# 矩形函数

${\displaystyle \mathrm {rect} (t)=\Pi (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}$

${\displaystyle \mathrm {rect} \left({\frac {t}{\tau }}\right)=u\left(t+{\frac {\tau }{2}}\right)-u\left(t-{\frac {\tau }{2}}\right)}$

${\displaystyle \mathrm {rect} (t)=u\left(t+{\frac {1}{2}}\right)-u\left(t-{\frac {1}{2}}\right)}$

${\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\,dt=1}$

${\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\mathrm {sinc} (f)}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt {2\pi }}}\cdot \mathrm {sinc} \left({\frac {\omega }{2}}\right)}$,

${\displaystyle \mathrm {tri} (t)=\mathrm {rect} (t)*\mathrm {rect} (t)}$

${\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}}\,}$

${\displaystyle M(k)={\frac {\mathrm {sinh} (k/2)}{k/2}}\,}$