Fourier Transform
Definition: F(ω) = ∫₋∞^∞ f(t)·e^(-jωt) dt | f(t) = (1/2π)∫₋∞^∞ F(ω)·e^(jωt) dω
Common Transform Pairs
| f(t) | F(ω) | Notes |
|---|---|---|
| δ(t) | 1 | Dirac delta |
| 1 | 2π·δ(ω) | Constant |
| u(t) (unit step) | π·δ(ω) + 1/(jω) | |
| e^(-at)·u(t), a>0 | 1/(a + jω) | Decaying exponential |
| e^(jω₀t) | 2π·δ(ω - ω₀) | Complex exponential |
| cos(ω₀t) | π[δ(ω-ω₀) + δ(ω+ω₀)] | |
| sin(ω₀t) | jπ[δ(ω+ω₀) - δ(ω-ω₀)] | |
| rect(t/T) | T·sinc(ωT/2) | Rectangle pulse |
| sinc(t) = sin(πt)/(πt) | rect(ω/2π) | Sinc function |
| e^(-a|t|), a>0 | 2a/(a² + ω²) | Two-sided exponential |
| e^(-at²) | √(π/a)·e^(-ω²/4a) | Gaussian |
| t·e^(-at)·u(t) | 1/(a+jω)² |
Properties
| Property | Time Domain | Frequency Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(ω) + bG(ω) |
| Time Shift | f(t - t₀) | F(ω)·e^(-jωt₀) |
| Frequency Shift | f(t)·e^(jω₀t) | F(ω - ω₀) |
| Time Scaling | f(at) | (1/|a|)·F(ω/a) |
| Duality | F(t) | 2π·f(-ω) |
| Differentiation | f'(t) | jω·F(ω) |
| Integration | ∫f(τ)dτ | F(ω)/(jω) + πF(0)δ(ω) |
| Convolution | f(t) * g(t) | F(ω)·G(ω) |
| Multiplication | f(t)·g(t) | (1/2π)F(ω) * G(ω) |
| Parseval's Theorem | ∫|f(t)|² dt | (1/2π)∫|F(ω)|² dω |
Discrete Fourier Transform (DFT)
DFT: X[k] = Σ(n=0 to N-1) x[n]·e^(-j2πkn/N)
IDFT: x[n] = (1/N) Σ(k=0 to N-1) X[k]·e^(j2πkn/N)
Key properties:
- Periodicity: X[k] = X[k+N]
- Conjugate symmetry (real input): X[N-k] = X*[k]
- FFT computes DFT in O(N log N) vs O(N²)