傅立葉變換的推導(dǎo)

2023-02-28 12:17 作者:啊嗚西嗚安 0人讀過(guò) | 我要投稿

三角函數(shù)系

首先引入傅立葉變換所使用的三角函數(shù)系

$%5C%7Bsin(0x)%2Ccos(0x)%2Csin(1x)%2Ccos(1x)%2Csin(2x)%2Ccos(2x)%2C%5Ccdots%20%2Csin(nx)%2Ccos(nx)%20%20%20%20%20%5C%7D$

這組三角函數(shù)系像波函數(shù)一樣具有正交性和完備性，但不是歸一的。對(duì)于正交性我們可以看到：

$%5Cbegin%7Balign%7D%0A%5Cint_%7B-%5Cpi%7D%5E%7B%5Cpi%7D%20sin(x)cos(x)dx%26%3D0%5C%5C%0A%5Cint_%7B-%5Cpi%7D%5E%7B%5Cpi%7D%20sin(nx)cos(mx)%26%3D0%5C%5C%0A%5Cint_%7B-%5Cpi%7D%5E%7B%5Cpi%7D%20cos(mx)cos(mx)%26%3D%5Cpi%0A%5Cend%7Balign%7D%0A%5Ctag%7B1%7D$

并且三角函數(shù)具有周期性，其周期 $T%3D2%5Cpi$ ，即 $f(x)%3Df(x%2B2%5Cpi)$ ，見(jiàn)圖1。

傅立葉發(fā)現(xiàn)周期為 $2%5Cpi$ 的函數(shù)可以展開(kāi)為一系列三角函數(shù)的和的形式，即

$%5Cbegin%7Balign%7D%0Af(x)%26%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20a_ncos(nx)%2B%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20b_nsin(nx)%5C%5C%0A%26%3Da_0%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20a_ncos(nx)%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20sin(nx)%0A%5Cend%7Balign%7D%0A%5Ctag%7B2%7D$

現(xiàn)在我們先來(lái)求系數(shù) $a_0$ ，對(duì)公式兩邊同時(shí)對(duì)x作積分，得

$%5Cbegin%7Balign%7D%0A%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)%20dx%26%3D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20a_0%20dx%2B%5Cint_%7B-%5Cpi%7D%5E%5Cpi%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20a_n%20cos(nx)dx%20%2B%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Db_nsin(nx)dx%5C%5C%0A%26%3D2%5Cpi%20a_0%2Ba_n%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20cos(0x)cos(nx)dx%2Bb_n%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20cos(0x)sin(nx)dx%5C%5C%0A%26%3D2%5Cpi%20a_0%0A%5Cend%7Balign%7D%0A%5Ctag%7B3%7D$

這里利用了三角函數(shù)的正交性，最后算得

$%5Cbegin%7Balign%7D%0Aa_0%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)dx%0A%5Cend%7Balign%7D%0A%5Ctag%7B4%7D$

這時(shí)，令 $a_0%5E%5Cprime%3D2a_0$ ，則公式(2)變?yōu)?/p>

$%5Cbegin%7Balign%7D%0Af(x)%3D%5Cfrac%7Ba_0%5E%5Cprime%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20a_n%20cos(nx)%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20b_n%20sin(nx)%2C%20a_0%5E%5Cprime%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)dx%0A%5Cend%7Balign%7D%0A%5Ctag%7B5%7D$

這里為了表示方便，把" $%5Cprime$ "丟掉，即

$%5Cbegin%7Balign%7D%0Af(x)%3D%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20a_n%20cos(nx)%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Db_n%20sin(nx)%2C%20a_0%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)dx%0A%5Cend%7Balign%7D%0A%5Ctag%7B6%7D$

接下來(lái)我們?cè)偾笙禂?shù) $a_n$ ，對(duì)公式(6)兩邊同乘cos(mx)，并對(duì)x進(jìn)行積分，得

$%5Cbegin%7Balign%7D%0A%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)cos(mx)dx%26%3D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Cfrac%7Ba_0%7D%7B2%7Dcos(mx)dx%2B%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_ncos(nx)cos(mx)dx%2B%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20b_nsin(nx)cos(mx)dx%5C%5C%0A%26%3D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20a_n%20cos(nx)%5E2%20dx%5C%5C%0A%26%3Da_n%20%5Cpi%0A%5Cend%7Balign%7D%0A%5Ctag%7B7%7D$

所以求得，

$%5Cbegin%7Balign%7D%0Aa_n%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%20%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)cos(nx)%20dx%0A%5Cend%7Balign%7D%0A%5Ctag%7B8%7D$

同理求得 $b_n%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)sin(nx)dx$ 。這樣就求完了周期為 $2%5Cpi$ 的函數(shù)f(x)的傅立葉展開(kāi)系數(shù)。

然而物理中的函數(shù)周期通常都不是 $2%5Cpi$ ，現(xiàn)在來(lái)求周期T=2L的函數(shù)f(t)的傅立葉展開(kāi)系數(shù)。對(duì)于函數(shù)f(t)，我們有 $f(t)%3Df(t%2B2L)$ 。為了把函數(shù)f(t)的周期變換到 $2%5Cpi$ ，這里利用換元法，令 $x%3D%5Cfrac%7B%5Cpi%7D%7BL%7Dt$ ，所以 $t%3D%5Cfrac%7BL%7D%7B%5Cpi%7Dx$ 。則有, $%5Cbegin%7Balign%7D%0Af(t)%3Df(%5Cfrac%7BL%7D%7B%5Cpi%7Dx)%3Dg(x)%3Df(%5Cfrac%7BL%7D%7B%5Cpi%7D(x%2B2%5Cpi))%3Dg(x%2B2%5Cpi)%3Df(t%2B2L)%0A%5Cend%7Balign%7D%0A%5Ctag%7B9%7D$

這樣就把周期為2L的函數(shù)f(t)變換成了周期為 $2%5Cpi$ 的函數(shù)g(x)。周期為 $2%5Cpi$ 的函數(shù)g(x)的展開(kāi)系數(shù)我們已經(jīng)求過(guò)了，即 $%5Cbegin%7Balign%7D%0Af(t)%26%3D%20%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Ba_ncos(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)%2Bb_nsin(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)%5D%5C%5C%0Aa_0%26%3D%5Cfrac%7B1%7D%7BL%7D%5Cint_%7B-L%7D%5EL%20f(t)dt%5C%5C%0Aa_n%26%3D%20%5Cfrac%7B1%7D%7BL%7D%20%5Cint_%7B-L%7D%5EL%20f(t)cos(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)dt%5C%5C%0Ab_n%26%3D%20%5Cfrac%7B1%7D%7BL%7D%5Cint_%7B-L%7D%5EL%20f(t)sin(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)dt%0A%5Cend%7Balign%7D%0A%5Ctag%7B10%7D$

做變換 $%5Cint_%7B-L%7D%5EL%20dt%20%5Cto%20%5Cint_0%5E%7B2L%7Ddt%20%5Cto%20%5Cint_0%5ET%20dt$ ，所以公式(10)可以寫為，

$%5Cbegin%7Balign%7D%0Af(t)%26%3D%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Ba_n%20cos(n%5Comega%20t)%2Bb_n%20sin(n%5Comega%20t)%5D%5C%5C%0Aa_0%26%3D%5Cfrac%7B2%7D%7BT%7D%5Cint_0%5ET%20f(t)dt%20%5C%5C%0Aa_n%26%3D%5Cfrac%7B2%7D%7BT%7Df(t)cos(n%5Comega%20t)dt%5C%5C%0Ab_n%26%3D%5Cfrac%7B2%7D%7BT%7D%20%5Cint_0%5ET%20f(t)sin(n%5Comega%20t)dt%0A%5Cend%7Balign%7D%0A%5Ctag%7B11%7D$

下面引入歐拉公式，

$%5Cbegin%7Balign%7D%0Ae%5E%7Bi%5Ctheta%7D%3Dcos(%5Ctheta)%2Bisin(%5Ctheta)%0A%5Cend%7Balign%7D%0A%5Ctag%7B12%7D$

所以 $%5Cbegin%7Balign%7D%0Acos(%5Ctheta)%3D%5Cfrac%7B1%7D%7B2%7D(e%5E%7Bi%5Ctheta%7D%2Be%5E%7B-i%5Ctheta%7D)%2C%20sin(%5Ctheta)%3D-%5Cfrac%7Bi%7D%7B2%7D(e%5E%7Bi%5Ctheta%7D-e%5E%7B-i%5Ctheta%7D)%0A%5Cend%7Balign%7D%0A%5Ctag%7B13%7D$

則公式(11)可以化為

$%5Cbegin%7Balign%7D%0Af(t)%26%3D%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Ba_n%5Cfrac%7B1%7D%7B2%7D(e%5E%7Bin%5Comega%20t%7D%2Be%5E%7B-in%5Comega%20t%7D)-%5Cfrac%7Bi%7D%7B2%7Db_n(e%5E%7Bin%5Comega%20t%7D-e%5E%7B-in%5Comega%20t%7D)%5D%5C%5C%0A%26%3D%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Ba_n-ib_n%7D%7B2%7De%5E%7Bin%5Comega%20t%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Cfrac%7Ba_n%2Bib_n%7D%7B2%7De%5E%7B-in%5Comega%20t%7D%0A%5Cend%7Balign%7D%0A%5Ctag%7B14%7D$

把公式(14)第二項(xiàng) $n%5Cto%20-n%2C%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Cfrac%7Ba_n%2Bib_n%7D%7B2%7De%5E%7B-in%5Comega%20t%7D%5Cto%20%5Csum_%7B-%5Cinfty%7D%5E%7B-1%7D%5Cfrac%7Ba_%7B-n%7D%2Bib_%7B-n%7D%7D%7B2%7De%5E%7Bin%5Comega%20t%7D$ ，公式(14)繼續(xù)化為，

$%5Cbegin%7Balign%7D%0Af(t)%26%3D%5Csum_%7B-%5Cinfty%7D%5E%5Cinfty%20C_ne%5E%7Bin%5Comega%20t%7D%2C%5C%5C%0AC_n%26%3D%5Cfrac%7Ba_0%7D%7B2%7D%2C%20n%3D0%2C%5C%5C%0AC_n%26%3D%5Cfrac%7Ba_n-ib_n%7D%7B2%7D%2C%20n%3D1%2C2%2C3%2C%5Ccdots%5C%5C%0AC_n%26%3D%5Cfrac%7Ba_%7B-n%7D%2Bib_%7B-n%7D%7D%7B2%7D%2C%20n%3D-1%2C-2%2C-3%2C%5Ccdots%0A%5Cend%7Balign%7D%0A%5Ctag%7B15%7D$

現(xiàn)在求系數(shù)C的具體形式，

$%5Cbegin%7Balign%7D%0An%26%3D0%2CC_n%3D%5Cfrac%7Ba_0%7D%7B2%7D%3D%5Cfrac%7B1%7D%7BT%7D%5Cint_0%5ET%20f(t)dt%5C%5C%0An%26%3E0%2C%20C_n%3D%5Cfrac%7B1%7D%7BT%7D%5Cint_0%5ET%20f(t)e%5E%7B-in%5Comega%20t%7Ddt%5C%5C%0An%26%3C0%2C%20C_n%3D%5Cfrac%7B1%7D%7BT%7D%5Cint_0%5ET%20f(t)e%5E%7B-in%5Comega%20t%7Ddt%0A%5Cend%7Balign%7D%0A%5Ctag%7B16%7D$

這時(shí)，顯然我們可以得出公式(15)為，

$%5Cbegin%7Balign%7D%0Af(t)%26%3Df(t%2BT)%5C%5C%0Af(t)%26%3D%5Csum_%7B-%5Cinfty%7D%5E%5Cinfty%20C_n%20e%5E%7Bin%5Comega%20t%7D%2C%5C%5C%0AC_n%26%3D%5Cfrac%7B1%7D%7BT%7D%5Cint_0%5ET%20f(t)e%5E%7B-in%5Comega%20t%7Ddt%0A%5Cend%7Balign%7D%0A%5Ctag%7B17%7D$

傅立葉變換

下面正式進(jìn)行傅立葉變換。每個(gè) $%5Comega$ 的間隔 $%5CDelta%20%5Comega$ 為 $%5CDelta%20%5Comega%20%3D(n%2B1)%5Comega%20-n%5Comega%3D%5Comega%3D%5Cfrac%7B2%5Cpi%7D%7BT%7D$ ，所以有 $%5Cfrac%7B1%7D%7BT%7D%3D%5Cfrac%7B%5CDelta%20%5Comega%20%7D%7B2%5Cpi%7D$ 。隨著周期T的變大， $%5CDelta%20%5Comega$ 越來(lái)越小，可以看作由離散變?yōu)檫B續(xù)。當(dāng) $T%5Cto%20%5Cinfty$ ，公式(17)e指數(shù)上的 $n%5Comega%20%5Cto%20w%5E%5Cprime$ ， $%5Cint_%7B-%5Cfrac%7BT%7D%7B2%7D%7D%5E%7B%5Cfrac%7BT%7D%7B2%7D%7Ddt%5Cto%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20dt$ ，有

$%5Cbegin%7Balign%7D%0Af(t)%26%3D%5Csum_%7B-%5Cinfty%7D%5E%5Cinfty%20%5Cfrac%7B1%7D%7BT%7D%5Cint_%7B-%5Cfrac%7BT%7D%7B2%7D%7D%5E%7B%5Cfrac%7BT%7D%7B2%7D%7Df(t)e%5E%7B-in%5Comega%20t%7Ddte%5E%7Bin%5Comega%20t%7D%5C%5C%0A%26%3D%5Csum_%7B-%5Cinfty%7D%5E%5Cinfty%20%5Cfrac%7B%5CDelta%20%5Comega%7D%7B2%5Cpi%7D%5Cint_%7B-%5Cfrac%7BT%7D%7B2%7D%7D%5E%7B%5Cfrac%7BT%7D%7B2%7D%7Df(t)e%5E%7B-in%5Comega%20t%7Ddte%5E%7Bin%5Comega%20t%7D%5C%5C%0A%26%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-i%5Comega%5E%5Cprime%20t%7Ddt%20e%5E%7Bi%5Comega%5E%5Cprime%20t%7Dd%5Comega%5E%5Cprime%0A%5Cend%7Balign%7D%0A%5Ctag%7B18%7D%20$

所以最后有

$%5Cbegin%7Balign%7D%0Af(t)%26%3D%5Cfrac%7B1%7D%7B2%5Cpi%7D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20F(%5Comega)e%5E%7Bi%5Comega%20t%7Dd%5Comega%7C%E9%80%86%E5%8F%98%E6%8D%A2%5C%5C%0AF(%5Comega)%26%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f(t)e%5E%7B-i%5Comega%20t%7Ddt%7C%E5%82%85%E7%AB%8B%E5%8F%B6%E5%8F%98%E6%8D%A2%0A%5Cend%7Balign%7D%0A%5Ctag%7B19%7D$

在物理中若要對(duì)滿足周期性邊界條件、正格矢、k空間的函數(shù)進(jìn)行傅立葉變換，只需要把公式(18)中的T代換成 $N%5COmega%2C%20%5Ctextbf%7Bl%7D%2C%5Ctextbf%7BK%7D$ 即可。詳細(xì)公式在李正中的《固體理論》第一章第五節(jié)中給出。

標(biāo)簽：學(xué)習(xí)筆記固體物理凝聚態(tài)物理傅立葉變換數(shù)學(xué)物理

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傅立葉變換的推導(dǎo)

三角函數(shù)系

傅立葉變換