Chapter 41
Fourier Transform as Frequency Decomposition

Flow Algebra

flow "Fourier Transform as Frequency Decomposition"

# ---------- CONCEPT ----------
# The Fourier Transform decomposes any signal into its constituent frequency components.
# It expresses structure as oscillation — translating from time (or space) to frequency (or wavenumber) domain.
# Each frequency is a rotational boundary e^{iωt}, and the transform measures how strongly each boundary contributes to the whole.
# It is the bridge between local variation and global pattern — a harmonic map of coherence.

# ---------- STATES ----------
state FourierSystem(F):
  TimeDomainSignal:
    description: "f(t) — a function representing amplitude over time (or position)."
  FrequencyDomainSpectrum:
    description: "F(ω) — complex-valued function describing amplitude and phase of each frequency component."
  ForwardTransform:
    description: "F(ω) = ∫_{−∞}^{∞} f(t) e^{−iωt} dt — projection of f onto rotational basis."
  InverseTransform:
    description: "f(t) = (1/2π) ∫_{−∞}^{∞} F(ω) e^{iωt} dω — reconstruction of signal from its frequency field."
  DiscreteTransform:
    description: "F[k] = Σ_{n=0}^{N−1} f[n] e^{−i2πkn/N} — DFT; sampled version of continuous transform."
  EnergyConservation:
    description: "Parseval’s theorem: ∫ |f(t)|² dt = (1/2π) ∫ |F(ω)|² dω — energy preserved across domains."

# ---------- PARAMETERS ----------
params:
  variable_time: "t ∈ ℝ"
  variable_frequency: "ω ∈ ℝ"
  sampling_period: "Δt, Δω (for discrete systems)"
  normalization: "Convention depends on field (physics, engineering, mathematics)."
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Forward: F(ω) = ∫ f(t) e^{−iωt} dt"
  - "Inverse: f(t) = (1/2π) ∫ F(ω) e^{iωt} dω"
  - "Linearity: Fourier transform is linear — transforms preserve superposition."
  - "Shift: f(t−t₀) ⇔ e^{−iωt₀} F(ω)"
  - "Modulation: e^{iω₀t} f(t) ⇔ F(ω−ω₀)"
  - "Scaling: f(at) ⇔ (1/|a|) F(ω/a)"
  - "Differentiation: dⁿf/dtⁿ ⇔ (iω)ⁿ F(ω)"
  - "Convolution: f∗g ⇔ F(ω)G(ω) — convolution in time becomes multiplication in frequency."
  - "Multiplication: f(t)g(t) ⇔ (1/2π)(F∗G)(ω)."
  - "Energy conservation: Parseval — ∫|f|²dt = (1/2π)∫|F|²dω."

# ---------- TRANSFORMS ----------
transforms:
  ForwardFourier(f):
    rule: "F(ω) = ∫_{−∞}^{∞} f(t) e^{−iωt} dt"
  InverseFourier(F):
    rule: "f(t) = (1/2π) ∫_{−∞}^{∞} F(ω) e^{iωt} dω"
  ShiftProperty(f,t₀):
    rule: "f(t−t₀) ⇔ e^{−iωt₀} F(ω)"
  ModulationProperty(f,ω₀):
    rule: "e^{iω₀t}f(t) ⇔ F(ω−ω₀)"
  DerivativeProperty(f,n):
    rule: "dⁿf/dtⁿ ⇔ (iω)ⁿF(ω)"
  ConvolutionProperty(f,g):
    rule: "(f∗g)(t) ⇔ F(ω)G(ω)"
  WindowedTransform(f,w):
    rule: "Short-time Fourier: F(ω,t₀)=∫ f(t)w(t−t₀)e^{−iωt}dt (local frequency structure)."
  PowerSpectrum(F):
    rule: "|F(ω)|² — intensity per frequency component."

# ---------- MONITORS ----------
monitors:
  EnergyBalance:
    measure: "∫|f(t)|²dt − (1/2π)∫|F(ω)|²dω"
    target: 0
    meaning: "Check Parseval consistency."
  PhaseSymmetry:
    meaning: "For real f(t): F(−ω) = F*(ω) (conjugate symmetry)."
  FrequencyResolution:
    measure: "Δω = 2π/T"
    meaning: "Resolution linked to signal duration T."
  TimeResolution:
    measure: "Δt = 1/(2B)"
    meaning: "Time–frequency uncertainty (Heisenberg limit)."
  PeriodicityDFT:
    meaning: "F[k] periodic in k with period N."
  AliasingCheck:
    meaning: "Ensure sampling rate > 2×max frequency (Nyquist condition)."

# ---------- EQUATIONS ----------
equations:
  - "F(ω) = \\int_{−∞}^{∞} f(t) e^{−iωt} dt"
  - "f(t) = \\frac{1}{2π} \\int_{−∞}^{∞} F(ω) e^{iωt} dω"
  - "f(t−t₀) ⇔ e^{−iωt₀}F(ω)"
  - "e^{iω₀t}f(t) ⇔ F(ω−ω₀)"
  - "dⁿf/dtⁿ ⇔ (iω)ⁿF(ω)"
  - "(f∗g)(t) ⇔ F(ω)G(ω)"
  - "Parseval: \\int|f(t)|²dt = \\frac{1}{2π}\\int|F(ω)|²dω"
  - "Power spectrum: S(ω) = |F(ω)|²"
  - "Heisenberg: Δt Δω ≥ 1/2"
  - "Discrete: F[k] = \\sum_{n=0}^{N−1} f[n] e^{−i2πkn/N}"

# ---------- NOTES ----------
notes:
  - "The Fourier Transform expresses every signal as a superposition of rotations — waves of different frequencies combining to form structure."
  - "Each term e^{iωt} is a rotational boundary; its amplitude and phase encode how much that frequency contributes to the total field."
  - "Inversion reconstructs the original function: integration of all rotations restores the full coherence of time-domain structure."
  - "Convolution in time corresponds to multiplication in frequency — spreading interactions become simple algebraic products."
  - "For real-valued signals, F(−ω)=F*(ω) — symmetry of coherence ensures mirrored frequency structure."
  - "Energy conservation (Parseval) shows that total coherence is preserved between domains — the Fourier Transform is an isometry of energy flow."
  - "In Flow Algebra, the Fourier Transform is a coherence decomposition: time becomes phase-space, curvature becomes frequency, and every oscillation is a boundary of the same flow viewed from another dimension."