Fourier Transform
Definition: F(ฯ) = โซโโ^โ f(t)ยทe^(-jฯt) dt | f(t) = (1/2ฯ)โซโโ^โ F(ฯ)ยทe^(jฯt) dฯ
Common Transform Pairs
| f(t) | F(ฯ) | Notes |
|---|---|---|
| ฮด(t) | 1 | Dirac delta |
| 1 | 2ฯยทฮด(ฯ) | Constant |
| u(t) (unit step) | ฯยทฮด(ฯ) + 1/(jฯ) | |
| e^(-at)ยทu(t), a>0 | 1/(a + jฯ) | Decaying exponential |
| e^(jฯโt) | 2ฯยทฮด(ฯ - ฯโ) | Complex exponential |
| cos(ฯโt) | ฯ[ฮด(ฯ-ฯโ) + ฮด(ฯ+ฯโ)] | |
| sin(ฯโt) | jฯ[ฮด(ฯ+ฯโ) - ฮด(ฯ-ฯโ)] | |
| rect(t/T) | Tยทsinc(ฯT/2) | Rectangle pulse |
| sinc(t) = sin(ฯt)/(ฯt) | rect(ฯ/2ฯ) | Sinc function |
| e^(-a|t|), a>0 | 2a/(aยฒ + ฯยฒ) | Two-sided exponential |
| e^(-atยฒ) | โ(ฯ/a)ยทe^(-ฯยฒ/4a) | Gaussian |
| tยทe^(-at)ยทu(t) | 1/(a+jฯ)ยฒ |
Properties
| Property | Time Domain | Frequency Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(ฯ) + bG(ฯ) |
| Time Shift | f(t - tโ) | F(ฯ)ยทe^(-jฯtโ) |
| Frequency Shift | f(t)ยทe^(jฯโt) | F(ฯ - ฯโ) |
| Time Scaling | f(at) | (1/|a|)ยทF(ฯ/a) |
| Duality | F(t) | 2ฯยทf(-ฯ) |
| Differentiation | f'(t) | jฯยทF(ฯ) |
| Integration | โซf(ฯ)dฯ | F(ฯ)/(jฯ) + ฯF(0)ฮด(ฯ) |
| Convolution | f(t) * g(t) | F(ฯ)ยทG(ฯ) |
| Multiplication | f(t)ยทg(t) | (1/2ฯ)F(ฯ) * G(ฯ) |
| Parseval's Theorem | โซ|f(t)|ยฒ dt | (1/2ฯ)โซ|F(ฯ)|ยฒ dฯ |
Discrete Fourier Transform (DFT)
DFT: X[k] = ฮฃ(n=0 to N-1) x[n]ยทe^(-j2ฯkn/N)
IDFT: x[n] = (1/N) ฮฃ(k=0 to N-1) X[k]ยทe^(j2ฯkn/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(Nยฒ)