# When might looking at rising edges perform better than cross-correlation?

A simple question as the title indicates:

Let us assume that we have two signals, measured from two different sensors that are close together. (For this application, the sensors are measuring blood/biological activity at different positions on an animal body).

I am wondering when, (if at all), measuring the rising edges of both signals might offer a better estimate of their relative time delays, VS doing a simple cross-correlation?

When might we expect this to be true, if ever?

I ask because I would like to know if looking at rising edges may indeed be optimal in some sense, instead of it being something 'ad-hocy' and possibly mis-leading in lieu of cross-correlation.

• In general, I guess, one can only talk about an optimal metric between sv X and Y if you know their distribution. Features could work better, and are used everywhere, but are also in a sense an admission that real underlying statistics are unknown. – Maurits Oct 1 '12 at 20:21
• @Maurits > "but are also in a sense an admission that real underlying statistics are unknown". Yes - that is a very good point, and I am inclined to agree. It thus perhaps stand to reason that using rising edges might be better than cross-correlations when channel and noise statistics are unknown, esp since coming from the gut of an animal! – Spacey Oct 1 '12 at 20:40

Using the rising edge is better than cross-correlation when there is the physical equivalent of an inductor in the system, which will introduce inertia into the system. That inertia will cause the output to rise gradually, then fall back down. Even in that case the cross-correlation might be more useful, but if you wanted to know when the input starts to affect the output the rise would be more accurate. The figure above is illustrative of what I'm talking about. The inductor (or "inertia introducing object") resists any change in the current flow, so initially it resists the flow causing an exponential rise (as opposed to a sharp rise that matches the presumed non-ideal impulse at the input), and then opposes the current stopping when the input ends, causing the exponential fall at the output.

If the input looks more-or-less like an impulse function, it will correlate most strongly with the output at the output's peak, at time 100. It's clear, though, that the input starts to affect the output at time 1. In different situations, either measurement could be the more useful one.

• Jim, by 'inductor in the system', I imagine this to manifest itself on a pulse as some sort of amplitude modulation? Can you please expand on what you mean here? TY. – Spacey Oct 2 '12 at 0:07
• I hope this edit helps. – Jim Clay Oct 2 '12 at 13:35
• Thanks Jim, and I see what you mean. I am trying to ascertain if, in cases where two signals whose time delay one wishes to measure, and where different channel effects imbue each signal differently, one might be better off looking at where rise times occur, than correlating them. – Spacey Oct 2 '12 at 13:46
• It looks like it might be the case, as your inertia post seems to indicate. It seems that the logic behind your post on inertia can be extended to any other types of 'misbehaving' signals, agree? – Spacey Oct 2 '12 at 13:47
• Yes. Inertia is one mechanism that can cause this behavior, but it wouldn't surprise me if there were other mechanisms besides that. – Jim Clay Oct 2 '12 at 14:25