[Optimization algorithm] Improved LMS algorithm-NLMS algorithm [Contains Matlab source code 631 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

clear all
close all
hold off%系统信道权数
sysorder = 5 ;%抽头数
N=1000;%总采样次数
inp = randn(N,1);%产生高斯随机系列
n = randn(N,1);
[b,a] = butter(2,0.25);
Gz = tf(b,a,-1);%逆变换函数
h= [0.0976;0.2873;0.3360;0.2210;0.0964;];%信道特性向量
y = lsim(Gz,inp);%加入噪声
n = n * std(y)/(10*std(n));%噪声信号
d = y + n;%期望输出信号
totallength=size(d,1);%步长
N=60 ; %60节点作为训练序列
%算法的开始
w = zeros ( sysorder , 1 ) ;%初始化
for n = sysorder : N 
u = inp(n:-1:n-sysorder+1) ;% u的矩阵
y(n)= w' * u;%系统输出
r(n)=u'*u;%自相关矩阵
e(n) = d(n) - y(n) ;%误差
fai=.0001;%修正参数，为防止u'*u过小导致步长值太大而设置的
if n < 20
mu=0.32;
else
mu=0.15;
end
w = w + mu * u * e(n)/(r(n)+fai) ;%迭代方程
end 
%检验结果
for n = N+1 : totallength
u = inp(n:-1:n-sysorder+1) ;
y(n) = w' * u ;
e(n) = d(n) - y(n) ;%误差
end 
hold on
plot(d)
plot(y,'r');
title('系统输出') ;
xlabel('样本')
ylabel('实际输出')
figure
semilogy((abs(e))) ;% e的绝对值坐标
title('误差曲线') ;
xlabel('样本')
ylabel('误差矢量')
figure%作图
plot(h, 'k+')

Three, running results

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Four, remarks