1 Holzlhner and Menyuk, CLEO 2003

1
Accurate Bit Error Rates from Multicanonical
Monte Carlo Simulations
CLEO 2003, CThJ3 Ronald Holzlöhner and Curtis
R. Menyuk University of Maryland Baltimore
County, Baltimore, MD
2
Resolving the Tails of a pdf
Typical Monte Carlo simulation results of the
received voltage
spaces
marks
Problem No hits in the tail regions Goal
Equal number of hits per histogram bin
Unbiased Monte Carlo cannot resolve the tails
3
Multicanonical Monte Carlo (MMC)
B. Berg and T. Neuhaus, Phys. Lett. 267, pp.
249253 (1991)
D. Yevick, Photon. Technol. Lett. 15, pp. 224226
(2003)

• MMC uses biased sampling,
• yields a uniform expected number of hits in each
  histogram bin,
• chooses samples from biased distribution,
• needs no a-priori knowledge of the bias

Uniform statistical uncertainty in histogram
4
Biased Sampling
unbiased Monte Carlo
biased Monte Carlo
The MMC bias is controlled by the Pkj
5
Metropolis Random Walk
N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44,
pp. 335341 (1949)
Random walk with small steps in the state space
k 1
V
V
k 2
Goal sample voltage contours evenly
6
Metropolis Random Walk
N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44,
pp. 335341 (1949)
Random walk with small steps in the state space
rejected step
k 1
V
V
k 2
Goal sample voltage contours evenly
7
Metropolis Random Walk
N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44,
pp. 335341 (1949)
Random walk with small steps in the state space
accepted step
k 1
acceptance decision based on
V
V
k 2
Goal sample voltage contours evenly
8
The MMC Algorithm
initialization Pkj1 1 / M, z i1 0 loop
over iterations j loop over noise
realizations i (1) randomly perturb
the current noise state z i1 z i ? z
(2) compute the receiver voltage and the bin
index k (3) accept or reject this
step (4) increment the histogram
update the weights Pkj1 end
Adjust bias to sample the voltages evenly
9
Test System
Submarine single-channel 10 Gb/s CRZ system, 6100
km, data 32 bit PRBS
916 ps/nm
916 ps/nm
34 map periods
post-compensation
pre-compensation
16.5 ps/nm-km
A
?2.5 ps/nm-km
20 km
25 km
45 km
45 km
45 km
We add biased ASE noise at every amplifier
10
Results
Average pdfs of the marks and spaces at the bit
slot centers, ca. 100,000 samples
unbiased Monte Carlo limit
Optimum BER 3.1 x 1012
Probability density
Voltage (normalized)
Agreement over 20 orders of magnitude for this
system !
R. Holzlöhner, et al., Photon. Technol. Lett.
14, pp. 10791081 (2002)
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Conclusions

• MMC is a good simulation method if the bias is
  unknown
• MMC and the Covariance Matrix Method agree over
  20 orders
• We validated the Covariance Matrix Method for
  this system

MMC might be the method of choice to simulate
rare events in complex systems
12
Metropolis Random Walk
N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44,
pp. 335341 (1949)
Goal Produce random walk of the biased
distribution
Detailed Balance
Zero net flow

We take small steps in the state space
13
Biased Sampling
unbiased Monte Carlo
biased Monte Carlo
The MMC bias is controlled by the Pkj
14
Resolving the Tails of a pdf
Typical Monte Carlo simulation results of the
received voltage
Probability density
Voltage
Voltage
Problem Relative uncertainty is proportional
to the pdf in Monte Carlo simulations
no hits in the tail regions at all Optimum
Equal number of hits per histogram bin
Unbiased Monte Carlo cannot resolve the tails
15
The MMC Algorithm
initialization Pkj1 1 / M, z i1 0 loop
over iterations j loop over noise
realizations i randomly perturb the
current noise state z i1 z i ? z
compute the receiver voltage and the bin index
kb accept or reject this step with
probability ?ab increment the histogram
Hkj ? Hkj 1 update the inverse weights
Pkj1 end
Simplified inverse weight update
Adjust bias to sample the voltages evenly
16
Metropolis Random Walk
N. Metropolis and S. Ulam, J. Am. Stat. Ass. 44,
pp. 335341 (1949)
Random walk with small steps in the state space
accepted step
k 1
k 2
acceptance decision based on
V
Goal sample voltage contours evenly