为信号一个振荡冲击的中心频率；

为其带宽。

的大小反映了信号的共振属性；

越大，信号的频率聚集性越好，具有越高的品质因子；反之，信号就具有低的共振属性。根据品质因子的定义可知：瞬态冲击信号为带宽信号，具有低的品质因子；持续振荡周期信号为窄带信号，具有高的品质因子，如下图所示。在用小波变换对待定信号进行分解时，小波基的选取对分解结果有着重要的影响，当待分析信号具有高的品质因子时，选取的小波基函数就需要有高的品质因子；反之亦然，当待分析信号同时含有高品质因子及低品质因子信号时，就需要用时构建两种不同品质因子的小波基函数对信号进行分解。传统的小波分析方法只能选取单一品质因子的小波基，共振稀疏分解提出了解决上述问题很好的方法。

为高通尺度因子，

为低通尺度因子，

为冗余度；子带信号

的采样频率为

;

的采样频率为

,

为原信号

的采样频率，共振稀疏分解利用图2（a）所示两通道分解滤波器组以迭代的方式实现信号的分解，L层共振稀疏分解如图3所示，图3中，

为信号经过第j层变换得到的高频系数；

为经过第j层变换得到的低频系数，

。

%  function：MP algorithm applied to decomposition of one signal named bat%% part 1:  read data (to be processed ) % load bat.dat
% bat = imread('bat.jpg');    % 读参考图像clc;
close all;
clear;
T=0.256;
fs=1000;
t=1/fs:1/fs:T;
[H,Lie]=size(t);
x=sin(2*pi*100*t);
bat=x;   %% part 2:
%% part 2.1  determine the decomposition parametersiterative_number=20;       %  the matching pursuit processing iterative number[a,N]=size(bat);%  the length of signal and the length of atoms;Nsignal_reconstruct=zeros(1,N);signal_r=bat;a_base=2;
j_min=0;
j_max=log2(N);
u_base=1/2;
p_min=0;
v_base=pi;
k_min=0;
w_base=pi/6;
i_min=0;
i_max=12;%% part 2.2  the matching pursuit process  for n=1:iterative_number%% the following program uses one subroutine to select the best atom;[proj,scale,translation,freq,phase]=select_best(signal_r,N,a_base,j_min,j_max,u_base,p_min,v_base,k_min,w_base,i_min,i_max);%% reconstruct the best atom from the parameters gotted by above subroutinet=0:N-1;t=(t-translation)/scale;g=(1/sqrt(scale))*exp(-pi*t.*t).*cos(freq*t+phase);g=g/sqrt(sum(g.*g));signal_reconstruct=signal_reconstruct+proj*g;signal_r=signal_r-proj*g;subplot(221);plot(bat);legend('original signal');subplot(222);plot(g);legend('best atom selected');subplot(223);plot(signal_r);legend('residual signal');subplot(224);plot(signal_reconstruct);legend('reconstructed signal');drawnow;disp('the MP process number or the number of atoms selected='),disp(n);
end

function [proj,scale,translation,freq,phase]=select_best(signal_r,N,a_base,j_min,j_max,u_base,p_min,v_base,k_min,w_base,i_min,i_max);%% this subroutine program(子程序)is to select in the dictionary the best atom suited the signal or the residual of the signal% INPUT
% the signal_r: the signal or the residual of the signal to be decomposed
% the N: the length of the signal or of the residual of the signal or the length of the atoms% parameters :the parameter to construct the dictionary , it has much influence on the speed of the decomposition
% a_base=2;
% j_min=0;
% j_max=log2(N);% OUTPUT
% proj:the projection of the signal or the residual of the signal on the best atom
% scale:the scale of the best atom(s in the formula)
% translation:the translation of the best atom(u in the formula)
% freq:the freq of the best atom(v in the formula)
% phase:the phase of the best atom(w in the formula)% proj_trans:to determine which projection is biggest
proj_trans=0;
proj=0;% size_dic is one parameter to show the size of the over-complete dictionary used
size_dic=0;% [hy,hx]=meshgrid(0:32-1,0:32-1);
% m=32;n=32;for j=j_min:j_maxfor p=p_min:N*2^(-j+1)for k=k_min:2^(j+1)for i=i_min:i_maxsize_dic=size_dic+1;s=a_base^j;u=p*s*u_base;v=k*(1/s)*v_base;w=i*w_base;t=0:N-1;t=(t-u)/s;g=(1/sqrt(s))*exp(-pi*t.*t).*cos(v*t+w);g=g/sqrt(sum(g.*g));proj_trans=sum(signal_r.*g);if abs(proj_trans)>abs(proj)proj=proj_trans;scale=s;translation=u;freq=v;phase=w;endendendend
enddisp('the size of the dictionary='),disp(size_dic);  % size_dic: the size of the dictionary
end%%%%%%%%%%%%%%%%%%%%%%%% end  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%