一覧へ戻るDSP & Compression

フーリエ変換の完全解剖：あらゆる波動を純粋な正弦波の和に分解する魔法

How Joseph Fourier's heat-equation breakthrough became the engine of JPEG compression, noise-canceling headphones, and Whisper AI spectrograms.

f^(ω)=∫−∞∞f(t)e−iωt dt  ⟺  f(t)=12π∫−∞∞f^(ω)eiωt dω\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt \iff f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\omega)e^{i\omega t}\,d\omega
Fourier Transform is simply an inner product (dot product) between signal f(t) and orthogonal complex sinusoids e^(iωt).
Convolution in the time domain becomes element-wise multiplication in the frequency domain, accelerating computation via FFT in O(N log N).
Discarding high-frequency coefficients imperceptible to human senses is the core mechanism of MP3 and JPEG compression.
⚡ この記事に登場する核心公式
#G04単位円上の三角関数と基本恒等式#C04微分積分学の基本定理と定積分#C07オイラーの公式と世界で最も美しい等式#C09フーリエ級数とフーリエ変換 (時間↔周波数)#L02ベクトルの内積(ドット積)とコサイン類似度

1. Orthogonality as a Frequency Scanner

∫−ππeinxe−imx dx=2π δnm\int_{-\pi}^{\pi} e^{inx} e^{-imx}\,dx = 2\pi\,\delta_{nm}

Complex exponentials of different frequencies are mutually orthogonal: their product integrates to zero over a period unless the frequencies match, acting as a mathematical resonance filter.

2. Fast Fourier Transform (FFT) & AI Spectrograms

Xk=∑n=0N−1xne−i2πkn/N(O(N2)→O(Nlog⁡N))X_k = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k n / N} \quad (\mathcal{O}(N^2) \to \mathcal{O}(N\log N))

The Cooley-Tukey FFT reduces complexity from O(N²) to O(N log N). Modern speech AI (like Whisper) converts waveforms into 2D Mel-spectrograms via Short-Time Fourier Transform before feeding them into Transformers.