一维傅里叶变换

任何一个周期函数都可以由一些不同频率的正弦（或余弦）函数线性叠加得到。

figure41.png

表示成数学表达式是这样子的：

$$f(x) = \frac{a_0}{2} +\sum_{n=1}^{\infty}[a_n \cos(n \omega x) + b_n \sin(n \omega x)]$$

其中

$$a_n = \frac{2}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) \cos(n \omega x)\, dx$$

$$b_n = \frac{2}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) \sin(n \omega x)\, dx$$

$$T = \frac{2\pi}{\omega}$$

这被称为傅里叶级数。由欧拉公式$$e^{i\theta}=cos\theta + sin\theta$$可将上述傅里叶级数公式变形为：

$$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in \omega x}$$

$$F(u)=\int_{-\infty}^{\infty}f(x)e^{-j2 \pi ux}dx$$ 

latex error! exitcode was 1 (signal 0), transscript follows:

This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=latex)
entering extended mode
(./latex_f76fb155bf9b5f9cd018ff3fa1f2c57ec39a891c_p.tex
LaTeX2e <2023-11-01> patch level 1
L3 programming layer <2024-01-22>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2023/05/17 v1.4n Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))

! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.6 $
     $c_n = \frac{1}{

! LaTeX Error: Command \mdseries invalid in math mode.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.8 }
     \int_{-\frac{T}{2}}^{\frac{T}{2}} f(x) e^{-in \omega x}\, dx$$
(/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-dvips.def)
No file latex_f76fb155bf9b5f9cd018ff3fa1f2c57ec39a891c_p.aux.

LaTeX Font Warning: Command \normalsize invalid in math mode on input line 8.

[1] (./latex_f76fb155bf9b5f9cd018ff3fa1f2c57ec39a891c_p.aux) )
(see the transcript file for additional information)
Output written on latex_f76fb155bf9b5f9cd018ff3fa1f2c57ec39a891c_p.dvi (1 page,
 712 bytes).
Transcript written on latex_f76fb155bf9b5f9cd018ff3fa1f2c57ec39a891c_p.log.

二维离散傅里叶变换

参考资源：

ch3n2k.com | Copyright (c) 2004-2020 czk.