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

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FORESTIERI: A FAST AND ACCURATE METHOD FOR EVALUATING JOINT SECOND-ORDER PMD STATISTICS 2947 Fig. 3. Joint pdf @Y A for (a) 1( a " and (b) 1( a 2". Contours are at IH , aIY PY FFFY IS. By using [21, eq. 10.1.48], we also have that (38) where are the spherical Bessel functions of the first kind. Substituting (37) and (38) in (10), we obtain where (39) (40) may be evaluated with the aid of [20, eq. 7.322]. 4 By using [21, eq. 10.1.45], (39) can be written as 4 Notice that a factor 2 is missing in the result in [20, eq. 7.322]. (41) and, taking the principal value, the square root appearing in (41) is equal to where so that (41) becomes (42) (43) (44) (45) . This is the expression to be used for evaluating the joint pdf when 2 . Notice that (45) could also be used for 2 , but, in this case, (11) turns out to be faster. This is because in (45) the various elementary functions must be reevaluated for each term in the series, whereas (22) only requires the evaluation of a few terms through simple recurrence relations, as already discussed. For larger values of , however, (11) would require
2948 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 11, NOVEMBER 2003 Fig. 4. Joint pdf @Y A (a) as a function of for a RY TY FFFY IT and (b) as a function of for a HY PY FFFY IH and 1( a 2", evaluated through (33). Circles are obtained through numerical integration of (10), solid lines by using (11). more terms than those needed in (45), which in most cases are in the order of to achieve the necessary precision. Also notice that the major contribution does not necessarily come from the first terms. Indeed, rewriting (45) as (46) where , and taking into account that, for increasing values of , in (44) asymptotically tends to , i.e., , we also have that , so that, in many cases, the most important terms are those for in the order of , or slightly less. Using (45), evaluation of (33) for the larger values of turns out to be about an order of magnitude slower than using (11) for the smaller values. For the larger values, evaluation of (33) through numerical integration of (10) turns out to be about an order of magnitude faster than for the smaller values. Therefore, using (45) for evaluating (33) is about orders of magnitude faster than using the integration method described at the end of Section II and even much faster with respect to other standard numerical integration methods. However, we point out that this discussion applies only when considering values for the joint pdf not smaller than about ; otherwise standard numerical integration methods slow down Fig. 5. Joint pdf @Y A (a) as a function of for a RY VY FFFY PV and (b) as a function of for a HY PY RY FFFY IH and 1( a 5", evaluated through (33). Circles are obtained through numerical integration of (10), and solid lines by using (45). considerably, and the values they provide are not reliable (as the curves become jagged and sometimes negative values are returned), whereas the methods presented in this and Section III do not suffer any slow down and can give reliable values down to about , i.e., the machine precision in IEEE Standard 754 double-precision arithmetic. In Fig. 5, , evaluated using (45) in (33), is plotted for 5 as a function of for several values of in (a), and as a function of for several values of in (b). Again, circles represents values obtained through numerical integration of (10). V. SUMMARY AND CONCLUSION The exact evaluation of the joint pdf of and , conditional upon , can only be performed by numerical methods, and, to this end, the double-integral formula (9) has been devised. It has been shown that the inner integral (10) has an analytic expression that can be exploited for a really efficient evaluation of the outer integral in (9) through (33), practically equivalent to a series expansion, as the step size (34) is to be halved only once in about 90% of cases, on average, to achieve four-digits accuracy when evaluating in (10) with sufficient accuracy; indeed, the fact that the step size is to be halved more than once may be taken as an indication of poor accuracy on .
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