Sanderson Maps Fourier Transform to Circular Center-of-Mass Machine

But what is the Fourier Transform? A visual introduction. — 3Blue1Brown

3Blue1Brown

Wrapping a time-domain signal around a circle at variable rates isolates constituent frequencies because non-matching tones cancel out while resonant frequencies shift the curve's center of mass.

AI-assisted summary of But what is the Fourier Transform? A visual introduction. — original source. Check the original for full context.

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Key ideas

  • The Fourier transform operates intuitively by wrapping a time-domain wave around a circle at varying winding rates and measuring the center of mass to detect resonant frequencies. (2:29)
  • Because the winding operation is linear, combined audio waves decompose into sharp, isolated spikes corresponding directly to each constituent pure frequency. (8:07)
  • The formal Fourier transform equation encapsulates circular winding using Euler's formula in the expression $g(t)e^{-2\pi i f t}$, omitting duration normalization so that sustained frequencies scale proportionally with signal length. (14:43)
  • Beyond audio filtering, the Fourier transform maps functions of time into complex-valued functions of frequency that encode both amplitude and phase across physics and pure mathematics. (17:08)

Detailed notes

## Background Published by Grant Sanderson on the YouTube channel 3Blue1Brown on January 26, 2018, this video provides an animated visual introduction to the Fourier transform. The video description frames the lesson as an intuitive exploration of frequency decomposition, made largely using Sanderson's custom Python animation library, manim.

## TL;DR The video demystifies the Fourier transform by presenting an intuitive visual metaphor: taking a time-domain wave and wrapping it around a circle at varying winding frequencies using a rotating vector. When the winding rate aligns with a frequency component present in the signal, the wound-up curve bunches toward one side, causing its center of mass to spike away from the origin. Sanderson formalizes this geometric mechanism into Euler's formula and complex numbers, showing how the term e^(-2πi f t) naturally handles clockwise circular rotation. Finally, by transitioning from the average center of mass to an unnormalized integral over time, he constructs the complete mathematical definition of the Fourier transform, explaining how it cleanly extracts individual frequencies and phase information from blended signals.

## Atmosphere & style The presentation carries 3Blue1Brown's signature calm, elegant, and intellectually satisfying tone, backed by gentle ambient music from Vincent Rubinetti. Sanderson guides the viewer through complex calculus and abstract algebra with patient, grounded explanations, transforming what is often seen as an intimidating formula into an intuitive geometric machine. The visual progression feels smooth and mesmerizing as tangled, chaotic loops suddenly snap into clean, resonant alignments, instilling a genuine sense of aesthetic and conceptual clarity.

## Bulletin - [02:29](https://www.youtube.com/watch?v=spUNpyF58BY&t=149s) The Fourier transform operates intuitively by wrapping a time-domain wave around a circle at varying winding rates and measuring the center of mass to detect resonant frequencies. - [08:07](https://www.youtube.com/watch?v=spUNpyF58BY&t=487s) Because the winding operation is linear, combined audio waves decompose into sharp, isolated spikes corresponding directly to each constituent pure frequency. - [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) The formal Fourier transform equation encapsulates circular winding using Euler's formula in the expression $g(t)e^{-2\pi i f t}$, omitting duration normalization so that sustained frequencies scale proportionally with signal length. - [17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) Beyond audio filtering, the Fourier transform maps functions of time into complex-valued functions of frequency that encode both amplitude and phase across physics and pure mathematics.

## Key takeaways - [00:04](https://www.youtube.com/watch?v=spUNpyF58BY&t=4s) A recorded audio signal combines individual frequency oscillations into a single net air pressure graph over time, mixing constituent notes together. - [01:22](https://www.youtube.com/watch?v=spUNpyF58BY&t=82s) Decomposing an aggregated sound wave back into pure frequencies resembles unmixing distinct colors of stirred paint. - [02:29](https://www.youtube.com/watch?v=spUNpyF58BY&t=149s) The winding machine wraps a time-domain signal around the origin of a 2D plane by matching a rotating vector's magnitude to the signal's value at each moment. - [03:24](https://www.youtube.com/watch?v=spUNpyF58BY&t=204s) The geometric appearance of the wound-up graph is governed by the difference between the signal's intrinsic frequency and the chosen winding frequency. - [04:20](https://www.youtube.com/watch?v=spUNpyF58BY&t=260s) When the winding frequency matches the signal frequency, peaks align symmetrically on one side of the circle, pulling the center of mass away from the origin. - [04:59](https://www.youtube.com/watch?v=spUNpyF58BY&t=299s) Tracking the horizontal coordinate of the center of mass against winding frequency produces an 'almost Fourier transform' marked by sharp resonant peaks. - [08:07](https://www.youtube.com/watch?v=spUNpyF58BY&t=487s) Due to linearity, computing the transform of a sum of signals is identical to summing the transforms of each individual signal. - [09:34](https://www.youtube.com/watch?v=spUNpyF58BY&t=574s) Transform linearity allows complex, blended audio waves to be cleanly unmixed into distinct frequency spikes. - [10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) Audio filtering isolates unwanted noise frequencies as distinct spikes in frequency space, allowing them to be suppressed before applying an inverse Fourier transform. - [11:49](https://www.youtube.com/watch?v=spUNpyF58BY&t=709s) The 2D plane is naturally formalized as the complex plane, where the center of mass is represented as a complex number with real and imaginary components. - [12:21](https://www.youtube.com/watch?v=spUNpyF58BY&t=741s) Euler's formula provides a compact mathematical notation, $e^{-2\pi i f t}$, to describe smooth, clockwise rotation at frequency $f$. - [13:52](https://www.youtube.com/watch?v=spUNpyF58BY&t=832s) Multiplying a signal $g(t)$ by $e^{-2\pi i f t}$ analytically models the physical process of winding the graph around a circle in the complex plane. - [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) The center of mass over a time window is formally computed as the continuous integral $\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t)e^{-2\pi i f t}\,dt$. - [15:32](https://www.youtube.com/watch?v=spUNpyF58BY&t=932s) The authentic Fourier transform drops the duration divisor $\frac{1}{t_2 - t_1}$, causing resonant peak magnitudes to scale linearly with how long a frequency persists. - [17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) The resulting Fourier transform $\hat{g}(f)$ maps frequency inputs to complex outputs whose magnitude represents frequency strength and whose angle represents phase. - [18:03](https://www.youtube.com/watch?v=spUNpyF58BY&t=1083s) Formal Fourier analysis evaluates limits spanning negative to positive infinity, underpinning advanced physics concepts including the Heisenberg uncertainty principle.

## Detailed walkthrough ### [00:04](https://www.youtube.com/watch?v=spUNpyF58BY&t=4s) Unmixing Sound: The Core Intuition of the Fourier Transform

The goal is to build an animated, visual understanding of one of the most vital concepts in mathematics: the Fourier transform. While the classical entry point is decomposing sound into constituent frequencies, the underlying idea stretches across wildly disparate fields of math and physics. As Grant Sanderson notes: “Really, it's crazy just how ubiquitous this idea is.”

To ground the concept, consider the sound of a pure concert A note at 440 beats (or cycles) per second. If you plot the air pressure next to a speaker or headphone as a function of time, it oscillates above and below its ambient equilibrium in a regular wave, completing 440 full oscillations every second. A lower note, such as a D, exhibits the exact same periodic wave shape, merely oscillating with fewer beats per second.

When two or more distinct notes are sounded simultaneously, the net pressure difference at any specific instant in time is simply the algebraic sum of the individual waves. At moments where the peaks align, the combined pressure surges; at other moments, a peak in one note encounters a trough in another, canceling out. As more notes are layered together, the combined pressure-versus-time graph rapidly becomes chaotic and irregular. Yet, despite how jagged and intricate the resulting wave appears, it contains only the underlying information of those few component frequencies. Because a physical microphone recording a performance registers only the total net air pressure at each moment, it captures only this aggregated sum. Decomposing that raw time-domain signal back into its pure, constituent frequencies presents a non-trivial challenge. Grant Sanderson summarizes the paradox: “Adding up those signals really mixes them all together, so pulling them back apart feels akin to unmixing multiple paint colors that have all been stirred up together.”

```mermaid flowchart TD accTitle: The Fourier Analysis Concept accDescr: Flowchart illustrating how mixed time-domain audio signals are decomposed into constituent frequencies using a winding machine.

subgraph TimeDomain["Time Domain (Combined Signal)"] A["Pure Note A (e.g., 440 Hz)"] --> C["Superposition: Mixed Waveform"] B["Pure Note D (Lower Pitch)"] --> C end

subgraph Machine["The Mathematical Machine"] C --> D["Wrap Wave Around Circle at Variable Winding Frequency"] D --> E["Calculate Center of Mass (x-coordinate)"] end

subgraph FrequencyDomain["Frequency Domain (Decomposition)"] E --> F["Fourier Peak at Note D Frequency"] E --> G["Fourier Peak at Note A Frequency"] end ```

[02:29](https://www.youtube.com/watch?v=spUNpyF58BY&t=149s) The Winding Machine: Wrapping Graphs Around a Circle

To separate the blended frequencies, the strategy is to design a mathematical mechanism that treats a specific target frequency fundamentally differently from all others. To trace the mechanics cleanly, start with a simple, pure test signal oscillating at a manageable rate of just 3 beats per second, plotted over a finite interval from $t = 0$ seconds to $t = 4.5$ seconds. The central conceptual trick is to take this linear wave graph and literally wrap it around a circle.

Concretely, this wrapping process is carried out by an imaginary rotating vector originating from the center of a two-dimensional plane. At every moment in time, the length of the vector equals the height of the signal graph at that precise timestamp. High points on the waveform cause the vector to extend far from the origin, while valleys keep it pulled in close. Simultaneously, the vector rotates around the origin at a chosen winding rate. For instance, if traversing 2 seconds of time along the original wave corresponds to completing one full $360^\circ$ circle, the vector rotates at a winding frequency of half a cycle per second ($0.5\text{ Hz}$).

Crucially, this setup involves two entirely distinct frequencies: - The intrinsic frequency of the signal itself (here, oscillating 3 times per second). - The winding frequency at which the graph is being wrapped around the origin.

Adjusting this winding frequency alters the visual geometry of the wound-up graph, producing shapes ranging from lopsided spirals to intricate circular petals. Grant Sanderson emphasizes: “And that choice of winding frequency determines what the wound up graph looks like.” To track these relationships visually, vertical reference markers spaced across the linear wave indicate intervals corresponding to one full revolution around the circle (for instance, markers spaced every 1.5 seconds when the winding period is 1.5 seconds). When the winding frequency is dialed until it matches the internal frequency of the wave (3 cycles per second), something dramatic happens: all the peaks of the wave line up on the right-hand side of the circle, while all the valleys concentrate on the left.

[04:59](https://www.youtube.com/watch?v=spUNpyF58BY&t=299s) Finding the Center of Mass and the "Almost Fourier Transform"

To turn this geometric alignment into a concrete metric, imagine the wound-up graph as a physical wire having uniform mass. Because mass tracks the wire's length, the center of mass of this wire provides an exact summary of where the curve concentrates in the plane. For the vast majority of winding frequencies, the peaks and troughs distribute fairly evenly around the circumference, meaning their pull balances out and the center of mass hovers very close to the central origin.

However, when the winding frequency matches the signal frequency (3 beats per second), the wire's long extensions consistently align toward the right, pulling the center of mass unusually far in that positive direction. Tracking only the horizontal ($x$) coordinate of this center of mass as a function of the winding frequency yields a new graph: - Near a winding frequency of 0, everything bunches up on the positive side because the original wave was shifted upward into strictly positive values, creating a large initial spike at zero. - As the winding frequency increases, the peaks distribute around the circle, dropping the center of mass toward zero, where it gently wobbles. - At exactly 3 cycles per second, the center of mass surges outward, creating a distinct, sharp peak on the plot.

Grant Sanderson describes this resulting plot: “This whole plot is what I'll call the almost Fourier transform of the original signal.” He clarifies that while the spike at zero is merely an artifact of shifting the wave upward above the horizontal axis to keep values intuitive, the defining feature of interest for decomposing frequencies is that resonant bump at 3. Running the same process on a pure 2-beat-per-second wave produces an identical characteristic spike centered cleanly at 2 cycles per second.

- The center of mass is fundamentally a two-dimensional coordinate $(x, y)$, but tracking just the horizontal $x$-coordinate is sufficient to demonstrate the resonance effect. - A signal oscillating symmetrically around zero (with negative values) eliminates the artificial zero-frequency spike, leaving only the resonant peak at the true frequency.

[08:07](https://www.youtube.com/watch?v=spUNpyF58BY&t=487s) Linearity and Unmixing Multiple Frequencies

The true power of this winding mechanism emerges when processing a combined wave made by adding a 2-cycle-per-second wave and a 3-cycle-per-second wave together. The combined wave looks noticeably distorted and lacks the clean simplicity of a single cosine curve. When wrapped around the circle, it generally appears as a tangled, chaotic nest of loops. Yet, as the winding dial sweeps through frequencies, the chaos momentarily crystallizes into an aligned loop at 2 cycles per second, drops back into disorder, and then aligns again at 3 cycles per second.

This occurs because of linearity: computing the center-of-mass winding transform on individual signals and adding those outputs together yields the exact same curve as adding the signals first and then running the transform. Grant Sanderson highlights the utility of this mathematical property: “Now this property makes things really useful to us, because the transform of a pure frequency is close to zero everywhere except for a spike around that frequency, so when you add together two pure frequencies, the transform graph just has these little peaks above the frequencies that went into it.”

Because the winding transform produces near-zero values across non-matching frequencies and produces distinct spikes only at resonant frequencies, the transform of an aggregated wave simply displays clean, isolated peaks matching each individual component. The machine successfully unmixes the blended signal, separating out the original ingredients just as one might hope to separate mixed paints. This operation serves immediate practical applications, such as audio editing, where an unwanted high-pitched drone in a time-domain recording can be isolated in the frequency spectrum and cleanly filtered out.

[10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) Filtering Frequencies and the Complex Plane

Viewing an audio signal purely as raw voltage supplied to a speaker millisecond by millisecond obscures its harmonic structure. When that signal is converted into frequency space via the Fourier transform, an annoying high-pitched noise ceases to be an intractable mess in time and instead isolates cleanly into an individual spike at a high frequency. Removing that nuisance is conceptually as simple as smushing down that unwanted spike. The resulting modified plot represents the Fourier transform of the clean audio, and an operation known as the inverse Fourier transform reconstructs the exact time-domain sound wave corresponding to that filtered profile. As Grant Sanderson notes when introducing this duality, “applying the Fourier transform to the Fourier transform gives you back something close to the original function.”

```mermaid flowchart LR accTitle: Audio Filtering with the Fourier Transform accDescr: Pipeline showing sound transformed to frequency space, filtered, and inverted back to audio. subgraph TimeDomain1["Time Domain"] A["Original Audio Signal<br/>(voltage vs. time)"] end subgraph FrequencyDomain["Frequency Domain"] B["Fourier Transform<br/>(frequency spectrum)"] C["Filtered Spectrum<br/>(smush high-pitch spike)"] end subgraph TimeDomain2["Clean Time Domain"] D["Inverse Fourier Transform<br/>(cleaned sound wave)"] end

A -->|"Fourier transform"| B B -->|"Filter noise"| C C -->|"Inverse transform"| D ```

Calling repeated Fourier transformation the exact path back to the original function is an instructive slight oversimplification, largely because tracking only the horizontal center of mass of the wound-up graph ignores critical geometry. A two-dimensional wound-up plot possesses both an $x$-coordinate and a $y$-coordinate. Rather than treating these coordinates as disconnected pairs of real numbers, mathematical convention packages them into the complex plane, where the center of mass is a single complex number composed of a real part and an imaginary part.

- Audio filtering operates by identifying specific frequency spikes, attenuating or zeroing them out, and synthesizing the remaining components back into the time domain. - The two-dimensional coordinates of the wound-up path map naturally to complex numbers of the form $x + iy$. - Complex numbers provide the natural mathematical language for describing circular motion, winding, and phase.

---

[12:21](https://www.youtube.com/watch?v=spUNpyF58BY&t=741s) Euler’s Formula and the Winding Expression

Complex numbers excel at modeling winding and circular rotation because of Euler's formula. When evaluating $e$ raised to an imaginary power, $e^{i\theta}$, the resulting value lands on the point located $\theta$ units counterclockwise along the unit circle starting from $(1, 0)$ on the positive real axis. A full revolution requires tracing the circle's circumference of $2\pi$, so a rotation at a standard rate of 1 cycle per second is written as $e^{2\pi i t}$, where $t$ denotes elapsed time. To adjust this pace to any desired frequency $f$, time is multiplied by $f$ in the exponent, yielding $e^{2\pi i f t}$. For instance, setting $f = \frac{1}{10}$ forces $t$ to advance all the way to 10 seconds before the exponent reaches $2\pi i$, thereby completing one gentle rotation every 10 seconds.

In Fourier analysis, the standard convention is to rotate clockwise rather than counterclockwise, requiring a negative sign inside the exponent. If a time-varying intensity signal $g(t)$ multiplies this rotating exponential, the unit-length rotating vector scales up and down according to the instantaneous amplitude of $g(t)$. The tip of this changing vector sketches out the wound-up signal in the complex plane across time. Sanderson marvels at the compactness of this description: “So think about it, this is awesome, this really small expression is a super elegant way to encapsulate the whole idea of winding a graph around a circle with a variable frequency, $f$.”

- Euler's formula establishes that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, placing values on a unit circle centered at the origin. - The term $e^{-2\pi i f t}$ governs clockwise rotation at a frequency of $f$ cycles per unit time. - Multiplying $g(t) \cdot e^{-2\pi i f t}$ scales the rotating complex exponential by the signal's value at time $t$, exactly implementing the winding machine.

---

[14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) From Center of Mass to the Actual Fourier Transform

To compute the center of mass of this wound-up curve mathematically, one can sample points at regular time intervals, sum their complex positions, and divide by the total sample count. Taking the limit as the sample points become infinitely dense replaces the discrete arithmetic average with an integral divided by the total duration: $\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t)e^{-2\pi i f t}\,dt$. While integrating a complex-valued function can appear intimidating to students rusty with calculus, its geometric meaning is completely unpretentious: it merely identifies the balance point of the wound curve in the complex plane.

```mermaid flowchart TD accTitle: Construction of the Fourier Transform Formula accDescr: Progression from discrete sampling to the continuous center of mass, and finally to the unnormalized Fourier integral. A["Sample Discrete Complex Points<br/>g(t) · e^(-2πi f t)"] --> B["Compute Sample Average<br/>(Sum of points / N)"] B --> C["Continuous Center of Mass<br/>(1 / Δt) ∫ g(t) e^(-2πi f t) dt"] C -->|"Drop the (1 / Δt) scale factor"| D["Honest Fourier Transform<br/>G(f) = ∫ g(t) e^(-2πi f t) dt"] ```

The transition from this physical center of mass to the actual, honest-to-goodness Fourier transform involves one crucial modification: dropping the division by the time interval length. The true Fourier transform is strictly the integral part, $\int g(t)e^{-2\pi i f t}\,dt$. Omitting the time divisor means the output scales proportionally with the duration over which a frequency persists. If a pure 2-beat-per-second signal is wound at 2 cycles per second, its center of mass remains anchored at the same offset point whether the clip lasts 3 seconds or 6 seconds; however, its Fourier transform magnitude doubles from a factor of 3 to a factor of 6. Meanwhile, non-matching frequencies loop evenly around the origin, experiencing near-complete cancellation as longer observation windows give the graph more opportunities to balance out.

- The center of mass of a continuous wound-up curve over an interval is $\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t)e^{-2\pi i f t}\,dt$. - The authentic Fourier transform omits the normalization factor $\frac{1}{t_2 - t_1}$. - Scaling with signal length ensures that sustained frequencies accumulate greater transform magnitudes, cleanly separating persistent pure tones from transient noise.

---

[17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) The Fourier Transform Function and Infinite Limits

Synthesizing all these moving parts reveals the structural definition of the transform. The Fourier transform maps a function of time, $g(t)$, into a brand-new function of frequency, universally denoted as $\hat{g}(f)$ (spoken as "$g$-hat"). The output of $\hat{g}(f)$ is a complex number whose distance from the origin encodes the strength of frequency $f$ within the original signal, while its angular orientation encodes the phase. Graphs depicting the transform often plot the real component (the horizontal balance coordinate), though graphing the imaginary component provides equal mathematical fidelity.

Although real-world audio engineering and data processing inherently operate across finite recording durations, formal mathematical theory defines the transform over the infinite horizon: $$\hat{g}(f) = \int_{-\infty}^{\infty} g(t)e^{-2\pi i f t}\,dt$$ This represents the limit of the integral evaluated across finite windows as the bounds expand toward negative and positive infinity. Reflecting on the synthesis, Sanderson emphasizes: “And out of context, you can imagine how seeing this formula would seem sort of daunting, but if you understand how exponentials correspond to rotation, how multiplying that by the function $g(t)$ means drawing a wound up version of the graph, and how an integral of a complex valued function can be interpreted in terms of a center of mass idea, you can see how this whole thing carries with it a very rich intuitive meaning.” He concludes by previewing that the Fourier transform reaches into far deeper areas of mathematics and physics—including the Heisenberg uncertainty principle—well beyond simple frequency extraction from audio signals.

- Input to $\hat{g}$: frequency $f$ (winding frequency). Output of $\hat{g}$: a complex number capturing amplitude and phase. - Practical signal analysis integrates over bounded windows $[t_1, t_2]$, whereas formal theory evaluates limits spanning $-\infty$ to $\infty$. - The machinery serves as the bedrock for wave mechanics, partial differential equations, and quantum uncertainty.

## Notable quotes > [10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) applying the Fourier transform to the Fourier transform gives you back something close to the original function. — Grant Sanderson

## Mentioned 3Blue1Brown, Grant Sanderson, Leonhard Euler, Werner Heisenberg, Vincent Rubinetti, Jane Street, Manim, Prajwal Souza, Omer Tuchfeld

## Video description An animated introduction to the Fourier Transform. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/fourier-thanks Learn more about Janestreet: https://janestreet.com/3b1b

Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4

Interactive made by a viewer inspired by this video: https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html

Also, take a look at this Jupyter notebook implementing this idea in a way you can play with: https://github.com/thatSaneKid/fourier/blob/master/Fourier%20Transform%20-%20A%20Visual%20Introduction.ipynb

Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Russian: xX-Masik-Xx Vietnamese: @ngvutuan2811

------------------ Animations largely made using manim, a scrappy open-source python library. https://github.com/3b1b/manim

If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------

3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).

If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended

Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown

[Source](https://www.youtube.com/watch?v=spUNpyF58BY)

中文简报

桑德森将傅里叶变换具象为圆周质心机制 以可变频率将时域信号缠绕在圆周上能够分离出各组成频率，因为非匹配音调会相互抵消，而共振频率则会使曲线的质心发生偏移。 该视频通过一种直观的视觉隐喻揭开了傅里叶变换的神秘面纱：利用旋转向量，以不同的缠绕频率将时域波形缠绕在圆周上。当缠绕速率与信号中存在的某个频率分量一致时，缠绕后的曲线就会聚集向某一侧，导致其质心明显偏离原点。桑德森随后将这一几何机制形式化为欧拉公式和复数表达，展示了表达式 e^(-2πi f t) 如何自然地处理顺时针圆周旋转。最后，通过从平均质心过渡到未归一化的时域积分，他构建出了傅里叶变换的完整数学定义，解释了它如何从混合信号中干净地提取出单个频率和相位信息。 傅里叶变换的直观运行方式是将时域波形以可变的缠绕速率缠绕在圆周上，并通过测量质心来检测共振频率。 (2:29) 由于缠绕操作是线性的，复合音频波形会分解为若干尖锐而独立的峰值，直接对应于各个组成的纯频率。 (8:07) 傅里叶变换的形式化方程在表达式 $g(t)e^{-2\pi i f t}$ 中利用欧拉公式概括了圆周缠绕，并且省略了持续时间归一化，使得持续存在的频率强度能够随信号长度成比例放大。 (14:43) 除了音频滤波，傅里叶变换还将时间函数映射为编码了振幅和相位的复数值频率函数，广泛应用于物理学与纯数学领域。 (17:08)

Reading summary

Grant Sanderson illustrates that the Fourier transform can be understood intuitively as a geometric winding machine that wraps a time-domain wave around a circle to isolate its component frequencies. By tracking where the resulting graph's center of mass concentrates as the winding frequency changes, the transform decomposes tangled wave superpositions into clean frequency peaks. Wrapping Waves to Detect Resonant Frequencies (2:29) When multiple pure sound waves combine, their air pressure variations add together algebraically into a complex, irregular waveform, making individual frequencies difficult to extract directly from time-domain data. To untangle these mixed signals, Sanderson introduces a visual mechanism that wraps a wave graph around the origin of a two-dimensional plane. An imaginary vector rotates around the center at an adjustable winding frequency, while its length is scaled continuously by the wave's amplitude at each moment in time. When the winding frequency does not match the wave's intrinsic rate, the plotted wire balances fairly evenly around the circumference, keeping its physical center of mass near the origin. However, when the winding frequency matches the wave's frequency—such as winding a 3-cycle-per-second wave at 3 cycles per second—the wave's peaks align on one side of the circle while its troughs concentrate on the other. Tracking this offset center of mass across various winding speeds generates what Sanderson calls the 'almost Fourier transform,' characterized by a distinct resonant spike at the exact frequency of the signal. Linearity and Frequency-Domain Filtering (8:07) The practical power of the winding mechanism rests on linearity: calculating the transform of individual frequencies and adding the results together yields the exact same curve as transforming an already aggregated wave. Because individual pure frequencies produce near-zero center-of-mass values everywhere except at their specific resonance spikes, running a blended signal through the machine separates it into isolated peaks corresponding directly to its original components. This separation underpins applications such as audio filtering. In a raw audio recording, an unwanted high-pitched drone appears as an intractable, chaotic distortion across the time domain. Once mapped into frequency space via the Fourier transform, that noise isolates cleanly into a single high-frequency spike. Attenuating or zeroing out that specific spike and applying the inverse Fourier transform reconstructs a clean time-domain sound wave free of the unwanted sound. Formulating Circular Motion with Euler's Formula (12:21) Tracking only the horizontal coordinate of the center of mass discards valuable geometric detail. To capture both coordinates, Fourier analysis frames the two-dimensional winding plane as the complex plane, representing positions as numbers of the form $x + iy$. Euler’s formula, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, provides the mathematical machinery for this representation by placing values at an angle of $\theta$ along a unit circle. By convention, Fourier analysis winds clockwise, expressed mathematically as $e^{-2\pi i f t}$, where $f$ represents the winding frequency in cycles per unit time and $t$ denotes elapsed time. Multiplying this rotating unit exponential by an amplitude function $g(t)$ dynamically scales the vector, tracing out the wound-up signal in the complex plane. Sanderson highlights that this compact product, $g(t)e^{-2\pi i f t}$, cleanly encapsulates the entire concept of wrapping a variable signal around a circle. The Fourier Integral and Duration Scaling (14:43) Calculating the center of mass mathematically involves taking the continuous average of the wound curve over a given interval: $\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t)e^{-2\pi i f t}\,dt$. The formal Fourier transform modifies this formulation by dropping the interval length divisor $\frac{1}{t_2 - t_1}$. Omitting this time division ensures that the output scales with duration; a persistent frequency held for six seconds accumulates twice the transform magnitude of one lasting only three seconds, whereas non-matching frequencies continue to cancel out near the origin. In formal mathematical theory, the integral is evaluated across infinite bounds, defining the transform as $\hat{g}(f) = \int_{-\infty}^{\infty} g(t)e^{-2\pi i f t}\,dt$. The resulting complex output $\hat{g}(f)$ encodes both the amplitude of frequency $f$ via its distance from the origin and the initial phase via its angular direction. Sanderson concludes that while the formal Fourier transform formula appears intimidating out of context, its components directly mirror the intuitive geometry of winding waves and locating centers of mass. He notes that the utility of this transformation reaches far beyond audio engineering, serving as a core foundation for differential equations, wave mechanics, and the Heisenberg uncertainty principle in quantum physics.

精炼阅读摘要

3Blue1Brown的Grant Sanderson指出，傅里叶变换的核心思想在于将复杂的时域叠加信号解构为其纯粹的组成频率。通过将一维波形在复平面上按可变频率“缠绕”成圆周几何图形，并利用质心偏移检测频率共振，数学上能够以高度直观的形式实现对混合信号的解构与滤波。 叠加难题与圆周缠绕机制 (2:29) 现实中的音频信号（例如440赫兹的标准音A与较低的D音同时演奏）在麦克风处仅记录为瞬时气压代数叠加后的总波形。不同频率波峰波谷的叠加会迅速形成极度杂乱的曲线，Sanderson将其比喻为把多种颜料混合在一起，逆向拆解极具难度。为了分离出各个组成频率，数学机制需要一种能够将特定目标频率与所有其他频率区分开来的处理手段。 该方案的核心直观设计是建立一个“缠绕机器”：在二维平面中引入一个自原点出发的旋转向量，其长度随着输入信号的瞬时幅度变化，并以给定的“缠绕频率”匀速旋转。此时时域波形被绘制成盘绕形状。系统中存在两个独立频率：波形固有的振动频率与坐标系的缠绕频率。当调节缠绕频率直至其与波形自身的频率完全吻合时，波形的所有波峰都会整齐地聚集在圆周的同一侧，波谷则集中在相反一侧，呈现出显著的不对称几何形态。 质心共振、线性性质与音频滤波 (4:59) 若将缠绕后的曲线视为一根质量均匀的导线，其质心位置便提供了量化几何聚集程度的物理指标。对于绝大多数不匹配的缠绕频率，曲线的峰谷均匀散布在圆周各处，拉力相互抵消，质心始终紧贴原点附近；唯有当缠绕频率精确匹配信号固有频率（例如每秒3次循环）时，质心才会向波峰聚集的一侧显著偏离。仅追踪该质心的水平横坐标随缠绕频率变化所形成的图表，被Sanderson称为原信号的“准傅里叶变换”。 得益于微积分与几何的线性性质，由多个纯频波（如2赫兹与3赫兹）叠加构成的复合信号，其缠绕图形的质心轨迹等同于各分量独立缠绕后质心轨迹的矢量和。由于非共振频率处的输出几乎为零，复合信号的变换图谱仅在各个组成频率处呈现出分立且清晰的尖峰。这种分离能力构成了音频滤波的理论基础：在频域中将代表噪音的高频尖峰压平，再通过傅里叶逆变换重构，即可完美还原出滤除杂音的时域音频。 欧拉公式与复平面数学表达 (12:21) 为了完整记录二维缠绕图形在平面上的全部几何信息，数学上使用复数坐标 $x + iy$ 替代单一的实数横坐标。欧拉公式确立了复指数与圆周运动的关系，表达式 $e^{i\theta}$ 描绘了单位圆上的点。若要实现每秒旋转 $f$ 圈且依照傅里叶分析的惯例采用顺时针旋转，指数项需调整为 $e^{-2\pi i f t}$。 当时间信号 $g(t)$ 与该旋转项相乘时，即构成了 $g(t)e^{-2\pi i f t}$。Sanderson强调，该简短的算式极为优雅地封装了整个缠绕机制：单位复指数项负责维持设定频率的顺时针匀速旋转，而信号函数 $g(t)$ 则同步缩放旋转向量的模长，从而在复平面上描摹出缠绕曲线的瞬时轨迹。 质心积分修正与完整傅里叶变换 (14:43) 在连续时间窗口 $[t_1, t_2]$ 内计算缠绕导线的质心，对应于复数函数积分后除以时间区间长度：$\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t)e^{-2\pi i f t}\,dt$。然而，从物理质心转向量学上真正的傅里叶变换时，必须舍弃时间跨度除数 $\frac{1}{t_2 - t_1}$。去掉该归一化系数后，积分结果的大小将与信号持续的时间成正比；持续发声的频率会随着时间推移累积出越来越高的尖峰，而不匹配的频率因反复在原点周围抵消而保持微弱，这使得长时间观测更易于区分稳定纯音与瞬时噪声。 在数学形式理论中，有限积分区间进一步拓展至正负无穷大，形成了标准定义式 $\hat{g}(f) = \int_{-\infty}^{\infty} g(t)e^{-2\pi i f t}\,dt$。该变换将时域函数 $g(t)$ 映射为频域函数 $\hat{g}(f)$，输出的复数不仅其模长对应着特定频率的振幅强弱，其旋转角度亦完整编码了该频率分量的相位信息。 Sanderson总结指出，尽管现实工程与音频处理只能在有限时间窗口内截取采样，但傅里叶变换的理论价值远超单纯的声音频率分解；它不仅是波动方程与偏微分方程的核心求解工具，其数学内核更直接贯穿了量子力学中的海森堡测不准原理等深层物理定律。

Detailed notes — zh

## Background 本内容由知名数学科普频道 3Blue1Brown（Grant Sanderson）于 2018 年 1 月 26 日在 YouTube 平台发布。视频主要通过生动的动态动画与几何视角，直观拆解傅里叶变换（Fourier Transform）的核心数学直觉与应用机制。 ## TL;DR 视频从声音信号的分解切入，将难以分离的复合波形类比为混合油漆，阐述了提取单一频率成分的核心问题。作者通过构建一台巧妙的“缠绕机器”，将时域信号动态缠绕在圆周上，并利用质心随缠绕频率变化的偏移规律精准定位各频率尖峰。随后，内容自然引入复平面与欧拉公式，将质心的几何概念严谨推导为定积分表达式，最终完整还原出标准傅里叶变换的数学定义及其物理意义。 ## Atmosphere & style 整部视频散发着 3Blue1Brown 标志性的优雅、沉静与启发性美感。伴随着轻柔灵动的背景音乐与精雕细琢的几何动画，原本枯燥晦涩的微积分与复分析公式被拆解得层次分明、如丝般顺滑。讲解语气温和且循循善诱，既充满对数学对称与统一性的真诚赞叹，又在关键概念处带有轻松幽默的互动感，给观众带来一种智力上被点亮的愉悦体验。 ## Bulletin - [02:40](https://www.youtube.com/watch?v=spUNpyF58BY&t=160s) 傅里叶变换的核心物理与几何机制，可直观理解为将时域信号作为可变半径并以特定频率缠绕在复平面圆周上，通过追踪图形质心的位移来识别固有频率。 - [02:40](https://www.youtube.com/watch?v=spUNpyF58BY&t=160s) 当缠绕频率与信号内部的实际频率精确匹配时，曲线的波峰波谷将在复平面一侧集中，使得质心大幅偏离原点并激发出显著尖峰。 - [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) 真正的连续傅里叶变换公式为 $\hat{g}(f) = \int g(t) e^{-2\pi i f t} dt$，它在数学本质上等于质心坐标乘以持续时间，从而能够线性反映特定频率分量的能量累积。 - [06:44](https://www.youtube.com/watch?v=spUNpyF58BY&t=404s) 基于线性叠加性，复杂混合信号在频域中表现为各个纯频分量尖峰的独立并存，这使得在音频中滤除特定噪声并借助逆变换重构波形变得极度直观且简便。 ## Key takeaways - [00:04](https://www.youtube.com/watch?v=spUNpyF58BY&t=4s) 纯音（如 440 Hz 的 A 音符）在时域中表现为标准正弦波，不同音符同时发声时会在空气中线性叠加形成复杂的混合波形。 - [00:04](https://www.youtube.com/watch?v=spUNpyF58BY&t=4s) 麦克风仅能记录每个时刻的总气压或电压变化，傅里叶变换的初衷正是从这种混合信号中反向分离出纯频分量。 - [02:40](https://www.youtube.com/watch?v=spUNpyF58BY&t=160s) 构建用于提取频率的“缠绕机器”时，包含两个独立频率：信号本身的固有振动频率与将波形绕圆周展开的旋转频率。 - [02:40](https://www.youtube.com/watch?v=spUNpyF58BY&t=160s) 在不匹配的缠绕频率下，波峰与波谷均匀散落于圆周各处，图形质心在原点附近徘徊；一旦频率对齐，波峰聚集会导致质心显著偏向一侧。 - [06:44](https://www.youtube.com/watch?v=spUNpyF58BY&t=404s) 将缠绕频率与质心横坐标绘制成图即可得到“近傅里叶变换”，对于在零上下对称振荡的纯音，图像仅在目标频率处形成尖峰。 - [06:44](https://www.youtube.com/watch?v=spUNpyF58BY&t=404s) 根据积分与变换的线性叠加性，先叠加信号再缠绕求质心，与各自变换后再相加完全等价，这保证了多频率混合信号在频域能被清晰剥离。 - [10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) 在音频编辑中，高频刺耳噪音在频域对应尖锐脉冲，直接平抑该尖峰并执行傅里叶逆变换即可无损消除噪声。 - [11:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=715s) 复平面是描述二维平面旋转的最优数学工具，质心因此可以完备地表示为包含实部（振幅/水平位移）和虚部（相位/垂直位移）的复数。 - [11:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=715s) 根据欧拉公式，逆时针单位旋转为 $e^{2\pi i f t}$，傅里叶变换依数学惯例采用顺时针旋转，因而指数引入负号写作 $e^{-2\pi i f t}$。 - [11:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=715s) 将时域强度函数 $g(t)$ 与顺时针旋转单位向量相乘，其几何意义就是动态缩放旋转矢量的长度，在复平面上精准绘出缠绕轨迹。 - [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) 离散质心求和在采样趋于无限时对应定积分除以时间跨度 $\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t) e^{-2\pi i f t} dt$，消除微积分复值积分的神秘感。 - [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) 真正的傅里叶变换移除了分母上的时间跨度，使得结果反映质心乘以持续时间，从而让持续更久、更确定的频率分量在频谱中展现更强的峰值。 - [17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) 在理论数学中，积分上下限从负无穷延伸至正无穷，其实际物理含义是考察有限时间区间无限扩大时的极限行为。 - [17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) 傅里叶变换将时间自变量函数 $g(t)$ 转化为频率自变量函数 $\hat{g}(f)$，其思想深植于现代信号处理、高阶数学乃至量子力学的不确定性原理。 ## Detailed walkthrough ### [00:04](https://www.youtube.com/watch?v=spUNpyF58BY&t=4s) 傅里叶变换的核心目标与声音的本质 本期内容旨在通过生动的动画形式，深入拆解数学中一个极其关键的概念——傅里叶变换（Fourier Transform）。对于初学者而言，这是一个直观的入门引导；而对于早已熟悉该概念的人来说，观察其所有组成部分的实际动态形态同样是一件极具启发性且富有趣味的事情。整个讲解从经典的声音频率分解入手，随后还会展现该思想如何跳出声音与频率的范畴，广泛延伸到看似毫不相干的数学乃至物理学各个分支。正如 3Blue1Brown 所感叹的那样：“Really, it's crazy just how ubiquitous this idea is.”（真的，这个概念的无处不在简直令人匪夷所思。） 以纯音 A 为例，它的振动频率为每秒 440 次（440 Hz）。如果测量耳机或扬声器旁边的气压随时间变化的关系，会发现气压在平衡位置上下振荡，形成每秒摆动 440 次的正弦波。像音符 D 这样更低沉的音符具有完全相同的结构，只是每秒振动次数较少。 当两个音符同时演奏时，气压与时间的图像会变成两者的叠加。在某些时刻，两列波的波峰重叠，导致极高的气压；在其他时刻，它们彼此抵消。最终得到的不是纯粹的正弦波，而是一条复杂得多的波形。若再加入更多音符，波形会越来越杂乱。然而，这种复杂性是表面上的，它本质上仅仅是几个纯频信号的组合。话筒在录制声音时只能捕捉每个时刻的总气压，即最终的叠加结果。因此，核心问题就在于：如何从这一复杂的混合信号中反向分解出构成它的纯频分量？这犹如要把已经彻底搅拌在一起的多色油漆重新分离开来。 ```mermaid flowchart LR accTitle: 声音合成与傅里叶分解流程 accDescr: 展示纯音频信号叠加为混合波形，再通过傅里叶变换还原出原始频率成分的过程。 subgraph SoundMix["声音叠加（正向）"] A["纯音 A (440 Hz)"] --> C["混合气压信号<br/>(时间域)"] B["纯音 D (较低频率)"] --> C end subgraph Decomposition["信号分解（逆向）"] C --> D["数学机器<br/>(傅里叶变换)"] D --> E["频率谱图<br/>(提取出 440 Hz 与 D 频率)"] end ``` --- ### [02:40](https://www.youtube.com/watch?v=spUNpyF58BY&t=160s) 缠绕机器与质心分析法 为了实现“分离油漆”的目标，需要构建一台能够对特定频率产生差异化响应的数学机器。首先考虑一个每秒仅振动 3 次的纯频信号，截取 0 秒到 4.5 秒的有限时间片段。核心思路在于：将该图像像绕线一样“缠绕”在一个圆周上。具体做法是引入一个旋转矢量，在每一个时间点，该矢量的长度等于信号在该时刻的函数值（高度）。波形的高点对应远离原点的位置，波形的低点则靠近原点。 缠绕过程中存在两个截然不同的频率： 1. 信号本身的固有频率（例如每秒上下振动 3 次）。 2. 将图像绕圆周缠绕的旋转频率（缠绕频率，例如设为每秒旋转半圈）。 随着缠绕频率的人为调节，缠绕图案的形态会发生剧烈变化。为了量化这种变化，可以将缠绕后的曲线想象成具有质量的金属丝，并追踪其“质心”（center of mass）。在绝大多数缠绕频率下，波峰与波谷均匀散布在圆周四周，导致质心基本停留在原点附近波动；然而，一旦缠绕频率与信号频率完全匹配（即每秒缠绕 3 圈），所有的波峰都会精准落在圆周的右侧，而所有的波谷则集中在左侧，使得整个物体的质心大幅向右偏移。 - 信号时间轴上的垂直标记线用来标示对应圆周完整旋转一周的时间跨度（例如间隔 1.5 秒代表缠绕一周耗时 1.5 秒）。 - 质心是一个二维点，需要两个坐标来确定；此处先简化记录其横坐标（x 坐标）。 - 缠绕频率为 0 时横坐标较高，因为所有点都挤在右侧；随着频率增加，质心回落到原点附近微幅颤动；当频率达到 3 Hz 时，x 坐标产生了一个极其显著的尖峰（spike）。 --- ### [06:44](https://www.youtube.com/watch?v=spUNpyF58BY&t=404s) “近傅里叶变换”与信号叠加分离 绘制出缠绕频率与质心横坐标的关系图像后，便得到了原始信号的“近傅里叶变换”（almost Fourier transform）。图像在零频率附近的隆起仅仅是因为原始余弦波被整体向上平移成了正值；如果信号本身是在零上下对称振荡并带有负值，那么整个图像将只在频率为 3 的位置出现唯一的尖峰。 这一机制的威力在于处理复合信号。当我们将一个 2 Hz 的波和一个 3 Hz 的波简单相加，送入缠绕机器中时，尽管轨迹在绝大多数频率下显得极其混沌混乱，但机器在旋转到 2 Hz 时会迅速规整化并产生一个尖峰，随后继续旋转回到混沌，在达到 3 Hz 时又再次精准对齐并激发出第二个尖峰。 这背后的根本数学原理在于线性叠加性：先将两个信号分别进行缠绕变换再将结果相加，与先将两个信号相加再进行缠绕变换，其结果完全一致。针对这一性质，3Blue1Brown 指出：“The attentive viewers among you might want to pause and ponder and convince yourself that what I just said is actually true.”（细心的观众不妨暂停思考一下，说服自己我刚才说的确实是真的。）由于单一纯频信号的变换结果除了目标频率处有尖峰外其余位置均接近于零，多个频率混合后的变换图像就清晰地在各个基频位置矗立起尖峰，从而完美实现了将混杂在一起的频率分离开来的目的。这种能力在实际应用中极其强大，例如在音频编辑中，工程师借此能够精准定位并滤除录音中恼人的高频噪音。 ### [10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) 信号滤波与傅里叶逆变换的直观理解 当我们观察麦克风录制的音频信号时，原始数据仅仅体现为扬声器电压随着时间每一毫秒的起伏变化。然而，在处理声音时，我们更希望从“频率”的视角来审视它。对该信号执行傅里叶变换后，原本混杂在声音中刺耳的高频蜂鸣声，会清晰地展现为高频区域的一个尖锐脉冲。想要过滤掉这个噪音，只需在频域图中将这个尖峰直接抹平（压下去）；此时所得到的图像，就是一段剥离了高频噪音、与原音频高度相似的声音信号的傅里叶变换结果。 从修改后的频域图像重构出声音，需要依赖“傅里叶逆变换”（Inverse Fourier Transform），它能反推出到底是哪种时域信号生成了当前的频谱。Grant Sanderson 强调：“Long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function.”（简而言之，对傅里叶变换的结果再施加一次傅里叶变换，就能得到接近原始函数的东西。）他随即笑着坦白，这种说法“Kind of, this is a little bit of a lie, but it's in the direction of truth”（在某种程度上算是一个小谎言，但它大致切中了本质真相）。之所以说是谎言，是因为到目前为止，视频所探讨的“缠绕图像质心 $x$ 坐标”还不是真正、完整的傅里叶变换。 ### [11:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=715s) 复平面与欧拉公式：用数学语言描述“旋转缠绕” 将信号缠绕在原点周围后，只记录质心的 $x$ 坐标显然只讲了一半的故事，因为整个图形是在二维平面中展开的，它同样拥有 $y$ 坐标。在数学中，处理二维旋转最优雅的工具便是复平面（Complex Plane），此时图形的质心自然成为一个同时包含实部（Real part）与虚部（Imaginary part）的复数。之所以引入复数，是因为它能以极其简洁优美的形式描述旋转运动。 欧拉公式指出，表达式 $e^{it}$ 表示从复平面实轴上的 $(1, 0)$ 出发，沿着半径为 1 的单位圆按逆时针方向走过 $t$ 个单位弧长所对应的点。若要表示每秒旋转 1 圈（即 1 个周期），可以将指数写为 $e^{2\pi i t}$，因为单位圆的周长恰好是 $2\pi$。如果希望调节旋转的快慢，只需在指数中乘以频率 $f$： - 当 $f = \frac{1}{10}$ 时，表达式变为 $e^{2\pi i (\frac{1}{10}) t}$，这意味着时间 $t$ 必须增加到整整 10 秒，指数才会积累到完整的 $2\pi i$，即向量每 10 秒才转动一整圈。 - 在傅里叶变换的标准数学约定中，旋转方向通常设定为**顺时针**，因此指数部分需要额外添上一个负号，写作 $e^{-2\pi i f t}$。 假设输入信号的强度随时间变化的函数为 $g(t)$。若将 $g(t)$ 乘以这个单位旋转向量 $e^{-2\pi i f t}$，其几何含义就是让原本长度为 1 的旋转向量根据函数值的大小进行动态伸缩。这个长度不断变动的旋转小向量，在复平面上精准地画出了前面所说的“缠绕图形”。Sanderson 惊叹道：“So think about it, this is awesome, this really small expression is a super elegant way to encapsulate the whole idea of winding a graph around a circle with a variable frequency, $f$.”（想想看，这太棒了，如此短小的表达式以极其优雅的方式浓缩了以可变频率 $f$ 将函数图像缠绕在圆周上的全部构想。） ```mermaid flowchart LR accTitle: 傅里叶变换公式的几何构造机制 accDescr: 展示从输入信号与复指数相乘得到缠绕图形，再通过积分与时间缩放形成完整傅里叶变换的过程。 subgraph WindingMachine["缠绕机制 (Winding)"] direction TB Sig["时域信号 g(t)"] Rot["顺时针旋转向量 e^(-2πi f t)"] Wound["缠绕图像向量<br/>g(t) · e^(-2πi f t)"] Sig --> Wound Rot --> Wound end subgraph Integration["质心与积分 (Center of Mass)"] direction TB Integ["积分累加<br/>∫ g(t) e^(-2πi f t) dt"] CoM["质心 (除以时间区间 Δt)"] Wound --> Integ Integ -.->|"除以时间区间"| CoM end subgraph Output["频域输出 (Fourier Output)"] direction TB FT["傅里叶变换结果 ĝ(f)<br/>保留积分，保留时间尺度放大效应"] Integ --> FT end ``` ### [14:43](https://www.youtube.com/watch?v=spUNpyF58BY&t=883s) 从离散质心到连续积分：真正的傅里叶变换 构建出缠绕向量后，下一步就是计算该图形的质心位置。在离散近似下，人们可以在原始信号中密集采样若干时间点，计算对应复数向量的总和，再除以采样点数 $N$。随着采样点无限密集，这种离散求和平均在极限下就演变成了微积分中的**定积分除以时间区间长度**： $$\frac{1}{t_2 - t_1} \int_{t_1}^{t_2} g(t) e^{-2\pi i f t} \, dt$$ 尽管对复值函数积分听起来令初学者生畏，但其底层物理意义毫无玄虚之处，它纯粹就是缠绕图形在复平面上的质心坐标。 此时，距离真正的、货真价实的傅里叶变换（honest-to-goodness Fourier transform）只剩下最后一步细微区别：**不要除以时间区间的长度**。真正的傅里叶变换仅仅保留积分部分： $$\hat{g}(f) = \int_{t_1}^{t_2} g(t) e^{-2\pi i f t} \, dt$$ 不除以时间区间意味着该值不再单纯代表几何质心，而是将质心乘以了时间跨度。这种设计蕴含着重要的物理意义： - 如果一个频率在信号中持续了 3 秒，其傅里叶变换在该频率处的峰值大小会被缩放 3 倍；如果持续了 6 秒，峰值就会被放大 6 倍。持续时间越长，该频率分量的确定性与能量累积就越强。 - 对于纯正的 2 Hz 信号，当缠绕频率同样设为 2 圈/秒时，质心固定在偏离原点的同一位置，信号持续越久，积分累加出的傅里叶变换模长就越大。 - 对于非匹配频率，即便频率只偏差一点点，随着时间区间拉长，旋转向量在圆周各方向上的分布越来越均匀，正负彼此抵消，整体积分值依旧保持在接近零的微弱水平。 ### [17:08](https://www.youtube.com/watch?v=spUNpyF58BY&t=1028s) 总结与展望：公式拆解与拓展应用 Sanderson 对构建出的整套体系进行了全面梳理。傅里叶变换将一个以时间 $t$ 为自变量的强度函数 $g(t)$，转化为一个以“缠绕频率 $f$”为自变量的新函数，数学上通常记作带有折音符号的 $\hat{g}(f)$。该函数的输出是一个复数，代表原信号中特定频率分量的强度与相位。 - 视频中前面所画的频域波形图，仅仅提取了这个复数输出的实部（即 $x$ 坐标）；如果需要更完备的信息，同样可以把虚部单独绘制出来。 - 脱离几何背景单看 $\int g(t)e^{-2\pi i f t} dt$ 这个公式会让人望而生畏，但只要明白指数项对应旋转、相乘对应缠绕图像、积分对应质心累积，公式背后的直观几何意义就会一目了然。 在实际的音频编辑等工程场景中，积分总是在有限的时间区间 $[t_1, t_2]$ 内进行；但在纯数学理论中，积分上下限通常被写作负无穷到正无穷（$-\infty$ 到 $\infty$）。其实际含义是考察所有可能的有限时间区间，并求解该积分在区间无限扩大时的极限。Sanderson 最后感叹，傅里叶变换的用途远不止从声音信号中提取频率，它在现代数学和物理学中有极深的应用（例如引申至后续视频中的测不准原理）。他打趣地建议观众保持关注，或者干脆“多刷几部 3Blue1Brown 的视频，好让 YouTube 推荐算法更积极地为你推送后续更新”。 ## Notable quotes > [10:55](https://www.youtube.com/watch?v=spUNpyF58BY&t=655s) Long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function. — Grant Sanderson ## Mentioned 3Blue1Brown, Grant Sanderson, YouTube, Vincent Rubinetti, Jane Street, Patreon, manim, Spotify, Bandcamp, Twitter, Facebook, Reddit ## Video description An animated introduction to the Fourier Transform. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/fourier-thanks Learn more about Janestreet: https://janestreet.com/3b1b Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4 Interactive made by a viewer inspired by this video: https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html Also, take a look at this Jupyter notebook implementing this idea in a way you can play with: https://github.com/thatSaneKid/fourier/blob/master/Fourier%20Transform%20-%20A%20Visual%20Introduction.ipynb Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Russian: xX-Masik-Xx Vietnamese: @ngvutuan2811 ------------------ Animations largely made using manim, a scrappy open-source python library. https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind. 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Source quotes

[10:55] Grant Sanderson: Long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function.

Source quotes

[10:55] Grant Sanderson: applying the Fourier transform to the Fourier transform gives you back something close to the original function.

Original transcript

[0:04] This right here is what we're going to build to this video, [0:07] a certain animated approach to thinking about a super important idea from math, [0:11] the Fourier transform. [0:13] For anyone unfamiliar with what that is, my number one goal [0:16] here is just for the video to be an introduction to that topic. [0:20] But even for those of you who are already familiar with it, [0:23] I still think that there's something fun and enriching about seeing what all of its [0:27] components actually look like. [0:29] The central example to start is going to be the classic one, [0:32] decomposing frequencies from sound. [0:34] But after that I also want to show a glimpse of how this idea extends well beyond [0:39] sound and frequency into many seemingly disparate areas of math, and even physics. [0:44] Really, it's crazy just how ubiquitous this idea is. [0:49] Let's dive in. [0:50] This sound right here is a pure A, 440 beats per second, [0:54] meaning if you were to measure the air pressure right next to your [0:58] headphones or your speaker as a function of time, [1:02] it would oscillate up and down around its usual equilibrium in this wave, [1:06] making 440 oscillations each second. [1:09] A lower pitch note, like a D, has the same structure, just fewer beats per second. [1:15] And when both of them are played at once, what do you think the resulting pressure vs. [1:19] time graph looks like? [1:22] Well, at any point in time, this pressure difference is going to [1:25] be the sum of what it would be for each of those notes individually, [1:29] which let's face it is kind of a complicated thing to think about. [1:33] At some points the peaks match up with each other, resulting in a really high pressure. [1:38] At other points they tend to cancel out. [1:41] And all in all, what you get is a wave-ish pressure vs. [1:44] time graph that is not a pure sine wave, it's something more complicated. [1:48] And as you add in other notes, the wave gets more and more complicated. [1:53] But right now, all it is is a combination of four pure frequencies, [1:57] so it seems needlessly complicated given the low amount of information put into it. [2:03] A microphone recording any sound just picks up on the air pressure [2:06] at many different points in time, it only sees the final sum. [2:10] So our central question is going to be how you can take a signal [2:14] like this and decompose it into the pure frequencies that make it up. [2:18] Pretty interesting, right? [2:20] Adding up those signals really mixes them all together, [2:23] so pulling them back apart feels akin to unmixing multiple paint colors that have [2:27] all been stirred up together. [2:29] The general strategy is going to be to build for ourselves a mathematical machine that [2:34] treats signals with a given frequency differently from how it treats other signals. [2:40] To start, consider simply taking a pure signal, [2:43] say with a lowly 3 beats per second, so we can plot it easily. [2:47] And let's limit ourselves to looking at a finite portion of this graph, [2:51] in this case the portion between 0 seconds and 4.5 seconds. [2:55] The key idea is going to be to take this graph and sort of wrap it up around a circle. [3:04] Concretely, here's what I mean by that. [3:07] Imagine a little rotating vector where at each point in time [3:10] its length is equal to the height of our graph for that time. [3:14] So high points of the graph correspond to a greater distance from the origin, [3:18] and low points end up closer to the origin. [3:22] And right now I'm drawing it in such a way that moving forward 2 [3:25] seconds in time corresponds to a single rotation around the circle. [3:29] Our little vector drawing this wound up graph is rotating at half a cycle per second. [3:35] So this is important, there are two different frequencies at play here. [3:38] There's the frequency of our signal, which goes up and down 3 times per second, [3:42] and then separately there's the frequency with which we're wrapping the graph [3:47] around the circle, which at the moment is half of a rotation per second. [3:51] But we can adjust that second frequency however we want. [3:54] Maybe we want to wrap it around faster? [3:58] Or maybe we go and wrap it around slower? [4:03] And that choice of winding frequency determines what the wound up graph looks like. [4:09] Some of the diagrams that come out of this can be pretty complicated, [4:12] although they are very pretty, but it's important to keep in mind that [4:15] all that's happening here is that we're wrapping the signal around a circle. [4:20] The vertical lines that I'm drawing up top, by the way, [4:23] are just a way to keep track of the distance on the original graph that corresponds to [4:27] a full rotation around the circle. [4:30] So lines spaced out by 1.5 seconds would mean [4:33] it takes 1.5 seconds to make one full revolution. [4:37] And at this point we might have some sort of vague sense that something special will [4:41] happen when the winding frequency matches the frequency of our signal, 3 beats per second. [4:46] All of the high points on the graph happen on the right side of the circle, [4:49] and all of the low points happen on the left. [4:52] But how precisely can we take advantage of that in [4:55] our attempt to build a frequency unmixing machine? [4:59] Well, imagine this graph is having some kind of mass to it, like a metal wire. [5:04] This little dot is going to represent the center of mass of that wire. [5:08] As we change the frequency and the graph winds up differently, [5:11] that center of mass kind of wobbles around a bit. [5:16] And for most of the winding frequencies, the peaks and valleys are all spaced out [5:19] around the circle in such a way that the center of mass stays pretty close to the origin. [5:26] But when the winding frequency is the same as the frequency of our signal, [5:30] in this case 3 cycles per second, all of the peaks are on the right, [5:34] and all of the valleys are on the left, so the center of mass is unusually far [5:38] to the right. [5:42] Here, to capture this, let's draw some kind of plot that keeps [5:45] track of where that center of mass is for each winding frequency. [5:49] Of course, the center of mass is a two-dimensional thing, [5:51] it requires two coordinates to fully keep track of, but for the moment, [5:54] let's only keep track of the x-coordinate. [5:57] So for a frequency of zero, when everything is bunched up on the right, [6:01] this x-coordinate is relatively high. [6:03] And then as you increase that winding frequency, [6:06] and the graph balances out around the circle, the x-coordinate of [6:10] that center of mass goes closer to zero, and it just kind of wobbles around a bit. [6:26] But then, at 3 beats per second, there's a spike, as everything lines up to the right. [6:44] This right here is the central construct, so let's sum up what we have so far. [6:47] We have that original intensity vs time graph, [6:50] and then we have the wound up version of that in some two-dimensional plane, [6:55] and then as a third thing, we have a plot for how the winding frequency influences [7:00] the center of mass of that graph. [7:03] And by the way, let's look back at those really low frequencies near zero. [7:07] This big spike around zero in our new frequency plot just [7:11] corresponds to the fact that the whole cosine wave is shifted up. [7:16] If I had chosen a signal that oscillates around zero, dipping into negative values, [7:21] then as we play around with various winding frequencies, [7:25] this plot of the winding frequency vs center of mass would only have a spike [7:30] at the value of 3. [7:32] But negative values are a little bit weird and messy to think about, [7:35] especially for a first example, so let's just continue thinking in terms of the [7:39] shifted up graph. [7:41] I just want you to understand that that spike around zero only corresponds to the shift. [7:45] Our main focus, as far as frequency decomposition is concerned, is that bump at 3. [7:51] This whole plot is what I'll call the almost Fourier transform of the original signal. [7:56] There's a couple small distinctions between this and the actual Fourier transform, [8:00] which I'll get to in a couple minutes, but already you might be able to [8:03] see how this machine lets us pick out the frequency of a signal. [8:07] Just to play around with it a little bit more, take a different Fourier signal, [8:11] let's say with a lower frequency of 2 beats per second, and do the same thing. [8:16] Wind it around a circle, imagine different potential winding frequencies, [8:20] and as you do that keep track of where the center of mass of that graph is, [8:24] and then plot the x coordinate of that center of mass as you adjust the winding frequency. [8:30] Just like before, we get a spike when the winding frequency is the same as [8:34] the signal frequency, which in this case is when it equals 2 cycles per second. [8:39] But the real key point, the thing that makes this machine so delightful, [8:43] is how it enables us to take a signal consisting of multiple frequencies and pick out [8:48] what they are. [8:49] Imagine taking the two signals we just looked at, [8:51] the wave with 3 beats per second and the wave with 2 beats per second, and add them up. [8:56] Like I said earlier, what you get is no longer a nice pure cosine wave, [8:59] it's something a little more complicated. [9:02] But imagine throwing this into our winding frequency machine. [9:06] It is certainly the case that as you wrap this thing around it looks a [9:09] lot more complicated, you have this chaos and chaos and chaos and chaos, [9:12] and then whoop, things seem to line up really nicely at 2 cycles per second. [9:16] Then as you continue on it's more chaos and more chaos and more chaos and chaos [9:20] and chaos and chaos, whoop, things nicely align again at 3 cycles per second. [9:23] And like I said before, the wound up graph can look kind of busy and complicated, [9:27] but all it is is the relatively simple idea of wrapping the graph around a circle. [9:31] It's just a more complicated graph and a pretty quick winding frequency. [9:36] Now what's going on here with the two different spikes is that if you were to [9:40] take two signals and then apply this almost Fourier transform to each of them [9:45] individually, and then add up the results, what you get is the same as if you [9:49] first added up the signals and then applied this almost Fourier transform. [9:55] And the attentive viewers among you might want to pause and ponder [9:58] and convince yourself that what I just said is actually true. [10:01] It's a pretty good test to verify for yourself that it's clear [10:05] what exactly is being measured inside this winding machine. [10:09] Now this property makes things really useful to us, [10:11] because the transform of a pure frequency is close to zero everywhere except [10:16] for a spike around that frequency, so when you add together two pure frequencies, [10:20] the transform graph just has these little peaks above the frequencies that went into it. [10:26] So this little mathematical machine does exactly what we wanted. [10:29] It pulls out the original frequencies from their jumbled up sums, [10:33] unmixing the mixed bucket of paint. [10:36] And before continuing into the full math that describes this operation, [10:40] let's just get a quick glimpse of one context where this thing is useful, sound editing. [10:44] Let's say that you have some recording and it's got an [10:47] annoying high pitch that you would like to filter out. [10:50] Well at first your signal is coming in as a function of various intensities over time, [10:55] different voltages given to your speaker from one millisecond to the next. [10:59] But we want to think of this in terms of frequencies. [11:02] So when you take the Fourier transform of that signal, [11:05] the annoying high pitch is going to show up just as a spike at some high frequency. [11:11] Filtering that out by just smushing the spike down, [11:13] what you'd be looking at is the Fourier transform of a sound that's just like your [11:18] recording, only without that high frequency. [11:21] Luckily there's a notion of an inverse Fourier transform that tells [11:24] you which signal would have produced this as its Fourier transform. [11:29] I'll be talking about that inverse much more fully in the next video, [11:32] but long story short, applying the Fourier transform to the Fourier [11:36] transform gives you back something close to the original function. [11:40] Kind of, this is a little bit of a lie, but it's in the direction of truth. [11:44] And most of the reason it's a lie is that I still have yet to [11:47] tell you what the actual Fourier transform is, [11:50] since it's a little more complex than this x-coordinate of the center of mass idea. [11:55] First off, bringing back this wound up graph and looking at its center of mass, [11:59] the x-coordinate is really only half the story, right? [12:02] I mean, this thing is in two dimensions, it's got a y-coordinate as well. [12:05] And as is typical in math, whenever you're dealing with something two-dimensional, [12:10] it's elegant to think of it as the complex plane, [12:12] where this center of mass is going to be a complex number that has both a real [12:16] and an imaginary part. [12:21] And the reason for talking in terms of complex numbers, [12:23] rather than just saying it has two coordinates, [12:25] is that complex numbers lend themselves to really nice descriptions of [12:29] things that have to do with winding and rotation. [12:32] For example, Euler's formula famously tells us that if you take e to some number times i, [12:37] you're going to land on the point that you get if you were to walk that number of [12:42] units around a circle with radius 1 counterclockwise starting on the right. [12:47] So imagine you wanted to describe rotating at a rate of 1 cycle per second. [12:54] One thing you could do is take the expression e to the 2 pi times i times t, [12:59] where t is the amount of time that has passed, since for a circle with radius 1, [13:04] 2 pi describes the full length of its circumference. [13:08] And this is a little dizzying to look at, so maybe you want to describe [13:12] a different frequency, something lower and more reasonable, [13:16] and for that you would just multiply that time t in the exponent by the frequency f. [13:21] For example, if f was 1 tenth, then this vector makes one full turn every 10 seconds, [13:27] since the time t has to increase all the way to 10 before the full exponent looks like 2 [13:33] pi i. [13:34] I have another video giving some intuition on why this is the [13:37] behavior of e to the x for imaginary inputs, if you're curious, [13:40] but for right now we're just going to take it as a given. [13:44] Now why am I telling you this, you might ask? [13:46] Well it gives us a really nice way to write down the idea [13:49] of winding up the graph into a single tight little formula. [13:53] First off, the convention in the context of Fourier transforms is to think about [13:58] rotating in the clockwise direction, so let's throw a negative sign up into that exponent. [14:04] Now take some function describing a signal intensity versus time, [14:08] like this pure cosine wave we had before, and call it g of t. [14:12] If you multiply this exponential expression times g of t, [14:16] it means that the rotating complex number is getting scaled up and down according to [14:21] the value of this function. [14:24] So you can think of this little rotating vector with [14:27] its changing length as drawing the wound up graph. [14:31] So think about it, this is awesome, this really small expression [14:35] is a super elegant way to encapsulate the whole idea of [14:38] winding a graph around a circle with a variable frequency, f. [14:43] And remember, the thing we want to do with this wound up graph is to track [14:47] its center of mass, so think about what formula is going to capture that. [14:51] Well, to approximate it at least, you might sample a whole bunch of times [14:55] from the original signal, see where those points end up on the wound up graph, [15:00] and then just take an average, that is, add them all together as complex numbers, [15:05] and then divide by the number of points you've sampled. [15:09] This will become more accurate if you sample more points which are closer together. [15:14] And in the limit, rather than looking at the sum of a whole bunch of [15:18] points divided by the number of points, you take an integral of this [15:21] function divided by the size of the time interval we're looking at. [15:25] The idea of integrating a complex valued function might seem weird, [15:29] and to anyone who's shaky with calculus maybe even intimidating, [15:32] but the underlying meaning here really doesn't require any calculus knowledge. [15:36] The whole expression is just the center of mass of the wound up graph. [15:41] So great, step by step, we have built up this kind of complicated but let's face it, [15:46] surprisingly small expression for the whole winding machine idea I talked about, [15:51] and now there is only one final distinction to point out between this and the actual [15:56] honest-to-goodness Fourier transform, namely, just don't divide out by the time interval. [16:02] The Fourier transform is just the integral part of this. [16:06] What that means is that instead of looking at the center of mass, [16:09] you would scale it up by some amount. [16:11] If the portion of the original graph you were using spanned 3 seconds, [16:15] you would multiply the center of mass by 3. [16:19] If it was spanning 6 seconds, you would multiply the center of mass by 6. [16:25] Physically, this has the effect that when a certain frequency persists for a long time, [16:30] then the magnitude of the Fourier transform at that frequency is scaled up more and more. [16:36] For example, what we're looking at here is how when you have a pure frequency of 2 [16:40] beats per second and you wind it around the graph at 2 cycles per second, [16:45] the center of mass stays in the same spot, just tracing out the same shape. [16:49] But the longer that signal persists, the larger the value of the Fourier transform [16:54] at that frequency. [16:56] For other frequencies, even if you just increase it by a bit, [16:59] this is cancelled out by the fact that for longer time intervals, [17:02] you're giving the wound-up graph more of a chance to balance itself around the circle. [17:08] That is a lot of different moving parts, so let's [17:11] step back and summarize what we have so far. [17:14] The Fourier transform of an intensity vs. [17:17] time function, like g of t, is a new function, which doesn't have time as an input, [17:22] but instead takes in a frequency, what I've been calling the winding frequency. [17:28] In terms of notation, by the way, the common convention is to [17:31] call this new function g-hat with a little circumflex on top of it. [17:35] The output of this function is a complex number, [17:38] some point in the 2d plane that corresponds to the strength of a given [17:43] frequency in the original signal. [17:46] The plot I've been graphing for the Fourier transform is just the real [17:49] component of that output, the x-coordinate, but you could also graph [17:53] the imaginary component separately if you wanted a fuller description. [17:57] And all of this is encapsulated inside that formula we built up. [18:01] And out of context, you can imagine how seeing this formula would seem sort of daunting, [18:07] but if you understand how exponentials correspond to rotation, [18:10] how multiplying that by the function g of t means drawing a wound up version of the [18:15] graph, and how an integral of a complex valued function can be interpreted in terms [18:20] of a center of mass idea, you can see how this whole thing carries with it a very rich [18:25] intuitive meaning. [18:27] And by the way, one quick small note before we can call this wrapped up. [18:30] Even though in practice, with things like sound editing, [18:33] you'll be integrating over a finite time interval, [18:36] the theory of Fourier transforms is often phrased where the bounds of this [18:40] integral are negative infinity and infinity. [18:43] Concretely, what that means is that you consider this expression [18:46] for all possible finite time intervals, and you just ask, [18:49] what is its limit as that time interval grows to infinity? [18:54] And man oh man, there is so much more to say. [18:57] So much, I don't want to call it done here. [18:58] This transform extends to corners of math well [19:01] beyond the idea of extracting frequencies from signal. [19:04] So the next video I put out is going to go through a couple of these, [19:06] and that's really where things start getting interesting. [19:10] So stay subscribed for when that comes out, or an alternate option [19:13] is to just binge on a couple 3Blue and Brown videos so that the [19:16] YouTube recommender is more inclined to show you new things that come out. [19:19] Really the choice is yours. [19:26] Thank you.