// minds behind maths

Joseph Fourier

1768–1830 · Mathematics, physics

French mathematician and physicist who showed that any periodic signal can be built from sine and cosine waves — the Fourier series — founding modern signal analysis.

Source: Britannica — Joseph Fourier

Formulas that trace back to Joseph Fourier

Custom Harmonics — Additive Synthesis y(t)=sum_i A_isin(n_iomega t+varphi_i)

Diffraction Patterns (Aperture → Light) I(theta) propto big|mathcal{F}{text{aperture}}big|^{2}

ECG Noise Filtering y = mathcal{F}^{-1}big[,mathcal{F}(x)cdot H(f),big]

Fourier Series building a square wave from sine waves

General Fourier Series (Your Coefficients) y(t)=frac{a_0}{2}+sum_{n=1}^{N}big[a_ncos(nomega t)+b_nsin(nomega t)big]

Half-Wave Rectified Sine Series y(t)=frac{A}{pi}+frac{A}{2}sin(omega t)-frac{2A}{pi}sum_{n=1}^{N}frac{cos(2nomega t)}{4n^{2}-1}

Heat Equation — Fourier's 1822 Problem u(x,t)=sum_{n}b_nsin(npi x),e^{-alpha (npi)^2 t}

Image Filtering (1D Pixel Row) text{low-pass / high-pass} = mathcal{F}^{-1}big[,mathcal{F}(x)cdot text{mask},big]

OFDM — 5G Sub-Carriers s(t)=sum_{k} b_kcos(2pi k,Delta f,t)

Sawtooth Wave Fourier Series y(t)=frac{2A}{pi}sum_{n=1}^{N}frac{(-1)^{n+1}}{n}sin(nomega t)

Sine & Cosine Building Blocks y(t) = Asin(2pi f t + varphi) text{and} Acos(2pi f t + varphi)

Square Wave Fourier Series y(t)=frac{4A}{pi}sum_{k=1,3,5,dots}^{N}frac{sin(komega t)}{k}

Triangle Wave Fourier Series y(t)=frac{8A}{pi^{2}}sum_{k=0}^{N}frac{(-1)^{k}}{(2k+1)^{2}}sin!big((2k+1)omega tbig)

Vowel Synthesis from Formants text{speech} = text{glottal source} ast text{vocal-tract filter}