科学与数学jezzamon.com编辑精选
傅里叶变换互动入门
An Interactive Introduction to Fourier Transforms
2019 — 2020，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这是 Jezzamon 制作的交互式傅里叶变换入门页面：它用可拖拽的动画、旋转的圆和滑块，演示如何把一段信号拆成一堆不同频率的正弦波，又如何把这些波重新叠加回原信号，并解释时域与频域之间的关系。页面面向没有信号处理背景的读者，强调动手玩耍式的直观理解。
为什么经久不衰：它把通常只出现在公式里的傅里叶分解变成了可以亲手拖动的图形，十几分钟就能建立起频率视角的直觉，因此多年来反复被推荐为最好的入门材料之一。
HN 评论精选
用卷积定理和移不变系统解释“为什么是正弦余弦”，而不是把它们当成凭空规定。
即使上过大学水平的信号处理课，我也没真正搞懂为什么偏偏要用正弦和余弦。所有傅里叶理论的讲解都一上来就说要把信号拆成正弦和余弦分量，也许还会说它们是最简单、最纯粹的频率实例，但这个想法仍然像是凭空掉下来的。我现在觉得我知道原因了：是卷积定理。做傅里叶变换时，我们把信号表示在一个新的基上，其中每个分量都会被线性移不变系统独立地变换——傅里叶基把卷积算子对角化了。更深层的原因是复指数函数平移只需乘一个常数。用更通俗的话说，这归结为理发店旋转灯柱的错觉：旋转一个螺旋形的三维曲线，看起来像在向上移动，而它的二维投影正是正弦和余弦。线性移不变系统非常常见，所以把信号拆成互不干扰、各自独立变换的部分很有用。
bonoboTP查看英文原文
Even after taking a college level signal processing course, I didn't really grok why we use sines and cosines. Any explanation of Fourier theory just starts with the idea to break up a signal into sine and cosine components. Perhaps they'd say those are the simplest, purest instantiation of a frequency. But the idea still just falls out of the sky. I believe I know now: it's because of the convolution theorem. When performing a Fourier transform, we represent the signal in a new basis, where each component will independently get transformed by linear shift invariant systems. Basically the Fourier basis diagonalizes the convolution operator. And the even deeper reason for that is that a complex exponential function can be shifted by simply multiplying it with a constant. In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are sines and cosines. And it turns out that linear, shift-invariant systems are really common (or are at least a good approximation of many common natural phenomena), so it's very helpful to break up a signal into pieces that each get independently transformed, without any interaction effects.
总结傅里叶变换的三大要点：能量守恒、可逆、幅度/相位带来的直觉。
在我看来傅里叶变换有三件很重要的事，两件是数学、一件是工程：一是它保持能量（帕塞瓦尔定理，变换后函数的范数与原函数相同）；二是存在逆变换可以还原原函数；三是一旦你领会到幅度和相位描述的是函数中的模式，你就能对变换以及如何把它当作分析和设计工具有强大的直觉。前两条说明变换不丢失信息，只是换一种方式看同一件事以获得更多洞见；第三条在工程课上强调得远远不够，我认为这正是很多人觉得控制论和信号处理像黑魔法的原因之一。一个自然的追问是：还有别的变换也保持能量且存在逆变换吗？答案是肯定的，而且有无穷多个；那第三条性质就更有用了——问题是这些变换里哪些真正有用。离散余弦变换就是一个例子，它在压缩、分类和机器学习（尤其是当代语音识别）中很关键。
holy_city查看英文原文
There are three really important things about the Fourier transform in my mind. Two are math and one is engineering. - the Fourier Transform preserves energy (Parceval's theorem, the norm of the transformed function is the same as the norm of the original) - there exists an inverse transform to get the original function back - once you grasp that magnitude/phase describe patterns in the function you can gain powerful intuition about the transform and how to use it as an analytical and design tool. Those first two properties tell us that the transform preserves information, basically it's another way of looking at the same thing to gain more insight without loss. The third is something not harped on enough in engineering courses, and failure to teach it in my mind is one reason so many people thing controls/signal processing is black magic. A big followup question here is, are there other transforms for which energy is preserved, and there exists an inverse? The answer is yes, there are infinitely many. The third property is more useful, which begs the question, which of those other transforms are useful? An example of this is a cousin of the Fourier Transform called the Discrete Cosine Transform which is critical in compression, classification, and machine learning (especially contemporary speech recognition). It's not as straightforward as Fourier, since the result isn't as obvious as breaking down the energy into patterns, but rather what it does is break the energy into correlated parts, in other words it preserves the energy while representing it in decorrelated bins. The strongest of those bins is the most important part, which is why compression technology works by taking the DCT and tossing out low magnitude components (it preserves the most important energy) while also showing how it can work for machine learning, where it decomposes the input information into an equivalent but decorrelated representation where inputs aren't shared for different parts of something like a neural net. There are other equally cool orthogonal transforms, I like the Hilbert transform myself because it can extract really useful info like signal envelopes and be used to make wacky noises, like a frequency shifter.
批评这类可视化只展示旋转圆，却无法解释傅里叶级数为何能逼近任意函数这一核心问题。
我很喜欢这些交互演示。但这类把一圈圈旋转的圆首尾相接、配上不同相位和幅度的视频和文章，总让我觉得缺了一件根本的事：这些构型究竟凭什么能逼近你任意给定的函数？在我看来这正是傅里叶变换的全部奥秘。旋转圆看起来新奇，却完全没说明傅里叶级数为什么收敛；正因此，我认为这个流行类比对初学者来说除了娱乐之外用处不大。当然，傅里叶级数收敛的细节正是经典调和分析的全部内容。一种粗略的理解方式是先采样，然后发现采样信号向量空间的 DFT 矩阵构成一组基；类似地，sin x 与 sin nx 在 0 到 2π 上的积分对所有 n 都是 0，而 {sin nx, cos nx} 的生成空间在某个函数空间里是稠密的——不过这对不懂的人来说只是数值上的巧合。我见过的每一篇这类文章在这方面都讲不下去，我觉得最有趣的部分反而被掩盖了。
enthdegree查看英文原文
love the interactive demos. one fundamental thing i always feel is missing with all these videos and articles about the spinny circles set end to end with different phases and amplitude is: why on earth do such configurations happen to have the capacity to approximate any function you prescribe?? to me this is the entire mystery behind fourier transforms. the spinny circles are kind of unusual to look at, but do nothing to illuminate to me why convergence of fourier series happens, and for this reason exactly i am of the opinion that this meme analogy is not useful for beginners beyond entertainment. of course the details for convergence of fourier series are the entire topic of classical harmonic analysis. one hand-wavy way to make sense of it is to first sample and then identify that the dft matrix for the vector space of sampled signals is a basis. kind of similarly, int dx sin x sin nx from 0 to 2pi is 0 for all n, and the span of {sin nx,cos nx} is somehow dense in some function space. although that isn't really very illuminating since to the uninformed it amounts to a numerical coincidence. every single article of this sort that i have seen falls flat in this respect and i feel like this most interesting part has been obscured.
指出人耳本身就是一台频谱分析仪，傅里叶分解在生物上也有对应。
有趣的事实：人类听觉系统的工作方式就是把 incoming（传入的）声音拆成频率——每根耳毛都有不同的固有共振模式，所以我们的大脑其实是在频率空间里处理信息的。
mensetmanusman查看英文原文
Fun fact: the human auditory system works by splitting incoming sounds into frequencies. Each hair has a different characteristic resonant mode, so our brain processes information in f-space. https://www.discovermagazine.com/mind/the-brain-ringing-in-the-ears-actually-goes-much-deeper-than-that
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