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

##

FT

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

\[X[k] = \sum_{n=0}^{N-1} x[n]e^{-jk\frac{2\pi}{N}n}\]