Comparison of Importance Sampling

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Comparison of Importance Sampling and
Multi-Canonical Monte Carlo Simulations in
Calculating the Performance of Optical Fiber
Communications Systems Curtis R. Menyuk and
Aurenice O. Lima University of Maryland
Baltimore County Computer Science and Electrical
Engineering Department Baltimore, MD 21250
with a focus on statistical errors and
applications to PMD
Currently searching for a job
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Collaborators Gino Biondini William Kath Ivan
Lima Brian Marks John Zweck
More details of our work on importance sampling
waspresented in a workshop in Brazil in
11/05 The slides and an 11-page bibliography can
be found at the URLhttp//www.photonics.umbc.edu
/Menyuk/Brazil-Workshop
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Importance Sampling Methods

• Standard (non-iterative) importance sampling
• Biasing is often not based on the quantity of
  interest
  
• Requires a priori knowledge of how to bias
• Multi-Canonical Monte Carlo method (iterative)
• Biasing is based on the quantity of interest Q
  in the case of PMD studies
• Requires no a priori knowledge of how to bias
• Is a little slower when both work

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Polarization mode dispersion
Random birefringence variations
lead to randomization of the polarization state
of light scale length 1100
meters
Nearby frequencies are randomized differently
scale length 10 10,000 km
The differential randomization leads to PMD!
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Polarization mode dispersion
At any frequency w
W Polarization dispersion vector
It is a polarization analog to group velocity An
arbitrary input state divides into polarization
states One is aligned with W One points in the
opposite direction
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Polarization mode dispersion
tD
Differential group delay
Explicitly,
for a pulse with a narrow bandwidth
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Compensators
Basic structure

• Tx/Rx model is needed. Rx model is critical
• A model for feedback is needed
• A single-section compensator is shown
• Three-section compensators are also modeled

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Compensators
Receiver model Feedback models

• Optimal compensation Best compensator
  configuration for fiber realization is found
• First-order compensation (1 section)First- and
  second-order compensation (3 sections)at the
  central frequency

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Compensators
Receiver model Feedback models
Neither corresponds to experimental
reality! These represent achievable limits (1) or
are useful for comparison to theory (2)
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Compensators
Biasing results First order compensator with
optimal biasing
First-order bias
First- and second-order bias
Why did this happen?
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Compensators
Biasing results First order compensator with
optimal biasing
Uncompensated
Single-section fixed compensator
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Compensators
Biasing results
Single-section variable compensator Center-frequen
cy compensation
Single-section variable compensator Optimal
compensation
Nearly horizontal lines indicate that penalty is
dominatedby second-order PMD
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Statistical Errors

• Monitoring statistical errors is critical!
• How we did it
• Used multiple importance sampling with 10 biases
• Divided
• Estimate the eye opening penalty m M / C

This estimator is biased
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Statistical Errors

• Monitoring statistical errors is critical!
• How we did it
• To reduce bias, we set
• We use the first order law of error propagation
• We plot contours of

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Multicanonical Monte Carlo

• With multi-section compensators Is biasing t
  and good enough?
• We could bias
• BUT
• The algebra would become very intricate
• We really want to bias the penalty
• and automate the search of the configuration
  space
• How do we do it?

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Multicanonical Monte Carlo

• With multi-section compensators
• We really want to bias the penalty
• and automate the search of the configuration
  space
• How do we do it? Our answer
• Multicanonical Monte Carlo
• An iterative learning procedure related to
  optimization
• like simulated annealing
• Falls into the framework of adaptive Monte Carlo
  methods

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Multicanonical Monte Carlo
Compensators
Single-section compensator
Uncompensated
MMC puts realizations in the regions that
dominatethe contribution to the penalty!
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Multicanonical Monte Carlo
Compensators
MMC Multicanonical Monte Carlo NI-IS
non-iterative importance sampling
Agreement between MMC and NI-ISis excellent!
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Multicanonical Monte Carlo
But what about statistical errors? This is
actually a very hard problem The samples are
correlated from bin to bin and iteration to
iteration Our answer The transmission matrix
method which is a variant of the bootstrap method
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Bootstrap method
Real world
Bootstrap world
Estimate statistic of interest
Bootstrap replications
empirical distribution estimated
from the observed samples x
F unknown distribution
x observed samples

• Cannot use standard error analysis to compute
  error in

• Can generate many bootstrap samples from
  by drawing n samples
• with replacement from x.

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Bootstrap method
General Procedure
(1) Generate B bootstrap samples x
(2) Generate B independent bootstrap samples
estimates of
(3) Estimate the error in using the standard
deviation formula on the bootstrap samples
where
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Transition matrix method

• The transition matrix method is divided in two
  parts

• Using a single standard MMC simulation Estimate
  the desired pdf and the
• 
  transition probability matrix

• Using the estimated transition matrix Generate
  pseudo-MMC simulations

• Transition probability matrix

• Nb x Nb matrix that contains the estimates of
  the transition probability ?i,j

Nb number of bins

• ?i,j probability that a sample in the bin i
  will move to bin j after a single
• step (perturbation) in the MMC
  algorithm

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Transition matrix method
(1) is an estimate of the transition matrix
from a single MMC simulation

• X1, , XB
• samples (DGD values) obtained from the
  pseudo-MMC simulations

(3) Each is a value for the probability of
the i-th bin

• With B independent , one can obtain an error
  estimate
• for in the DGD pdf using standard
  deviation formula

where
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Transition matrix method
Validation of the approach
Relative variation of the pdf of the normalized
DGD for 15-section and 80-section PMD emulators
Relative variation
(i) Circles transition matrix method with
a single MMC simulation (ii) Solid 103 MMC
simulations (iii) Dashed confidence interval
for (i) (iv) Squares transition matrix method
with a single MMC simulation (v) Dot-dashed
103 MMC simulations
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Multi-section compensators
Comparison of MMC and NI-IS
(i) Uncompensated case (ii) Single-section PMD
compensator (iii) Three-section PMD compensator
Symbols Results with IS Solid lines Results
with MMC Error bars confidence interval for MMC
results (Transition Matrix method)
MMC and NI-IS are in good agreement! ? Biasing
first and second order PMD is sufficient
WHY?
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Multi-section compensators
ANSWER Higher-orders of PMD and penalties
after compensationare correlated with the first
two orders of PMD before compensation
(i) Solid conditional expectation of the
residual eye-opening penalty after compensation.
Penalty (from bottom to top) at
0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 dB
(ii) Dashed joint pdf of first- and
second-order PMD of the transmission line
Joint pdf (from bottom to top) at 3 x
10-n, n1, , 7, and 10-m, m1, , 11
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Conclusions

• By using bootstrap, it is possible to calculate
  statistical errors with the multicanonical Monte
  Carlo method
• Biasing first- and second-order PMD is sufficient
  because of higher-order correlations