自适应傅里叶分解(AFD)算法及其实现
AFD是一种自适应信号分解方法，能够根据信号特性自动选择最佳的基函数。
算法原理
核心思想
AFD通过迭代过程将信号分解为一组自适应选择的单分量函数，这些函数在Hardy空间中具有最优的能量聚集特性。
MATLAB实现
1. 基础AFD算法
function [components, residues, parameters] = adaptive_fourier_decomposition(signal, max_iter, tolerance)% 自适应傅里叶分解% 输入: signal - 输入信号, max_iter - 最大迭代次数, tolerance - 收敛容差% 输出: components - 分解分量, residues - 残差, parameters - 分解参数N = length(signal);t = (0:N-1)/N; % 归一化时间components = zeros(max_iter, N);residues = zeros(max_iter, N);parameters = zeros(1, max_iter);% 初始化残差residues(1, :) = signal;for k = 1:max_iter% 计算当前残差的傅里叶变换R_k = fft(residues(k, :));% 寻找最优参数 a_k (在单位圆内)a_candidates = linspace(0.01, 0.99, 100);best_a = 0;max_inner_product = 0;for a = a_candidates% 生成自适应基函数basis_func = generate_basis_function(a, t, N);% 计算内积inner_prod = abs(sum(residues(k, :) .* conj(basis_func)));if inner_prod > max_inner_productmax_inner_product = inner_prod;best_a = a;endendparameters(k) = best_a;% 生成最优基函数e_a = generate_basis_function(best_a, t, N);% 计算分解系数c_k = sum(residues(k, :) .* conj(e_a)) / sum(abs(e_a).^2);% 提取分量components(k, :) = c_k * e_a;% 更新残差if k < max_iterresidues(k+1, :) = residues(k, :) - components(k, :);end% 检查收敛if norm(residues(k, :)) < tolerancecomponents = components(1:k, :);residues = residues(1:k, :);parameters = parameters(1:k);break;endend
endfunction basis_func = generate_basis_function(a, t, N)% 生成自适应傅里叶基函数% e_a(t) = sqrt(1-|a|^2)/(1 - a_bar * exp(2*pi*i*t))basis_func = zeros(1, N);for n = 1:Nbasis_func(n) = sqrt(1 - abs(a)^2) / (1 - conj(a) * exp(2*pi*1i*t(n)));end
end
2. 信号重构
function reconstructed_signal = afd_reconstruction(components, parameters, t)% AFD信号重构% 输入: components - 分解分量, parameters - 分解参数, t - 时间向量% 输出: reconstructed_signal - 重构信号[K, N] = size(components);reconstructed_signal = zeros(1, N);for k = 1:K% 使用存储的参数重新生成基函数e_a = generate_basis_function(parameters(k), t, N);% 计算系数c_k = sum(components(k, :) .* conj(e_a)) / sum(abs(e_a).^2);% 累加重构信号reconstructed_signal = reconstructed_signal + c_k * e_a;end
end
3. 完整示例
% 完整AFD示例
clear; clc;% 生成测试信号
fs = 1000; % 采样频率
t = 0:1/fs:1-1/fs; % 1秒时间向量
N = length(t);% 多分量信号
f1 = 5; % 5Hz
f2 = 20; % 20Hz
f3 = 50; % 50Hzsignal = 2*sin(2*pi*f1*t) + 1.5*cos(2*pi*f2*t) + sin(2*pi*f3*t) + 0.5*randn(1,N);% AFD分解
max_iter = 10;
tolerance = 1e-6;[components, residues, parameters] = adaptive_fourier_decomposition(signal, max_iter, tolerance);% 信号重构
reconstructed_signal = afd_reconstruction(components, parameters, t);% 计算重构误差
reconstruction_error = norm(signal - real(reconstructed_signal)) / norm(signal);% 结果显示
figure('Position', [100, 100, 1200, 800]);% 原始信号与重构信号
subplot(3, 2, 1);
plot(t, signal, 'b', 'LineWidth', 1.5);
hold on;
plot(t, real(reconstructed_signal), 'r--', 'LineWidth', 1);
legend('原始信号', '重构信号');
title('原始信号 vs 重构信号');
xlabel('时间 (s)'); ylabel('幅值');
grid on;% 重构误差
subplot(3, 2, 2);
plot(t, signal - real(reconstructed_signal), 'g', 'LineWidth', 1);
title(['重构误差, RMSE: ', num2str(reconstruction_error)]);
xlabel('时间 (s)'); ylabel('误差');
grid on;% 分解分量
subplot(3, 2, 3);
for k = 1:size(components, 1)plot(t, real(components(k, :)), 'LineWidth', 1);hold on;
end
title('AFD分解分量');
xlabel('时间 (s)'); ylabel('幅值');
grid on;% 残差序列
subplot(3, 2, 4);
residue_norms = zeros(1, size(residues, 1));
for k = 1:size(residues, 1)residue_norms(k) = norm(residues(k, :));
end
semilogy(1:length(residue_norms), residue_norms, 'ro-', 'LineWidth', 2);
title('残差能量衰减');
xlabel('迭代次数'); ylabel('残差能量 (对数)');
grid on;% 参数分布
subplot(3, 2, 5);
plot(parameters, 'bo-', 'LineWidth', 2, 'MarkerSize', 6);
title('自适应参数 a_k');
xlabel('分量索引'); ylabel('参数值');
grid on;% 频谱比较
subplot(3, 2, 6);
f = (0:N-1)*fs/N;
Y_orig = abs(fft(signal));
Y_recon = abs(fft(real(reconstructed_signal)));
plot(f(1:N/2), Y_orig(1:N/2), 'b', 'LineWidth', 1.5);
hold on;
plot(f(1:N/2), Y_recon(1:N/2), 'r--', 'LineWidth', 1);
legend('原始频谱', '重构频谱');
title('频谱比较');
xlabel('频率 (Hz)'); ylabel('幅值');
grid on;fprintf('AFD分解完成:\n');
fprintf('信号长度: %d\n', N);
fprintf('分解分量数: %d\n', size(components, 1));
fprintf('最终重构误差: %.6f\n', reconstruction_error);
4. 改进的快速AFD算法
function [components, energies] = fast_adaptive_fourier_decomposition(signal, target_components)% 快速自适应傅里叶分解% 基于能量准则选择最重要的分量N = length(signal);t = (0:N-1)/N;residual = signal;components = zeros(target_components, N);energies = zeros(1, target_components);for comp_idx = 1:target_components% 计算残差的频谱R_freq = fft(residual);% 寻找能量最大的频率区域[~, dominant_freq] = max(abs(R_freq(1:floor(N/2))));% 转换为自适应参数 (映射到单位圆内)a_k = 0.9 * exp(2*pi*1i*(dominant_freq-1)/N);% 生成基函数basis_func = generate_basis_function(a_k, t, N);% 正交投影coeff = sum(residual .* conj(basis_func)) / sum(abs(basis_func).^2);% 存储分量components(comp_idx, :) = coeff * basis_func;energies(comp_idx) = abs(coeff)^2 * sum(abs(basis_func).^2);% 更新残差residual = residual - components(comp_idx, :);% 提前终止条件if norm(residual) < 1e-6components = components(1:comp_idx, :);energies = energies(1:comp_idx);break;endend
end
参考代码 自适应傅里叶分解算法 www.3dddown.com/cna/79065.html
算法特点
优势
- 自适应性: 基函数根据信号特性自动调整
- 快速收敛: 通常比传统傅里叶分解需要更少的分量
- 能量集中: 分量具有更好的时频聚集性
- 正交性: 分解分量在Hardy空间中正交
应用领域
- 信号去噪和压缩
- 故障诊断和特征提取
- 生物医学信号处理
- 通信信号分析