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Symbolic Fourier transform. Symbolic Fourier transform and inverse Fourier transform (table + rules).
The public entry points are the Ex methods fourier_transform,
inverse_fourier_transform, fourier_transform_with and
inverse_fourier_transform_with; the convention is selected with
FourierConvention.
§Conventions
FourierConvention | Forward F(ω) = | Inverse f(t) = |
|---|---|---|
NonUnitaryAngular | ∫ f(t) e^{−iωt} dt | (1/2π) ∫ F(ω) e^{iωt} dω |
UnitaryAngular | (1/√(2π)) ∫ f(t) e^{−iωt} dt | (1/√(2π)) ∫ F(ω) e^{iωt} dω |
Ordinary | ∫ f(t) e^{−2πiνt} dt | ∫ F(ν) e^{2πiνt} dν |
Everything is computed in the non-unitary angular convention and then
converted: F_unitary = F/√(2π) and F_ordinary(ν) = F(2πν).
§Table (non-unitary angular)
f(t) | F(ω) | condition |
|---|---|---|
δ(t) | 1 | |
1 | 2π δ(ω) | |
H(t) | π δ(ω) + 1/(iω) | |
sign(t) | 2/(iω) | |
1/t | −iπ sign(ω) | |
|t| | −2/ω² | |
H(t+a) − H(t−a) (rect) | 2 sin(aω)/ω | |
e^{−a|t|} | 2a/(a² + ω²) | a > 0 |
e^{−at²} | √(π/a) e^{−ω²/(4a)} | a > 0 |
tⁿ e^{−at} H(t) | n!/(iω + a)^{n+1} | a > 0 |
tⁿ e^{at} H(−t) | (−1)ⁿ n!/(a − iω)^{n+1} | a > 0 |
cos(ω₀t), sin(ω₀t) | π[δ(ω−ω₀) + δ(ω+ω₀)], iπ[δ(ω+ω₀) − δ(ω−ω₀)] | |
sin(at)/t (sinc) | π [H(ω+a) − H(ω−a)] | a > 0 |
Rules: linearity, time shift f(t−t₀) → e^{−iωt₀}F(ω), modulation
e^{iω₀t} f(t) → F(ω−ω₀), scaling f(at) → F(ω/a)/|a|, derivative
f'(t) → iω F(ω), and multiplication by t: t f(t) → i F'(ω).
Piecewise inputs with linear conditions in t are converted to
Heaviside form first.
Symbols other than the transform variables are treated as real
parameters; conditions such as a > 0 are checked through the
assumption system (Assumption::Positive) and an unprovable condition is
an error rather than a guess.
Results containing δ(ω − c)·g(ω) are simplified to δ(ω − c)·g(c).
Enums§
- Fourier
Convention - Normalisation convention of the Fourier transform pair.