R. Holzlhner,1 C. R. Menyuk,1,2

1
Direct Calculation of the Noise Evolution in a
Highly Nonlinear Transmission System Using the
Covariance Matrix
CLEO 2002, CThG5

• R. Holzlöhner,1 C. R. Menyuk,1,2
• V. S. Grigoryan,3,1 W. L. Kath 4,1

1University of Maryland Baltimore County,
Baltimore, MD 2 PhotonEx Corporation, Maynard,
MA 3 CeLight Corporation, Silver Spring, MD 4
Northwestern University, Evanston, IL
2
Calculating Bit Error Rates
BERs are commonly calculated from Monte Carlo
simulation results using Gaussian extrapolation
Log scale
Linear scale
Monte Carlo Gaussian fit
spaces (Zeros)
102
decision level
marks (Ones)
probability density

decision level
1015
0
voltage
voltage
But voltage probability densities are not
Gaussian Marcuse 1990
3
Outline
Goal

• Accurate calculation of BER vs. decision level
• Linearize noise propagation
• Include signal-noise interactions and data
  modulation
• Model a realistic electrical receiver filter

Approach
Calculate the multivariate Gaussian noise pdf of
the optical field Justification Noise-noise
interaction in the fiber is negligible
4
Linearizing the NLS
Nonlinear Schrödinger equation with ASE noise
added Gaussian white noise
Now set
noise-free signal
accumulated noise
Doobs Theorem du is multivariate Gaussian
distributed
5
Noise Covariance Matrix

Covariance matrix
Multivariate Gaussian distribution of a
6
How to Compute the Covariance Matrix
Solve the linearized homogeneous propagation
equation
ASE
ASE
G
ASE input
0
L
But ODE is stiff due to dispersion. Solution
perturbative approach
Compute ? by perturbing each of the N frequency
modes separately
7
Strong Jitter Distorts the Gaussian pdf
Small phase jitter
Large phase jitter
Im
Im
ak
ak
Ak
Ak
pdf ak
?f
0
0
Re
Re
Separate phase and timing jitter from
Phase jitter rotates signal around origin,
distorting the Gaussian pdf
8
Test System 10 Gb/s DMS over 24,000 km
R.-M. Mu et al., IEEE J. Sel. Topics Quant.
Electronics 6, 248257 (2000)
AO switch
A
N
A
2.8 nm OBF
EDFA
N
N
N
N 4 ? 25 100 km DSF, D ? 1.1 ps/nm-km A?2 ?
3.5 km SMF, D 16 ps/nm-km
Highly nonlinear system, hence stringent test of
our approach
9
Accurate Probability Density Functions
Low-pass filtered receiver voltage of 8-bit
sequence 11101000
Minimum Accurate BER 1.1 x 10 9
spaces
marks
Probability Density
Monte Carlo
Minimum Gaussian BER 3.36 x 10 14 Q 7.49
Gaussian fit
Linearization
Optimum decision level
Voltage (normalized)
Probability density functions deviate strongly
from Gaussians in tails
10
Conclusions / Current position

• Linearization method works in a highly nonlinear
  system
• Method is robust and completely deterministic
• Critical step phase and timing jitter
  separation
• Computational cost equal to 2N Monte Carlo noise
  realizations
• Accurate pdfs in a chirped RZ system Poster 44