# Calculating cross-correlation from two auto-correlations

Started by 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?

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?
>
> 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
```