EMD 分解

今天看了些EMD信号分解方面的东西，matlab官网上有个Hilbert-Huang Transform的代码，代码效率极高啊，人家3句语句就解决了一个大问题，很牛啊！还有一个GRilling的EMD工具箱，好多文件，功能应该相当强大。

这里研究了研究matlab官网的代码，加了些注释、功能演示，效果如下

原始信号由3个正弦信号加噪声组成，如下





下面为做EMD分解的结果









第三次分解信号的瞬时频率如下





第四次分解信号的Hilbert分析





具体代码如下

test.m文件

clc

clear all

close all

% [x, Fs] = wavread('Hum.wav');

% Ts = 1/Fs;

% x = x(1:6000);

Ts = 0.001;

Fs = 1/Ts;

t=0:Ts:1;

x = sin(2*pi*10*t) + sin(2*pi*50*t) + sin(2*pi*100*t) + 0.1*randn(1, length(t));

imf = emd(x);

plot_hht(x,imf,1/Fs);

k = 4;

y = imf{k};

N = length(y);

t = 0:Ts:Ts*(N-1);

[yenvelope, yfreq, yh, yangle] = HilbertAnalysis(y, 1/Fs);

yModulate = y./yenvelope;

[YMf, f] = FFTAnalysis(yModulate, Ts);

Yf = FFTAnalysis(y, Ts);

figure

subplot(321)

plot(t, y)

title(sprintf('IMF%d', k))

xlabel('Time/s')

ylabel(sprintf('IMF%d', k));

subplot(322)

plot(f, Yf)

title(sprintf('IMF%d的频谱', k))

xlabel('f/Hz')

ylabel('|IMF(f)|');

subplot(323)

plot(t, yenvelope)

title(sprintf('IMF%d的包络', k))

xlabel('Time/s')

ylabel('envelope');

subplot(324)

plot(t(1:end-1), yfreq)

title(sprintf('IMF%d的瞬时频率', k))

xlabel('Time/s')

ylabel('Frequency/Hz');

subplot(325)

plot(t, yModulate)

title(sprintf('IMF%d的调制信号', k))

xlabel('Time/s')

ylabel('modulation');

subplot(326)

plot(f, YMf)

title(sprintf('IMF%d调制信号的频谱', k))

xlabel('f/Hz')

ylabel('|YMf(f)|');

findpeaks.m文件

function n = findpeaks(x)

% Find peaks. 找极大值点，返回对应极大值点的坐标

n = find(diff(diff(x) > 0) < 0); % 相当于找二阶导小于0的点

u = find(x(n+1) > x(n));

n(u) = n(u)+1; % 加1才真正对应极大值点

% 图形解释上述过程

% figure

% subplot(611)

% x = x(1:100);

% plot(x, '-o')

% grid on

%

% subplot(612)

% plot(1.5:length(x), diff(x) > 0, '-o')

% grid on

% axis([1,length(x),-0.5,1.5])

%

% subplot(613)

% plot(2:length(x)-1, diff(diff(x) > 0), '-o')

% grid on

% axis([1,length(x),-1.5,1.5])

%

% subplot(614)

% plot(2:length(x)-1, diff(diff(x) > 0)<0, '-o')

% grid on

% axis([1,length(x),-1.5,1.5])

%

% n = find(diff(diff(x) > 0) < 0);

% subplot(615)

% plot(n, ones(size(n)), 'o')

% grid on

% axis([1,length(x),0,2])

%

% u = find(x(n+1) > x(n));

% n(u) = n(u)+1;

% subplot(616)

% plot(n, ones(size(n)), 'o')

% grid on

% axis([1,length(x),0,2])

plot_hht.m文件

function plot_hht(x,imf,Ts)

% Plot the HHT.

% :: Syntax

% The array x is the input signal and Ts is the sampling period.

% Example on use: [x,Fs] = wavread('Hum.wav');

% plot_hht(x(1:6000),1/Fs);

% Func : emd

% imf = emd(x);

for k = 1:length(imf)

b(k) = sum(imf{k}.*imf{k});

th = unwrap(angle(hilbert(imf{k}))); % 相位

d{k} = diff(th)/Ts/(2*pi); % 瞬时频率

end

[u,v] = sort(-b);

b = 1-b/max(b); % 后面绘图的亮度控制

% Hilbert瞬时频率图

N = length(x);

c = linspace(0,(N-2)*Ts,N-1); % 0:Ts:Ts*(N-2)

for k = v(1:2) % 显示能量最大的两个IMF的瞬时频率

figure

plot(c,d{k});

xlim([0 c(end)]);

ylim([0 1/2/Ts]);

xlabel('Time/s')

ylabel('Frequency/Hz');

title(sprintf('IMF%d', k))

end

% 显示各IMF

M = length(imf);

N = length(x);

c = linspace(0,(N-1)*Ts,N); % 0:Ts:Ts*(N-1)

for k1 = 0:4:M-1

figure

for k2 = 1:min(4,M-k1)

subplot(4,2,2*k2-1)

plot(c,imf{k1+k2})

set(gca,'FontSize',8,'XLim',[0 c(end)]);

title(sprintf('第%d个IMF', k1+k2))

xlabel('Time/s')

ylabel(sprintf('IMF%d', k1+k2));

subplot(4,2,2*k2)

[yf, f] = FFTAnalysis(imf{k1+k2}, Ts);

plot(f, yf)

title(sprintf('第%d个IMF的频谱', k1+k2))

xlabel('f/Hz')

ylabel('|IMF(f)|');

end

end

figure

subplot(211)

plot(c,x)

set(gca,'FontSize',8,'XLim',[0 c(end)]);

title('原始信号')

xlabel('Time/s')

ylabel('Origin');

subplot(212)

[Yf, f] = FFTAnalysis(x, Ts);

plot(f, Yf)

title('原始信号的频谱')

xlabel('f/Hz')

ylabel('|Y(f)|');

emd.m文件

function imf = emd(x)

% Empiricial Mode Decomposition (Hilbert-Huang Transform)

% EMD分解或HHT变换

% 返回值为cell类型，依次为一次IMF、二次IMF、...、最后残差

x = transpose(x(:));

imf = [];

while ~ismonotonic(x)

x1 = x;

sd = Inf;

while (sd > 0.1) || ~isimf(x1)

s1 = getspline(x1); % 极大值点样条曲线

s2 = -getspline(-x1); % 极小值点样条曲线

x2 = x1-(s1+s2)/2;

sd = sum((x1-x2).^2)/sum(x1.^2);

x1 = x2;

end

imf{end+1} = x1;

x = x-x1;

end

imf{end+1} = x;

% 是否单调

function u = ismonotonic(x)

u1 = length(findpeaks(x))*length(findpeaks(-x));

if u1 > 0

u = 0;

else

u = 1;

end

% 是否IMF分量

function u = isimf(x)

N = length(x);

u1 = sum(x(1:N-1).*x(2:N) < 0); % 过零点的个数

u2 = length(findpeaks(x))+length(findpeaks(-x)); % 极值点的个数

if abs(u1-u2) > 1

u = 0;

else

u = 1;

end

% 据极大值点构造样条曲线

function s = getspline(x)

N = length(x);

p = findpeaks(x);

s = spline([0 p N+1],[0 x(p) 0],1:N);

FFTAnalysis.m文件

% 频谱分析

function [Y, f] = FFTAnalysis(y, Ts)

Fs = 1/Ts;

L = length(y);

NFFT = 2^nextpow2(L);

y = y - mean(y);

Y = fft(y, NFFT)/L;

Y = 2*abs(Y(1:NFFT/2+1));

f = Fs/2*linspace(0, 1, NFFT/2+1);

end

HilbertAnalysis.m文件

% Hilbert分析

function [yenvelope, yf, yh, yangle] = HilbertAnalysis(y, Ts)

yh = hilbert(y);

yenvelope = abs(yh); % 包络

yangle = unwrap(angle(yh)); % 相位

yf = diff(yangle)/2/pi/Ts; % 瞬时频率

end