Tom wrote:
If you don't put in the (1-beta) part then you get an offset ie you have to
re-scale afterwards to get the right
answer.

On the contrary, it is impossible to get the right answer if you DO put
in the 1-beta part.

.... if the question being answered is the one the original poster asked:
how to correlate two sequences using the fft.

Or perhaps I am also guilty of assuming too much. Perhaps the OP
wanted the single-valued correlation : cov(x,y) / sqrt( var(x)*var(y) )
But I don't think so. Covariance and variance operations are already
linear complexity. I can't see how FFTs would make them any faster.
I'm pretty sure the OP wanted cross-correlation when he asked for
correlation.

There is a very specific definition for cross-correlation.
See http://mathworld.wolfram.com/Cross-Correlation.html
or http://cnx.rice.edu/content/m10686/latest/
Notice the pure integration and infinite bounds.

The formula you posted may have uses for continuous (open loop)
processing, but it is NOT cross-correlation.

I can only assume the question for your "right answer" is "how do I get
a time-decaying approximation of cross-correlation". That question was
never asked. You made a good suggestion that was certainly topical,
considering more people read a thread than just the few persons writing
it. But the OP had signals "up to 45 seconds long", far from infinite.

-- Mark