在有限长度数位滤波器的设计中，最小平均平方误差(英文：least mean-square error)是所设计的滤波器响应与理想滤波器响应在欲探讨的频段中的最小误差。由于所设计的数位滤波器的长度不是无限长，所以在设计上和完美的滤波器会存在误差。是讯号工程师会关心设计的滤波器的响应重点。此平均误差利用统计学的方均值来计算求得。
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运用定义，将
取代
做变数变换：
，
![{\displaystyle {\begin{aligned}MSE&=\int _{-1/2}^{1/2}|R(F)-H_{d}(F)|^{2}dF\\&=\int _{-1/2}^{1/2}|\sum _{n=1}^{k}s[n]cos(2\pi nF)-H_{d}(F)|^{2}dF\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/49705efe107f85eaf22a9e96b97a637f33737eb6)
因为绝对值内函数是以时刻变数构成的实函数，所以可将绝对值平方直接写成两个实函数部分相乘。
![{\displaystyle MSE=\int _{-1/2}^{1/2}\left(\sum _{v=0}^{k}s[v]cos(2\pi vF)-H_{d}(F)\right)\left(\sum _{\tau =0}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F)\right)dF}](//wikimedia.org/api/rest_v1/media/math/render/svg/5034bd3dc6c927f3def4f371720b47fa2e7f9ca1)
将MSN对s{n]做偏微分，
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[n]}}&=\int _{-1/2}^{1/2}cos(2\pi nF){\Bigl (}\sum _{\tau =0}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F){\Bigr )}dF\\&+\int _{-1/2}^{1/2}{\Bigl (}\sum _{v=0}^{k}s[v]cos(2\pi vF)-H_{d}(F){\Bigr )}dF\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/15351ea780c33ccd5372058ad0a90610678d0403)
将上式第一个等号后面的两个
项合并，因为引数起始值和最末项值相同。
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[n]}}&=2\sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}cos(2\pi \tau F)cos(2\pi nF)dF\\&-2\int _{-1/2}^{1/2}H_{d}(F)cos(2\pi nF)dF,\forall n=0,1,2,...k\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2b2cc7142e8c02a26d16ff3faea9576d9f61ca41)
上式等号右边的两项中，都各有将余弦函数从-1/2积分到1/2，又因为有两个自变数系数不同的余弦函数相乘，整个计算结果要分开讨论。
与
的项次
与
相异或相同，因为函数的正交性，会有不同的积分效果。此可由三角函数的积化合差验证出来。
故可得：
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[0]}}&=2s[0]\times 1-2\int _{-1/2}^{1/2}H_{d}(F)\times 1dF=0,\forall n=0\\{\frac {\partial MSE}{\partial s[n]}}&=s[n]-2\int _{-1/2}^{1/2}cos(2\pi nF)H_{d}(F)dF=0,\forall n\neq 0\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1d22b300f924575896c41946090f830f7e978284)
MSE 是设计上滤波器响应函数和理想函数之间的误差，吾人会希望这个误差越小越好，即是设计的响应要和理想的响应，在所有时刻n上的趋势一样。
故使：![{\displaystyle {\partial MSE \over \partial s[n]}=0}](//wikimedia.org/api/rest_v1/media/math/render/svg/0b7c98d5dd8adc97dc740ce8e14e0a46f32f25ac)
for all n.
可得：
![{\displaystyle {\begin{aligned}s[0]=\int _{-1/2}^{1/2}H_{d}(F)dF\\s[n]=2\int _{-1/2}^{1/2}cos(2\pi nF)H_{d}(F)dF\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1bdc3ffa0fc200c17eab5ba185c3c8bb47aaaa24)
观察上式，若我们要得到滤波器的最小方均误差，则要将理想滤波器频域响应对余弦函数做多项式内积。
最后将![{\displaystyle s[n]}](//wikimedia.org/api/rest_v1/media/math/render/svg/ffe878c033cc70d1aa3ffbd156394791aa8dffde)
的值带入![{\displaystyle h[k]}](//wikimedia.org/api/rest_v1/media/math/render/svg/e944606b94261ca96a82ff41b4b8aade7ff14acc)
：
![{\displaystyle {\begin{cases}h[k]=s[0]\\h[k+n]=s[n]/2,h[k-n]=s[n]/2,&{\text{for }}n=1,2,3,...,k\\h[n]=0,{\text{ for }}n<0{\text{ and }}n\geq N.\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/cd061e044f96f2d6ce30cd1941e524952976681d)
![{\displaystyle h[n]}](//wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46)
即为欲设计的滤波器脉冲响应式。
在数位滤波器设计中，可以利用权重函数
去设计定频段更接近理想值。则MSE则表示为：
![{\displaystyle MSE=\int _{-1/2}^{1/2}W(F)|R(F)-H_{d}(F)|^{2}dF=\int _{-1/2}^{1/2}W(F){\Bigl (}\sum _{-1/2}^{1/2}s[\tau ])cos(2\pi \tau F)-H_{d}(F){\Bigr )}dF}](//wikimedia.org/api/rest_v1/media/math/render/svg/aab6073034b36ef5117b49d092f7087485de8784)
MSE对s[n]的微分值为零可改写为：
![{\displaystyle {\frac {\partial MSE}{\partial s[n]}}=2\int _{-1/2}^{1/2}W(F)cos(2\pi nF){\biggl (}\sum _{tau}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F){\biggr )}dF}](//wikimedia.org/api/rest_v1/media/math/render/svg/3ddebcd48e9f4aa3d8b732945507f2e82d982c97)
![{\displaystyle =\sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF-\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)df=0,n\in [0,k]}](//wikimedia.org/api/rest_v1/media/math/render/svg/8a5218b532c85eb8fe771e355c9f88ad6634d5db)
![{\displaystyle \Rightarrow \sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF=\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)dF}](//wikimedia.org/api/rest_v1/media/math/render/svg/86be578fdd088653a941349d7672ef995adab9ff)
可再将上式写成矩阵式：
![{\displaystyle {\begin{bmatrix}A[0,0]&A[0,1]&A[0,2]&...&A[0,k]\\A[1,0]&A[1,1]&A[1,2]&...&A[1,k]\\\vdots &\vdots &\vdots &\ddots &\vdots \\A[k,0]&A[k,1]&A[k,2]&\cdots &A[k,k]\end{bmatrix}}{\begin{bmatrix}s[0]\\s[1]\\\vdots \\s[k]\end{bmatrix}}={\begin{bmatrix}B[0]\\B[1]\\\vdots \\B[k]\end{bmatrix}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/742b8ea8e354484800a371823e77ab779e5624f4)
A矩阵每一行表示不同代表的数值，每一列代表不同n代表的数值；s向量与B向量每一列表示不同n代表的数值。
其中：![{\displaystyle A[n,\tau ]=\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF,B[n]=\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)dF}](//wikimedia.org/api/rest_v1/media/math/render/svg/5a94abeabd6f537e75786bf97510597b9080401a)
经由此举阵式可解出带有权重函数设计的。
最后将带入：
即是欲设计的滤波器响应。
![{\displaystyle R(F)=\sum _{n=0}^{k}s[n]cos(2\pi nF)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bcbf5711d4989437c9fbc8c5d9ff8503a75902af)
![{\displaystyle H_{d}(f)=\sum _{n}h[n]e^{-j2\pi f \over f_{s}}=H_{d}(f)=\sum _{n}h[n]e^{-j2\pi nF}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a8b50a66c25204b9d0d86f8faae6bafeba01404a)
![{\displaystyle {\begin{aligned}MSE&=\int _{-1/2}^{1/2}|R(F)-H_{d}(F)|^{2}dF\\&=\int _{-1/2}^{1/2}|\sum _{n=1}^{k}s[n]cos(2\pi nF)-H_{d}(F)|^{2}dF\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/49705efe107f85eaf22a9e96b97a637f33737eb6)
![{\displaystyle MSE=\int _{-1/2}^{1/2}\left(\sum _{v=0}^{k}s[v]cos(2\pi vF)-H_{d}(F)\right)\left(\sum _{\tau =0}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F)\right)dF}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5034bd3dc6c927f3def4f371720b47fa2e7f9ca1)
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[n]}}&=\int _{-1/2}^{1/2}cos(2\pi nF){\Bigl (}\sum _{\tau =0}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F){\Bigr )}dF\\&+\int _{-1/2}^{1/2}{\Bigl (}\sum _{v=0}^{k}s[v]cos(2\pi vF)-H_{d}(F){\Bigr )}dF\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/15351ea780c33ccd5372058ad0a90610678d0403)
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[n]}}&=2\sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}cos(2\pi \tau F)cos(2\pi nF)dF\\&-2\int _{-1/2}^{1/2}H_{d}(F)cos(2\pi nF)dF,\forall n=0,1,2,...k\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2b2cc7142e8c02a26d16ff3faea9576d9f61ca41)
![{\displaystyle {\begin{aligned}{\frac {\partial MSE}{\partial s[0]}}&=2s[0]\times 1-2\int _{-1/2}^{1/2}H_{d}(F)\times 1dF=0,\forall n=0\\{\frac {\partial MSE}{\partial s[n]}}&=s[n]-2\int _{-1/2}^{1/2}cos(2\pi nF)H_{d}(F)dF=0,\forall n\neq 0\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1d22b300f924575896c41946090f830f7e978284)
![{\displaystyle {\partial MSE \over \partial s[n]}=0}](http://wikimedia.org/api/rest_v1/media/math/render/svg/0b7c98d5dd8adc97dc740ce8e14e0a46f32f25ac)
![{\displaystyle {\begin{aligned}s[0]=\int _{-1/2}^{1/2}H_{d}(F)dF\\s[n]=2\int _{-1/2}^{1/2}cos(2\pi nF)H_{d}(F)dF\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1bdc3ffa0fc200c17eab5ba185c3c8bb47aaaa24)
![{\displaystyle s[n]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ffe878c033cc70d1aa3ffbd156394791aa8dffde)
![{\displaystyle h[k]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e944606b94261ca96a82ff41b4b8aade7ff14acc)
![{\displaystyle {\begin{cases}h[k]=s[0]\\h[k+n]=s[n]/2,h[k-n]=s[n]/2,&{\text{for }}n=1,2,3,...,k\\h[n]=0,{\text{ for }}n<0{\text{ and }}n\geq N.\end{cases}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cd061e044f96f2d6ce30cd1941e524952976681d)
![{\displaystyle h[n]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46)
![{\displaystyle MSE=\int _{-1/2}^{1/2}W(F)|R(F)-H_{d}(F)|^{2}dF=\int _{-1/2}^{1/2}W(F){\Bigl (}\sum _{-1/2}^{1/2}s[\tau ])cos(2\pi \tau F)-H_{d}(F){\Bigr )}dF}](http://wikimedia.org/api/rest_v1/media/math/render/svg/aab6073034b36ef5117b49d092f7087485de8784)
![{\displaystyle {\frac {\partial MSE}{\partial s[n]}}=2\int _{-1/2}^{1/2}W(F)cos(2\pi nF){\biggl (}\sum _{tau}^{k}s[\tau ]cos(2\pi \tau F)-H_{d}(F){\biggr )}dF}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3ddebcd48e9f4aa3d8b732945507f2e82d982c97)
![{\displaystyle =\sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF-\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)df=0,n\in [0,k]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/8a5218b532c85eb8fe771e355c9f88ad6634d5db)
![{\displaystyle \Rightarrow \sum _{\tau =0}^{k}s[\tau ]\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF=\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)dF}](http://wikimedia.org/api/rest_v1/media/math/render/svg/86be578fdd088653a941349d7672ef995adab9ff)
![{\displaystyle {\begin{bmatrix}A[0,0]&A[0,1]&A[0,2]&...&A[0,k]\\A[1,0]&A[1,1]&A[1,2]&...&A[1,k]\\\vdots &\vdots &\vdots &\ddots &\vdots \\A[k,0]&A[k,1]&A[k,2]&\cdots &A[k,k]\end{bmatrix}}{\begin{bmatrix}s[0]\\s[1]\\\vdots \\s[k]\end{bmatrix}}={\begin{bmatrix}B[0]\\B[1]\\\vdots \\B[k]\end{bmatrix}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/742b8ea8e354484800a371823e77ab779e5624f4)
![{\displaystyle A[n,\tau ]=\int _{-1/2}^{1/2}W(F)cos(2\pi nF)cos(2\pi \tau F)dF,B[n]=\int _{-1/2}^{1/2}W(F)H_{d}(F)cos(2\pi nF)dF}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5a94abeabd6f537e75786bf97510597b9080401a)
