傅立叶变换的推导

2023-02-28 12:17 作者:啊呜西呜安 0人读过 | 我要投稿

三角函数系

首先引入傅立叶变换所使用的三角函数系

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

这组三角函数系像波函数一样具有正交性和完备性，但不是归一的。对于正交性我们可以看到：

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

并且三角函数具有周期性，其周期 $T%3D2%5Cpi$ ，即 $f(x)%3Df(x%2B2%5Cpi)$ ，见图1。

傅立叶发现周期为 $2%5Cpi$ 的函数可以展开为一系列三角函数的和的形式，即

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

现在我们先来求系数 $a_0$ ，对公式两边同时对x作积分，得

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

这里利用了三角函数的正交性，最后算得

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

这时，令 $a_0%5E%5Cprime%3D2a_0$ ，则公式(2)变为

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

这里为了表示方便，把" $%5Cprime$ "丢掉，即

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

接下来我们再求系数 $a_n$ ，对公式(6)两边同乘cos(mx)，并对x进行积分，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 $b_n%3D%5Cfrac%7B1%7D%7B%5Cpi%7D%5Cint_%7B-%5Cpi%7D%5E%5Cpi%20f(x)sin(nx)dx$ 。这样就求完了周期为 $2%5Cpi$ 的函数f(x)的傅立叶展开系数。

然而物理中的函数周期通常都不是 $2%5Cpi$ ，现在来求周期T=2L的函数f(t)的傅立叶展开系数。对于函数f(t)，我们有 $f(t)%3Df(t%2B2L)$ 。为了把函数f(t)的周期变换到 $2%5Cpi$ ，这里利用换元法，令 $x%3D%5Cfrac%7B%5Cpi%7D%7BL%7Dt$ ，所以 $t%3D%5Cfrac%7BL%7D%7B%5Cpi%7Dx$ 。则有, $%5Cbegin%7Balign%7D%0Af(t)%3Df(%5Cfrac%7BL%7D%7B%5Cpi%7Dx)%3Dg(x)%3Df(%5Cfrac%7BL%7D%7B%5Cpi%7D(x%2B2%5Cpi))%3Dg(x%2B2%5Cpi)%3Df(t%2B2L)%0A%5Cend%7Balign%7D%0A%5Ctag%7B9%7D$

这样就把周期为2L的函数f(t)变换成了周期为 $2%5Cpi$ 的函数g(x)。周期为 $2%5Cpi$ 的函数g(x)的展开系数我们已经求过了，即 $%5Cbegin%7Balign%7D%0Af(t)%26%3D%20%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Ba_ncos(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)%2Bb_nsin(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)%5D%5C%5C%0Aa_0%26%3D%5Cfrac%7B1%7D%7BL%7D%5Cint_%7B-L%7D%5EL%20f(t)dt%5C%5C%0Aa_n%26%3D%20%5Cfrac%7B1%7D%7BL%7D%20%5Cint_%7B-L%7D%5EL%20f(t)cos(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)dt%5C%5C%0Ab_n%26%3D%20%5Cfrac%7B1%7D%7BL%7D%5Cint_%7B-L%7D%5EL%20f(t)sin(%5Cfrac%7Bn%5Cpi%7D%7BL%7Dt)dt%0A%5Cend%7Balign%7D%0A%5Ctag%7B10%7D$

做变换 $%5Cint_%7B-L%7D%5EL%20dt%20%5Cto%20%5Cint_0%5E%7B2L%7Ddt%20%5Cto%20%5Cint_0%5ET%20dt$ ，所以公式(10)可以写为，

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

下面引入欧拉公式，

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

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

则公式(11)可以化为

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

把公式(14)第二项 $n%5Cto%20-n%2C%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20%5Cfrac%7Ba_n%2Bib_n%7D%7B2%7De%5E%7B-in%5Comega%20t%7D%5Cto%20%5Csum_%7B-%5Cinfty%7D%5E%7B-1%7D%5Cfrac%7Ba_%7B-n%7D%2Bib_%7B-n%7D%7D%7B2%7De%5E%7Bin%5Comega%20t%7D$ ，公式(14)继续化为，

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

现在求系数C的具体形式，

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

这时，显然我们可以得出公式(15)为，

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

傅立叶变换

下面正式进行傅立叶变换。每个 $%5Comega$ 的间隔 $%5CDelta%20%5Comega$ 为 $%5CDelta%20%5Comega%20%3D(n%2B1)%5Comega%20-n%5Comega%3D%5Comega%3D%5Cfrac%7B2%5Cpi%7D%7BT%7D$ ，所以有 $%5Cfrac%7B1%7D%7BT%7D%3D%5Cfrac%7B%5CDelta%20%5Comega%20%7D%7B2%5Cpi%7D$ 。随着周期T的变大， $%5CDelta%20%5Comega$ 越来越小，可以看作由离散变为连续。当 $T%5Cto%20%5Cinfty$ ，公式(17)e指数上的 $n%5Comega%20%5Cto%20w%5E%5Cprime$ ， $%5Cint_%7B-%5Cfrac%7BT%7D%7B2%7D%7D%5E%7B%5Cfrac%7BT%7D%7B2%7D%7Ddt%5Cto%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20dt$ ，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

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

在物理中若要对满足周期性边界条件、正格矢、k空间的函数进行傅立叶变换，只需要把公式(18)中的T代换成 $N%5COmega%2C%20%5Ctextbf%7Bl%7D%2C%5Ctextbf%7BK%7D$ 即可。详细公式在李正中的《固体理论》第一章第五节中给出。

标签：学习笔记固体物理凝聚态物理傅立叶变换数学物理

傅立叶变换的推导

三角函数系

傅立叶变换