Effect of record length on the correlation of complex exponentials

Resolution of two sinusoidal signals of frequencies f
1
and f
2
requires a time record of length at least
1/|f_{1}-f_{2}|
. Resolution of two decaying exponentials of complex frequencies s
1
= σ
1
+jω
1
and s
2
= σ
2
+jω
2
, with
\sigma_{2} \leq \sigma_{1}
, by correlation coefficient is independent of the ratios σ
2
/σ
1
, and (ω
1
- ω
2
)/σ
1
for a time record of length greater than
3/|\sigma_{1}|
. This is also the condition for near-orthogonalization of a set of complex exponentials, with small error.