User:Vanished user fijw983kjaslkekfhj45/DTFT Tables

Definition

 * $$X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}.$$

Inverse transform

 * $$x[n] = \frac{1}{2 \pi}\int_{-\pi}^{\pi} X(\omega)\cdot e^{i \omega n} \, d \omega$$

Table of discrete-time Fourier transforms

 * $$n \!$$ is an integer representing the discrete-time domain (in samples)
 * $$\omega \!$$ is a real number in $$(-\pi,\ \pi)$$, representing continuous angular frequency (in radians per sample).
 * The remainder of the transform $$(|\omega| > \pi \,)$$ is defined by: $$X(\omega + 2\pi k) = X(\omega)\,$$
 * $$u[n] \!$$ is the discrete-time unit step function
 * $$\operatorname{sinc}(t) \!$$ is the normalized sinc function
 * $$\delta (\omega) \!$$ is the Dirac delta function
 * $$\delta [n] \!$$ is the Kronecker delta $$\delta_{n,0} \!$$
 * $$ \operatorname{rect}(t) $$ is the rectangle function for arbitrary real-valued t:
 * $$\mathrm{rect}(t) = \sqcap(t) = \begin{cases}

0          & \mbox{if } |t| > \frac{1}{2} \\[3pt] \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\[3pt] 1          & \mbox{if } |t| < \frac{1}{2} \end{cases} $$
 * $$\operatorname{tri}(t) $$ is the triangle function for arbitrary real-valued t:
 * $$\operatorname{tri}(t) = \land (t) =

\begin{cases} 1 + t; & - 1 \leq t \leq 0 \\ 1 - t; & 0 < t \leq 1 \\ 0 & \mbox{otherwise} \end{cases} $$