A fast and accurate method for evaluating joint second-order PMD ...

FORESTIERI: A FAST AND ACCURATE METHOD FOR EVALUATING JOINT SECOND-ORDER PMD STATISTICS 2949
Evaluation of the inner integral in (10) is the most
critical, as it is to be evaluated with very high accuracy for the
tails of the joint pdf, and we had to use a value for accuracy
as small as in the numerical integration method described
at the end of Section II to obtain reliable values down to
. We also noticed that the higher the accuracy in evaluating
, the lesser the number of times that step size
in (33) is to be halved to achieve a given number of reliable
digits for the joint pdf. The very high accuracy needed for
is easily and efficiently achieved through the expansions
reported in Sections II–IV. Until a closed-form expression
is not available, a fast and accurate method for evaluating the
joint second-order PMD statistics is a prerequisite for a full analytical
evaluation of the PMD impact on the performance of
compensated systems and may also be a useful tool for validating
IS techniques (see Appendix C).
APPENDIX A
In this appendix, we show the derivation of (9), (11), and (22)
mentioned in the paper. The exact expression for
can be obtained by direct antitransformation of (7) as
(A1)
where is as in (8). Changing to spherical polar coordinates
yields
(A2)
and the integral with respect to is easily performed, yielding
(A3)
From (A3), we obtain the joint pdf of and
as
(A4)
from which, by the change of variable and after some
algebra, we obtain (9).
By using the expansion , the integral
in (10) can be evaluated as
Letting and as in (12), we start from [20, eq. 6.677.6]
and, by repeated differentiation with respect to , obtain
(A5)
(A6)
(A7)
which, substituted in (A5), gives (11).
Thus, we simply need an explicit expression for the
right-hand-side term in (A7). To obtain it, we use the rule
for the th derivative of a composite function. Letting
(such that ), if
, then [20]
where
(A8)
(A9)
Given , where
, , we apply the previous rule
to obtain the th derivative of as
where
By using the formula [20]
(A10)
(A11)
(A12)
where denotes the Pochhammer’s symbol, defined as [21]
(A13)