科学与数学betterexplained.com编辑精选
傅里叶变换交互式入门
An Interactive Guide to the Fourier Transform (2012)
2012 — 2025，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
BetterExplained 的经典科普长文，用交互式可视化把傅里叶变换从“绕圈的叠加”讲到公式的直观含义，避开密集的数学推导。文章自 2012 年起多次登上 HN 首页，是理解傅里叶变换直觉的常被推荐的入门读物。
为什么经久不衰：它把一个被公式埋没的核心数学思想转成可交互的直觉，十几年来仍是初学者理解傅里叶变换的最佳起点之一。
HN 评论精选
推荐配套的进阶书籍，并给出“轮子频闪”式的傅里叶系数直觉
我一直在思考如何把 FFT 应用到神经网络，这篇文章来得正是时候。如果想看更深入的讲解，我推荐《Practical Signal Processing》这本书，其中第 4 章“解缠绕”的直觉讲解是我见过最好的。换个说法：当轮子转得太快看起来像在倒转时，如果它看起来静止不动，那就在那个旋转频率上存在一个傅里叶系数——至少我认为是这样，用类比很难做到精确。
sillysaurusx查看英文原文
I've been thinking a lot about how to apply FFTs to neural networks, so this is timely; thank you. If you're looking for deeper explanations, I recommend "Practical Signal Processing": https://www.amazon.com/Practical-Signal-Processing-Mark-Owen/dp/1107411823 Many of the chapters are here: https://battle.shawwn.com/sdb/books/PSP/ but buy the book if you can afford to. In particular, Chapter 4's "untwisting" intuition is the best I've found: https://imgur.com/a/r7KvO60 Another way of phrasing it: You know that effect when a wheel is spinning so fast, it looks like it's going backwards? If it looks like it's standing still, there's a Fourier coefficient at that rotational frequency. ... Or at least, I think there is. It's hard to be precise with analogies.
反驳“数学是障碍”的流行说法，认为数学记号本身才是精确理解的来源
文章开头说“不幸的是，其含义被埋在密集的公式里”。我对此类说法总是持怀疑态度：很多文章都有这种模式——‘不幸’这个概念用数学表达，让我们来解决这个问题。我认为恰恰相反：数学和数学记号在工程教育中过早、过轻易地被抛弃，导致了粗糙且不精确的定义和理解。应该改成“幸运的是，其含义铭刻在密集的公式中”。
etamponi查看英文原文
From the beginning of the article: > The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations The post does a great job at explaining what the Fourier transform means, but I am always skeptical of sentences like this. It's a pattern found in many other posts too: "unfortunately" this concept is expressed in math form, let's try to solve this problem. I find that it's exactly the other way around. Math and math notation is dismissed too early and too easily even in engineering studies, resulting in shoddy and imprecise definitions and understandings. > The Fourier Transform is one of deepest insights ever made. Fortunately, the meaning is engraved in dense equations
批评科普式直觉讲解会带来误导，强调正交基与谐波限制才是正确理解
这类科普式讲解看完后你对数学的理解并不会更好，反而得到一种含糊且有误导性的认识。文章说傅里叶变换“测量每一种可能的循环”，这是错误的思考方式：傅里叶基是一组相互正交的正弦波，只是恰好覆盖了你感兴趣的频率范围。如果选一个不是采样周期谐波的正弦信号，变换结果会是一团难以解读的“涂抹”。相比之下相位信息是连续的，可能比频率信息更有意思。“圆周路径”的可视化也很有误导性。
geokon查看英文原文
At the end of these types of popsci explanations you still can't grok the math any better and you end up with a hand-wave-y misleading understanding of what's going on. It starts off by saying that the Fourier Transform "measures every possible cycle". This is an wrong approach to thinking of the Fourier Transform. If you think in these terms you will quickly get lost. The Fourier basis is a set of sinusoids that are mutually orthogonal (ie. none can be represented as a combination of the others). They just happen to generally cover the frequency ranges you're interested in (though this depends on your problem). If you choose some random sinusoidal signal that isn't a harmonic of your sampling period, then the Fourier Transform gives you back a mess that's hard to interpret (how a off-harmonic sinusoid gets "smeared" across the basis is actually something I frankly still don't quite understand https://ccrma.stanford.edu/~jos/mdft/Frequencies_Cracks.html ) This is in contrast to the phase information that you get that is continuous. Each frequency has two associated basis vectors and they can give you any phase offset. The frequencies are only set harmonics, but the phase can be any value. This is the result of how the basis is setup. If anything, the phase information you get from the Fourier transform is more interesting than the frequency information you can glean. The "Circular Paths" visualization is also highly misleading. The complex plane is a crutch that gives you pretty pictures. It pretends to be like a 2D plane but it's actually fundamentally distinct from a Cartesian/"normal" X-Y plane b/c complex numbers can be multiplied while normal [x,y] pairs can not (b/c the operation is simply undefined - you can't multiply two points on a map). The product of two complex numbers is giving you another complex number.. but good luck visualizing where that point ends up! You'll notice all complex plane examples stay on the unit circle b/c you get a "rotation" but it's an edge case. The whole 2D analogy is setting you up to be very confused
推荐 3Blue1Brown 的视频作为另一种优质直觉讲解
那 3Blue1Brown 关于傅里叶变换的视频呢？我觉得它也相当简洁直观，但似乎没有出现在这个讨论帖里。
itissid查看英文原文
What about the 3 blue 1 brown video on it? I think that is pretty concise and intuitive too but does not seem to appear on this thread https://www.youtube.com/watch?v=spUNpyF58BY&ab_channel=3Blue1Brown
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