The Fourier Transform: Complex Plane Winding, Epicycles & Spectral Decomposition
Decompose any time-domain signal into rotating complex exponentials via Euler's formula and inner products.
The Continuous Fourier Transform maps a square-integrable time-domain signal f(t) in L^2(R) onto its complex frequency-domain spectrum F(xi) by projecting f(t) against orthogonal complex sinusoids e^{-i 2 pi xi t}.
Multiplying a real signal f(t) by e^{-i 2 pi xi t} winds the signal's graph around the origin of the complex plane at winding frequency xi. For almost all frequencies, the wound curve balances symmetrically around the origin and its center of mass cancels to zero; when xi matches a true constituent frequency of f(t), all peaks align on one side of the complex plane and the center of mass spikes.
Fourier Transform & Epicycles (FourierEpicyclesScene)
Step-by-Step Symbolic LaTeX Derivation
Follow each algebraic transformation from first principles to computational implementation.
Euler's Complex Exponential Basis
Each complex exponential simultaneously probes both cosine (even/in-phase) and sine (odd/quadrature) components via a single complex multiplication.
Discrete Fourier Transform (DFT) & Orthogonality
The N-th roots of unity form an orthogonal basis over C^N, enabling exact inversion.
Cooley-Tukey Radix-2 FFT Recurrence
Splitting x_n into even and odd indices halves the problem size at each level, reducing O(N^2) to O(N log N).
Why does multiplying f(t) by e^{-2*pi*i*xi*t} and integrating isolate frequency xi?
Why does the Fourier Transform produce complex numbers for a purely real signal?
For a real signal, the magnitude |F(xi)| encodes the amplitude of frequency xi and the angle arg(F(xi)) encodes its phase offset (the balance between cosine and sine).
What is the Convolution Theorem?
Convolution in the time domain equals point-wise multiplication in the frequency domain: F{f * g} = F(xi) . G(xi), allowing O(N^2) convolutions to be computed in O(N log N) via FFT.