Performance evaluation and parameter optimization of wavelength division multiplexing networks with adaptive importance sampling techniques
ABSTRACT In this paper new adaptive importance sampling techniques are
applied to the performance evaluation and parameter optimization of
wavelength division multiplexing (WDM) network impaired by crosstalk in
an optical crossconnect. Worstcase analysis is carried out including
all the beat noise terms originated by inband crosstalk. Both input
signal hypotheses are considered. The accurate biterrorrate estimates,
which are obtained in short runtimes, indicate that the influence of
crosstalk is much lower than that predicted by previous analyses. This
finding has a strong impact on the design of WDM networks. Besides, a
method is used to optimize the detection threshold, which turns out to
improve the system performance significantly. The presented techniques
also allow us to determine the power penalty due to the introduction of
additional WDM channels

Article: Optical Signal Processing; a novel approach to modifying digital information in optical domain
[Show abstract] [Hide abstract]
ABSTRACT: It is concluded that the OSP offers a new and alternative method for manipulating and modifying data in the optical domain. Under certain circumstances during signal processing, a number of signal properties such as wavelength and modulation format can be preserved. Interesting applications can be found in wavelength routed networks in which the OSP can be applied as a wavelength independent access mechanism. Additional experimental research is needed to decide whether these advantages counterbalance the disadvantages of this access mechanism.01/1998;
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PERFORMANCE EVALUATION AND PARAMETER OPTIMIZATION OF
WAVELENGTH DIVISION MULTIPLEXING NETWORKS WITH
ADAPTIVE IMPORTANCE SAMPLING TECHNIQUES
David Remondo, Rajan Srinivasan, Victor F. Nicola, Wim C. van Etten, and Henk E.P. Tattje
University of Twente
Department of Electrical Engineering
Postbus 217
7500 AE Enschede
The Netherlands
ABSTRACT
In this paper new adaptive importance sampling
techniques are applied to
evaluation and parameter optimization of a
wavelength division multiplexing (WDM) network
impaired by crosstalk in an optical crossconnect.
Worstcase analysis is carried out including all the
beat noise terms originated by inband crosstalk.
Both input signal hypotheses are considered. The
accurate biterrorrate estimates, which are obtained
in short runtimes, indicate that the influence of
crosstalk is much lower than that predicted by
previous analyses. This finding has a strong impact
on the design of WDM networks. Besides, a method
is used to optimize the detection threshold, which
turns out to improve the system performance
significantly. The presented techniques also allow us
to determine the power penalty due to the
introduction of additional WDM channels.
the performance
I. INTRODUCTION
In wavelength division multiplexing (WDM) systems
several information channels can be transmitted along
the same optical fiber by using different wavelengths.
The main advantage of WDM is that communication
networks can be easily reconfigured to adapt to varying
traffic demands, without changing the physical layout.
A fundamental element in WDM networks is the all
optical wavelength router, also called optical cross
connect. For this purpose, an arrayedwaveguide grating
(AWG) seems to he a good candidate (see, e.g. [l]). In
this device, however, there will be crosstalk
components originating from different information
streams. The performance of WDM networks can he
significantly degraded by this disturbance [Z].
The application of analytical techniques to the
performance evaluation of WDM systems impaired by
crosstalk is usually very difficult and often requires
excessive simplification of the system model. On the
other hand, building a hardware prototype is expensive,
timeconsuming and relatively inflexible. Owing to
these difficulties, computer simulation represents an
attractive alternative.
The hiterrorrate (BER) is a fundamental performance
parameter. The values of interest in optical
communications are very small. Unfortunately, Monte
Carlo simulation requires large runtimes to yield
accurate BER estimates. Therefore, it is desirable to
find efficient variancereduction techniques, such as
those derived from importance sampling (IS), that lead
to simulation speedup.
IS has found application in a variety of fields, such as
optical fiber communications [3][4], reliability 151,
queuing [61, detection [7][8], fading channels [9][10],
and other issues in digital communications [111[151.
IS involves running a Monte Carlo simulation where
probability density functions (pdf s) are employed that
are different from the actual ones, so that the
probability that an error arises during simulation
increases. An unbiased BER estimate is then obtained
by weighing the results with the likelihood ratios of the
actual to the IS densities.
The principle of IS is simple, hut its efficient
application to particular systems is a research issue. The
researcher must decide which type of IS pdf to use, and
then has to find the parameters of the pdf that yield a
minimum estimator variance.
In general, the performance of the IS estimator closely
depends on the choice of the IS pdfs and their
parameters. Two main methodologies have been
developed for the optimization of IS parameters:
adaptive techniques [9], [Ill, [16], and techniques
based on Large Deviations Theory [17]. The advantages
of the former are its generality and applicability to a
wide range of systems. The latter often requires difficult
analysis that is possible only for relatively simple
systems.
In this paper ihe search for optimal IS parameter values
is made with new adaptive techniques based on
stochastic Newton recursions. The techniques require
some additional analytical work, but robust and easyto
implement algorithms result. Therefore, simulation mn
time is traded for algorithm design effort. Furthermore,
a conditioning technique, referred to as the gmethod
[7], is combined with the adaptive IS algorithm, so that
knowledge of the distribution of the underlying random
variables is more fully exploited. A related technique is
0780349849/98/$10.00 01998 IEEE.
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used for the optimization of system parameters. It
involves the minimization, through simulation, of a
suitable stochastic objective function with respect to
parameters of interest 171. All these IS techniques are
briefly explained in Section U.
We use the presented IS techniques to determine the
BER degradation due to crosstalk in the AWG. The
system model is described in Section nI. In practical
situations, outband crosstalk can be neglected with
respect to inband crosstalk due to the demultiplexing
process made before the receiver [Z]. Worstcase
analysis is carried out including all the beat noise terms.
Moreover, both input signal hypotheses are considered.
Results are compared with the commonly used
Gaussian approximation [2] and a recently developed
Chemoff bound [IS]. Furthermore, we present novel
results on optimization of detector threshold setting,
which tuns out to have a relevant impact on system
perfomiance. Besides, an accurate assessment is given
of the power penalty due to the introduction of
additional WDM channels.
1 1 . IMPORTANCE SAMPLING
A. Basics of IS
Consider estimating the quantity G = E{g(X)}< +,
where g(X) is a realvalued function. For notational
convenience, we assume that X is a random variable
with density f . The extension to random vectors is
straightforward. An unbiased IS estimator d of G is
given by
6=L$g(X,)W(X,,@),
K ti
X , f.(x,@),
(1)
where f. denotes a biasing family of densities
parameterized by 8, the function W is the likelihood
ratio W (x, 8)= f (x)/ f . (x, 8) used as a weighing
function, and K is the IS simulation length. The
notation X  f denotes that X is drawn from a
distribution with density f . The estimator variance is
given by
1
K1
where
I(8)=E{g2(X)W(X,8))
= K{g' (X )W' ( X , e)}
and E. denotes expectation with respect to f . . If g()
represents the indicator or some event, say {X zr},
then G = P(X 2 r) and d is an estimator of a tal
probability.
varG=[I(8)G2],
(2)
(3)
The first step in the application of IS is to select a
family of densities f.(x,8) that enhances the tail
probability in an adequate manner. In an application, 8
could represent a set of parameters. Once f.(x,S) is
chosen, the IS problem centers around determining the
value of 8 that minimizes the variance in (2) or
equivalently I @ ) in (3).
B. Adaptive IS
The algorithmic minimization of I(8) can be done in
the followisg way. From (3) we have
I@)= E{g2 (X)W'(X,8)}
= E. {g2 (X )W'(X, 8)W (X , B)},
where prime indicates derivative with respect to 8.
Similarly,
~"(~)=~,{g'(X)W"(X,~)W(X,8)). (5)
Estimators of these derivatives can he set up as
(4)
~ ( ~ ) = ~ ~ g ' ( X , ) W ( X , , B ) W ' ( X , , B X
K 11
X , f. , 16)
and
~ . ( ~ ) = ~ ~ P ' ( X , ) W ( X , , B ) W " ( X , , B ) ,
11
X ,  f. . (7)
We can now use a root finding algorithm in the form of
stochastic Newton formula recursions to estimate an
optimum 8 , Such an algorithm is given by
where the rate factor 6, controls convergence speed
and noisiness. As is typical of stochastic approximation
procedures, convergence of
characterized by a small random oscillation around the
optimum. For a large class of IS problems the function
I(8) has a single minimum and the algorithm can
locate it. The function I(8) does not need to be convex.
Other numerical methods are available (e.g. 191, [ill)
that can be combined with IS simulation procedures to
minimize the estimator variance. On the other band, the
Brent's method and the Golden Section Search method,
which are meant for
minimization, do not yield SatisfactoIy results.
this algorithm is
deterministic function
C. The gMethod
In some applications, the system performance can be
characterized
as a probability
p, = P(Z + X 2 r), where 2 is a random variable with
known density, and X represents a random variable or
function of random variables. The variable z
represents some system parameter, for example, a
in the form
Page 3
2347
threshold level in a digital receiver. It is assumed here
that Z and X are independent. Then we can write
p, = E{P(Z 2 Tx
where g,(x)=P(ZTzn) is a continuous function of
I . In analogy with (l), we have the IS estimator Bq of
P, as
I x)}= E{~,(x)},
(9)
The estimator exploits knowledge of the density of Z ,
with IS being performed on X . In contrast to this is the
normal IS estimator given by
z,  f,.G,s), x,  fx.(x.@),
where l,(y)=l for y T T and l,(y)=O
is usually called the indicator function). Here IS is
performed on 2 and X . It has heen shown 171 that for
any biasing scheme the estimator in (10) yields a
smaller variance than that in (1 1).
(11)
otherwise (this
D. Optimization o f System Parameters
The differentiability, with respect to T , of the estimator
in (10) permits optimization of the system parameters to
achieve a desired performance. Suppose that q is the
desired value of performance probability p, , which is
obtained at T = T~ < +m . To estimate T~ we form the
stochastic objective function
J(T) [i%
and minimize J ( ) with respect to z using the
 4 1 '
(12)
This approach was proposed in 171 as the inverse IS
problem. On the other hand, if p, represents an error
probability in a communication system that is to be
minimized. then the algorithm
can be used. The estimates of the derivatives in these
algorithms can be obtained from (10).
111. SYSTEM MODEL
Consider the schematic of a 4x4 AWG in Fig. 1.
There are 4 nodes connected to the crossconnect. Each
node includes a multiwavelength transmitter (4 light
Sources and a multiplexer) and a multiwavelength
receiver (a demultiplexer and 4 photodetectors). The
router can send any wavelength from any input port to
any output port 121.
Worstcase analysis implies considering that the
interfering channels are in the ONstate. The outband
crosstalk is neglected. The phase of the desired optical
signal is assumed to he zero without any loss of
generality, and the phases of the interfering signals are
independent and uniformly distributed in [0,2n). The
extinction ratio is assumed infinite. The optical field of
the desired input channel is
s,(t)=a,Excos(kf,f).
where a? E {OJ}
is the information hit, E is the pulse
amplitude, and T is the symbol period. Each of the
M  1 interfering channels has an optical field
sm(t) = &amE x cos(kf,t + $* (t)).
The factor E accounts for the amount of crosstalk.
Within the symbol period, the phase Gm(t) is assumed
to be constant, i.e.
@ m ( f ) =
implies, under both signal
a,=l, m = 2 ,..., M .
O < t < T ,
(15)
(16)
$m. Worstcase analysis
hypotheses, that
Fig. 1. Schematic of a 4x4 AWG. Thick and thin lines
indicate signal and crosstalk components, respectively.
Only one wavelength is shown.
The photocurrent generated by a photodiode with unity
quantum efficiency at one of the outputs of an M XM
AWG will he 121
..
Ed . = I
a E Z +t&a,E2xcos(@m)
2
n=*
+ CEE*XCOS(@~~~~)+(MI)E+~,,
Y
EZ
2
m=*
nm
(17)
where n, is the additive white Gaussian noise (AWGN)
of the receiver, which is independent of the signal and
the crosstalk components. The second term in (17) is
the signalcrosstalk heat noise and the third term is the
crosstakcrosstalk beat noise.
IV. ANALYSIS AND RESULTS
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2348
1
2
=e&
The usual Gaussian approximation [Z] assumes that the
third and fourtb terms in (17) can be neglected (small
E). When the decision threshold a is set at half the
ONsignal output current (symmetric setting, i.e.
z = E' 14 ), the system BER can be assumed to be equal
to half the probability that the ONsymbol is detected
erroneously. The Gaussian approximation is then given
bv
~:
where 0; is the variance of the receiver noise. The
Chernoff bound [ 181 is
a 
her‘
,
(18)
a
2
+ (M ~)E~E'
, (19)
where Io() is the zeroorder modified Bessel function
of the first kind.
In contrast with the two approximate methods in (18)
and (19), the IS experiments include all the terms in
(17). Moreover, error probabilities are obtained for both
the ON (a,=l) and the OFF (a,=O) signal
hypotheses. Estimating the error probability for the OFF
case represents a challenge because this probability
possesses a very low BER floor when a = E' 14 .
The system was simulated with modified biasing
densities for the M1 phases of the interfering
components. All modified phase densities were
identical Gaussian pdf's, with means at n, and with
common variance to be determined w i t h the stochastic
Newton formula (8). In this way, the probability
densities of the phases are concentrated in the region
where the second term in (17) yields the largest
negative values and the third term yields the largest
positive values. Under hypothesis a, = 1, the second
term is much more significant than the third term, so
that smaller values of id will become more probable.
When a, = 0, the second term in (17) is zero, therefore
the described modification of the phase densities will
tend to increase the third term and thereby id , The tails
outside the interval [ O , h ) will only affect the
estimator accuracy when the error probability is very
high.
The gmethod is applied to the AWGN component, n, ,
hence reducing the IS parameter optimization problem
to one dimension: the variance of the modified phase
densities. The function defined in Section IIC becomes
E2
2
 (M1)E
11
where a = 1 for the ON hypothesis and a = 1 for the
OFF hypothesis. The weighing function and its two first
derivatives can be easily found analytically.
Let us first consider a 4channel WDM router w i t h a
symmetric threshold setting. In Fig. 2 we show the
BER, for a particular value of the receiver noise
variance U:, as a function of the crosstalktosignal
ratio XSR=lOxlog~. The different curves were
obtained w i t h the Gaussian approximation (18), the
Chernoff bound (19), and our IS techniques. All the IS
values shown where obtained with the same rate factor
Se, and yielded an accuracy better than k 3% for 95%
confidence level. A runtime of 10 seconds per value on
a Pentium PC was sufficient to achieve this accuracy.
This high accuracy is maintained through low BER
values, indicating that the IS strategy is close to the
optimum.
10l'
50
4 5
40 35 30
XSR (dB)
25 20 15
Fig. 2. BER curves obtained with the Gaussian
approximation in [2] (GA), the Chernoff bound in [ 181
(CB), and our adaptive IS techniques.
As expected, the Gaussian approzmauon results shown
in Fig. 2 coincide w i t h the IS results for very low
crosstalk levels. The lack of tightness of the Chernoff
bound can be observed: this upper bound is more than
one order of magnitude above the true BER at practical
crosstalk levels (around XSR = 25dB ). The practical
implication of the results in Fig. 2 is that our techniques
allow the network designer to employ optical cross
Page 5
2349
connects with almost twice as large crosstalk levels
than those predicted by the approximation methods.
The performed IS experiments indicated that, with the
symmetric threshold setting used, the probability of
error for the OFF signal hypothesis is much smaller
than that of the ON case. Therefore, threshold
optimization can he expected to improve the BER
significantly. We find the optimal threshold for a given
XSR and AWGN level by applying the technique
described in Section ILD.
The impact of the threshold setting can be appreciated
in Fig. 3. Shown in this figure are BER curves for
symmetric threshold as well as for optimal thresholds
obtained at three XSR values: 20 dB, 25 dB, and 30
dB. The number of channels and the AWGN level are
the same as in Fig. 2. The influence of the threshold
setting is quite significant. We also ohserve that the
tolerable crosstalk level increases further by 3 dB, for a
wide range of XSR values. This is equivalent to a 5 dB
improvement with respect to the value predicted by the
Chemoff hound.
An important parameter in WDM networks is the
number of channels. In Fig. 4 we can appreciate the
power penalty due to the introduction of an additional
channel. The curves were obtained with XSR = 25 dB
and the threshold being optimized at SNR = 22 dB for
each of the curves. In this example, the introduction of
a fifth channel requires additional SNR of ahout 1 dB to
maintain the BER at IO'.
Fig. 3. Effect of threshold optimization. The threshold
was optimized at the indicated XSR values.
, o . , s 
18
19
20
21
22
23
SNR (dB)
Fig. 4. Impact of the number of channels.
I
V. CONCLUSIONS
We investigated the performance degradation in a
WDM network due to crosstalk in an AWG working as
an optical crossconnect. Worstcase analysis was
carried out and, in contrast with the approximation
methods in the literature, all the beatnoise terms in
(17) were included.
Appropriate IS strategies were developed that give
accurate BER estimates in short simulation runtimes.
After conditioning with the gmethod, the optimization
of IS parameters was done using stochastic Newton
recursions. BER estimates were obtained with high
accuracy in 10 seconds runtime, for practical values OF
system parameters. The IS results indicate that, at
practical XSR levels, more than twice as much crosstalk
can be tolerated than predicted by the approximation
methods.
Because both input signal hypotheses are included,
different threshold settings could he considered, which
turned out to have a strong impact on the system
performance. Due to the effectiveness of the IS
techniques, a minimum search algorithm can he used
along with IS to perform threshold optimization. This
resulted in a further relaxation of 3 d B for the crosstalk
requirements.
Finally, the IS techniques also proved to be useful for
the determination of the power penalty due to the
introduction of additional WDM channels.
ACKNOWLEDGEMENTS
This work was supported in part by the Netherlands
Organization for Scientific Research (NWO).
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