Meta-learning (MAML) for FxLMS Initialization

It realizes the modified Model-Agnostic Meta-Learning to accelerate the convergence of the FxLMS algorithm in the adaptive ANC system.

The program implements a modified Model-Agnostic Meta-learning (MAML) algorithm in Matlab. This modified MAML algorithm takes the forgetting factor to alleviate the initial-zero-input issue of the adaptive filters. This work uses the proposed method to seek the best initial control filter for the adaptive active noise control system based on its noise dataset. With barely any additional computations, the modified MAML algorithm can assist the filtered reference square (FxLMS) algorithm in achieving fast convergence. The numerical simulation demonstrates that the FxLMS algorithm with the MAML initialization converges much faster than that with normal zero initialization when dealing with raw aircraft noise, as shown in Figure S3.

Figure S3： Aircraft noise reduction performance of the FxLMS algorithms with zero and MAML initializations.

The measured primary and secondary paths are available on GitHub. It also gives a detailed introduction to the code of the Meta-Learning algorithm. The reference paper is "Fast adaptive active noise control based on Modified model-agnostic meta-learning algorithm."

Code Explanation

• Main_tst_function.m: the main function is utilized to test the proposed modified MAML algorithm.
• MAML_Nstep_forget.m: the matlab code of the modified MAML algorithm.
• FxLMS.m: the matlab code of the FxLMS algorithm.
• 707_Sound_for_Simulation.mat: the raw data of an aircrat noise.
• \path: the measured primary and secondary paths.

The explanation of Main_tst_function.m

The following code snippet offers a concise overview of Main_tst_function.m, which serves as the main function for evaluating the effectiveness of the proposed MAML method. For this numerical simulation, three distinct broadband sounds are utilized to train the MAML algorithm and obtain a single initial control filter. Subsequently, the aforementioned initial control is employed within an FxLMS algorithm to effectively eliminate actual aircraft noise. In comparison to zero initialization, the MAML technique can significantly enhance the convergence speed of the conventional FxLMS approach.

Contents:

• Clean the memory and worksapace
• Configure the system simulation condition
• Build the broad band noise for training set
• Randomly sampling the noise tracks to build dataset for the MAML algorithm
• Using Modified MAML algorithm to get the best initial control filter
• Testing aircraft noise cancellation by using MAML initial control filter

Clean the memory and worksapace

Code snippet to clean workspace and set initial conditions.

%% Clean the memory and worksapace 
close all ;
clear     ;
clc       ;

Configure the system simulation condition

This code snippet provides the system configuration for the numerical simulation. In this program, fs and T denote the system sampling rate and the simulation duration, respectively. LenN repesents the length of the control filter.

%% Configure the system simulation condition 
fs  =   16000    ; % The system sampling rate.
T   =   3        ; % The duration of the simulation.
t   =   0:1/fs:T ; 
N   =   length(t); % The number of the data.

Len_N = 512      ; % Seting the length of the control filter.
%<<===Progress bar===>> 
f = waitbar(0,'Please wait...');
pause(.5)

Build the broad band noise for training set

This section of the program produces filtered references, disturbances, and primary noises. These are then used in the modified MAML algorithm to obtain the initial control filter. The main sounds consist of three distinct broadband noises, as depicted in Figure S1.

%% Build the broad band noise for training set
%<<===Progress bar===>> 
waitbar(0.25,f,'Build the broad band noise for training set');
pause(1)
%<<===Progress bar===>> 
% Loading path 
load('path\P1.mat')    ;
load('path\S11.mat')   ;
Pri_path = conv(P1,S11);

Track_num = 3         ; % Seting the number of the track for the trainning noise. 
if exist('Primary_noise.mat', 'file') == 2
    disp('Primary_noise exists in the current path.\n');
    % Loading the primary noise 
    load('Primary_noise.mat');
    load('Disturbance.mat')  ;
    load('Reference.mat')    ;
else
Noise     = randn(N,1);
% filter 
filter_1 = fir1(512,[0.05 0.25]);
filter_2 = fir1(512,[0.20 0.55]) ;
filter_3 = fir1(512,[0.5,0.75]) ;
% Primary noise 
Pri_1 = filter(filter_1,1,Noise) ;
Pri_2 = filter(filter_2,1,Noise) ;
Pri_3 = filter(filter_3,1,Noise) ;
% Drawing fiture 
data = [Pri_1,Pri_2,Pri_3];
figure ;
len_fft = length(Pri_1)   ;
len_hal = round(len_fft/2);
title('Frequency spectrum of primary noises')
for ii = 1:3
    freq = 20*log(abs(fft(data(:,ii))));
    subplot(3,1,ii);
    plot(0:(fs/len_fft):(len_hal-1)*(fs/len_fft), freq(1:len_hal));
    grid on   ;
    title("The "+num2str(ii)+"th primary noise")
    xlabel('Frequency (Hz)')
end
% Save primary noise into workspace 
save('Primary_noise.mat','Pri_1','Pri_2','Pri_3');
% Generating Distrubance 
Dis_1 = filter(Pri_path,1,Pri_1);
Dis_2 = filter(Pri_path,1,Pri_2);
Dis_3 = filter(Pri_path,1,Pri_3);
% Save distrubancec into workspace 
save('Disturbance.mat','Dis_1','Dis_2','Dis_3');
% Genrating Filtered reference signal 
Rf_1 = filter(S11,1,Pri_1);
Rf_2 = filter(S11,1,Pri_2);
Rf_3 = filter(S11,1,Pri_3);
% Save filter reference signal into workspace 
save('Reference.mat','Rf_1','Rf_2','Rf_3');
end

Figure S1: The frequency spectrum of the primary noises for the MAML algorithm.

Randomly sampling the noise tracks to build dataset for the MAML algorithm

This part of the program randomly samples the previously generated three types of noise tracks and their corresponding reference signals and disturbances to form the dataset. The length of the sampled vector equals that of the control filter, and these samples will be used in the proposed MAML algorithm to get the best initial control filter.

%% Radomly sampling the noise tracks to build dataset for the MAML algorithm
%<<===Progress bar===>> 
waitbar(0.5,f,'Radomly sampling the noise tracks to build dataset for the MAML algorithm');
pause(1)
%<<===Progress bar===>> 
if exist('Sampe_data_N_set.mat', 'file') == 2
    disp('Sampe_data_N_set in the current path.\n');
    load('Sampe_data_N_set.mat');
else
N_epcho  = 4096 * 80                   ; % Setting the number of the epcho 
Trac     = randi(Track_num,[N_epcho,1]); % Randomly choosing the different tracks. 
Len_data = length(Dis_1)               ;
% Seting the N steps 
len   = 2*Len_N -1 ;
Fx_data = zeros(Len_N,N_epcho);
Di_data = zeros(Len_N,N_epcho);
Ref_data = [Rf_1,Rf_2,Rf_3]   ;
Dis_data = [Dis_1,Dis_2,Dis_3];
for jj = 1:N_epcho
    End = randi([len,Len_data]);
    Di_data(:,jj) = Dis_data(End-511:End,Trac(jj));
    Fx_data(:,jj) = Ref_data(End-511:End,Trac(jj));
end
save('Sampe_data_N_set.mat','Di_data','Fx_data');
end

Using Modified MAML algorithm to get the best initial control filter

This segment of the program applies the proposed modified MAML algorithm to compute the best initial control filter for the conventional FxLMS algorithm. The step size of the FxLMS algorithm, the learning rate of the MAML algorithm, and the forgetting factor are set to $0.003$, $0.5$, and $0.99$, respectively. Figure S2 illustrates the proposed MAML algorithm's learning curve and demonstrates the proposed algorithm's convergence.

%<<===Progress bar===>> 
waitbar(0.75,f,'Using Modified MAML algorithm to get the best initial control filter');
pause(1)
%<<===Progress bar===>> 
if exist('Weigth_initiate_Nstep_forget.mat', 'file') == 2
    disp('Weigth_initiate_Nstep_forget in the current path.\n');
    load('Weigth_initiate_Nstep_forget.mat');
else
% Create a MAML algorithm
a  = MAML_Nstep_forget(Len_N);
N  = size(Di_data,2)         ; % The number of the sample in training set.
Er = zeros(N,1)              ; % Residual error vector
% Seting the step size for the embeded FxLMS algorithm 
mu    = 0.0003              ;
% Seting the forget factor 
lamda = 0.99                ;
% Seting the learning for MAML 
epslon = 0.5 ;
% Runing the MAML algorithm 
for jj = 1:N
    [a, Er(jj)] = a.MAML_initial(Fx_data(:,jj),Di_data(:,jj),mu,lamda,epslon);
end
% Drawing the residual error of the Modified MAML algorihtm 
figure    ;
plot(Er)  ;
grid on   ;
title('Leanring curve of the modified MAML algorithm');
xlabel('Epoch');
ylabel('Residual error');
% Getting the optimal intial control filter
Wc = a.Phi ;
% Saving the best initial control filter into workspace
save('Weigth_initiate_Nstep_forget.mat','Wc');
end

Figure S2: The learning curve of the modified MAML algorithm.

Testing aircraft noise cancellation by using MAML initial control filter

The program employs the FxLMS algorithms, utilizing zero initialization and MAML initialization, to reduce the impact of aircraft noise. Figure S3 illustrates the error signal produced by the FxLMS algorithm during both the on and off states. It has been observed that the utilization of MAML initialization significantly enhances the convergence rate of the adaptive algorithm in comparison to the conventional zero initialization approach.

%% Testing aircraft noise cancellation by using MAML initial control filter 
%<<===Progress bar===>> 
waitbar(0.95,f,'Testing aircraft noise cancellation by using MAML initial control filter');
pause(1)
%<<===Progress bar===>> 
% Loading aricrat noise data 
aircrat = load('707_Sound_for_Simulation.mat');
% Building primary noise
Pri_1   = aircrat.PilingNoise(1:153945)       ; 
% Generating the disturbacne 
Dis_1   = filter(Pri_path,1,Pri_1)            ;
% Generating the filter reference             
Rf_1    = filter(S11,1,Pri_1)                 ;
% Runging the FxLMS with the zero-initialization control filter    
Wc_initial = zeros(Len_N,1);
muw        = 0.00001 ; % Step size of All FxLMS algorithms. 
Er         = FxLMS(Len_N, Wc_initial, Dis_1, Rf_1, muw);
% Runging the FxLMS with the MAML-initialization control filter    
Wc_initial = Wc;
Er1        = FxLMS(Len_N, Wc_initial, Dis_1, Rf_1, muw);
figure 
% Drawing the figures of the MAML and FxLMS 
plot((0:length(Dis_1)-1)*(1/fs),Dis_1,(0:length(Er)-1)*(1/fs),Er,(0:length(Er1)-1)*(1/fs),Er1);
title('Aircraft noise cancellation')
xlabel('Time (seconds)')
ylabel('Error signal')
legend({'ANC off','FxLMS with zero-initialization','FxLMS with MAML-initialization'})
grid on;
%<<===Progress bar===>> 
waitbar(1,f,'Finishing');
pause(1)
%<<===Progress bar===>> 
%-------------------------------end----------------------------------------

Figure S3: Aircraft noise reduction performance of the FxLMS algorithms with the zero and MAML initializations.

Summray

The document provides a detailed introduction to implementing a modified Model-Agnostic Meta-learning (MAML) algorithm in Matlab. This modified MAML algorithm takes the forgetting factor to alleviate the initial-zero-input issue of the adaptive filters. This work uses the proposed method to seek the best initial control filter for the adaptive active noise control system based on its noise dataset. With barely additional computations, the modified MAML algorithm can assist the filtered reference square (FxLMS) algorithm in achieving fast convergence. The numerical simulation demonstrates that the FxLMS algorithm with the MAML initialization converges much faster than that with normal zero initialization when dealing with raw aircraft noise.