Fourier Transform Reference
Continuous and discrete Fourier transform definitions, properties, and common pairs.
Reference
Definitions
- CTFT
- X(f) = ∫ x(t) · e^{−j2πft} dt
- DTFT
- X(e^{jω}) = Σ x[n] · e^{−jωn}
- DFT
- X[k] = Σ_{n=0}^{N−1} x[n] · e^{−j2πkn/N}
- IDFT
- x[n] = (1/N) Σ_{k=0}^{N−1} X[k] · e^{j2πkn/N}
- FFT
- O(N log N) algorithm for computing the DFT
Properties
| Property | x(t) ↔ X(f) |
|---|---|
| Linearity | a·x + b·y ↔ a·X + b·Y |
| Time shift | x(t − t₀) ↔ X(f) · e^{−j2πf t₀} |
| Frequency shift | x(t) · e^{j2πf₀t} ↔ X(f − f₀) |
| Time scaling | x(a·t) ↔ (1/|a|) X(f/a) |
| Convolution ↔ product | x ⊛ y ↔ X · Y |
| Product ↔ convolution | x · y ↔ X ⊛ Y |
| Parseval (energy) | ∫|x(t)|² dt = ∫|X(f)|² df |
| Duality | x(t) ↔ X(f) implies X(t) ↔ x(−f) |
Common pairs
| x(t) | X(f) |
|---|---|
| δ(t) (impulse) | 1 |
| 1 | δ(f) |
| cos(2πf₀t) | (δ(f − f₀) + δ(f + f₀)) / 2 |
| rect(t) | sinc(f) |
| sinc(t) | rect(f) |
| e^{−at}u(t) (a > 0) | 1 / (a + j2πf) |
| e^{−πt²} | e^{−πf²} |
| u(t) (step) | 1 / (j2πf) + ½δ(f) |
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