# 傅立叶变换家族中的关系

## 家族中各个变换的定义

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- ${\displaystyle X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\,dt}$ ${\displaystyle {\bar {X}}(f)=\sum _{n=-\infty }^{+\infty }x[n]\ e^{-i2\pi fnT}}$

- ${\displaystyle X[k]={\frac {1}{T_{0}}}\int _{T_{0}}{\bar {x}}(t)\;e^{-i{\frac {2\pi k}{T_{0}}}t}\,dt}$ ${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\;e^{-i{\frac {2\pi }{N}}kn},\quad k=0,\dots ,N-1.}$

• ${\displaystyle x[n]}$${\displaystyle X[k]}$ 都为无限序列，其采样间隔，即间隔时间和间隔频率分别为 ${\displaystyle T}$${\displaystyle f_{0}=1/T_{0}}$
• ${\displaystyle {\bar {x}}(t)}$${\displaystyle {\bar {X}}(f)}$ 都为周期函数，且时间周期和频率周期分别为 ${\displaystyle T_{0}}$${\displaystyle f_{s}=1/T}$
• ${\displaystyle x_{n}}$${\displaystyle X_{k}}$ 都为有限序列，且序列长度都为 ${\displaystyle N}$

## 关系推导所需的公式

1. Dirac comb函数的傅里叶变换

Dirac comb函数的定义为

${\displaystyle \Delta _{T}(t){\stackrel {\text{def}}{=}}\sum _{n=-\infty }^{\infty }\delta (t-nT)}$

${\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }e^{i{\frac {2\pi k}{T}}t}\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)=\sum _{n=-\infty }^{\infty }e^{-i2\pi nTf}}$

2. 傅里叶变换的卷积定理（convolution theorem）

{\displaystyle {\begin{aligned}x_{1}(t)\ast x_{2}(t)&\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad X_{1}(f)\cdot X_{2}(f)\\x_{1}(t)\cdot x_{2}(t)&\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad X_{1}(f)\ast X_{2}(f)\end{aligned}}}

3. 泊松求和公式（Poisson summation formula）

{\displaystyle {\begin{aligned}1.\qquad &\sum _{n=-\infty }^{\infty }x(t-nT_{0})={\frac {1}{T_{0}}}\sum _{k=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}\right)e^{i{\frac {2\pi k}{T_{0}}}t}\\2.\qquad &\sum _{n=-\infty }^{\infty }x(nT)e^{-i2\pi nTf}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X\left(f-{\frac {k}{T}}\right)\end{aligned}}}

${\displaystyle \sum _{n=-\infty }^{\infty }x(nT)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X\left({\frac {k}{T}}\right)}$

## 各种变换之间的关系

• ${\displaystyle {\tilde {X}}_{k}}$ 为由FS和DTFT推导DFT得到的DFT'频域形式，与传统DFT的频域 ${\displaystyle X_{k}}$ 有关系： ${\displaystyle X_{k}=T_{0}{\tilde {X}}_{k}}$
• 图中粗的双箭头（${\displaystyle \leftrightarrow }$）表示每个函数和其变换之间的联系；

• 如果时域进行周期扩展，则频域为采样；如果时域进行采样，则频域为周期扩展；
• 一个转换中，周期扩展的周期与采样的间隔有倒数关系；
• 频域的周期扩展或者采样，都有一个周期或采样间隔作系数；

### 由DTFT推导DFT

{\displaystyle {\begin{aligned}x[n]&\quad {\stackrel {\mathcal {DTFT}}{\longleftrightarrow }}\quad {\bar {X}}(f)\\\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)&\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad {\bar {X}}(f)\end{aligned}}}

{\displaystyle {\begin{aligned}&\left(\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right)\ast \left(\sum _{n=-\infty }^{\infty }\delta (t-nT_{0})\right)\\&=\sum _{n=-\infty }^{\infty }\sum _{i=-\infty }^{\infty }x(nT)\cdot \delta (t-nT-iNT)\\&=\sum _{n=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }x(nT-iNT)\right)\delta (t-nT)\end{aligned}}}

{\displaystyle {\begin{aligned}x_{n}&=\sum _{i=-\infty }^{\infty }x(nT-iNT)=\sum _{i=-\infty }^{\infty }x[n-iN]\\&=\sum _{i=-\infty }^{\infty }x(nT-iT_{0})\end{aligned}}}

{\displaystyle {\begin{aligned}&{\bar {X}}(f)\cdot {\frac {1}{T_{0}}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T_{0}}}\right)\\&={\frac {1}{T_{0}T}}\left(\sum _{k=-\infty }^{\infty }X\left(f-{\frac {k}{T}}\right)\right)\cdot \left(\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T_{0}}}\right)\right)\\&={\frac {1}{T_{0}T}}\sum _{k=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }X\left(f-{\frac {i}{T}}\right)\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)\right)\\&={\frac {1}{T_{0}T}}\sum _{k=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}-{\frac {i}{T}}\right)\right)\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)\end{aligned}}}

${\displaystyle {\tilde {X}}_{k}={\frac {1}{T_{0}T}}\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}-{\frac {i}{T}}\right)={\frac {1}{T_{0}}}{\bar {X}}\left({\frac {k}{T_{0}}}\right)}$

### 由FS推导DFT

{\displaystyle {\begin{aligned}{\bar {x}}(t)&\quad {\stackrel {\mathcal {FS}}{\longleftrightarrow }}\quad X[k]\\{\bar {x}}(t)&\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad \sum _{k=-\infty }^{\infty }X[k]\cdot \delta \left(f-{\frac {k}{T_{0}}}\right).\end{aligned}}}

{\displaystyle {\begin{aligned}&{\bar {x}}(t)\cdot \sum _{n=-\infty }^{\infty }\delta (t-nT)\\&=\left(\sum _{n=-\infty }^{\infty }x(t-nT_{0})\right)\cdot \left(\sum _{n=-\infty }^{\infty }\delta (t-nT)\right)\\&=\sum _{n=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }x(t-iT_{0})\cdot \delta (t-nT)\right)\\&=\sum _{n=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }x(nT-iT_{0})\right)\cdot \delta (t-nT)\end{aligned}}}

${\displaystyle x_{n}=\sum _{i=-\infty }^{\infty }x(nT-iT_{0})={\bar {x}}(nT)}$

{\displaystyle {\begin{aligned}&\left(\sum _{k=-\infty }^{\infty }X[k]\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)\right)\ast \left({\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\right)\\&={\frac {1}{T_{0}T}}\sum _{k=-\infty }^{\infty }\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}\right)\cdot \delta \left(f-{\frac {k}{T_{0}}}-{\frac {iN}{T_{0}}}\right)\\&={\frac {1}{T_{0}T}}\sum _{k=-\infty }^{\infty }\left(\sum _{i=-\infty }^{\infty }X\left({\frac {k-iN}{T_{0}}}\right)\right)\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)\end{aligned}}}

{\displaystyle {\begin{aligned}{\tilde {X}}_{k}&={\frac {1}{T_{0}T}}\sum _{i=-\infty }^{\infty }X\left({\frac {k-iN}{T_{0}}}\right)={\frac {1}{T}}\sum _{i=-\infty }^{\infty }X[k-iN]\\&={\frac {1}{T_{0}T}}\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}-{\frac {i}{T}}\right)\end{aligned}}}

### CFT与DFT的关系

{\displaystyle {\begin{aligned}{\tilde {X}}_{k}&={\frac {1}{T_{0}}}\sum \limits _{n=0}^{N-1}\ x_{n}e^{-{\frac {2\pi i}{N}}kn}\\x_{n}&=T\sum \limits _{k=0}^{N-1}\ {\tilde {X}}_{k}e^{{\frac {2\pi i}{N}}kn}\end{aligned}}}

${\displaystyle x_{n}\quad {\stackrel {\mathcal {DFT'}}{\longleftrightarrow }}\quad {\tilde {X}}_{k}}$

${\displaystyle X_{k}=T_{0}{\tilde {X}}_{k}.}$

${\displaystyle \sum _{n=-\infty }^{\infty }x_{n}\cdot \delta (t-nT)\quad {\stackrel {\mathcal {F}}{\longleftrightarrow }}\quad {\begin{matrix}\displaystyle {\sum _{k=-\infty }^{\infty }{\tilde {X}}_{k}\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)}\\\displaystyle {={\frac {1}{T_{0}}}\sum _{k=-\infty }^{\infty }X_{k}\cdot \delta \left(f-{\frac {k}{T_{0}}}\right)}\end{matrix}}}$

${\displaystyle x_{n}=\sum _{i=-\infty }^{\infty }x(nT-iT_{0}){\begin{matrix}\displaystyle {\quad {\stackrel {\mathcal {DFT'}}{\longleftrightarrow }}\quad {\tilde {X}}_{k}={\frac {1}{T_{0}T}}\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}-{\frac {i}{T}}\right)}\\\displaystyle {\quad {\stackrel {\mathcal {DFT}}{\longleftrightarrow }}\quad X_{k}={\frac {1}{T}}\sum _{i=-\infty }^{\infty }X\left({\frac {k}{T_{0}}}-{\frac {i}{T}}\right)}\end{matrix}}}$

## 参考文献

1. Oppenheim, Alan V.; Schafer, R. W.; and Buck, J. R., (1999). Discrete-time signal processing, Upper Saddle River, N.J. : Prentice Hall. ISBN 0137549202
2. Sklar, B., (2001). Digital Communications: Foundamentals and Applicatons, 2nd Edition, Prentice Hall PTR. ISBN 0130847887