[足式机器人]Part2 Dr. CAN学习笔记 - Ch03 傅里叶级数与变换

发布时间：2024年01月08日

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Dr. CAN学习笔记-Ch03 傅里叶级数与变换

1. 三角函数的正交性
2. 周期为 $2\pi$ 的函数展开为傅里叶级数
3. 周期为 $2 L$ 的函数展开
4. 傅里叶级数的复数形式
5. 从傅里叶级数推导傅里叶变换FT
6. 总结

1. 三角函数的正交性

三角函数系 ：集合 $\left\{ \sin nx,\cos nx \right\} n=0,1,2,\cdots$
正交：
$\int_{-\pi}^{\pi}{\sin nx\sin mx}\mathrm{d}x=0,n\ne m \\ \int_{-\pi}^{\pi}{\sin nx\cos mx}\mathrm{d}x=0,n\ne m \\ \int_{-\pi}^{\pi}{\cos nx\sin mx}\mathrm{d}x=0,n\ne m$

积化和差： $\Rightarrow \int_{-\pi}^{\pi}{\frac{1}{2}\left[ \cos \left( n-m \right) x+\cos \left( n+m \right) x \right]}\mathrm{d}x=\frac{1}{2}\frac{1}{\left( n-m \right)}\sin \left( n-m \right) x\mid_{-\pi}^{\pi}+\frac{1}{2}\frac{1}{\left( n+m \right)}\sin \left( n+m \right) x\mid_{-\pi}^{\pi}$
$\int_{-\pi}^{\pi}{\cos mx\cos mx}\mathrm{d}x=\pi$

2. 周期为 $2\pi$ 的函数展开为傅里叶级数

$T=2\pi :f\left( x \right) =f\left( x+2\pi \right)$

$f\left( x \right) =\sum_{n=0}^{\infty}{a_{\mathrm{n}}\cos nx}+\sum_{n=0}^{\infty}{b_{\mathrm{n}}\sin nx}=a_0\cos ox+\sum_{n=1}^{\infty}{a_{\mathrm{n}}\cos nx}+b_0\sin 0x+\sum_{n=1}^{\infty}{b_{\mathrm{n}}\sin nx}=a_0+\sum_{n=1}^{\infty}{a_{\mathrm{n}}\cos nx}+\sum_{n=1}^{\infty}{b_{\mathrm{n}}\sin nx}$

找 $a_0$ :
$\int_{-\pi}^{\pi}{f\left( x \right)}\mathrm{d}x=\int_{-\pi}^{\pi}{a_0}\mathrm{d}x+\int_{-\pi}^{\pi}{1\cdot \sum_{n=1}^{\infty}{a_{\mathrm{n}}\cos nx}}\mathrm{d}x+\int_{-\pi}^{\pi}{1\cdot \sum_{n=1}^{\infty}{b_{\mathrm{n}}\sin nx}}\mathrm{d}x \\ =a_0\int_{-\pi}^{\pi}{}\mathrm{d}x+0+0=a_0\cdot 2\pi$
$\Rightarrow a_0=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f\left( x \right)}\mathrm{d}x$
找 $a_n$ :
$\int_{-\pi}^{\pi}{f\left( x \right) \cos mx}\mathrm{d}x=\int_{-\pi}^{\pi}{a_0}\cos mx\cdot 1\mathrm{d}x+\int_{-\pi}^{\pi}{\sum_{n=1}^{\infty}{a_{\mathrm{n}}\cos nx\cos mx}}\mathrm{d}x+\int_{-\pi}^{\pi}{\sum_{n=1}^{\infty}{b_{\mathrm{n}}\sin nx\cos mx}}\mathrm{d}x=\int_{-\pi}^{\pi}{a_{\mathrm{n}}\cos nx\cos nx}\mathrm{d}x=a_{\mathrm{n}}\pi$
$\Rightarrow a_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \cos nx}\mathrm{d}x$
找 $b_n$ :
$\int_{-\pi}^{\pi}{f\left( x \right) \sin .mx}\mathrm{d}x\Rightarrow b_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \sin nx}\mathrm{d}x$

$\Rightarrow f\left( x \right) =f\left( x+2\pi \right) ,T=2\pi \begin{cases} f\left( x \right) =\frac{a_0}{2}+\sum_{n=0}^{\infty}{a_{\mathrm{n}}\cos nx}+\sum_{n=0}^{\infty}{b_{\mathrm{n}}\sin nx}\\ a_0=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f\left( x \right)}\mathrm{d}x\\ a_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \cos nx}\mathrm{d}x\\ b_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \sin nx}\mathrm{d}x\\ \end{cases}$

3. 周期为 $2 L$ 的函数展开

$f\left( t \right) =f\left( t+2L \right)$ , 换元： $x=\frac{\pi}{L}t,t=\frac{L}{\pi}x$
$f\left( t \right) =f\left( \frac{L}{\pi}x \right) =g\left( x \right) ,g\left( x+2\pi \right) =f\left( \frac{L}{\pi}\left( x+2\pi \right) \right) =f\left( \frac{L}{\pi}x+2L \right) =f\left( \frac{L}{\pi}x \right) =g\left( x \right)$

$g\left( x \right) =\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left[ a_{\mathrm{n}}\cos nx+b_{\mathrm{n}}\sin nx \right]} \\ a_0=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f\left( x \right)}\mathrm{d}x,a_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \cos nx}\mathrm{d}x,b_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \sin nx}\mathrm{d}x$

$\rightarrow x=\frac{\pi}{L}t\Rightarrow \cos nx=\cos \frac{n\pi}{L}t,\sin nx=\sin \frac{n\pi}{L}t,g\left( x \right) =f\left( t \right) \\ \int_{-\pi}^{\pi}{}\mathrm{d}x=\int_{-\pi}^{\pi}{}\mathrm{d}\frac{\pi}{L}t\Rightarrow \frac{1}{\pi}\int_{-\pi}^{\pi}{}\mathrm{d}x=\frac{1}{L}\int_{-L}^L{}\mathrm{d}t$

$\Rightarrow f\left( t \right) =\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left[ a_{\mathrm{n}}\cos \frac{n\pi}{L}t+b_{\mathrm{n}}\sin \frac{n\pi}{L}t \right]},a_0=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\mathrm{d}t,a_{\mathrm{n}}=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\cos \frac{n\pi}{L}t\mathrm{d}t,b_{\mathrm{n}}=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\sin \frac{n\pi}{L}t\mathrm{d}t$
在这里插入图片描述

4. 傅里叶级数的复数形式

$f\left( t \right) =\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left[ a_{\mathrm{n}}\frac{1}{2}\left( e^{inwt}+e^{-inwt} \right) -b_{\mathrm{n}}\frac{1}{2}\left( e^{inwt}-e^{-inwt} \right) \right]}=\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left[ \frac{a_{\mathrm{n}}-ib_{\mathrm{n}}}{2}e^{inwt}+\frac{a_{\mathrm{n}}+ib_{\mathrm{n}}}{2}e^{-inwt} \right]} \\ =\sum_{n=0}^0{\frac{a_0}{2}e^{inwt}}+\sum_{n=1}^{\infty}{\frac{a_{\mathrm{n}}-ib_{\mathrm{n}}}{2}e^{inwt}}+\sum_{n=-\infty}^{-1}{\frac{a_{\mathrm{n}}+ib_{\mathrm{n}}}{2}e^{inwt}} \\ =\sum_{n=-\infty}^{\infty}{C_{\mathrm{n}}e^{inwt}},C_{\mathrm{n}}=\begin{cases} \frac{a_0}{2}\,\,n=0\\ \frac{a_{\mathrm{n}}-ib_{\mathrm{n}}}{2}\,\,n=1,2,3,\cdots\\ \frac{a_{\mathrm{n}}+ib_{\mathrm{n}}}{2}\,\,n=-1,-2,-3,\cdots\\ \end{cases}\begin{array}{c} \rightarrow \frac{1}{T}\int_0^T{f\left( t \right)}\mathrm{d}t\\ \rightarrow \frac{1}{T}\int_0^T{f\left( t \right)}\left( \cos nwt-i\sin nwt \right) \mathrm{d}t=\frac{1}{T}\int_0^T{f\left( t \right)}\left( \cos \left( -nwt \right) +i\sin \left( -nwt \right) \right) \mathrm{d}t=\frac{1}{T}\int_0^T{f\left( t \right)}e^{-inwt}\mathrm{d}t\\ \rightarrow \frac{1}{T}\int_0^T{f\left( t \right) e^{-inwt}}\mathrm{d}t\\ \end{array} \\ \Rightarrow f\left( t \right) =\sum_{-\infty}^{\infty}{C_{\mathrm{n}}e^{inwt}},C_{\mathrm{n}}=\frac{1}{T}\int_0^T{f\left( t \right) e^{-inwt}}\mathrm{d}t$

Euler’s Formula

5. 从傅里叶级数推导傅里叶变换FT

$f_{\mathrm{T}}\left( t \right) =f\left( t+T \right)$
$f_{\mathrm{T}}\left( t \right) =\sum_{-\infty}^{\infty}{C_{\mathrm{n}}e^{inw_0t}}$ , 基频率： $w_0=\frac{2\pi}{T}$ , 定义函数: $C_{\mathrm{n}}=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}{f_{\mathrm{T}}\left( t \right) e^{-inwt}}\mathrm{d}t$

非周期，一般形式： $T\rightarrow \infty$
$\underset{T\rightarrow \infty}{\lim}f_{\mathrm{T}}\left( t \right) =f\left( t \right) ,\varDelta w=\left( n+1 \right) w_0-nw_0=w_0=\frac{2\pi}{T}\,\,T\nearrow \varDelta w\searrow$

$f_{\mathrm{T}}\left( t \right) =\sum_{-\infty}^{\infty}{\left( \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}{f_{\mathrm{T}}\left( t \right) e^{-inw_0t}}\mathrm{d}t \right) e^{inw_0t}},\frac{1}{T}=\frac{\varDelta w}{2\pi} \\ \Rightarrow f_{\mathrm{T}}\left( t \right) =\sum_{-\infty}^{\infty}{\left( \frac{\varDelta w}{2\pi}\int_{-\frac{T}{2}}^{\frac{T}{2}}{f_{\mathrm{T}}\left( t \right) e^{-inw_0t}}\mathrm{d}t \right) e^{inw_0t}},T\rightarrow \infty :\begin{cases} \int_{-\frac{T}{2}}^{\frac{T}{2}}{}\mathrm{d}t\rightarrow \int_{-\infty}^{\infty}{}\mathrm{d}t\\ nw_0\rightarrow w\\ \sum_{-\infty}^{\infty}{\varDelta w}\rightarrow \int_{-\infty}^{\infty}{}\mathrm{d}w\\ \end{cases} \\ \Rightarrow f\left( t \right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}{\left( \int_{-\infty}^{\infty}{f\left( t \right) e^{-iwt}}\mathrm{d}t \right)}e^{iwt}\mathrm{d}w,\int_{-\infty}^{\infty}{f\left( t \right) e^{-iwt}}\mathrm{d}t=F\left( w \right) \\ \Rightarrow f\left( t \right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}{F\left( w \right)}e^{iwt}\mathrm{d}w$

$F\left( w \right) =\int_{-\infty}^{\infty}{f\left( t \right) e^{-iwt}}\mathrm{d}t$ : FT 傅里叶变换

$f\left( t \right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}{F\left( w \right)}e^{iwt}\mathrm{d}w$ : 逆变换

在这里插入图片描述

6. 总结

三角函数的正交性：
$\left[ 0,1,\sin x,\cos x,\sin 2x,\cos 2x,\cdots ,\sin nx,\cos nx \right] ,n=0,1,2,\cdots$
$\int_{-\pi}^{\pi}{\sin nx\sin mx}\mathrm{d}x=0,n\ne m \\ \int_{-\pi}^{\pi}{\sin nx\sin mx}\mathrm{d}x=0,n\ne m \\ \int_{-\pi}^{\pi}{\sin nx\cos mx}\mathrm{d}x=0,n\ne m$

周期 $2\pi$ :
$f\left( x \right) =f\left( x+2\pi \right)$
$f\left( x \right) =\sum_{n=0}^{\infty}{a_{\mathrm{n}}\cos nx}+\sum_{n=0}^{\infty}{b_{\mathrm{n}}\sin nx}\gets f\left( x \right) =\frac{a_0}{2}+\sum_{n=0}^{\infty}{a_{\mathrm{n}}\cos nx}+\sum_{n=0}^{\infty}{b_{\mathrm{n}}\sin nx}$
$a_0=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f\left( x \right)}\mathrm{d}x,a_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \cos nx}\mathrm{d}x,b_{\mathrm{n}}=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \sin nx}\mathrm{d}x$

周期 $2 L$ :
$T=2L,f\left( t \right) =f\left( t+2L \right) ,x=\frac{\pi}{L}t$
$f\left( t \right) =\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left[ a_{\mathrm{n}}\cos \frac{n\pi}{L}t+b_{\mathrm{n}}\sin \frac{n\pi}{L}t \right]}$
$a_0=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\mathrm{d}t,a_{\mathrm{n}}=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\cos \frac{n\pi}{L}t\mathrm{d}t,b_{\mathrm{n}}=\frac{1}{L}\int_{-L}^L{f\left( t \right)}\sin \frac{n\pi}{L}t\mathrm{d}t$

复指数：
$f\left( t \right) =\sum_{-\infty}^{\infty}{C_{\mathrm{n}}e^{inw_0t}},w_0=\frac{2\pi}{T},C_{\mathrm{n}}=\frac{1}{T}\int_0^T{f\left( t \right) e^{-inw_0t}}\mathrm{d}t$

FT ：
$f\left( t \right) =f\left( t+T \right) ,T\rightarrow \infty$ , $f\left( t \right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}{\left( \int_{-\infty}^{\infty}{f\left( t \right) e^{-iwt}}\mathrm{d}t \right)}e^{iwt}\mathrm{d}w$
$\begin{array}{c} FT\rightarrow F\left( w \right) =\int_{-\infty}^{\infty}{f\left( t \right) e^{-iwt}}\mathrm{d}t\\ IFT\rightarrow \left( t \right) =\frac{1}{2\pi}\int_{-\infty}^{\infty}{F\left( w \right)}e^{iwt}\mathrm{d}w\\ \end{array}$
Laplace : $F\left( s \right) :\int_{-\infty}^{\infty}{f\left( t \right) e^{-st}}\mathrm{d}t$

文章来源:https://blog.csdn.net/LiongLoure/article/details/135453891
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