How to keep one signal fixed in dynamic time warping (dtw)?

Consider two discrete time uniformly sampled signals to be analyzed offline: the reference signal r(k) and the measured signal y(k), where k is the time index. Assume that r(k) and y(k) are equal except for a nonconstant delay in time. More precisely: r(k) = y(k + d(k)), where d(k) is the delay of y(k) versus r(k) at time k. I am interested in determinig d(k) for all k.

To my understanding,

[dist, kr, ky] = dtw(r, y)

stretches both signals such that the distance between r(kr) and y(ky) is minimal. However, I do not want to stretch the reference signal r(k) at all, it shall remain untouched. More precisely, I want to enforce: kr = 1:length(r) such that i can infer d(k) from ky.

Do you know a way to accomplish this enforcement?

Also, I'm not restricted to use the dtw function in this context. In case you know about a better approach to the problem stated above, please feel free to mention it (as an alternative, I will look into a cross correlation based approach)!

From what I understand, the finddelay and alignsignals function only determines a constant delay.

I highly appreciate your efforts!

Answer 1

[Mathieu NOE]

hello

in case that may answer the question this is a home made buffered xcorr approach

hope it helps !

clc
clearvars

samples = 1000;
dt = 0.01;
Fs = 1/dt;
t = (0:samples-1)*dt;

%% sinus waveforms
% reference signal
f = 3;
ph_ref = 2*pi*f*t+1;
sig_ref = sin(ph_ref);
% introduce some variable time delay in second signal 
td = 1.5*sin(t/max(t)*pi)/(2*pi*f);
sig_meas = sin(ph_ref+td*2*pi*f);

%--------------------------------------------------------------------------
% using xcorrr-function to fit signals on x-axis
%--------------------------------------------------------------------------

% do the xcorr on data buffer to get a delay trend vs time
buffer = 64; 
Overlap = 0.95;
[time,t_a] = myxcorr(sig_ref,sig_meas, Fs, buffer, Overlap);
t_aa = interp1(time,t_a,t,'linear','extrap'); % variable time delay must be computed for all time values

%--------------------------------------------------------------------------
% plotting the results
%--------------------------------------------------------------------------

figure(1)
subplot(211),plot(t, sig_ref,'b',t, sig_meas,'r')
legend('ref', 'measure')
subplot(212),plot(t,td,'b', t, t_aa,'r')
title('time delay')
legend('theoretical', 'measured')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function  [time,t_a] = myxcorr(x,y, Fs, buffer, Overlap)

	%    compute running xcorr with overlap 
	samples = length(x);
	offset = fix((1-Overlap)*buffer);
	segments = 1+ fix((samples-buffer)/offset); % Number of windows
	
        for ci=1:segments
            start = 1+(ci-1)*offset;
            stop = start+buffer-1;
            [c_a, lag_a] = xcorr(x(start:stop),y(start:stop));
            % % c_a = c_a/max(c_a); % not needed
            [~, i_a] = max(c_a);
            t_a(ci) = lag_a(i_a)/Fs; % lag in seconds
        end
	% time vector 
	% time stamps are defined in the middle of the buffer
	time = ((0:segments-1)*offset + round(buffer/2))/Fs;

end

[Mathieu NOE]

hello again

problem solved ?