#C09 · Calculus & Analysis · Lv.5View in 3D Mind-Map

Fourier Series & Fourier Transform (FFT)

Joseph Fourier, Cooley & Tukey · 1822 CE

\hat{f}(\omega) = \int_{-infty}^{\infty} f(t)\,e^{-i\omega t}\,dt

f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\omega)\,e^{i\omega t}\,d\omega

Decomposes any complex time-domain signal into its constituent sinusoidal frequencies via Euler's complex exponential e^(-iωt)—powering Wi-Fi, MP3, JPEG, MRI, and audio AI.

🧮 Live Formula Calculator & Plotter

Freq ω₁

Harmonic ω₂

Harmonic Amp A₂

f(t) = \sin(1t) + 0.4\sin(3t), \quad E = 1^2 + 0.4^2 = 1.16

파르세발 에너지 ∑A_k²

1.1600

기본파 ω₁=1 비중

86.2%

고조파 ω₂=3 진폭

0.40

Prerequisite Path (22 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 4#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 5#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 6#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 7#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 8#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 9#G022D/3D Euclidean Distance Formula

d(P

STEP 10#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 11#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 12#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 13#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 14#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 15#C03Product Rule & Chain Rule (Backpropagation Core)

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

STEP 16#A09Complex Numbers & Polar Plane

z = a + bi = r(\cos\theta + i\sin\theta)

STEP 17#G06Trigonometric Addition & Double-Angle Formulas

\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

STEP 18#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 19#C05Integration by Substitution & by Parts

\int u\

STEP 20#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

STEP 21#C07Euler's Formula & Euler's Identity

e^{i\theta} = \cos\theta + i\sin\theta

STEP 22#C09Fourier Series & Fourier Transform (FFT)

\hat{f}(\omega) = \int_{-infty}^{\infty} f(t)\

Geometric & Intuitive Insight

과일 스무디(복합 파동 f(t))를 마시고 그 안에 딸기·바나나·망고(주파수 ω 성분)가 각각 몇 그램씩 들어있는지 레시피(스펙트럼 f̂(ω))를 정확히 역추적해 내는 수학적 프리즘입니다!

Step-by-Step Derivation

Step 1.1단계: 서로 다른 주파수의 삼각함수(또는 복소지수 e^(iωt))는 한 주기 적분 시 내적이 0이 되는 직교성(Orthogonality)을 가집니다.
Step 2.2단계: 신호 f(t)에 e^(-iωt) 를 곱해 전체 시간 구간에서 정적분하면, 주파수 ω와 일치하는 공명 성분만 살아남고 나머지는 상쇄됩니다.
Step 3.3단계: 이로써 시간 영역 신호 f(t)가 주파수 영역 스펙트럼 f̂(ω)로 변환됩니다.

AI & Coding Real-World Use

Speech AI (Whisper Mel-Spectrograms), MP3/JPEG compression, 5G/Wi-Fi OFDM, and MRI reconstruction

#Audio AI Spectrogram#5G/Wi-Fi OFDM#JPEG/MP3 Compression

const dftBin = (sig: number[], k: number) => sig.reduce((s, x, n) => s + x * Math.cos(2*Math.PI*k*n/sig.length), 0);

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