One approach is to find the max of the signals correlation. Alternatively, there is a finddelay function (cf doc) Here is some code for you using max of correlation.
Thsi will work on simple signal. Depending on the amount of noise and quality of the data, this approach may not be so robust in practice.
clear all
close all
clc
% 1) Générate noisy signal
s=0.01;
t=0:s:10;
TheoreticalDelay=0.5;
w1=4;
w2=10;
mu=0.5;
f=@(t) sin(w1*t).*cos(w2*t)+randn(size(t))*mu;
x1=f(t);
x2=f(t-TheoreticalDelay);
% 2) Compute correlation and find delay
[C21,lags]=xcorr(x2,x1);
[~,iDelay]=max((C21))
EstimatedDelay=lags(iDelay)*s;
fprintf('Theoretical delay : %.2f s\n',TheoreticalDelay);
fprintf('Estimated delay : %.2f s\n',EstimatedDelay);
figure;
subplot(3,1,1);
plot(t,x1,'r',t,x2,'b');
grid on;legend('x1','x2');xlabel('Time(s)');
subplot(3,1,2);
plot(lags*s,C21);
hold on;
plot(lags(iDelay)*s,C21(iDelay),'r*');
legend('correlation','max');
grid on;
xlabel('Time(s)');
subplot(3,1,3);
plot(t+EstimatedDelay,x1,'r',t,x2,'b');
grid on;legend('x1 sync with x2','x2');xlabel('Time(s)');
