Compensate delay of non-symmetric (non-linear phase) filter

Question

Lets assume I want to apply a matched filter h which is non-symmetric to my signal x and the output is y in matlab:

y=filter(h,1,x);

Now I am interested in comparing the information between my filtered signal yand my input x. For this I want to compensate for the delay of the filter h.

When googling this topic I found the following tutorial which handles linear-phase fir filters. Obviously in such a case the group delay is $\frac{N-1}{2}$ given that the filter has the length $N$.

%this is the example from matlab
delay = mean(grpdelay(h,1,128)); 
x = x(1:end-delay);
y(1:delay) = [];

However, I am aware that the group delay of a non symmetric filter is frequency depending and I am not sure how to proceed to align my signals in time to compare them. As I am interested in real time application the function filtfilter is not applicable.

So my question is, how do I estimate in the best way the time delay to align (sync?) my signals in time given the coefficients of a non-symmetric filter h?

delay = .... h ....? 
x = x(1:end-delay);
y(1:delay) = [];

The desired output should look like the red curve in the following figure

Here the signal in blue is the input signal x and in green the filtered signal y. However, I would like to to compensate for the delay of the green signal so that the input and output align in time as in the case of the red curve.

Answer 1

If your filters have a low pass characteristic (which at least the one in your example appears to have), then a possibly useful measure for the delay is the center of gravity of the impulse response:

$$\hat{\tau}=\frac{\displaystyle\sum_nnh[n]}{\displaystyle\sum_nh[n]}\tag{1}$$

For more general impulse responses, other than low pass, you can modify $(1)$ by either taking magnitudes or squares of $h[n]$. Note that $(1)$ is the filter's group delay evaluated at DC.