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

FORESTIERI: A FAST AND ACCURATE METHOD FOR EVALUATING JOINT SECOND-ORDER PMD STATISTICS 2943
Fig. 1. Pdf of for a 15-, 20-, and 30-ps mean DGD. Circles are obtained
from (2), and solid lines from (3).
where means expectation, is the mean DGD. A comparison
of (2) and (3) is shown in Fig. 1.
Even if the pdf of all the quantities appearing in (1) is
known, this is not enough as, for averaging, the joint pdf
is needed. 1 As and is known [14],
we focus on , which is the unknown quantity we need.
Following [15], we let
and denote by
the normalized PMD vector and its derivative, respectively.
Parameter is the standard deviation of each one of the
components of the dispersion vector . In the reference frame
whose first axis is coincident with , the characteristic function
of the conditional joint pdf
of the components 2 is [15]
where
is the three-dimensional (3-D) transform
variable and .
Obtaining a closed-form expression for by analytical
inversion of (7) proved to be a formidable task, and on the other
side, a numerical inversion through a 3-D fast Fourier transform
1For brevity, the pdf arguments are sometimes omitted.
2In the following ‘1“ denotes transpose.
(5)
(6)
(7)
(8)
(FFT) is not feasible as, by simply and optimistically using 64
points one-dimensional (1-D) FFTs, a total of about 7 000 000
multiplications should be carried out, which does not exactly
account for a fast and efficient evaluation.
Analytical approximations for the joint pdf of and are
reported in [14] and [15], but, even if second-order moments are
correctly reproduced, they greatly underestimate the tails
and therefore are not suitable for PMD-induced penalties evaluation
in compensated systems.
Regarding the inversion of (7), as shown in Appendix A,
after changing to spherical polar coordinates, the
circular symmetry of one of the angle coordinates allows that
angle dimension to be easily integrated in closed form, and,
after a suitable transformation, the joint pdf of and
, conditional upon , can be written as 3
where
(10)
Hence, to evaluate (9), we have to numerically perform a twodimensional
(2-D) integral, but a very difficult one, as both
and the integrand in (10) have an oscillatory nature.
Indeed, evaluation of through (9) over a grid
of 100 100 points, covering the tails down to about ,
requires from a few hours to several days of computing time
on a fast processor, depending on . The higher the value of
with respect to , the lesser the required time to obtain a
prescribed accuracy. The inner integral (10) is the most critical
one, and we tried several numerical integration methods to
evaluate it, including evaluation through FFT, but the most efficient
turned out to be a recursive one based on the determination
of a set of subintervals , , of (0,1)
such that , , for which 8- and 16-points Gauss
quadrature formulas give results differing by less than a prescribed
accuracy . Denoting by , the approximate
value of the integral on interval given by the -point
Gauss formula, (10) is approximated as ,
where , for ,
and are found by the following recursive procedure: i) initially
; ii) for , is successively set to
, , until the prescribed
accuracy is attained in ; and iii) the procedure completes
when accuracy is obtained for .
III. EVALUATING THE JOINT PDF FOR
As already pointed out in Section II, the numerical evaluation
of (9) is not a trivial task. In this section, we will develop a
method that turns out to be very efficient in the case
, other than evidencing a key feature of the inner integral
in (10).
3 We moved the conditioning on from the subscript to avoid clutter.
(9)