Signal correlation vs Pearson correlation between signals

I have some data which are the sets of values uniformly distributed along some measure (simple example - along the time intervals). I tested both the coefficients and found the difference, some times essential. F.e. if both the data are noise-like, signal r is about 0.7, and Pearson r - about 0.25.

Which coefficient should be considered better and why?

I compute the signal correlation coefficient as $$r=\frac{\sum_ix_iy_i}{\sqrt{\sum_ix_i^2}\sqrt{\sum_iy_i^2}}$$

and Pearson correlation coefficient as $$r=\frac{\sum_i(x_i-m_x)(y_i-m_y)}{\sqrt{\sum_i(x_i-m_x)^2}\sqrt{\sum_i(y_i-m_y)^2}}$$

There are two fragments of pair of signals:

Fragment 1. Two signals with similar Signal- and Pearson- coefficients

Fragment 1 http://net2ftp.ru/node0/nf05@ngs.ru/Signals1.png

Fragment 2. Two noise-like signals: Signal r = 0.7, Pearson r = 0.2

Fragment 2 http://net2ftp.ru/node0/nf05@ngs.ru/Signals2.png

PS I do not need to performe cross-corelation, convolution etc.

• I think it's clear that both measures are the same when the signals are zero mean. If they're not zero mean, then which is better is completely dependant on what you want/can measure. For example, if you don't know the signals' means, you may be limited to your first coefficient. – MBaz Mar 21 '16 at 3:01
• Of course you are right. It iis easy to calculate the means, and they are obviously non-zero. But does that mean, that I can not use signal correlation? I have added two images of pair of signals. The goal is to say how similar the signals are. – user3446122 Mar 24 '16 at 4:27
• is DC considered part of what the measure of "similarity" is? or is it unimportant if $x_i$ has one DC value while $y_i$ is identical in appearance, but is offset by a different DC value? – robert bristow-johnson Mar 24 '16 at 4:55
• Actually these signals are very far from electrical, but it doesn't matter. Yes, DC values are unimportant. Only the shapes signify. For instance, I have to identify that signal-1 is more or less similar to signal-0, than signal-2. Is it right to say, that signal correlation is inalienable from DC? – user3446122 Mar 24 '16 at 6:05
• robert bristow-johnson, you right question gives me a right answer. To use the signal correlation I should to "normalize" data. Thank you. – user3446122 Mar 24 '16 at 6:13