DSP#3 Fourier tranforms summary
Summary so far.
| Continuous-time | Discrete-time | |
|---|---|---|
| Periodic | Fourier series | Discrete fourier transform (DFT) |
| Aperiodic | Fourier transform | Discrete-time Fourier transform (DTFT) |
| Aperiodic (general) | Laplace transform | Z-transform |
##
Fourier series
Synthesis
\[x(t) = \sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0t}\]Analysis
\[a_k = \frac{1}{T} \int_0^T x(t)e^{-jk\omega_0t}dt\]Fourier transform
Synthesis
\[x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega)e^{j\omega t}d\omega\]Analysis
\[X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega_0t}dt\]Laplace transform
\[X(s) = \int_{-\infty}^{\infty}x(t)e^{-st}dt\]Equals to fourier transform when \(s=j\omega_0\).
DTFT
Synthesis
\[x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega)e^{j\omega n}d\omega\]Analysis
\[X(\omega) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\]z-transform
\[X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}\]Equals to DTFT when \(z=e^{j\omega}\).
DFT
Synthesis
\[x[n] = \frac{1}{N} \sum_{n=0}^{N-1}X[k]e^{jk \frac{2\pi}{N}n}\]Analysis
\[X[k] = \sum_{n=0}^{N-1} x[n]e^{-jk\frac{2\pi}{N}n}\]