If you can't find a module that does cross-correlation, you can always use one of the many FFT implementations - in pseudo-code:

# input arrays are @f and @g, this calculates @cc which is the cross-c
+orrelation of these.
@Ff = fft(@f);
@Fg = fft(@g);
@Fcc = complex_conjugate(@Ff) * @Fg;
@cc = inverse_fft(@Fcc);
[download]

This makes use of the fourth of the properties listed here. If your input arrays are purely real, I think it may be possible to optimise this further by performing their FFTs in a single FFT calculation, with a bit of clever maths.

I did calculation of a cross-correlation using this method some years ago, but it wasn't in Perl, unfortunately.

Found one!

In Math::FFT the author has provided an easy way to use FFTs to calculate cross-correlation:

   $fft1 = new Math::FFT($data1);
   $fft2 = new Math::FFT($data2);
   $corr = $fft1->correl($fft2);
[download]

Look under Applications - Correlation in the documention of Math::FFT.

--

"Any sufficiently analyzed magic is indistinguishable from science" - Agatha Heterodyne

I disagree, and so do Wikipedia and Wolfram...

--

"Any sufficiently analyzed magic is indistinguishable from science" - Agatha Heterodyne

I checked your references and they agree with me: Both clearly state that the second series (and third for wolfram) is derived from the first; the wolfram version has a total of three series but in which the two complex functions f and g are both derived from the same starting series. In both cases there is only one driving variable, everything else is derived. If you start with two series, you want ordinary correlation, not cross-correlation.

One world, one people

use Statistics::Basic::Correlation;

sub xcorr {
    # given two sets and a phase increment, return
    # at which phase the best correlation is found
    # and what that value is
    my ($xref, $yref, $incdeg ) = @_;
    my $maxcorr = -1;

    $incdeg ||= 1; default 1 degree of phase.
    die "sets of unequal size" if ( $#$xref != $#$yref );

    my $indexinc = $incdeg * ( 1 + $#$xref );
    my $iters = 360 / $incdeg;  
    { use integer;
      $indexinc /= 360;
      $iters++;
    }
    $indexinc ||=1;

    my @copy = @$yref;
    my $where = 0;
    my $match = 0;
    for (my $i = 0; $i < $iters; $i++) {
        unshift @copy, ( pop @copy ) for ( 1 .. $indexinc );
        my $res = correlate( $xref, \@copy );
        if ( $res > $maxcorr ) {
            $maxcorr = $res;
            $match = $where;
        }
        $where += $incdeg;            
    }
    return ( $match, $maxcorr );
}
[download]

More update:

- the arrays are passed by reference i.e. \@x and \@y

- the approach does not take into account potential change of amplitude of one of the signals relative to the other, which is not taken into account by correlation alone. To compensate if necessary, find s(x)/s(y) where s() is standard deviation found using Statistics::Basic::StdDev and multiply all @copy by that before iterating through the phase shifts.