eksmoother.m

用于心电信号滤波

function [Xhat,Phat,X,PSmoothed,a] = EKSmoother(Y,X0,P0,Q,R0,Wmean,Vmean,Inits,VarWinlen1,tau,gamma,VarWinlen2,varargin)
%
% The Extended Kalman Filter (EKF) and Extended Kalman Smoother (EKS) for
% noisy ECG observations.
%
% [Xekf,Pekf,Xeks,Peks,a] = EKSmoother(Y,X0,P0,Q,R,Wmean,Vmean,Inits,VarWinlen1,tau,gamma,VarWinlen2,flag),
%
% inputs:
% Y: matrix of observation signals (samples x 2). First column corresponds
% to the phase observations and the second column corresponds to the noisy
% ECG
% X0: initial state vector
% P0: covariance matrix of the initial state vector
% Q: covariance matrix of the process noise vector
% R: covariance matrix of the observation noise vector
% Wmean: mean process noise vector
% Vmean: mean observation noise vector
% Inits: filter initialization parameters
% VarWinlen1: innovations monitoring window length
% tau: Kalman filter forgetting time. tau=[] for no forgetting factor
% gamma: observation covariance adaptation-rate. 0<gamma<1 and gamma=1 for no adaptation
% VarWinlen2: window length for observation covariance adaptation
% flag (optional): 1 with waitbar / 0 without waitbar (default)
%
% outputs:
% Xekf: state vectors estimated by the EKF (samples x 2). First column
% corresponds to the phase estimates and the second column corresponds to
% the denoised ECG
% Pekf: the EKF state vector covariance matrix (samples x 2 x 2)
% Xeks: state vectors estimated by the EKS (samples x 2). First column
% corresponds to the phase estimates and the second column corresponds to
% the denoised ECG
% Peks: the EKS state vector covariance matrix (samples x 2 x 2)
% a: measure of innovations signal whiteness
%
%
% Open Source ECG Toolbox, version 1.0, November 2006
% Released under the GNU General Public License
% Copyright (C) 2006  Reza Sameni
% Sharif University of Technology, Tehran, Iran -- LIS-INPG, Grenoble, France
% [e-mail]

% This program is free software; you can redistribute it and/or modify it
% under the terms of the GNU General Public License as published by the
% Free Software Foundation; either version 2 of the License, or (at your
% option) any later version.
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
% Public License for more details. You should have received a copy of the
% GNU General Public License along with this program; if not, write to the
% Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
% MA  02110-1301, USA.

%//////////////////////////////////////////////////////////////////////////
plotflag = 0;
if(nargin==13),
    plotflag = varargin{1};
end
if(plotflag==1),
    wtbar = waitbar(0,'Forward filtering in progress. Please wait...');
end
%//////////////////////////////////////////////////////////////////////////
% Initialization
StateProp(Inits);           % Initialize state equation
ObservationProp(Inits);     % Initialize output equation
Linearization(Inits);       % Initialize linearization

%//////////////////////////////////////////////////////////////////////////
Samples = length(Y);
L = length(X0);
Pminus = P0;
Pplus = zeros(L,L);
Xminus = X0;
Xbar = zeros(L,Samples);
Pbar = zeros(L,L,Samples);
Xhat = zeros(L,Samples);
Phat = zeros(L,L,Samples);
%//////////////////////////////////////////////////////////////////////////
% For innovation monitoring
mem2 = zeros(VarWinlen2,size(Y,2)) + R0(2,2);
mem1 = ones(VarWinlen1,size(Y,2));

%//////////////////////////////////////////////////////////////////////////
% Forgetting factor
fs = Inits(end); % the last init is fs
dt = 1/fs;
if(~isempty(tau))
    alpha = exp(-dt/tau);
else
    alpha = 1;
end

%//////////////////////////////////////////////////////////////////////////
R = R0;
% Filtering
for k = 1 : Samples,

    % This is to prevent 'Xminus' mis-calculations on phase jumps
    if(abs(Xminus(1)-Y(k,1))>pi)
        Xminus(1) = Y(k,1);
    end

    % Store results
    Xbar(:,k) = Xminus';
    Pbar(:,:,k) = Pminus';

    XX = Xminus;
    PP = Pminus;
    for jj = 1:size(Y,2);
        % Measurement update (A posteriori updates)
        Yminus = ObservationProp(XX,Vmean);
        YY = Yminus(jj);
        [CC,GG] = Linearization(XX,Vmean,1);                                % Linearized observation eq.
        C = CC(jj,:);
        G = GG(jj,:);

        K = PP*C'/(C*PP*C' + alpha*G*R(jj,jj)*G');                          % Kalman gain
        PP   = ( (eye(L)-K*C)*PP*(eye(L)-K*C)' + K*G*R(jj,jj)*G'*K' )/alpha;% Stabilized Kalman cov. matrix
        XX = XX + K*(Y(k,jj)-YY);                                           % A posteriori state estimate
    end
    % Monitoring the innovation variance
    inovk = Y(k,:)-Yminus';
    Yk = C*Pminus*C'+G*R*G';
    mem1 = [inovk.^2/Yk ; mem1(1:end-1,:)];
    mem2 = [inovk.^2 ; mem2(1:end-1,:)];

    a(k,:) = mean(mem1,1);

    R(2,2) = gamma*R(2,2) + (1-gamma)*mean(mem2(:,2));

    Xplus = XX;
    Pplus = (PP + PP')/2;

    Xminus = StateProp(Xplus,Wmean);                                        % State update
    [A,F] = Linearization(Xplus,Wmean,0);                                   % Linearized equations
    Pminus = A*Pplus*A' + F*Q*F';                                           % Cov. matrix update

    % Store results
    Xhat(:,k) = Xplus';
    Phat(:,:,k) = Pplus';

    if(plotflag==1 && mod(k,Samples/5)==0)
        waitbar(k/Samples,wtbar);
    end
end

%//////////////////////////////////////////////////////////////////////////
if (plotflag == 1),
    waitbar(0,wtbar,'Backward smoothing in progress. Please wait...');
end

% Smoothing
PSmoothed = zeros(size(Phat));
X = zeros(size(Xhat));
PSmoothed(:,:,Samples) = Phat(:,:,Samples);
X(:,Samples) = Xhat(:,Samples);
for k = Samples-1 : -1 : 1,
    [A,F] = Linearization(Xhat(:,k),Wmean,0);
    S = Phat(:,:,k) * A' * inv(Pbar(:,:,k+1));
    X(:,k) = Xhat(:,k) + S * (X(:,k+1) - Xbar(:,k+1));
    PSmoothed(:,:,k) = Phat(:,:,k) - S * (Pbar(:,:,k+1) - PSmoothed(:,:,k+1)) * S';

    if(plotflag==1 && mod(k,Samples/5)==0)
        waitbar(1-k/Samples,wtbar);
    end
end

if (plotflag == 1),
    close(wtbar);
end

%//////////////////////////////////////////////////////////////////////////
Xhat = shiftdim(Xhat,1);
Phat = shiftdim(Phat,2);
% Xbar = shiftdim(Xbar,1);
% Pbar = shiftdim(Pbar,2);
X = shiftdim(X,1);
PSmoothed = shiftdim(PSmoothed,2);

%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
function xout = StateProp(x,u,W)

% Make variables static
persistent tetai alphai bi fs w dt;

% Check if variables should be initialized
if nargin==1,
    % mean of the noise parameters
    % Inits = [alphai bi tetai w fs];
    L = (length(x)-2)/3;
    alphai = x(1:L);
    bi = x(L+1:2*L);
    tetai = x(2*L+1:3*L);
    w = x(3*L+1);
    fs = x(3*L+2);

    dt = 1/fs;
    return
end

xout(1,1) = x(1) + w*dt;                                                    % teta state variable
if(xout(1,1)>pi),
    xout(1,1) = xout(1,1) - 2*pi;
end

dtetai = rem(xout(1,1) - tetai,2*pi);
xout(2,1) = x(2) - dt*sum(w*alphai./(bi.^2).*dtetai.*exp(-dtetai.^2./(2*bi.^2))); % z state variable

%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
function y = ObservationProp(x,v)

% Check if variables should be initialized
if nargin==1,
    return
end

% Calculate output estimate
y = zeros(2,1);
y(1) = x(1) + v(1);   % teta observation
y(2) = x(2) + v(2);   % amplidute observation


%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
%//////////////////////////////////////////////////////////////////////////
function [M,N] = Linearization(x,WVmean,flag)

% Make variables static
persistent tetai alphai bi fs w dt L;

% Check if variables should be initialized
if nargin==1,
    % Inits = [alphai bi tetai w fs];
    L = (length(x)-2)/3;
    alphai = x(1:L);
    bi = x(L+1:2*L);
    tetai = x(2*L+1:3*L);
    w = x(3*L+1);
    fs = x(3*L+2);

    dt = 1/fs;
    return
end
% Linearize state equation
if flag==0,
    dtetai = rem(x(1) - tetai,2*pi);

    M(1,1) = 1;                                                                     % dF1/dteta
    M(1,2) = 0;                                                                     % dF1/dz

    M(2,1) = -dt*sum( w*alphai./(bi.^2).*(1 - dtetai.^2./bi.^2).*exp(-dtetai.^2./(2*bi.^2)) ) ;    % dF2/dteta
    M(2,2) = 1 ;                                                                    % dF2/dz

    % W = [alpha1, ..., alpha5, b1, ..., b5, teta1, ..., teta5, omega, N]
    N(1,1:3*L) = 0;
    N(1,3*L+1) = dt;
    N(1,3*L+2) = 0;

    N(2,1:L) = -dt*w./(bi.^2).*dtetai .* exp(-dtetai.^2./(2*bi.^2));
    N(2,L+1:2*L) = 2*dt.*alphai.*w.*dtetai./bi.^3.*(1 - dtetai.^2./(2*bi.^2)).*exp(-dtetai.^2./(2*bi.^2));
    N(2,2*L+1:3*L) = dt*w*alphai./(bi.^2).*exp(-dtetai.^2./(2*bi.^2)) .* (1 - dtetai.^2./bi.^2);
    N(2,3*L+1) = -sum(dt*alphai.*dtetai./(bi.^2).*exp(-dtetai.^2./(2*bi.^2)));
    N(2,3*L+2) = 1;

    % Linearize output equation
elseif flag==1,
    M(1,1) = 1;
    M(1,2) = 0;
    M(2,1) = 0;
    M(2,2) = 1;

    N(1,1) = 1;
    N(1,2) = 0;
    N(2,1) = 0;
    N(2,2) = 1;
end