也可以将它定义为 r e c t ( ± 1 / 2 ) {\displaystyle \mathrm {rect} (\pm 1/2)} 的值为 0、1 或者未定义的值,另外也可以用 单位阶跃函数 u ( t ) {\displaystyle u(t)} 来定义: r e c t ( t τ ) = u ( t + τ 2 ) − u ( t − τ 2 ) {\displaystyle \mathrm {rect} \left({\frac {t}{\tau }}\right)=u\left(t+{\frac {\tau }{2}}\right)-u\left(t-{\frac {\tau }{2}}\right)} 矩形函数 矩形函数的定义为, r e c t ( t ) = Π ( t ) = { 0 if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 {\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}}} 或者, r e c t ( t ) = u ( t + 1 2 ) − u ( t − 1 2 ) {\displaystyle \mathrm {rect} (t)=u\left(t+{\frac {1}{2}}\right)-u\left(t-{\frac {1}{2}}\right)} 矩形函数归一化: ∫ − ∞ ∞ r e c t ( t ) d t = 1 {\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\,dt=1} 矩形函数的傅立叶变换, ∫ − ∞ ∞ r e c t ( t ) ⋅ e − i 2 π f t d t = sin ( π f ) π f = s i n c ( f ) {\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\mathrm {sinc} (f)} 或用用归一化Sinc函数表示为: 1 2 π ∫ − ∞ ∞ r e c t ( t ) ⋅ e − i ω t d t = 1 2 π ⋅ s i n c ( ω 2 ) {\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)} , 我们可以将三角形函数定义为两个矩形函数的卷积: t r i ( t ) = r e c t ( t ) ∗ r e c t ( t ) {\displaystyle \mathrm {tri} (t)=\mathrm {rect} (t)*\mathrm {rect} (t)} 如果将矩形函数当作一个概率分布函数,那么它的特征函数是, φ ( k ) = sin ( k / 2 ) k / 2 {\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}}\,} 并且它的动差生成函数为, M ( k ) = s i n h ( k / 2 ) k / 2 {\displaystyle M(k)={\frac {\mathrm {sinh} (k/2)}{k/2}}\,} 其中 s i n h ( t ) {\displaystyle \mathrm {sinh} (t)} 是双曲正弦函数。 Remove ads参见 傅立叶变换 方波 三角形函数 Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for FirefoxRemove ads
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