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Calculating cross-correlation from two auto-correlations

Started by karame83 June 27, 2008
Hello,

I have two autocorrelation sequences of unknown signals A and B, and I
need to calculate cross-correlation between these two. Is there any way to
do that without performing spectral factorization?

Thanks in advance.
Tony


On 27 Jun, 13:02, "karame83" <in_touch...@hotmail.com> wrote:
> Hello, > > I have two autocorrelation sequences of unknown signals A and B, and I > need to calculate cross-correlation between these two. Is there any way to > do that without performing spectral factorization?
You can't reconstruct the time domain sequence from its autocorrelation function since infinitely many time domain sequences share the same autocorrelation function. Since you can't reconstruct the time sequences, you can't compute their cross correlation. The best you might achieve is to compoute the cross magnitude spectrum, and maybe possibly the cross correlations between the corresponding minimum phase sequences. This will, of course give the wrong answers if at least one of the sequences was non-minimum phase. Rune
On Jun 27, 6:02 am, "karame83" <in_touch...@hotmail.com> wrote:
> Hello, > > I have two autocorrelation sequences of unknown signals A and B, and I > need to calculate cross-correlation between these two. Is there any way to > do that without performing spectral factorization? > > Thanks in advance. > Tony
As Rune Allnor has already noted, what you are asking is not possible. There *is* one measure that you can compute: the sum of the squares of the magnitudes of the crosscorrelation sequence equals the inner product of the autocorrelation sequences. That is, sum |C_{x,y}[n]|^2 = sum C_{x,x}[n](C_{y,y}[n])* where C_{x,y}[n] is the n-th term of the crosscorrelation sequence, and the sum is over all values of n. (When x and y are the same, C_{x,y} = C_{x,x} becomes the autocorrelation function). What this *can* give you is a lower bound on the maximum magnitude of C_{x,y}[n]. If the right side of the above equation has value K, then the average value of |C_{x,y}[n]|^2 is K/N where N is the number of terms in the sum, and hence there is at least one value of n for which |C_{x,y}[n]| is at least as large as sqrt(K/N), and so the maximum crosscorrelation magnitude must be at least as large as sqrt(K/N). Hope this helps --Dilip Sarwate
Thanks a lot for your answers, this helps for sure!

Tony