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

FORESTIERI: A FAST AND ACCURATE METHOD FOR EVALUATING JOINT SECOND-ORDER PMD STATISTICS 2951
letting , , and taking into account (9), it can
be shown that
where
(C1)
(C2)
(C3)
By the change of variable and using [20, eq. 6.677.6],
in (C3) can be integrated in closed form, and we see that it is
independent from
Substituting (C4) in (C2) and, in turn, this in (C1), we obtain
where
(C4)
(C5)
(C6)
is as in (45) for , i.e., . Multiplying
(C5) by the Maxwellian pdf finally gives
(C7)
where the integral can be evaluated as done in (33) with an initial
step size of .
Fig. 6 shows contour plots of the denormalized joint pdf
(C8)
for 3.257 ps, evaluated through (C7), and should be
compared with Fig. 4 in [23], reporting the same quantity at the
Fig. 6. Contour plots of the joint pdf of and for a mean DGD of
3.257 ps. Starting from the inner, the contours are at IH with a 1.5, 1.75,
2, 2.25, 2.5, 3, 4, 5, 6, 8, 10, 15, 20, 25, and 30.
same contour levels. A close examination reveals that the accordance
is excellent until , but afterwards, the joint pdf
obtained through IS starts underestimating the true pdf progressively
more as the contour level lowers. The contours below
in Fig. 6 required quadruple precision to be computed
(notice that (C6) can be used for values smaller than when
using a higher precision arithmetic, but only about the first 16
significant digits are correct).
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