# 从傅里叶级数到傅里叶变换（连续、离散）

## 傅里叶级数

\displaystyle f(t)=\frac{a_0}{2}+\sum_{n=1}^\infty[a_n\cos(nw_0t)+b_n\sin(nw_0t)]

\displaystyle \int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)dt=\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}\frac{a_0}{2}dt=\frac{\pi}{w_0}a_0\\{}\\ \Rightarrow a_0=\frac{w_0}{\pi}\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)dt=\frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)dt\\{}\\ \int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)\cos(nw_0t)dt=\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}a_n\cos^2(nw_0t)dt=\frac{\pi}{w_0}a_n\\{}\\ \Rightarrow a_n=\frac{w_0}{\pi}\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)\cos(nw_0t)dt=\frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)\cos(nw_0t)dt\\{}\\ \int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)\sin(nw_0t)dt=\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}b_n\sin^2(nw_0t)dt=\frac{\pi}{w_0}b_n\\{}\\ \Rightarrow b_n=\frac{w_0}{\pi}\int_{-\frac{\pi}{w_0}}^{\frac{\pi}{w_0}}f(t)\sin(nw_0t)dt=\frac{2}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)\sin(nw_0t)dt

\displaystyle e^{jx}=\cos(x)+j\sin(x)

\displaystyle \cos(nw_0t)=\frac{e^{jnw_0t}+e^{-jnw_0t}}{2}\\{}\\ \sin(nw_0t)=\frac{e^{jnw_0t}-e^{-jnw_0t}}{2j}

\displaystyle f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left(\frac{a_n-jb_n}{2}e^{jnw_0t}+\frac{a_n+jb_n}{2}e^{-jnw_0t}\right)\\{}\\ F_n=\frac{a_n-jb_n}{2}\Rightarrow F_{-n}=\frac{a_n+jb_n}{2},F_0=\frac{a_0}{2}\\{}\\ \Rightarrow f(t)=F_0+\sum_{n=1}^\infty\left(F_ne^{jnw_0t}+F_{-n}e^{-jnw_0t}\right)=\sum_{n=-\infty}^{\infty}F_ne^{hnw_0t}

a_n,b_n代到F_n中：

\displaystyle F_n=\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)e^{-jnw_0t}dt

## 连续傅里叶变换

\displaystyle \lim_{T_0\to\infty}F_nT_0=\lim_{T_0\to\infty}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)e^{-jnw_0tdt}=\int_{-\infty}^\infty f(t)e^{-jwt}dt

F(w)=\lim_{T_0\to\infty}F_nT_0得到傅里叶正变换

\displaystyle F(w)=\int_{-\infty}^\infty f(t)e^{-jwt}dt

\displaystyle \lim_{T_0\to\infty}F_n=\lim_{T_0\to\infty}\frac{F(w)}{T_0}=\lim_{T_0\to\infty}\frac{F(w)w_0}{2\pi}=\frac{F(w)dw}{2\pi}

\displaystyle f(t)=\lim_{T_0\to\infty}\sum_{n=-\infty}^\infty F_ne^{jnw_0t}=\frac{1}{2\pi}\int_{-\infty}^\infty F(w)e^{jwt}dw

## 离散傅里叶变换

w_0=\frac{2\pi}{T_0}代入到F_n

\displaystyle F_n=\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)e^{-jnw_0t}dt=\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}f(t)e^{-j2\pi\frac{nt}{T_0}}dt

\displaystyle F(u)=\frac{1}{N}\sum_{x=0}^Nf(x)e^{-j2\pi\frac{ux}{N}},u=0,1,\cdots,N-1

\displaystyle F(u)=\sum_{x=0}^Nf(x)e^{-j2\pi\frac{ux}{N}},u=0,1,\cdots,N-1

\displaystyle f(x)=\frac{1}{N}\sum_{u=0}^NF(u)e^{j2\pi\frac{ux}{N}},x=0,1,\cdots,N-1

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Coder @ Galois

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