[Optimization algorithm] Improved LMS algorithm [including Matlab source code 630 period]

1. Introduction

Least Mean Squares (LMS, Least Mean Squares) is the most basic adaptive filtering algorithm.
The LMS algorithm is a commonly used algorithm in adaptive filters. The difference from the Wiener algorithm is that the coefficients of the system change with the input sequence. In the Wiener algorithm, a section of the autocorrelation function of the input sequence is intercepted to construct the optimal coefficient of the system. The LMS algorithm is implemented by continuously correcting the initialized filter coefficients according to the minimum mean square error criterion. Therefore, theoretically speaking, the performance of the LMS algorithm is better than Wiener under the same conditions. However, the LMS is adjusted gradually under the initial value, so before the system is stable, there will be a period of adjustment time. The adjustment time is controlled by the step size factor. Within a certain range, the larger the step size factor, the smaller the adjustment time, and the step size factor The maximum value of is the trace of R. LMS adopts the principle of minimum square error instead of the principle of minimum mean square error, and the basic signal relationship is as follows:
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Second, the source code

 %一种变步长lms 算法
clear all
close all
hold off    
sysorder = 5 ; %抽头数

t = 0:1/5000:1-0.0001;
s = sin(2*pi*t); %不带噪声的信号
N = size(t);
M = length(t);
% %计算自相关矩阵的最大特征值
% xx=rcorr(length(s),s);
% [V,D]=eig(xx);
% Dmax=max(max(D));

n = randn(N);   %产生高斯白噪声
[b,a] = butter(2,0.25);
Gz = tf(b,a,-1); %逆变换函数
y = lsim(Gz,s);  %加入正弦信号
z = n * std(y)/(10*std(n));  %标准差为0的噪声信号
x = s + z;      %实际输入信号
dd = s; %期望输出信号

%算法的开始
w = zeros(1,sysorder);%权向量初始化
umax=0.45;   %设置步长最大值
umin=0.0003; %设置步长最小值

for i=1:2
          u = umax;
          y(i) = x(i:i+4)* w' ; %系统输出

          e(i) = dd(i) - y(i);%误差
  
          w = w + u * e(i) * x(i); %迭代方程
end

for i = 3:M-4

     if ((u >= umin)&&(u <= umax))

       y(i) = x(i:i+4)* w' ;  %系统输出
 
       e(i) = dd(i) - y(i);  %误差

       w = w + u * e(i) * x(i);  %迭代方程

       u = ((e(i) * e(i-1)/((dd(i))^2)))* umax;  %步长更新公式
    end
   
    if    u > umax
          u = umax;
          
          y(i) = x(i:i+4)* w' ; %系统输出
   
          e(i) = dd(i) - y(i);   %误差

           w = w + u * e(i) * x(i);  %迭代方程
           
           u = ((e(i) * e(i-1)/((dd(i))^2)))* umax;  %步长更新公式
    end
    
   if     u < umin
          u = umin;

          y(i) = x(i:i+4)* w' ;  %系统输出

          e(i) = dd(i) - y(i); %误差

             w = w + u * e(i) * x(i);  %迭代方程

             u = ((e(i) * e(i-1)/((dd(i))^2)))* umax; %步长更新公式
   end

end

 %计算均方误差 
st = 0;
 i = 1;

while i <= M-4       
      a = (e(i))^2;
     st = st+a;
    out = (1/i)*st;
    yy(1,i) = out;
    i = i+1;
  
end

% %作图
hold on;
%不带噪声的信号
subplot(3,1,1)
plot(s)      
title('纯净的输入信号')
 %噪声
subplot(3,1,2)
plot(z,'g')     
title('噪声')
 %带噪声的输入信号  
subplot(3,1,3)
plot(x,'c')      
title('带噪声的输入信号')
figure

Three, running results

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Four, remarks