Time delay / lag estimation non periodic signals (and periodic signals) - matlab

Question

I'm working on aligning different measurements from sensors. Some of these are periodic and I just used the maximum of the cross correlations and it worked fine. Now I have a couple of non periodic signals similar to ramp/sigmoids/step/hill functions that I want to align, but for these the cross correlation fails miserably (giving me always the maximum at lag 0).
What is the approach for these kind of signals?
Ideal the approach would work for both signals without prior knowledge which one I'm encountering.
Here is an example (with noise)

Answer 1

One possible approach is to take your aperiodic signal and coerce it into a periodic signal.
One way to do this is to first normalize your signal, and then append an inverted version of your signal (1 - normalizedSignal) to your signal. This makes it a periodic signal which then should be able to be fed into cross-correlation analysis relatively easily.
Here is an example I whipped up using an inverted sigmoid shifted in time.
function aperiodicxcorr()
% Time step at which to sample the sigmoid
dt = 0.1;
t = -10:dt:5;
% Artificial lags to apply to the second and third signals
actualLag2 = 3;
actualLag3 = 5;
% Now create signals that are negative sigmoids with delays
S1 = -sigmoid(t);
S2 = -sigmoid(t + actualLag2);
S3 = -sigmoid(t + actualLag3);
% Normalize each sigmal
S1 = normalize(S1);
S2 = normalize(S2);
S3 = normalize(S3);
% Concatenate the inverted signal with signal to make it periodic
S1 = cat(2, 1-S1, S1);
S2 = cat(2, 1-S2, S2);
S3 = cat(2, 1-S3, S3);
% Retrieve lag (in samples)
[corr2, lag2] = computeLag(S1, S2);
[corr3, lag3] = computeLag(S1, S3);
% Convert lags to time by multiplying by time step
lag2 = lag2 * dt;
lag3 = lag3 * dt;
fprintf('Lag of S2: %0.2f (r = %0.2f)\n', lag2, corr2);
fprintf('Lag of S3: %0.2f (r = %0.2f)\n', lag3, corr3);
end
function [corr, lag] = computeLag(A, B)
[corr, lags] = xcorr(A, B, 'coeff');
[corr, ind] = max(corr);
lag = lags(ind);
end
function data = normalize(data)
data = data - min(data(:));
data = data ./ max(data(:));
end
function S = sigmoid(t)
S = 1 ./ (1 + exp(-t));
end
The modification to the signal that I discussed, looks like this for the above code.
And the result of the fprintf statements at the bottom are:
Lag of S2: 3.00 (r = 1.00)
Lag of S3: 5.00 (r = 1.00)
And these match up with the specified lags.
The drawback of this is that it won't work for signals which are already periodic. That being said, periodicity is relatively easy to check (particularly for a normalized signal) by comparing the first and last values of your signal and ensuring that they are within a specified tolerance of one another.