Transcript Estimation of the Four‑Wave Mixing Distortion Statistics

Performance Estimation of
Bursty Wavelength Division
Multiplexing Networks
I. Neokosmidis, T. Kamalakis and T. Sphicopoulos
University of Athens
Department of Informatics and Telecommunications
Email: [email protected]
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Introduction
People tend to calculate the performance of an
WDM network assuming worst case scenarios:
 Optical Sources always on (no bustiness)
 Phase difference between signals is zero (max
interference)
Etc…
What happens in more “average” cases?
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Bursty Traffic
IP over WDM
 exponential growth of IP traffic (almost
doubles every six months)
 WDM is a promising technology (high capacity)
Why IP directly over WDM?
 Lack of optical random access memories
(RAMs) required for all-optical packet
switching
 Need for infrastructures / schemes in
order to “route” IP packets without optical
buffering
 Multiprotocol Lambda Switching
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Bursty Traffic
 The label of the packets is the wavelength on
which they are transmitted
 MPS network forwards and labels the IP
packets according to their FEC
 Each wavelength can be modeled as an
M/G/1 system (short-range dependence)
 The burstiness of each wavelength is
characterized by the traffic load ρ
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Bursty Traffic
 Inside a silent period (between two packets), the
power of the source can be assumed zero
 Silent periods can be considered as series of “0”s
 Within a packet, the “1”s and the “0”s appear with
equal likelihood
 The probability, Ppacket(t), that at any given time t,
a packet is being transmitted equals ρ
 The traffic load ρ essentially determines the
statistics of the bits
T
t1
t2
t3
t4
0 1 0 1
0 1 0 1
0 1 0 1
0 1 0 1
t1+t2+t3+t4=ρT
OFF
p0=1-ρ/2
ON
p1=ρ/2
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Modelling Bursty Traffic
Under the M/G/1 assumption:
p0  P bit "0" is transmitted  1   / 2
p1  P bit "1" is transmitted   / 2
How does this affect the statistics of signal
dependent noises (FWM, inband crosstalk,…)
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Four Wave Mixing (FWM)
FWM is due to Kerr nonlinearity
Generation of a fourth signal: f1+f2-f3=f4
FWM is very useful in wavelength conversion
In a WDM system, some of the products may
coincide with the wavelength channels
 This causes nonlinear crosstalk between the
WDM channels
 FWM-induced distortion is therefore signal
dependent!




COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Four Wave Mixing (FWM)
Im 
d pqr
1
B
B
B
cos pqr   n 

p q r
3 pqr
pn qn
2

 

d pqr
d pqr
1
 1

I s    B p Bq Br
cos pqr     B p Bq Br
sin  pqr 
pn qn
3 pqr
pn qn
 3 rpqr



 n
  r n

2
You can calculate the value of the FWM-induced distortion if
you have the values of the bits being transmitted in all
channels (Bp) and their phases (θp)
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Inband Crosstalk
 It is due to filtering
imperfections at
optical cross-connects
 It is at the same
wavelength as the
signal
 It cannot be removed
using additional
filtering
Node 4
Node 3
Node 1
Node 2
λ1
λ1
λ1 λ1
2x2
Switch
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Inband Crosstalk
T
M M
1
2
S   A(t ) dt   Dkn e j  k  n 
20
k 0 n 0
Dkn 
Bk ck Bn cn
*
g
(
t
)
g
(t )dt
k
n

2
T
You can calculate the value of the inband crosstalk field if you
have the values of the bits being transmitted in all channels
(Bp) and their phases (θp)
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Similarities…
In both cases you can calculate the value of the noise field if
you have the values of the bits being transmitted in all
channels (Bp) and their phases (θp)
z1
z2
F()
Y
z2N
RVs whose PDF we
want to compute
RVs with known PDF
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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How to model?
 Standard Monte Carlo
 requires an excessive number of samples
(~10/EP)
 MultiCanonical Monte Carlo
 increases the occurrence of samples in the
tail regions of the PDF (faster)
 it can easily be implemented in any
general-purpose programming language
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Multicanonical Sampling
 Calculation of the PDF of a random variable Y
which depends on random variables z1,…zN
through Y=f(z1,…,zN)
 In the first iteration, standard MC is
performed
 On each iteration i, the estimated PDF of Y is
stored in the variables Pik
 A sample of Y is calculated by randomly
selecting zi using the Metropolis algorithm
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Multicanonical Sampling
 At the end of the iteration the values of Pki
are updated according to the MCMC
recurrence relations
 Pki are normalized such that their sum with
respect to k is equal to unity
 The process is repeated until the PDF
reaches sufficiently low values
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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System Parameters
Symbol
Quantity
Values

nonlinear coefficient
2.4 (Watt×km)-1
c
speed of light in vacuum
3×108 m/sec
λ
Wavelength
1.55μm
D
fiber chromatic dispersion
coefficient
2 ps/km/nm
Δf
channel spacing
50GHz
α
The fiber loss coefficient
0.2 dB/km
L
total fiber length
80 km
Leff
effective length
21.169 km
R
receiver responsivity
1.28 A/W
N
number of channels
16
M
number of interferers
10
c02
# of photoelectrons in the signal
at the receiver
100
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10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
10
-16
0,2
FWM Pin=8dBm
MCMC
Gaussian approximation
In-band crosstalk SXR=12dB
MCMC
Gaussian approximation
0,4
0,6
0,8
Traffic load 
1,0
(a)
-9
20
18
Pin (dBm)
16
BER=10
FWM-MCMC
FWM-Gaussian approximation
Crosstalk-MCMC
Crosstalk-Gaussian approximation
20
18
16
14
14
12
12
10
10
8
8
6
6
4
0,2
0,4
0,6
Traffic load 
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
0,8
4
1,0
(b)
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SXR (dB)
 For the case of the
crosstalk noise, the
Gaussian model does
not provide an accurate
estimate of the BER
especially for small
values of the traffic load
 The error is much more
pronounced in the case
of FWM noise.
 The Gaussian
approximation cannot
predict the maximum
power that the system
can tolerate.
BER
Are the noises Gaussian?
Calculation of the FWM PDF
 As the traffic becomes heavier, the average power
at each wavelength is increased
 An increment of the traffic load leads to a
broadening of the PDFs
1
1
=
=
=
=
=
-2
10
-4
10
0.2
0.4
0.6
0.8
1
10
-2
10
-4
10
-6
10
-8
 = 0.2
 = 0.4
 = 0.6
 = 0.8
=1
pdf
pdf
-6
10
-8
10
10
10
10
-10
-12
10
-10
-14
10
-12
-30
-20
-10
0
Im
10
20
30
(a)
0
100
200
Is
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
300
400
(b)
17
Inband Crosstalk PDF
-1
10
-2
10
-4
10
-6
10
-8
bs=0
 = 0.2
 = 0.4
 = 0.6
 = 0.8
=1
bs=1
10
-10
10
-12
0
50
100
150
200
S (photoelectrons)
250
300
(a)
1
BER
 The pdf of the decision
variable is strongly
dependent on the value of ρ
 As the traffic becomes
lighter, smaller BER values
are obtained for the same
SXR
 For light traffic, more nodes
can be concatenated in the
network
 There is a strong
dependence between the
system performance and the
SXR
pdf (photoelectrons )
1
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
 = 0.2
 = 0.4
 = 0.6
 = 0.8
=1
11
12
13
14
15
16
SXR (dB)
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
(b)
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Packet Error Rates
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
0,2
In-band crosstalk SXR=12dB
MCMC, Short Packets
MCMC, Long Packets
Gaussian, Short Packets
Gaussian, Long Packets
0,4
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
0,2
0,6
0,8
Traffic load 
1
PER
 The performance of the higher
layers can be quantified in terms
of the packet error rate
 PER=1-(1-BER)k
 k=256bytes=2048bits (short
packets) and
k=1500bytes=12000bits (long
packets)
 The PER has almost the same
behaviour as the BER (PERkBER)
 The PER is higher for longer
packets (segmentation)
 Erroneous receptions could cause
packet retransmissions and/or
loss of quality of service
 Inaccuracy of the Gaussian model,
especially in the case of FWM
noise
PER
1
1,0
(a)
FWM, Pin=8dBm
MCMC, Short Packets
MCMC, Long Packets
Gaussian, Short Packets
Gaussian, Long Packets
0,4
0,6
Traffic load 
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
0,8
1,0
(b)
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Channel Traffic Load
Distribution
1,2
Traffic load 
1,0
0,8
0,6
0,4
0,2
0,0
0
2
4
6
8
10
12
14
16
BER
Channels
10
-6
10
-8
10
-10
10
-12
10
-14
10
-16
18
(a)
FWM, Pin=7.6dBm
=0.6
0
2
4
6
8
10
12
Channel Number
mean()=0.6
14
16
18
(c)
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Conclusions
 The MCMC method is used to study the statistical
behavior of FWM and inband crosstalk taking into
account the impact of traffic burstiness in an IP
over MPλS-based WDM network
 The MCMC method is proved to be more efficient
(faster) and accurate
 The performance of such systems is very
sensitive to the traffic load
 The Gaussian approximation does not yield
accurate results
 Careful traffic engineering can improve the
system performance in terms of the BER
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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Thank you!
Email: [email protected]
COMMUNICATION SYSTEMS, NETWORKS AND DIGITAL SIGNAL PROCESSING 19-21 July, 2006
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