# 傅立葉變換

「傅立葉變換」一詞既指變換操作本身（將函數 ${\displaystyle f}$ 進行傅立葉變換），又指該操作所生成的複數函數（${\displaystyle {\hat {f}}}$${\displaystyle f}$ 的傅立葉變換）。

## 定義

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx}$ξ為任意實數

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi i\xi x}\,d\xi }$x為任意實數。

## 傅立葉變換的不同變種

${\displaystyle {\hat {f}}(\omega )=\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx.}$

${\displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbf {R} ^{n}}{\hat {f}}(\omega )e^{i\omega \cdot x}\,d\omega .}$

${\displaystyle {\hat {f}}(\omega )={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx}$
${\displaystyle f(x)={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}{\hat {f}}(\omega )e^{i\omega \cdot x}\,d\omega .}$

普通頻率ξ（ 赫茲） ${\displaystyle \displaystyle {\hat {f}}_{1}(\xi )\ {\stackrel {\mathrm {def} }{=}}\ \int _{\mathbf {R} ^{n}}f(x)e^{-2\pi ix\cdot \xi }\,dx={\hat {f}}_{2}(2\pi \xi )=(2\pi )^{n/2}{\hat {f}}_{3}(2\pi \xi )}$${\displaystyle \displaystyle f(x)=\int _{\mathbf {R} ^{n}}{\hat {f}}_{1}(\xi )e^{2\pi ix\cdot \xi }\,d\xi \ }$ ${\displaystyle \displaystyle {\hat {f}}_{2}(\omega )\ {\stackrel {\mathrm {def} }{=}}\int _{\mathbf {R} ^{n}}f(x)e^{-i\omega \cdot x}\,dx\ ={\hat {f}}_{1}\left({\frac {\omega }{2\pi }}\right)=(2\pi )^{n/2}\ {\hat {f}}_{3}(\omega )}$ ${\displaystyle \displaystyle f(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbf {R} ^{n}}{\hat {f}}_{2}(\omega )e^{i\omega \cdot x}\,d\omega \ }$ ${\displaystyle \displaystyle {\hat {f}}_{3}(\omega )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}f(x)\ e^{-i\omega \cdot x}\,dx={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{1}\left({\frac {\omega }{2\pi }}\right)={\frac {1}{(2\pi )^{n/2}}}{\hat {f}}_{2}(\omega )}$ ${\displaystyle \displaystyle f(x)={\frac {1}{(2\pi )^{n/2}}}\int _{\mathbf {R} ^{n}}{\hat {f}}_{3}(\omega )e^{i\omega \cdot x}\,d\omega \ }$

### 傅立葉級數

${\displaystyle f(x)=\sum _{n=-\infty }^{\infty }F_{n}\,e^{inx},}$

${\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos(nx)+b_{n}\sin(nx)\right]}$

### 離散傅立葉變換

${\displaystyle x_{n}=\sum _{k=0}^{N-1}X_{k}e^{-i{\frac {2\pi }{N}}kn}\qquad n=0,\dots ,N-1}$

## 常用傅立葉變換表

### 函數關係

${\displaystyle \displaystyle f(x)\,}$ ${\displaystyle \displaystyle {\hat {f}}(\xi )=}$

${\displaystyle \displaystyle \int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx}$

${\displaystyle \displaystyle {\hat {f}}(\omega )=}$

${\displaystyle \displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-i\omega x}\,dx}$

${\displaystyle \displaystyle {\hat {f}}(\nu )=}$

${\displaystyle \displaystyle \int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx}$

101 ${\displaystyle \displaystyle a\cdot f(x)+b\cdot g(x)\,}$ ${\displaystyle \displaystyle a\cdot {\hat {f}}(\xi )+b\cdot {\hat {g}}(\xi )\,}$ ${\displaystyle \displaystyle a\cdot {\hat {f}}(\omega )+b\cdot {\hat {g}}(\omega )\,}$ ${\displaystyle \displaystyle a\cdot {\hat {f}}(\nu )+b\cdot {\hat {g}}(\nu )\,}$ 線性
102 ${\displaystyle \displaystyle f(x-a)\,}$ ${\displaystyle \displaystyle e^{-2\pi ia\xi }{\hat {f}}(\xi )\,}$ ${\displaystyle \displaystyle e^{-ia\omega }{\hat {f}}(\omega )\,}$ ${\displaystyle \displaystyle e^{-ia\nu }{\hat {f}}(\nu )\,}$ 時域平移
103 ${\displaystyle \displaystyle e^{2\pi iax}f(x)\,}$ ${\displaystyle \displaystyle {\hat {f}}\left(\xi -a\right)\,}$ ${\displaystyle \displaystyle {\hat {f}}(\omega -2\pi a)\,}$ ${\displaystyle \displaystyle {\hat {f}}(\nu -2\pi a)\,}$ 頻域平移，變換102的頻域對應
104 ${\displaystyle \displaystyle f(ax)\,}$ ${\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)\,}$ ${\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}$ ${\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\nu }{a}}\right)\,}$ 在時域中定標。如果${\displaystyle \displaystyle |a|\,}$值較大，則${\displaystyle \displaystyle f(ax)\,}$會收縮到原點附近，而${\displaystyle \displaystyle {\frac {1}{|a|}}{\hat {f}}\left({\frac {\omega }{a}}\right)\,}$會擴散並變得扁平。當${\displaystyle \displaystyle |a|\,}$趨向無窮時，${\displaystyle \displaystyle f(ax)\,}$成為狄拉克δ函數
105 ${\displaystyle \displaystyle {\hat {f}}(x)\,}$ ${\displaystyle \displaystyle f(-\xi )\,}$ ${\displaystyle \displaystyle f(-\omega )\,}$ ${\displaystyle \displaystyle 2\pi f(-\nu )\,}$ 傅立葉變換的二元性性質。這裡${\displaystyle {\hat {f}}}$的計算需要運用與傅立葉變換那一列同樣的方法。通過交換變數${\displaystyle x}$${\displaystyle \xi }$${\displaystyle \omega }$${\displaystyle \nu }$得到。
106 ${\displaystyle \displaystyle {\frac {d^{n}f(x)}{dx^{n}}}\,}$ ${\displaystyle \displaystyle (2\pi i\xi )^{n}{\hat {f}}(\xi )\,}$ ${\displaystyle \displaystyle (i\omega )^{n}{\hat {f}}(\omega )\,}$ ${\displaystyle \displaystyle (i\nu )^{n}{\hat {f}}(\nu )\,}$ 傅立葉變換的微分性質
107 ${\displaystyle \displaystyle x^{n}f(x)\,}$ ${\displaystyle \displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}{\hat {f}}(\xi )}{d\xi ^{n}}}\,}$ ${\displaystyle \displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\omega )}{d\omega ^{n}}}}$ ${\displaystyle \displaystyle i^{n}{\frac {d^{n}{\hat {f}}(\nu )}{d\nu ^{n}}}}$ 變換106的頻域對應
108 ${\displaystyle \displaystyle (f*g)(x)\,}$ ${\displaystyle \displaystyle {\hat {f}}(\xi ){\hat {g}}(\xi )\,}$ ${\displaystyle \displaystyle {\sqrt {2\pi }}{\hat {f}}(\omega ){\hat {g}}(\omega )\,}$ ${\displaystyle \displaystyle {\hat {f}}(\nu ){\hat {g}}(\nu )\,}$ 記號${\displaystyle \displaystyle f*g\,}$表示${\displaystyle f}$${\displaystyle g}$的摺積—這就是摺積定理
109 ${\displaystyle \displaystyle f(x)g(x)\,}$ ${\displaystyle \displaystyle ({\hat {f}}*{\hat {g}})(\xi )\,}$ ${\displaystyle \displaystyle ({\hat {f}}*{\hat {g}})(\omega ) \over {\sqrt {2\pi }}\,}$ ${\displaystyle \displaystyle {\frac {1}{2\pi }}({\hat {f}}*{\hat {g}})(\nu )\,}$ 變換108的頻域對應。
110 ${\displaystyle \displaystyle f(x)}$是實變函數 ${\displaystyle \displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}}\,}$ ${\displaystyle \displaystyle {\hat {f}}(-\omega )={\overline {{\hat {f}}(\omega )}}\,}$ ${\displaystyle \displaystyle {\hat {f}}(-\nu )={\overline {{\hat {f}}(\nu )}}\,}$ 埃爾米特對稱。${\displaystyle \displaystyle {\overline {z}}\,}$表示複共軛
111 ${\displaystyle \displaystyle f(x)}$是實偶函數 ${\displaystyle \displaystyle {\hat {f}}(\omega )}$, ${\displaystyle \displaystyle {\hat {f}}(\xi )}$${\displaystyle \displaystyle {\hat {f}}(\nu )\,}$都是實偶函數
112 ${\displaystyle \displaystyle f(x)}$是實奇函數 ${\displaystyle \displaystyle {\hat {f}}(\omega )}$, ${\displaystyle \displaystyle {\hat {f}}(\xi )}$${\displaystyle \displaystyle {\hat {f}}(\nu )}$都是奇函數
113 ${\displaystyle \displaystyle {\overline {f(x)}}}$ ${\displaystyle \displaystyle {\overline {{\hat {f}}(-\xi )}}}$ ${\displaystyle \displaystyle {\overline {{\hat {f}}(-\omega )}}}$ ${\displaystyle \displaystyle {\overline {{\hat {f}}(-\nu )}}}$ 複共軛，110的一般化

### 平方可積函數

${\displaystyle g(t)\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}\mathrm {d} \omega \,}$
${\displaystyle G(\omega )\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}\mathrm {d} t\,}$
${\displaystyle G(f)\!\equiv }$

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}\mathrm {d} t\,}$
10 ${\displaystyle \mathrm {rect} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} \left({\frac {f}{a}}\right)}$ 矩形脈衝和歸一化的sinc函數
11 ${\displaystyle \mathrm {sinc} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {rect} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {rect} \left({\frac {f}{a}}\right)\,}$ 變換10的頻域對應。矩形函數是理想的低通濾波器，sinc函數是這類濾波器對反因果衝擊的響應。
12 ${\displaystyle \mathrm {sinc} ^{2}(at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {tri} \left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {tri} \left({\frac {f}{a}}\right)}$ tri三角形函數
13 ${\displaystyle \mathrm {tri} (at)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi a^{2}}}}\cdot \mathrm {sinc} ^{2}\left({\frac {\omega }{2\pi a}}\right)}$ ${\displaystyle {\frac {1}{|a|}}\cdot \mathrm {sinc} ^{2}\left({\frac {f}{a}}\right)\,}$ 變換12的頻域對應
14 ${\displaystyle e^{-\alpha t^{2}}\,}$ ${\displaystyle {\frac {1}{\sqrt {2\alpha }}}\cdot e^{-{\frac {\omega ^{2}}{4\alpha }}}}$ ${\displaystyle {\sqrt {\frac {\pi }{\alpha }}}\cdot e^{-{\frac {(\pi f)^{2}}{\alpha }}}}$ 高斯函數${\displaystyle \exp(-\alpha t^{2})}$的傅立葉變換是他本身.只有當${\displaystyle \mathrm {Re} (\alpha )>0}$時，這是可積的。
15 ${\displaystyle e^{iat^{2}}=\left.e^{-\alpha t^{2}}\right|_{\alpha =-ia}\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cdot e^{-i\left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cdot e^{-i\left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}}$ 光學領域應用較多
16 ${\displaystyle \cos(at^{2})\,}$ ${\displaystyle {\frac {1}{\sqrt {2a}}}\cos \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle {\sqrt {\frac {\pi }{a}}}\cos \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
17 ${\displaystyle \sin(at^{2})\,}$ ${\displaystyle {\frac {-1}{\sqrt {2a}}}\sin \left({\frac {\omega ^{2}}{4a}}-{\frac {\pi }{4}}\right)}$ ${\displaystyle -{\sqrt {\frac {\pi }{a}}}\sin \left({\frac {\pi ^{2}f^{2}}{a}}-{\frac {\pi }{4}}\right)}$
18 ${\displaystyle \mathrm {e} ^{-a|t|}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {a}{a^{2}+\omega ^{2}}}}$ ${\displaystyle {\frac {2a}{a^{2}+4\pi ^{2}f^{2}}}}$ a>0
19 ${\displaystyle {\frac {1}{\sqrt {|t|}}}\,}$ ${\displaystyle {\frac {1}{\sqrt {|\omega |}}}}$ ${\displaystyle {\frac {1}{\sqrt {|f|}}}}$ 變換本身就是一個公式
20 ${\displaystyle J_{0}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2\cdot \mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ J0(t)0階第一類貝索函數
21 ${\displaystyle J_{n}(t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {(-i)^{n}T_{n}(\omega )\mathrm {rect} \left({\frac {\omega }{2}}\right)}{\sqrt {1-\omega ^{2}}}}}$ ${\displaystyle {\frac {2(-i)^{n}T_{n}(2\pi f)\mathrm {rect} (\pi f)}{\sqrt {1-4\pi ^{2}f^{2}}}}}$ 上一個變換的推廣形式; Tn (t)第一類切比雪夫多項式
22 ${\displaystyle {\frac {J_{n}(t)}{t}}\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {i}{n}}(-i)^{n}\cdot U_{n-1}(\omega )\,}$

${\displaystyle \cdot \ {\sqrt {1-\omega ^{2}}}\mathrm {rect} \left({\frac {\omega }{2}}\right)}$

${\displaystyle {\frac {2\mathrm {i} }{n}}(-i)^{n}\cdot U_{n-1}(2\pi f)\,}$

${\displaystyle \cdot \ {\sqrt {1-4\pi ^{2}f^{2}}}\mathrm {rect} (\pi f)}$

Un (t)第二類切比雪夫多項式

### 分布

${\displaystyle g(t)\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,}$
${\displaystyle G(\omega )\!\equiv \!}$

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,}$
${\displaystyle G(f)\!\equiv }$

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
23 ${\displaystyle 1\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega )\,}$ ${\displaystyle \delta (f)\,}$ ${\displaystyle \delta (\omega )}$代表狄拉克δ函數分布.這個變換展示了狄拉克δ函數的重要性：該函數是常函數的傅立葉變換
24 ${\displaystyle \delta (t)\,}$ ${\displaystyle {\frac {1}{\sqrt {2\pi }}}\,}$ ${\displaystyle 1\,}$ 變換23的頻域對應
25 ${\displaystyle e^{iat}\,}$ ${\displaystyle {\sqrt {2\pi }}\cdot \delta (\omega -a)\,}$ ${\displaystyle \delta (f-{\frac {a}{2\pi }})\,}$ 由變換3和24得到.
26 ${\displaystyle \cos(at)\,}$ ${\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!+\!\delta (\omega \!+\!a)}{2}}\,}$ ${\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!+\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2}}\,}$ 由變換1和25得到，應用了歐拉公式${\displaystyle \cos(at)=(e^{iat}+e^{-iat})/2.}$
27 ${\displaystyle \sin(at)\,}$ ${\displaystyle {\sqrt {2\pi }}{\frac {\delta (\omega \!-\!a)\!-\!\delta (\omega \!+\!a)}{2i}}\,}$ ${\displaystyle {\frac {\delta (f\!-\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})\!-\!\delta (f\!+\!{\begin{matrix}{\frac {a}{2\pi }}\end{matrix}})}{2i}}\,}$ 由變換1和25得到
28 ${\displaystyle t^{n}\,}$ ${\displaystyle i^{n}{\sqrt {2\pi }}\delta ^{(n)}(\omega )\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}\delta ^{(n)}(f)\,}$ 這裡, ${\displaystyle n}$是一個自然數. ${\displaystyle \delta ^{(n)}(\omega )}$是狄拉克δ函數分布的${\displaystyle n}$階微分。這個變換是根據變換7和24得到的。將此變換與1結合使用，我們可以變換所有多項式
29 ${\displaystyle {\frac {1}{t}}\,}$ ${\displaystyle -i{\sqrt {\frac {\pi }{2}}}\operatorname {sgn} (\omega )\,}$ ${\displaystyle -i\pi \cdot \operatorname {sgn} (f)\,}$ 此處${\displaystyle \operatorname {sgn} (\omega )}$符號函數；注意此變換與變換7和24是一致的.
30 ${\displaystyle {\frac {1}{t^{n}}}\,}$ ${\displaystyle -i{\begin{matrix}{\sqrt {\frac {\pi }{2}}}\cdot {\frac {(-i\omega )^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn} (\omega )\,}$ ${\displaystyle -i\pi {\begin{matrix}{\frac {(-i2\pi f)^{n-1}}{(n-1)!}}\end{matrix}}\operatorname {sgn} (f)\,}$ 變換29的推廣.
31 ${\displaystyle \operatorname {sgn} (t)\,}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}\cdot {\frac {1}{i\ \omega }}\,}$ ${\displaystyle {\frac {1}{i\pi f}}\,}$ 變換29的頻域對應.
32 ${\displaystyle u(t)\,}$ ${\displaystyle {\sqrt {\frac {\pi }{2}}}\left({\frac {1}{i\pi \omega }}+\delta (\omega )\right)\,}$ ${\displaystyle {\frac {1}{2}}\left({\frac {1}{i\pi f}}+\delta (f)\right)\,}$ 此處${\displaystyle u(t)}$單位階躍函數;此變換根據變換1和31得到.
33 ${\displaystyle e^{-at}u(t)\,}$ ${\displaystyle {\frac {1}{{\sqrt {2\pi }}(a+i\omega )}}}$ ${\displaystyle {\frac {1}{a+i2\pi f}}}$ ${\displaystyle u(t)}$單位階躍函數，且${\displaystyle a>0}$.
34 ${\displaystyle \sum _{n=-\infty }^{\infty }\delta (t-nT)\,}$ ${\displaystyle {\begin{matrix}{\frac {\sqrt {2\pi }}{T}}\end{matrix}}\sum _{k=-\infty }^{\infty }\delta \left(\omega -k{\begin{matrix}{\frac {2\pi }{T}}\end{matrix}}\right)\,}$ ${\displaystyle {\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta \left(f-{\frac {k}{T}}\right)\,}$ 狄拉克梳狀函數英語Dirac comb——有助於解釋或理解從連續到離散時間的轉變.

### 二元函數

400 ${\displaystyle \displaystyle f(x,y)}$ ${\displaystyle \displaystyle {\hat {f}}(\xi _{x},\xi _{y})=}$
${\displaystyle \displaystyle \iint f(x,y)e^{-2\pi i(\xi _{x}x+\xi _{y}y)}\,dx\,dy}$
${\displaystyle \displaystyle {\hat {f}}(\omega _{x},\omega _{y})=}$
${\displaystyle \displaystyle {\frac {1}{2\pi }}\iint f(x,y)e^{-i(\omega _{x}x+\omega _{y}y)}\,dx\,dy}$
${\displaystyle \displaystyle {\hat {f}}(\nu _{x},\nu _{y})=}$
${\displaystyle \displaystyle \iint f(x,y)e^{-i(\nu _{x}x+\nu _{y}y)}\,dx\,dy}$
401 ${\displaystyle \displaystyle e^{-\pi \left(a^{2}x^{2}+b^{2}y^{2}\right)}}$ ${\displaystyle \displaystyle {\frac {1}{|ab|}}e^{-\pi \left(\xi _{x}^{2}/a^{2}+\xi _{y}^{2}/b^{2}\right)}}$ ${\displaystyle \displaystyle {\frac {1}{2\pi \cdot |ab|}}e^{\frac {-\left(\omega _{x}^{2}/a^{2}+\omega _{y}^{2}/b^{2}\right)}{4\pi }}}$ ${\displaystyle \displaystyle {\frac {1}{|ab|}}e^{\frac {-\left(\nu _{x}^{2}/a^{2}+\nu _{y}^{2}/b^{2}\right)}{4\pi }}}$
402 ${\displaystyle \displaystyle \mathrm {circ} ({\sqrt {x^{2}+y^{2}}})}$ ${\displaystyle \displaystyle {\frac {J_{1}\left(2\pi {\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}\right)}{\sqrt {\xi _{x}^{2}+\xi _{y}^{2}}}}}$ ${\displaystyle \displaystyle {\frac {J_{1}\left({\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}\right)}{\sqrt {\omega _{x}^{2}+\omega _{y}^{2}}}}}$ ${\displaystyle \displaystyle {\frac {2\pi J_{1}\left({\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}\right)}{\sqrt {\nu _{x}^{2}+\nu _{y}^{2}}}}}$

400： 變數ξxξyωxωyνxνy為實數。 對整個平面積分。

401： 這兩個函數都是高斯分布，而且可能不具有單位體積。

402： 此圓有單位半徑，如果把circ（t）認作階梯函數u(1-t); Airy分布用J1（1階第一類貝索函數）表達。（Stein & Weiss 1971，Thm. IV.3.3）

### 三元函數

${\displaystyle \mathrm {circ} ({\sqrt {x^{2}+y^{2}+z^{2}}})}$ ${\displaystyle 4\pi {\frac {\sin[2\pi f_{r}]-2\pi f_{r}\cos[2\pi f_{r}]}{(2\pi f_{r})^{3}}}}$ 此球有單位半徑；fr是頻率矢量的量值{fx,fy,fz}.

## 參考資料

### 文內資料引用

1. ^ 楊毅明. 數字信號處理（第2版）. 北京: 機械工業出版社. 2017年: 第25、29頁. ISBN 9787111576235.