1、c TINEIA TELECOMMUNICATIONS SYSTEMS BULLETIN Guideline for the Statistical Specification of Polarization Mode Dispersion on Optical Fiber Cables TSBl07 NOVEMBER 1999 TELECOMMUNICATIONS INDUSTRY ASSOCIATION Rcprtsenhg the tcleammuniutions industry ia assouation with the Elechonic hdustries Alliance E
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5、ch action, TIA/EIA does not assume any liability to any patent owner, nor does it assume any obligation whatever to parties adopting the Standard, Publication, or Bulld = 1) To find the maximum DGD at a given probability level from a given PMD value, d, compute the value of S that satisfies equation
6、 6 for the desired probability and multiply this value of S times d to obtain the maximum DGD value. The following table has some S values along with the associated probabilities. 7 TINEIA-TSB-107 - 3.0 3.1 Table I Probability based on 4.2E-05 2.OE-05 I Wavelength Average I 3.2 3.3 I S I Probabilitv
7、 I 9.2E-06 4.1 E-06 3.775 6.5E-08 3.9 2.OE-08 4.0 7.4E-09 4.1 2.7E-09 4.3 3.3E-10 4.4 I 1.1E-IO I 4.5 I 3.7E-1 I I I Note: If the PMD value is defined as the root mean square (rms) of the DGD values, the constant 4h should be replaced with 312 in equations 4 and 5. 4. Definition and calculations for
8、 Method I The maximum link PMD coefficient for a given process distribution, PMDQ, is defined in terms of a small probability value, Q, and an assumed number of cable sections in the link, M, such that if dc-link is a possible link PMD coefficient: Pr(dc-link PMD,) e Q for M or more cable sections (
9、7) Note: This reads as: The probability that a link PMD coefficient is greater than PMDQ is less than Q. The requirement is expressed by stipulating M and Q and requiring that PMDQ be less than some value, PMDmm. The default values are: Q 100ppm M 20 PMDQ I PMD, = 0.5 psldkrn 8 E b057743 0000455 709
10、 W TINEIA-TSB-I 07 Note: For Q=lOO ppm, PMDQ is the 99.99 percentile of the link PMD coefficients. There are three techniques that may be used to calculate PMDQ: Monte Carlo, Gamma model, and Generalized Central Limit Theorem 7,8,9,1 O. The Monte Carlo technique makes no assumptions about the distri
11、bution. It cannot, however, be used to extrapolate beyond the empirically sampled data. The gamma model allows extrapolation by using a model that has two parameters that can be related to average and standard deviation. The gamma model is more suited to PMD distributions than the gaussian distribut
12、ion because the gamma model is skewed right and produces only non-negative outcomes. The Generalized Central Limit Theorem model has three parameters that allow the skew to be set independently from average and standard deviation. At the I O0 ppm probability level one may question whether any modeli
13、ng can be effective with the finite sample sizes that are practical. A more appropriate way of viewing the requirement is that while the form of the criterion is in terms of probability, the net effect is a boundary on the combination of process distribution parameters. To illustrate, consider the f
14、ollowing example based on gaussian assumptions: The requirement is that the probability be less than 1 O0 ppm that a sample mean of M values, x, , exceeds a given maximum, X,: Pr(X, A-,) 5. If the inequality is not verified, repeat the Monte Carlo using a larger value of M. 4.2.2 Maximum likelihood
15、estimate This technique does not require the use of Monte Carlo, so the M parameter is set to one. It does require an iterative optimization procedure. Let d,-i represent N measurements on individual cable sections. Calculate the quadrature average, v, as: Choose the value of a that maximizes the fo
16、llowing expression by iterative evaluation of the function or by iterative reduction of its derivative with respect to a to zero : Using the computed value of a, obtained from maximizing 14, calculate as: a =- V2 4.3 Generalized central limit theorem This technique might be considered as “model inde
17、pendent” because it is derived from the Central Limit Theorem 9,10, adjusted to include a skew term. Let d,-i be N measured values on individual cable sections. Calculate the following moments 12 lN P3 = - c (4-i - PI l3 N - 1 The cumulative probability density function for a link of M concatenated
18、cable sections is given by: Where Note: Equations 20a and 20b are just the expressions for a gaussian probability density function and its integral, or cumulative probability density function. Note: The probability density function (histogram representation) may be obtained by taking the derivative
19、of equation 19 with respect to u. The value for PMDQ is determined by setting it equal to the value of u that satisf es: The value of PMDQ can be approximated by: Where ZQ satisfies: (zQ)= I- Q Note: For Q=100 ppm, ZQ = 3.72. 13 TINEIA-TSB-107 5. Assessment of Method 1 vs. DGD Method 1 results in a
20、reduction in the variability of the link PMD coefficient. Compared to worst case approach it should provide either some reduction in the estimated maximum DGD or some decrease in the probability of exceeding a fixed maximum DGD value. Figure 3 illustrates three cases: - The worst case assumption, th
21、at the distribution of link PMD coefficients is a “spike” or dirac function; - A distribution of the PMD coefficients of individual cable sections, with moderate probability of exceeding the worst case; - The distribution of link PMD coefficients, with very low probability of exceeding the worst cas
22、e. PMD Coefficient Distribution O 25 0.2 5r a, 3 U u. 0.15 e! 0.1 0.05 O O 0.1 0.2 0.3 0.4 0.5 0.6 ps/sqrt( kirn) Figure 3 The traditional means of determining the maximum DGD, DGD probability level, PDGD, and a worst case PMD coefficient is to: defined by a - - - Select the desired probability leve
23、l from Table 1. Multiply the associated value of S with the maximum PMD coefficient Multiply the result with the square root of the link length Taking the substantially decreased probability that a link PMD coefficient exceeds PMD max into account, as illustrated in Figure 3, would surely either dec
24、rease DGD, or the associated probability of exceeding a pre-fixed value of DGD,. 14 TINEIA-TSB-107 The problem is illustrated in Figure 4, which shows a series of Gamma distributions of link PMD coefficient. All these distributions meet the default criterion of Method I. The far right distribution i
25、s approaching the worst case dirac function, which would, in fact pass the Method 1 criterion. Clearly the far left distribution would yield better DGD performance than the far right distribution. The Method 1 criterion, while suitable as being a measurable attribute, does not provide accurate infor
26、mation with regard to a key system design attribute: DGD. Various passing distributions 0.06 a, 2 0.05 0.04 t $! 0.03 m iu 0.02 PL 0.01 O 3 .- U O 0.1 0.2 0.3 0.4 0.5 0.6 Concatenated link PMD coefficient Figure 4 If one were to take an extreme view, the probability associated with multiplying the P
27、MDm, value by an S multiplier should be somewhat larger than the probability found in Table I, which represents a “pure” worst case. This is due to the fact that there is theoretically some “tail” of the statistical representation that extends beyond PMD, while the pure worst case has no such tail.
28、Using a Maxwell adjustment factor S = 3, for example, in the pure worst case yields probability: PDGD = 4.2. IO- . The statistical worst case has been estimated as PED = I .4*1 O-4. This same S multiplier applied to a maximum cable PMD link coefficient of 0.5 psldkm yields a maximum DGD value of 30
29、ps over 400 km. In effect, the application of the S multiplier resumes the assumption that the link PMD distribution is a spike. 5 Using the S multiplier approach with PMDm, does not reflect the realistic improvements that could be garnered from a statistical design approach. This is especially true
30、 when one considers the realistic needs of system design. According to the ITU recommendation G.691 IO, an end-to-end DGD value of 30 ps will induce a maximum of 1 dB receive sensitivity penalty for an NRZ 15 TINEIA-TSB-107 system operating at 10 Gbis. If the actual DGD exceeds this value the system
31、 could become unavailable. Various means of converting the DGD probability (PDGD) to system unavailability have been discussed in standards development groups. While none have been agreed, it has become clear that a PDGD value on the order of less than is desirable. Since the link DGD and probabilit
32、y include both optical fiber cable and components, the maximum DGD and probability for optical fiber cable must be reduced from the levels discussed above. IEC 61282-3 provides some guidance on these issues. Inclusion of components with PMD into the system will induce a need to: - Reduce the optical
33、 fiber cable induced portion of DGDm, to a value less than 30 ps to allow components with PMD. - Reduce the optical fiber cable DGD probability, PDGD, to a value that is half of what is desired for the combined link. 6. Definition and calculations for Method 2 Method 2 is introduced to solve the pro
34、blems mentioned in clause 5. The specific values that are introduced were chosen to provide: - A statistical requirement that is nearly the same as the Method 1 specification - An indication of operability of 10 Gbis systems over 400 km. 6.1 Definition Method 2 is expressed in terms of a maximum DGD
35、 value, DGDm, and the probability that a DGD on a given link and wavelength exceeds this maximum. This probability is specified with a maximum, PDGDmax. A reference link length, Ld, and assumed cable length, Lcab, are also stipulated. The combination of these lengths implies a value of M, the number
36、 of concatenated cable sections, following 3.1. The Method 2 probability criterion is stated as: Provide a distribution of PMD coefficient values so a concatenated link of length, Ld, composed of individual cable sections of length, Lcab in length, yields a probability, PDGD, less than PDGDmax, that
37、 a given IinWwavelength DGD value exceeds DGD m. Default values for the four defining variables are: TINEIA-TSB-107 Note that for Method 1, the probability level is set and PMDQ is calculated - and required to be less than PMDm,. For Method 2, DGD, is set and PDGD is calculated - and required to be
38、less than PDGDma. The reference link length is taken as 400 km to match certain ITU system assumptions I I. The I O km cable section length is taken as a conservative estimation because most installed lengths are from 2 km to 4 km. The values for DGDm, and PDGDma are derived from the discussion in c
39、lause 5. The value of DGD, can safely be used for links of any length that is less than the reference length. That is, the probability that DGD, is exceeded reduces with decreasing link length if the cable length assumption is maintained. For links with longer lengths, the DGD limit should be increa
40、sed in proportion to the square root of the ratio of link length to reference length. 6.2 Calculation principle - convolution The calculation principle is derived from extending the worst case approach discussed in clause 5. With this approach, the link PMD value is assumed to be a dirac function an
41、d the DGD distribution is represented as a Maxwell distribution. The probability that the Maxwell distribution exceeds DGD yields PWD. These distributions are represented in Figure 5. Worst case approach assumption 0.07 0.06 005 0.04 al J g! IL 9 003 f 002 .- c E 0.01 O: O 10 20 30 40 DGDIPMD link v
42、alue Figure 5 Note: Though not shown, the dirac function illustrated in Figure 5 extends to a relative frequency value of 1 .O. Suppose the link PMD value could be represented by two dirac functions, each with a magnitude of 0.5. This would represent a situation where half the links 17 TINEIA-TSB-10
43、7 were at one value and the other half at another value. The DGD probability density function of the combined distribution would be the weighted total of the two individual Maxwell distributions. Figure 6 illustrates this case. Convolution of two diracs 0.07 0.06 1 010: 0.03 i 0.02 0.01 O O 10 20 30
44、 40 DGDIPMD value (ps) Figure 6 Note: Though not shown, the twodirac functions representing PMD value distributions extend to relative frequency values of 0.5. In this example, the probability of DGD exceeding 30 ps is reduced by just a little less than a factor of two. Full convolution extends this
45、 notion to a complete distribution of link PMD coefficients. For the Monte Carlo technique, the histogram of link PMD coefficients may be thought of as a collection of dirac functions. For the continuous models, the probability density function is reduced to a histogram by integrating the curve over
46、 the region that is represented as a single histogram bin. The probability that DGD, is exceeded is calculated for each of the histogram bins (using the bin maximum). The weighted total yields PDGD. 6.3 Convolution calculation Define the jth histogram bin for link PMD coefficients between dj-1 and d
47、j and the relative frequency of coefficient values falling in that range as pj. This histogram should be of a resolution that is compatible with the measurement resolution, typically 0.001 psldkm. Calculate M=L,f/L,b. Compute the relative frequency values of the histogram according to the instructio
48、ns following, for one of the three characterization techniques. Compute P DGD as: 18 W b057Li3 00004b5 b58 TINEIA-TSB-I 07 Note: Pmax(S) is defined in equation 6. do = O. 6.3.1 Monte Carlo technique Using the computed value of M from 6.3, complete the Monte Carlo calculation described in 4.1. Typica
49、lly far fewer iterations are required for this calculation than are required for determination of PMDQ. Using a value, K = 32000, will generally be sufficient for stable estimates. The reason for this is that the Method 2 calculation is a weighted average. 6.3.2 Gamma model technique Using the parameter values determined with 4.2, compute the relative frequency of the jth bin as: 6.3.3 Generalized limit theorem technique Using the parameter values determined with 4.3, compute the relative frequency of the jth bin as: 6.4 Statistical equivalence of Method 2 to Method 1
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