Effect of record length on the correlation of complex exponentials

The correlation coefficient is one measure of how well two signals can be resolved. The effect of record length on the correlation of complex exponentials is examined. For two decaying exponentials of complex frequencies
s_ {1} = \sigma_{1} + j\omega_{1}
and
s_ {2} = \sigma_{2} + j\omega_{2}
with
\sigma_{2} > \sigma_{1}
, it is shown that a finite time record length
\Delta
may be considered as though it were infinite, provided
\Delta > 2/ |\sigma_{1}|
. This is also the condition for near-orthogonalization of a set of complex exponentials, with small error.