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From: Demus on 3 Mar 2010 15:47
Hi,

Having the amplitude spectrum of a sign I have previously aimed to assign a
phase to the corresponding signal such that the sum of cross correlations
(at timeshift zero) with a number of other signals is maximized. In matlab
it would be expressed as:

max z = sum(sum(conj(S).*X_1) + sum(conj(S).*X_2) + ... +
sum(conj(S).*X_n))
= sum(sum(|S|*exp(-i*q).*X_1) + sum(|S|*exp(-i*q).*X_2) + ... +
sum(|S|*exp(-i*q).*X_n))

where q is the vector with phases to be determined.

After some basic manipulations I believe I end up with the simple result
q_max = phase(X_1 + X_2 + ... + X_n)

Hope that makes sense at least!

Anyway, now I would like to maximize the sum of absolute values of cross
correlations at zero time shift, i.e.
max z = sum(|sum(|S|*exp(-i*q).*X_1)| + |sum(|S|*exp(-i*q).*X_2)| + ... +
|sum(|S|*exp(-i*q).*X_n)|)
with respect to q.

With the first result in mind, would it be "enough" to try all 2^n
combinations of signs for X_i or is there some well known/obvious simpler
approach. I mean, this is in essence almost a fundamental problem. I think
it's equivalent to having a sum of vectors and choosing the sign of each
individual vector as to maximize the absolute value of the resultant.

Any ideas are more than welcome... I'm kind of stuck.
From: Demus on 3 Mar 2010 16:30
The first sentence should obviously read
"Having the amplitude spectrum of a signal..."
From: Rune Allnor on 3 Mar 2010 16:34
On 3 Mar, 21:47, "Demus" <sode...(a)hotmail.com> wrote:

> With the first result in mind, would it be "enough" to try all 2^n
> combinations of signs for X_i or is there some well known/obvious simpler
> approach. I mean, this is in essence almost a fundamental problem. I think
> it's equivalent to having a sum of vectors and choosing the sign of each
> individual vector as to maximize the absolute value of the resultant.
>
> Any ideas are more than welcome... I'm kind of stuck.

I have seen articles where these kinds of things were approached
by a Genetic Algorithm approach. I *think* it was Porsani and
Ursin who rote something about reconstructing mized-phase signals
from autorcorrelation functions, some time in the late '90s.
As I remember it, the article appeared in Geophysics, but I
might be wrong.

Rune
From: Demus on 4 Mar 2010 14:45

>I have seen articles where these kinds of things were approached
>by a Genetic Algorithm approach. I *think* it was Porsani and
>Ursin who rote something about reconstructing mized-phase signals
>from autorcorrelation functions, some time in the late '90s.
>As I remember it, the article appeared in Geophysics, but I
>might be wrong.
>
>Rune
>
Might that have been at your Alma Mater, a friend of yours?
http://cat.inist.fr/?aModele=afficheN&cpsidt=2182345

Anyway, that makes sense.

Thanks for the tip.
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