Correlation meter

I have three questions regarding the correlation meters used in audio.

1. How are they calculated? I have heard correlation is performed by convolving one signal with the time flipped version of a other, but this seems impossible in real time.

2. How does this differ from the pearson's correlation coefficient used in statistics?

3. Sometimes the correlation meter takes a round shape representation. Is there any place to read more about the exact details on how this representation works?

Thankyou.

Let me answer in slightly different order:

3) It seems you're really asking for a "Goniometer", which is a 2 channel oscilloscope with the left and right signal components plotted in two orthogonal directions on the scope screen. These directions are typically chosen so that the signal channels make an angle of 45 degrees to the horizontal.

1) The correlation meter is often part of a Goniometer display and displays a value between -1 and +1 indicating the relative correlation between left and right channel. It is calculated using three different low pass filters acting on the left channel squared, the right channel squared and the product of the left and right channel. In math terms:

$$c(t) = \frac{H_l ( L(t) \cdot R(t) )}{\sqrt{H_l( L(t)^2 ) \cdot H_l(R(t)^2)}}$$

where $L(t)$ and $R(t)$ are the channel signals and $H_l$ is the low pass filter. The low pass filter mimics a windowed integral, and the correlation is measured as average over the integration time. The equivalent integration time for audio use is typically in the range of 10 to 100 milliseconds, which gives a low pass cutoff of around 100 to 10 Hz.

2) And this definition coincides with Pearson's definition for a time range of length $T$ of the two channels exactly if the low pass filter used above is a box-filter with support of duration $T$.

• Wow, this is super informative! Jun 28 '14 at 22:28
• @Jazzmaniac Thankyou! That equation does look a lot like Pearson's correlation. But would the method of phase flipped convolution between the signals yield the same results, as described by DSP guide by smith (provided that same time constant was used): dspguide.com/ch7/3.htm ? How would you transform the signal produced by the convolution into a single number between -1 and 1? Or are these just fundamentally completely different operations having the same name?
– Tony
Jun 29 '14 at 9:51
• @Tony, the difference between a correlation sequence that you mention and the correlation coefficient is that the sequence really calculates the correlation using different time lags between the two signals to be compared. So you get a correlation coefficients for every possible time difference. The correlation meter only displays the correlation without lag. In addition, the correlation meter uses normalized (or relative) correlation, which is divided by the norm of the signals involved. Jun 29 '14 at 11:23
• Sorry, I made a mistake in the definition of $c(t)$ which I have just fixed. The denominator must contain the product of the norms of the channels. The version I had before didn't have the square and square roots. Also, in order to avoid division by zero you can add a very small positive constant to the denominator. Jun 29 '14 at 11:27
• @Jazzmaniac: It's not 100% correct, because linear correlation coefficient can be calculated only at one position, yielding normalized value of correlation. Shifting is optional.
– jojek
Jun 29 '14 at 12:38