describe the seemingly complex red curve, we only need to report the composition of each of its kind blue curve "ingredients" also on it!
The red curve = (0.5, 0.2, 0.1, 0.08, ......)
This is the Fourier transform.
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现在你看出傅里叶变换的好处来了没有？明明是一条复杂的曲线，可是我们只需要用几个数字就可以描写它！
这就是数字音乐的原理。那些标准化的简单音调都是大家约定好的，所以只需要记录一个声音分解成简单音调的成分值就行。而且因为特别高频和特别低频的声音人的耳朵是听不见的，所以标准化简单音调并没有无限多个，我们只需要用很有限的一组数字就能描写一段时间内的一个复杂声音……这就是最基本的 WAVE 音频格式。把 WAVE 文件里的信息再做一些压缩处理，就是 MP3。JPG 图像的原理也是类似的，只要把时间上的波动改成空间上的波动就行。
傅里叶变换并不要求你记录的这一段信息具有周期性。任何形状的线条都可以用那些标准化的简单曲线合成出来，哪怕只有一个周期也可以做，是不是“波动”并不重要。
那些“标准化的简单音调”都是如何选取的呢？这其中有一些讲究，要求“不重不漏”。所谓不漏，就是它们组合在一起必须在一定的分辨率之内，能覆盖耳朵能听见的所有频率；所谓不重，就是它们互相之间不能有重叠。比如你不能说这一个简单蓝色曲线又可以用其他几个简单蓝色曲线合成出来 —— 那样的话傅里叶变换的解就不是唯一的了。
这些标准化的简单事物是一个傅里叶变换的基石，你可以把它们想象成“维度”。复杂事物就好像是由那一大堆简单事物构成的多维空间中的一点，每一种简单事物的成分就构成了这个复杂事物的坐标。为了保证坐标系统的清爽，各个维度之间应该是互相垂直（数学语言叫“正交”）的关系，也就是谁也不能覆盖和取代谁。
数学概念就说到这里，下面说意义。
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你每一次对着菜谱做菜，都是在做傅里叶变换。
菜谱说，用这个、这个和这个食材，什么时候加多少盐，什么时候放多少水……那些食材、盐和水，就是傅里叶变换中的那些“简单的标准化事物”。
菜谱无需告诉你牛肉是什么东西、西蓝花是什么东西、盐和水又是什么东西，大家约定俗成都知道它们是什么东西。菜谱只需要把成分告诉你就行。
这说明什么呢？说明如果一个社会有一个大家约定俗成的、标准化的简单事物话语体系，我们的交流就会非常方便。这也说明，要想让交流方便和高效，你就必须得有一个约定俗成的、标准化的简单事物话语体系。
比如古代行军打仗有个最原始的密码系统。事先约定二十个字，每个字代表一个意思。通信的时候写一首诗，比如其中有一句是“大漠孤烟直”。收信人一看“大”字上盖了个章，而知道事先的约定是“大”的意思是要求增兵，就知道你想说什么。
没有这个标准化的约定，我们就无法有效交流。请问谁能用语言精确描写前面图中那条曲线呢？了解一个领域，就得了解这个领域的话语体系。
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现实中使用的傅里叶变换，总是失真的。理论上有无限个标准化简单音调，但是现实中我们只用有限个数字描写一个声音，这是因为那些不易分辨的、或者振幅特别低的音调都被省略了。所以对数字化声音来说，你得知道你面临下面这几个限制 ——
1. 你发不出不能用我们选取的那几个标准化音调描写的声音；
2. 你的声音的特别细微之处，将会被忽略；
3. 所有能传播的声音都是规定好的单纯声音的排列组合而已。
要不怎么福柯说，“人类的一切知识都是通过‘话语’获得的，任何脱离‘话语’的东西都是不存在的。”
这就意味着，在傅里叶变换的视角下，这个世界并没有什么新鲜的东西。
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比如有一天你做了一个梦。你觉得这个梦太精彩了，就把它写成了一个小说，你认为这要是拍成电影肯定能火！你兴冲冲地把它拿个一个编剧朋友看，结果他说，你这不就是《罗生门》× 0.5 + 《哈姆雷特》× 0.2 + 《侏罗纪公园》× 0.3 吗？
他给你的剧情做了个傅里叶变换。
现在的情况是凡是能想到的剧情，可能都已经被人拍过了。我以前专门写文章说过 [3]，TV Tropes 这个网站列举了所有的剧情桥段。
你所谓的创造，通常只不过是已知的、标准化的简单事物的排列组合而已。
这就是为什么成熟的领域里搞“纯创新”那么难。如果这个领域已经形成了自己特有的话语体系 —— 也就是说都用上傅里叶变换了 —— 你首先要做的大概是学习这个话语体系。
Fortunately, however, the real world is not necessarily a fully digital closed system [4], perhaps Fourier transform after all can not put the whole world to standardization.
Note
[1] Wu Jun · 40 talking about information theory, "10 equivalence: how the information is compressed? "
[2] The main example of this section and pictures from AATISH BHATIA, The Math Trick Behind MP3s, JPEGs, and Homer Simpson's Face, Nautilus, JUN 10, 2019.
[3] "never thought": plot will ruin your life.
[4] Japanese elite class in the second quarter, "Gödel's incompleteness theorem view of the world."