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LMS Algorithm: A MATLAB Tutorial for Adaptive Filtering

The LMS algorithm is one of the most widely used adaptive filter algorithms. It is based on the method of steepest descent, which updates the filter coefficients by minimizing the mean square error between the desired signal and the filter output. The LMS algorithm can be applied to various problems such as noise cancellation, system identification, channel equalization, and adaptive beamforming.

In this article, we will explain the basic principles of the LMS algorithm and show how to implement it in MATLAB. We will also demonstrate its performance on a simple system identification example, where we use the LMS algorithm to estimate the coefficients of an unknown linear filter. We will compare the results with the theoretical predictions and analyze the effects of different parameters such as step size, filter length, and input signal characteristics.

Background

The LMS algorithm belongs to a class of adaptive filters that use a finite impulse response (FIR) structure. An FIR filter is a linear system that can be represented by a vector of coefficients w = [w0,w1,...,wM-1]T, where M is the filter length. The filter output y(n) at time n is given by

y(n) = w0u(n) + w1u(n-1) + ... + wM-1u(n-M+1)

= wTu(n)

where u(n) = [u(n),u(n-1),...,u(n-M+1)]T is the input signal vector at time n.

The goal of adaptive filtering is to adjust the filter coefficients w in order to achieve a desired output d(n) for a given input u(n). The difference between the desired output and the filter output is called the error signal e(n), which is defined as

e(n) = d(n) - y(n)

The LMS algorithm updates the filter coefficients w at each time step n by moving them in the direction of the negative gradient of the mean square error (MSE) function E[e2(n)], which is given by

w(n+1) = w(n) + mu*u(n)*e*(n)

where mu is the step size parameter that controls the speed of convergence and e*(n) is the complex conjugate of e(n).

Implementation

To implement the LMS algorithm in MATLAB, we need to define some parameters and variables such as the filter length M, the step size mu, the input signal u, and the desired signal d. We also need to initialize the filter coefficients w and the error signal e. Here is an example code that implements a simple LMS algorithm for system identification:

% Define parameters

M = 5; % Filter length

mu = 0.1; % Step size

N = 1000; % Number of samples

% Generate input signal

u = randn(N,1); % White Gaussian noise

% Generate desired signal

h = 0.5.^[0:4]; % Unknown system coefficients

d = conv(h,u); % Filtered input signal

% Initialize variables

w = zeros(M,1); % Filter coefficients

e = zeros(N,1); % Error signal

% LMS algorithm loop

for n = M:N

% Input signal vector

uvec = u(n:-1:n-M+1);

% Filter output

y = w'*uvec;

% Error signal

e(n) = d(n) - y;

% Coefficient update

w = w + mu*uvec*e'(n);

end

Results

To evaluate the performance of the LMS algorithm, we can plot the learning curve, which shows how the MSE changes over time. The MSE can be estimated by taking the average of e2(n) over a large number of samples. We can also plot the final filter coefficients w and compare them with the true system coefficients h. Here is an example code that generates these plots:

% Estimate MSE ec8f644aee