IEC TR 61282-3:2006

Fibre optic communication system design guides - Part 3: Calculation of link polarization mode dispersion

This technical report provides guidelines for the calculation of polarisation mode dispersion (PMD) in fibre optic systems. It accomodates the statistical variation of PMD and differential group delay (DGD) of optical fibre cables and optical components in an optical link. Example calculations are given, including concatenating individual optical fibre cables. The calculations describe first order PMD only.

General Information

• Status: Published
• Publication Date: 24-Oct-2006
• ICS: 33.180.01 - Fibre optic systems in general
• Technical Committee: SC 86C - Fibre optic systems, sensing and active devices
• Drafting Committee: WG 1 - TC 86/SC 86C/WG 1
• Current Stage: PPUB - Publication issued
• Start Date: 25-Oct-2006
• Completion Date: 31-Oct-2006

Relations

• Revises: IEC TR 61282-3:2002 - Fibre optic communication system design guides - Part 3: Calculation of polarization mode dispersion
• Effective Date: 05-Sep-2023

Standards Content (Sample)

TECHNICAL IEC
REPORT TR 61282-3
Second edition
2006-10
Fibre optic communication system design guides –
Part 3:
Calculation of link polarization mode dispersion
Reference number
IEC/TR 61282-3:2006(E)
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TECHNICAL IEC
REPORT TR 61282-3
Second edition
2006-10
Fibre optic communication system design guides –
Part 3:
Calculation of link polarization mode dispersion

© IEC 2006 ⎯ Copyright - all rights reserved
No part of this publication may be reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying and microfilm, without permission in writing from the publisher.
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For price, see current catalogue

– 2 – TR 61282-3 © IEC:2006(E)
CONTENTS
FOREWORD.3
INTRODUCTION.5

1 Scope.6
2 Basic design models for total system PMD performance.6
2.1 Notation.6
2.2 Calculation of system PMD .7
3 Calculation of cabled fibre PMD .9
3.1 General .9
3.2 Method 1: Calculating PMD , the PMD link design value.11
Q
3.3 Method 2: Calculating the probability of exceeding DGD .14
max
3.4 Equivalence of methods.18
4 Calculation of optical component PMD .20
5 Summary of acronyms and symbols .20

Annex A (informative) PMD concatenation fundamentals .22
Annex B (informative) Combining Maxwell extrema from two populations.26
Annex C (informative) Worked example.30
Annex D (informative) Relationship of probability to system performance .31
Annex E (informative) Concatenation experiment.32

Bibliography .34

Figure 1 – Various passing distributions .15
Figure 2 – Worst case approach assumption .17
Figure 3 – Convolution of two Diracs .17
Figure 4 – Equivalence envelopes for method 1/2 defaults.19
Figure A.1 – Sum of randomly rotated elements.25
Figure A.2 – Sum of randomly rotated elements.25

Table 1 – Probability based on wavelength average.9
Table 2 – Acronyms and definitions .21
Table 3 – Symbols and clause of definition .21
Table E.1 – Concatenation math verification .33

TR 61282-3 © IEC:2006(E) – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 3: Calculation of link polarization mode dispersion

FOREWORD
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The main task of IEC technical committees is to prepare International Standards. However, a
technical committee may propose the publication of a technical report when it has collected
data of a different kind from that which is normally published as an International Standard, for
example "state of the art".
IEC 61282-3, which is a technical report, has been prepared by subcommittee 86C: Fibre optic
systems and active devices, of IEC technical committee 86: Fibre optics.
This second edition cancels and replaces the first edition published in 2002. It is a technical
revision that includes the following significant changes:
a) the title has been changed to better reflect its applicability to links;
b) Equations (1) and (2) were simplified in order to align with agreements in the ITU-T.

– 4 – TR 61282-3 © IEC:2006(E)
The text of this technical report is based on the following documents:
Enquiry draft Report on voting
86C/701/DTR 86C/720/RVC
Full information on the voting for the approval of this technical report can be found in the report
on voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts of the IEC 61282 series, published under the general title Fibre optic communication
system design guides, can be found on the IEC website.
The committee has decided that the contents of this publication will remain unchanged until the
maintenance result date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
A bilingual version of this publication may be issued at a later date.

TR 61282-3 © IEC:2006(E) – 5 –
INTRODUCTION
Polarization mode dispersion (PMD) is usually described in terms of a differential group delay
(DGD), which is the time difference between the principal states of polarization of an optical
signal at a particular wavelength and time. PMD in cabled fibres and optical components
causes an optical pulse to spread in the time domain, which may impair the performance of
a fibre optic telecommunication system, as defined in IEC 61281-1.

– 6 – TR 61282-3 © IEC:2006(E)
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 3: Calculation of link polarization mode dispersion

1 Scope
This part of IEC 61282 provides guidelines for the calculation of polarization mode dispersion
(PMD) in fibre optic systems to accommodate the statistical variation of PMD and differential
group delay (DGD) in optical fibre cables and components.
This technical report describes methods for calculating PMD due to optical fibre cables and
optical components in an optical link. The calculations are compatible with those documented
in the outdoor optical fibre cable specification IEC 60794-3. Example calculations are given to
illustrate the methods for calculating total optical link PMD from typical cable and optical
component data. The calculations include the statistics of concatenating individual optical fibre
cables drawn from a specified distribution. The calculations assume that all components have
PMD equal to the maximum specified value.
The calculations described cover first order PMD only. The following subject areas are
currently beyond the scope of this technical report, but remain under study:
– calculation of second and higher order PMD;
– accommodation of components with polarization dependent loss (PDL) – if it is assumed
that PDL is negligible in optical fibre cables;
– system impairments (power penalty) due to PMD;
– interaction with chromatic dispersion and other nonlinear effects.
Measurement of PMD is beyond the scope of this technical report. Methods of measurement of
PMD of optical fibre and cable are given in IEC 60793-1-48. The measurement of optical
amplifier PMD is in IEC 61290-11-1. The measurement of component PMD is in
IEC 61300-3-32. Measurement of link PMD is given in 61280-4-4. A general theory and
guidance on measurements is given in 61282-9.
2 Basic design models for total system PMD performance
2.1 Notation
For cabled fibre and components with randomly varying DGD, the PMD frequency domain
measurement is based on averaging the individual DGD values for a range of wavelengths. The
probability density function of DGD values is known to be Maxwell for fibre, and is assumed to
be Maxwell, in effect, for components. The single parameter for the Maxwell distribution scales
with the PMD value.
For long fibre and cable (typically longer than 500 m to 1 000 m), the PMD value is divided by
the square root of the length to obtain the PMD coefficient. For components, the PMD value is
reported without normalization. The following terms and meanings will be used to distinguish
the various expressions:
– DGD value The differential group delay at a time and wavelength (ps)
– PMD value The wavelength average of DGD values (ps)
– PMD coefficient The length normalised PMD (ps/sqrt(km))
– DGD coefficient The length normalised DGD (ps/sqrt(km))

TR 61282-3 © IEC:2006(E) – 7 –
NOTE The term “DGD coefficient” is used only in some of the calculations. The physical square root length
dependence of the PMD value does not apply to DGD.
Deterministic components are those for which the DGD may vary with wavelength, but not
appreciably with time. The variation in wavelength may be complex, depending on the number
and characteristics of the sub-components within. For these types of components, either the
maximum DGD is reported or the wavelength average is reported as the PMD value. For
components with multiple paths, such as an optical demultiplexer, the maximum DGD of the
different paths should be reported as the PMD value.
2.2 Calculation of system PMD
PMD values of randomly varying elements can be added in quadrature. Annex A shows the
basis of this, as well as one basis for concluding that the Maxwell distribution is appropriate to
describe the distribution of DGD values. Annex A describes the concatenation in terms of the
addition of rotated polarization dispersion vectors (pdv) which are, for randomly varying
components, assumed to be random in magnitude and direction over both time and
wavelength.
For deterministic components, the evolution of the pdv with wavelength may be quite complex,
but for each wavelength, there is a value that does not vary appreciably with time. Analysis of
the relationships in Annex A shows that deterministic components that are randomly aligned in
combination with random elements behave like random components.
For randomly varying components such as fibre, the statistics of DGD variation imply that there
is little wavelength dependence of the PMD value. This leads to an equivalence between PMD
measurement methods such as Jones Matrix Eigenanalysis (JME) and interferometric methods
(IT) where the wavelength ranges of the two are different. For deterministic elements, there
can be distinct dependence of both the DGD and PMD on the wavelength range. Therefore for
these elements, when doing calculations which combine both randomly varying and
deterministic elements, the combined values are only representative of the wavelength overlap.
The relationships of Annex A also show an analysis for an assumption that the deterministic
components are randomly aligned. For this assumption, the DGD values are time randomised
across the wavelengths by the fibre. The random alignment of these components with respect
to the other elements leads to the following conclusions for deterministic components.
– The quadrature addition of PMD values can be used to calculate the contribution to system
PMD.
– The Maxwell distribution can conservatively be used to describe the variation in DGD
across time and wavelength.
The following two subclauses provide equations to calculate: a) the maximum PMD value for
the link, b) the maximum DGD value for the link. In both cases, the maximum is defined in
terms of a probability level that takes into account the statistics of the concatenation of
individual cables drawn from a specified distribution of optical fibre cable. For maximum DGD,
these statistics are combined with the Maxwell statistics of DGD variation. Clause 3 provides
methods of calculating the relevant statistics for the contribution of optical fibre cable, which
are used in combination with the component values below.
2.2.1 Link maximum PMD
The total maximum PMD value of a fibre optic system including optical fibre cables and other
optical components is given by the following:
1/ 2
⎡ ⎤
2 2
PMD = L PMD + PMD (1)
⎢ ⎥
tot link Q ∑ Ci
⎢ ⎥
⎣ ⎦
i
– 8 – TR 61282-3 © IEC:2006(E)
where
PMD is the total link PMD value (ps);
tot
PMD is the link design value of the concatenated optical fibre cable (ps/√km);
Q
L is the link length (km);
link
th
PMD is the PMD value of the i optical component (ps);
Ci
The link design value, PMD , (see 3.1) defines a maximum in terms of the probability, Q, for
Q
links with at least M individual cable sections.
NOTE The PMD parameter is not related to the Q factor used in bit error ratio calculations.
Q
For a link for which the individual cabling sections have been measured, the term L PMD
link Q
th
can be replaced by PMD , where PMD is the PMD value (ps) of the i section.
i
∑ i
i
The validity of these equations has been demonstrated empirically for systems composed of
concatenated optical fibre cables [2].
2.2.2 Calculation of system maximum DGD
The total maximum DGD value of a fibre optic system including optical fibre cable and other
optical components is given by one of the following:
1/ 2
⎡ ⎤
2 2 2
DGD =⎢DGD + S PMD ⎥ (2)
max tot maxF Ci
∑
⎢ ⎥
⎣ i ⎦
where
DGD is the maximum link DGD (ps);
max tot
DGD is the maximum concatenated optical fibre cable DGD (ps) (see below);
maxF
S is the Maxwell adjustment factor (see below);
th
PMD is the PMD value of the i component (ps);
Ci
For a statistical specification of optical fibre cable, the maximum DGD is defined by a
probability, P and reference length (see 3.2). It is computed from the convolution of the
F,
distribution of the concatenated link PMD distribution and the Maxwell distribution of DGD
values. For a link that has been measured, such as described below Equation (1), the term
1/ 2
⎡ ⎤
2 2
DGD is replaced by S PMD .
⎢ ⎥
maxF ∑ i
⎢ ⎥
⎣ ⎦
i
The S parameter relates to the probability, P , that a random component DGD value exceeds
C
S⋅PMD , assuming the Maxwell distribution. Table 1 shows the relationship of S to probability
C
when the PMD value is defined as the wavelength average.

TR 61282-3 © IEC:2006(E) – 9 –
Table 1 – Probability based on wavelength average
S Probability
3,0 4,2E-05
3,1 2,0E-05
3,2 9,2E-06
3,3 4,1E-06
3,4 1,8E-06
3,5 7,7E-07
3,6 3,2E-07
3,7 1,3E-07
3,775 6,5E-08
3,8 5,1E-08
3,9 2,0E-08
4,0 7,4E-09
4,1 2,7E-09
4,2 9,6E-10
4,3 3,3E-10
4,4 1,1E-10
4,5 3,7E-11
Annex B shows that the probability that a system DGD value, DGD , exceeds DGD is
tot maxtot
bounded by the sum of the two probabilities as:
P()DGD > DGD ≤ P + P (3)
tot maxtot F C
NOTE The notation P( ) indicates a probability statement relative to the inequality within the parenthesis.
The above equations are applicable to all links with length less than the reference length. An
adjustment for longer lengths is included in 3.2. Equation (2) is relevant for the assumption that
deterministic components are randomly aligned. The multiplication of the deterministic PMD
values with the S parameter treats these elements as though their DGD values are distributed
as Maxwell – a conservative assumption that allows the quadrature addition.
Equation (3) illustrates that the total probability of exceeding some overall maximum can be
bounded by an addition that does not depend on the relative magnitude of DGDmax and
F
S⋅PMD . Given an overall probability target, one approach is to allocate half the overall allowed
C
probability to fibre and half to components. Annex C provides a worked example.
3 Calculation of cabled fibre PMD
3.1 General
PMD is a stochastic attribute that varies in magnitude randomly over time and wavelength. The
variation in the DGD value is described by a Maxwell probability density function that can be
characterised by a single parameter, the PMD value (see Equation (16) in 3.3.1). This
parameter may be the average of the DGD values measured across a wavelength band, or it
may be the r.m.s. value of these DGD values, depending on the definition chosen. For mode
coupled fibre, the PMD coefficient is the PMD value divided by the square root of length.

– 10 – TR 61282-3 © IEC:2006(E)
In accordance with the outdoor Sectional Specification for outdoor optical fibre cable,
IEC 60794-3, the PMD of cabled fibre should be specified/characterised on a statistical basis,
not on an individual fibre basis. Two methods for this specification are proposed: Method 1 can
be used to obtain PMD , used in 2.2.1, and Method 2 can be used to obtain DGD and P ,
Q maxF F
used in 2.2.2.
In the ITU Recommendation G.652 (and others), Method 1 forms a normative requirement and
Method 2 is used to determine functionality for system performance, which is specified in terms
of DGD in Recommendations such as G.691 or G.959.1.
maxtot
Subclause 3.4 shows how specification values for each method can be selected so the two
methods are nearly equivalent.
Method 1 relies on the fact that the mean PMD coefficient of an optical link is the root mean
square (quadrature average) of the mean PMD coefficients of the cabled fibres comprising the
link. Method 2 assumes the same relationship.
th
Let x and L be the PMD coefficient (ps/√km) and length, respectively, of a fibre in the i cable
i i
in a concatenated link of N cables. The PMD coefficient, x (ps/√km), of this link is:
N
1/ 2
N
⎡ ⎤
L x
⎢ ⎥
1/ 2
∑ i i
N
⎡ ⎤
⎢ ⎥
i=1 2
x = = L x (4)
⎢ ⎥
N ⎢ ⎥ ∑ i i
N
L
⎢ ⎥
Link
⎢ ⎥
⎣ i=1 ⎦
L
∑ i
⎢ ⎥
⎣ i=1 ⎦
If it is assumed that all cable section lengths are less than some common value, L , and
Cab
simultaneously reducing the number of assumed cable sections to M = L /L then, for a
Link Cab,
link comprised of equal-length cables, L = L Equation (4) becomes
i cable,
1/ 2 1/ 2
M M
⎡ ⎤ ⎡ ⎤
L 1
Cab 2 2
x ≤ x =⎢ x ⎥ =⎢ x ⎥ (5)
N M i i
∑ ∑
L M
⎢ Link ⎥ ⎢ ⎥
⎣ i=1 ⎦ ⎣ i=1 ⎦
The variation in the concatenated link PMD coefficient, x , will be less than the variation in the
M
individual cable sections, x , because of the averaging of the concatenated fibres.
i
Method 1 should be used with Equation (1) of 2.2.1. In Method 1, the manufacturer supplies a
maximum PMD link design value, PMD , that serves as a statistical upper bound for the PMD
Q
coefficient of the concatenated fibres comprising an optical cable link. For this case, the upper
bound for the PMD value of the concatenation of optical fibre cables, PMD , in Equation (1)
FTot
becomes:
PMD = PMD L (6)
FTot Q Link
Unless otherwise specified in the cabled fibre detail specification, the PMD link design value
shall be less than 0,5 ps/√km, and the probability that a PMD coefficient of a link comprised of
–4
at least 20 cables will exceed the link design value shall be less than 10 . The link design
value shall be computed using a method agreed upon between the buyer and cable
manufacturer (see 3.2 for examples).

TR 61282-3 © IEC:2006(E) – 11 –
Because Method 1 provides a statistical upper bound on the PMD of concatenated links,
approved PMD measurement methods can be used on installed cable links to determine
whether their PMD complies with the statistical upper bound stated by the manufacturer.
Furthermore, the upper bound can be used to compute the effect of the link PMD on
the performance of any type of transmission system and is a more realistic indication of the
maximum PMD likely to be encountered in a concatenated link than the value that would be
obtained using a worst-case PMD value.
Method 2 should be used with Equations (2) and (3) of 2.2.2. Method 2 combines the PMD
density function of the concatenated links with the Maxwell probability density function of
DGD values to compute an estimate of the probability that the DGD of a concatenated link at
a given wavelength exceeds a specified value for a defined reference link.
The specification is that the probability that the DGD over the link exceeds a given value,
DGDmax , shall be less than some maximum, P . One useful reference system consists of
F F
a concatenated link of 400 km comprised of forty 10 km cable sections. For such a reference
–8
link, a value such as DGD = 25 ps for P ≤ 6,5×10 has been used to justify the maximum
maxF F
1,2
value on PMD of 0,5 ps/km . For other lengths and bit rates, different values have been
Q
selected in the ITU Recommendations and IEC Specifications.
NOTE Subclause 3.4 shows conditions under which the specifications of the two methods are nearly equivalent.
3.2 Method 1: Calculating PMD , the PMD link design value
Q
3.2.1 Determining the probability distribution of the link PMD coefficients
Equation (5) shows that the PMD coefficient, x , of a particular concatenated link can be
M
derived from the PMD coefficients of the individual cable sections, x , comprising that link. The
i
probability distribution of the link PMD coefficients depends on the distribution of the cable
PMD coefficients and the number of cable sections comprising the link.
The following clauses describe three methods that can be used to estimate the distribution of
the link PMD coefficients. One method is numerical [1] , and two are analytic [4]. Of the two
analytic methods, the first assumes a specific analytic function for the distribution of the cable
PMD coefficients, while the second method makes no such assumptions but invokes an
extension of the central limit theorem.
3.2.1.1 Monte Carlo numeric method [1]
The Monte Carlo method can be used to determine the probability density, f , of the
link
concatenated link PMD coefficients without making any assumption about its form. This method
simulates the process of building links by sampling the measured cable population repeatedly.
PMD coefficients are measured on a sufficiently large number of cabled fibres so as to
characterise the underlying distribution. This data is then used to compute the PMD coefficient
for a single fibre-path in a concatenated link.
Computation of the link PMD coefficient is made by randomly selecting M values from the
measured cabled PMD coefficients, and adding them on an r.m.s. basis (in quadrature)
according to Equation (5). The computed link PMD coefficient is placed in a table or a
histogram of values derived from other random samplings. The process is repeated until a
sufficient number of link PMD values has been computed to produce a high density
(0,001 ps/√km) histogram of the concatenated link PMD coefficient distribution. If used directly,
without any additional characterization, the number of resamples should be at least 100 000.
___________
Figures in square brackets refer to the Bibliography.

– 12 – TR 61282-3 © IEC:2006(E)
Because of the central limit theorem, the histogram of link PMD coefficients will tend to
converge to distributions that can be described with a minimum of two parameters. Hence, the
histogram can be fit to a parametric distribution that enables extrapolation to probability levels
that are smaller than what would be implied by the sample size. The two parameters will
invariably represent two aspects of the distributions: the central value and the variability about
the central value. A choice of probability distributions can be made on the basis of the shape of
the histogram. Typical distributions could include lognormal (the log of the link PMD
coefficients is Gaussian) or one that is derived from the Gamma distribution.
3.2.1.2 Gamma distribution analytic method [4]
The Gamma family of distributions can often be used to represent the distributions of both the
measured cable PMD coefficients and the link PMD coefficients. If one assumes that the
square of the measured cable PMD coefficients, x , is distributed as a Gamma random variable,
i
the probability density of the cabled PMD coefficients is given by
α 2α −1
2β x
f ()x;α, β = exp()− βx (7)
cable
Γ()α
where x is a possible value of the cable PMD coefficient, Γ( ) is the Gamma function, and the
two parameters α and β control the shape of the density. Standard fitting techniques, such as
the method of maximum likelihood, can be used to fit Equation (7) to measured cable PMD
data to find values for α and β.
The probability density of the link PMD coefficients, x , of M concatenated equal cable lengths
M
has the same form as Equation (7), but with α and β replaced by Mα and Mβ:
Mα
2Mα −1
2()Mβ x
f()x;α, β, M = exp()− Mβx (8)
Link
()
Γ Mα
Consequently, the α and β parameters found by fitting Equation (7) to the measured cable
PMD coefficients can be used in Equation (8) to describe the probability density of the link PMD
coefficients.
3.2.1.3 Model-independent analytic method [4]
A more general alternative to the one described in 3.2.1.2 can be used that does not make any
assumptions regarding the form of the density function that describes the measured PMD
coefficients of the cabled fibre.
After measuring the PMD coefficients, x , on N cabled fibres, compute the mean, variance and
i
third moment of their squares
N N N
2 3
1 1 1
2 2 2
μ = x μ = ()x − μ μ = ()x − μ (9)
1 i 2 i 1 3 i 1
∑ ∑ ∑
N N − 1 N − 1
i=1 i=1 i=1
Let x be a random variable representing the link PMD coefficient of a fibre-path formed from
M
the concatenation of M equal-length cables, and let u be a possible value of x . Invoking the
M
extended central limit theorem [5], it can be shown that the distribution of the link PMD
coefficients is approximated by:
t
f ()u; M = Φ[]z(u) + φ[]z()u[]1− z()u (10)
Link
1/ 2
M
TR 61282-3 © IEC:2006(E) – 13 –
where
z
⎛ ⎞
1 z
⎜ ⎟
φ()z = exp − Φ()z = φ(y)dy
∫
⎜ ⎟
2π
⎝ ⎠
−∞
and
1/ 2
⎛ M ⎞ μ
2 3
() ⎜ ⎟()
z u = u − μ , t =
⎜ ⎟
3 / 2
μ
⎝ 2⎠ 6μ
Differentiating Equation (10) with respect to u provides an approximation to the link PMD
coefficient probability density function.
3.2.2 Determining the value of PMD
Q
The density functions found for the link PMD coefficients using one of the three methods
described in 3.2.1 can now be used to compute the PMD link design value. For a concatenated
link comprised of M cables, the PMD link design value, PMD , is defined as the value that the
Q
link PMD coefficient, x , exceeds with probability Q:
M
P(x > PMD ) = Q (11)
M Q
It follows, that for N > M, the probability that x exceeds PMD is less than Q:
N Q
P(x > PMD ) < Q (12)
N Q
For discussion purposes, an assumption is made that N ≥ 20 (the link contains at least
–4
20 cables) and that Q = 10 (the probability that the link PMD exceeds the PMD design value
is less than 0,0001). The following subclauses discuss how PMD can be found using the cable
Q
PMD density functions obtained in 3.2.1.
3.2.2.1 Determining the PMD link design value from the Monte Carlo density of 3.2.1.1
–4
To obtain probability levels of Q = 10 using a pure numeric approach requires Monte Carlo
simulations of at least 10 samples. Once this is complete, PMD can be interpolated from the
Q
associated cumulative probability density function.
Alternatively, the histogram of the link PMD coefficients can be fit with a parametric distribution
to enable extrapolation to lower probability levels than the measurement resampling would
otherwise allow. A choice of probability distributions can be made on the basis of the shape of
the histogram. Typical distributions could include lognormal (the log of the link PMD
coefficients is Gaussian) or one that is derived from the Gamma distribution. After the function
th
is fit, the value for PMD at the Q quantile can be computed.
Q
3.2.2.2 Determining the PMD link design value from the Gamma density of 3.2.1.2
–4
A good approximation for the link PMD coefficient, x , for M cables at the 10 quantile is given
Q
by:
2,004 + 0,975 Mα
PMD = (13)
Q
Mβ
where the α and β parameters were those found in 3.2.1.2.

– 14 – TR 61282-3 © IEC:2006(E)
For the 288 randomly selected scaled cabled fibres reported in [6], α = 0,979 and β = 48,6, and
PMD = 0,20 ps/√km.
Q
3.2.2.3 Determining the PMD link design value using the model-independent method
of 3.2.1.3
The moments computed in 3.2.1.3 can be used to compute the link PMD coefficient, x . For a
Q
th
link comprised of M cables, x at the Q quantile can be approximated by [5]:
Q
1/ 2
1/ 2
⎡ ⎤
μ
⎛ μ ⎞
2 3 2
⎢ ⎥
PMD = μ + z ⎜ ⎟ + ()z −1 (14)
Q 1 Q Q
⎜ ⎟
M 6Mμ
⎢ ⎥
⎝ ⎠
⎣ ⎦
th
where z is the Q quantile of the standard normal distribution. For N > M = 20 cables and
Q
–4
Q=10 , z = 3,72, the PMD design value becomes:
Q
1/ 2
⎡ μ ⎤
PMD = μ + 0,832 μ + 0,107 (15)
Q 1 2
⎢ ⎥
μ
⎣ 2⎦
For the 288 randomly selected scaled cabled fibres reported in [6],
−2 −4 −5
μ = 2,2×10  μ = 7,43×10 μ = 8,26×10
1 2 3
and Equation (15) produces PMD = 0,23 ps/√km.
Q
3.3 Method 2: Calculating the probability of exceeding DGD
max
PMD induced impairment of an optical signal occurs when the DGD at the signal’s wavelength
is too high. Since DGD varies randomly with time and wavelength, some means of imposing an
upper limit, defined in terms of a low probability value, is necessary for system design. This
upper limit is usually associated with a receiver sensitivity penalty. The probability can be
associated with a potential PMD-induced impairment time (min/year/circuit). (See Annex D.)
One means of calculating an upper limit on DGD is to multiply the upper limit on the PMD value
by a Maxwell adjustment factor, i.e. 3 (see Table 1 of 2.2.2). This could also be done with the
upper limit represented by PMD . When this is done, one is in effect assuming that the bulk of
Q
the distribution is very close to the upper limit. In reality, the bulk of the distribution is usually
well away from the upper limit. Method 2 is intended to provide metrics and methods to take
this into account. Because Method 2 takes into account the statistics of the individual optical
fibre cables and their concatenation, as well as the combined statistics of DGD variation, the
values calculated for system design, Equation (2), are substantially reduced from the “worst-
case” values – both in the value of maximum DGD and the probability of exceeding it.
The following figure shows several distributions of concatenated link PMD coefficient. Each
–4
distribution just passes the default criteria of M = 20, Q = 10 , and PMD ≤ 0,5 ps/√km.
Q
TR 61282-3 © IEC:2006(E) – 15 –
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0 0,1 0,2 0,3 0,4 0,5 0,6
Concatenated link PMD coefficient
IEC  1896/02
NOTE The above distributions are representative of the Gamma type distribution defined in 3.2.1.2.
Figure 1 – Various passing distributions
The leftmost distributions should provide better DGD performance than the rightmost
distributions. Method 2 assigns value to producing a distribution that is more to the left.
3.3.1 provides a means to link Method 1 and Method 2 so that, for most practical situations,
passing the default Method 1 criterion will imply passing a default Method 2 criterion.
3.3.1 Combining link and Maxwell variations
DGD coefficient (ps/√km) values, X , vary randomly with time and wavelength according to the
M
Maxwell probability density function:
3 / 2
⎡ ⎤
⎛ ⎞
X ⎛ X ⎞
4 4
M M
⎜ ⎟
⎢ ⎥
f()X ; x = 2 exp − ⎜ ⎟ (16)
Max M M
⎜ ⎟
⎜ 2 ⎟
Γ()3 / 2 π x
⎢ ⎥
πx ⎝ M ⎠
⎝ ⎠
M
⎣ ⎦
where x is the PMD coefficient of a concatenated link comprised of M cables as given by
M
Equations (4) or (5). The distribution of DGD values (over the length) is obtained by multiplying
the DGD coefficient values with the square root of the link length.
To combine the variations in the concatenated link PMD coefficient with the Maxwell variation
into a value to be used in a system design, a reference link is defined. Performance on the
reference link can then be generalised to other links. The reference link is defined with two
parameters, the overall link length, L , and the cable section length, L , which is assumed
REF Cab
to be constant for all cable sections.
Let f (xi) be the discretised probability density function (histogram) of the values of the
Link
concatenated link PMD coefficient values defined by the analysis of the distribution of
the measured PMD coefficient values and Equation (4). Any of the methods of 3.2.1 for
determining the probability density function of the concatenated link can be used.
Relative frequecy
– 16 – TR 61282-3 © IEC:2006(E)
Let X be some DGD coefficient value (ps/√km) that is to be used in system design. The
max
probability, P , of exceeding X is:
F max
⎡ ⎤
4⎛ X ⎞
M
⎜ ⎟
⎢ ⎜ ⎟ ⎥
X π x
⎝ ⎠
⎡ max ⎤ M
3 / 2−1
⎢ ⎥
y
⎢ ⎥
P = f ()x 1− f (Y; x)dY = f ()x 1− exp()− y dy  (17)
⎢ ⎥
F Link i Max i Link i
∑ ∑
∫ ∫
⎢ ⎥ ()
Γ 3 / 2
⎢ ⎥
i i
⎣ 0 ⎦ 0
⎢ ⎥
⎢ ⎥
⎣ ⎦
NOTE The rightmost integral is just the standard gamma function that can be calculated within many
spreadsheets.
The maximum DGD over the link derived from optical fibre cable, DGDmax , is the product of
F
the square root of the reference link length and X :
Max
DGD = X L (18)
max Max REF
F
For Method 1, the probability is pre-set and the associated PMD value is calculated and
Q
required to be less than a specified value. For Method 2, DGD is pre-set and the
maxF
probability value, P is calculated and required to be less than a specified value.
F
For link lengths less than the reference length, the DGD and probability relationship will be
conservative as long as either the installed lengths are less than L or the cable lengths
Cab
measured to obtain the distribution are less than L The reduction in averaging because of
Cab.
the reduced number of cable lengths is offset by the decrease in overall length. For link lengths
greater than the reference length, the maximum DGD to optical fibre cable should be adjusted
as:
L
Link
DGD = DGD (19)
adj max
F
F
L
REF
3.3.2 Convolution: Theory of Method 2
The calculation principle is derived from extending the worst case approach. With this
approach, the link PMD distribution is assumed to be a Dirac function and the DGD distribution
is represented as a Maxwell distribution. The probability that the Maxwell distribution exceeds
DGD yields P . These distributions are represented in Figure 2.
maxF F
TR 61282-3 © IEC:2006(E) – 17 –
0,07
0,06 Link PMD
Link DGD
0,05
0,04
0,03
0,02
0,01
0 1020 3040
DGD/PMD link value
IEC  1897/02
Figure 2 – Worst case approach assumption
NOTE Though not shown, the Dirac function illustrated in Figure 2 extends to a relative frequency value of 1,0.
Suppose the link PMD distribution could be represented by two Dirac functions, each with a
magnitude of 0,5. This would represent a situation where half the links 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 3
illustrates this case.
0,07
0,06
Link PMD A
0,05
Link DGD A
Link PMD B
0,04
Link DGD B
0,03
Link DGD tot
0,02
0,01
010 20 30 40
DGD/PMD value (ps)
IEC  1898/02
Figure 3 – Convolution of two Diracs
NOTE Though not shown, the two Dirac functions representing PMD value distributions in Figure 3 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, compared to the result associated with Figure 2.
Relative frequency
Relative frequency
– 18 – TR 61282-3 © IEC:2006(E)
Full convolution extends this notion to a complete distribution of link PMD coefficients. For the
Monte Carlo technique, the histogram of link PMD coefficients may be considered as a
collection of Dirac functions. For the continuous models, the probability density function is
reduced to a histogram by integrating the curve over the region that is represented as a single
histogram bin. The probability that DGD is exceeded is calculated for each of the
maxF
histogram bins (using the bin maximum). The weighted total yields P .
F
3.4 Equivalence of methods
Method 1 might be considered most practical for commercial specification because it can be
interpreted in terms of the defined measurements. Method 2 provides the most direct
information on the possible signal impairments. This subclause shows how the two methods
can be compared and establishes near equivalence of the default specifications.
The method for determining equivalence of statistical criteria relies on a parametric model and
is based on the following.
– A process can be characterised by parameters relevant to an assumed parametric
distribution type.
– Given these parameters, any statistical criterion can be evaluated to determine whether the
process distribution is conforming or not.
– For each criterion, the mathematical space of all possible parameters can be segmented
into two regions: conforming and not.
– The boundary between the two regions will form a curve, or envelope, in at least two
dimensions. Parameter values falling on one side of the envelope are conforming. Those on
the other side are not.
– Processes that are on the conforming side of the envelopes of two criteria pass both
criteria. Processes that are on the non-conforming side of the envelopes of two criteria fail
both criteria.
− Criteria that pass and/or fail the same process distributions are considered equivalent. In
this case, the two envelopes will overlay one another.
3.4.1 Equivalence of the default specifications
Figure 4 shows the envelopes for the
...