What is correlation

Quote

What is correlation?

When we say that our instruments "find the Doppler shift of the backscattered acoustic echo", we are not really being very specific.   Indeed, there are many ways to find the frequency shift of an echo.  In the case of Nortek - and the Vector - we use a method commonly referred to as the "autocovariance method".  

In this method, we do not directly calculate a frequency shift. Iinstead we we calculate the phase shift between two signals coming back from the water:

Signal 1: s1
Signal 2: s2

If we are two find the phase change between the two signal (assuming s1 and s2 complex, which they are), we find:

Covar=<[s1 * conj(s2)>      - conj means complex conjugation

We then normalize:

Corr=Covar/sqrt(<s1(0)**2><s2(0)**2>

and the phase change dteta can be expressed as:

dteta=atan(imag(Corr)/real(Corr))

and "the correlation" we output from the Vector is:

Cor = sqrt(imag(Corr)**2 + real(Corr)**2) or the "norm" of Corr.

This is all standard stuff and can be found in most university books on digital signal processing.

I usually think of the correlation ("Cor") as a measure of how similar the two signals are. If the two time series have nothing in common, the value is 0 . If the two time series are exactly the same, the value is 1 (or 100%)    .  

- Atle Lohrmann

Am I not able to see things or are there missing bits in this. I have looked at it several times and can not see where s1 and s2 are used. What are S1 and S2, returns from just two particles or 1/2 of the particles in an averaging segment?

I am trying to see how amplitude, velocity and correlation are all linked. This hints at how phase or velocity is linked to correlation, but it is fuzzy as it appears to be missing things.

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Hi Again,

Is it possible to be more specific about what s1 and s2 really are? For example, if I am sampling at 64Hz on the 1 m/s range, I am pinging at 125 Hz so I get roughly 2 returns. I would suspect that s1 and s2 are the two returns. Is that true? Since I am not pinging at 128 Hz there is a subtle problem here that all data reported will not be in pairs. What happens then?

Another example is when I am change the sample rate to 4 Hz, and 1 m/s. I am still pinging at 125 Hz so I now have up to 31 returns per data reported. How are s1 and s2 formed?

Thanks, this will be greatly helpful for me to understand what is going on.

Cheers,

Dave