Relationship of Q penalty to eye-closure penalty for NRZ and RZ signals with signal-dependent noise

Abstract

The relationship between Q penalty (QP) and eye-closure penalty (ECP) is examined for distorted signals in the presence of signal-dependent noise. A simple model is developed to describe the behavior of return-to-zero (RZ) modulation formats and is compared with a model for nonreturn-to-zero (NRZ) formats. The accuracy of the analysis is investigated with extensive simulations, and the numerical results from analysis and simulation are found to be in generally good agreement. Experimental measurements of distortion caused by uncompensated dispersion also show agreement with the simulation results and model predictions. The simplified models allow a means to budget QPs from distortion effects in a straightforward manner during network design for different modulation formats. The analysis predicts a smaller Q penalty as a function of ECP for RZ modulation formats in comparison with NRZ and smaller relative penalties for RZ formats with narrower pulsewidths.

Full text

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 6, JUNE 2005
2031
Relationship of ? Penalty to Eye-Closure Penalty for
NRZ and RZ Signals With Signal-Dependent Noise
John D. Downie, Member, OSA
Abstract—Therelationshipbetween
sure penalty (ECP) is examined for distorted signals in the pres-
ence of signal-dependent noise. A simple model is developed to
describe the behavior of return-to-zero (RZ) modulation formats
and is compared with a model for nonreturn-to-zero (NRZ) for-
mats. The accuracy of the analysis is investigated with extensive
simulations, and the numerical results from analysis and simula-
tion are found to be in generally good agreement. Experimental
measurements of distortion caused by uncompensated dispersion
also show agreement with the simulation results and model predic-
tions. Thesimplified modelsallowameansto budgetQPs fromdis-
tortion effects in a straightforward manner during network design
fordifferentmodulationformats.Theanalysispredictsasmaller
penalty as a function of ECP for RZ modulation formats in com-
parison with NRZ and smaller relative penalties for RZ formats
with narrower pulsewidths.
penalty(QP)andeye-clo-
Index Terms—Distortion, eye-closure penalty (ECP),
(QP), signal-dependent noise (SDN).
penalty
I. INTRODUCTION
N
mission systems. It is usually based on the split-step Fourier
propagation technique [1]. While accurate, this type of simu-
lation can also be lengthy and computationally expensive with
the presence of noise from optical amplifiers in the system. It is
for this reason that it is sometimes desirable to obtain estimates
of the effects of impairments that cause signal distortion by the
relatively simple evaluation of the eye closure in the absence
of noise. Signal distortion can be assessed in this manner by
the simpler propagation of optical channels without noise for
impairments such as fiber dispersion and without any actual
distance propagation at all for some effects such as optical fil-
tering. In fact, the effects and impairments of passage through
multiple optical filters in a transparent optical network have
been extensively studied with simulation and analysis in the
recent past by evaluation of the eye-closure penalty (ECP)
[2]–[8]. In most of these cases, the results have been either left
in terms of the ECP or equated to
penalty (PP) [8] for signal-independent noise.
In general, however, the relationship between QP and ECP
depends strongly on the noise statistics of the signal at the re-
ceiver. For a detected signal that is dominated by thermal noise
in the photodetector, it is straightforward to show that these
UMERICAL simulation is an effective and well-used
means of predicting the performance of optical trans-
penalty (QP) [2] or power
Manuscript received August 27, 2004; revised November 23, 2004.
The author is with Corning, Inc., Science and Technology, Corning, NY
14831 USA.
Digital Object Identifier 10.1109/JLT.2005.849899
penalties are indeed essentially equal. However, in recent work,
it has been demonstrated experimentally that for nonreturn-to-
zero (NRZ) signals in the presence of signal-dependent noise
(SDN) such as produced in an optically preamplified receiver,
the QP is not necessarily linearly dependent on ECP and is
also greater than the ECP [9], [10]. In this case, the greater
signal–spontaneous beat noise of the signal “zero” values when
in a distorted state (because they have a larger value than in the
undistorted state) causes a larger effect to the signal
causedsimplybyasmallereyeopening.Thisproducesagreater
QP than ECP for signals with SDN, and this fact must be ac-
counted for in estimates of the system performance when as-
sessing the impact of impairments such as optical filtering or
dispersion. This effect has been previously noted in models of
the PP for distorted NRZ signals in the presence of signal-de-
pendent noise [11]–[13].
This paper describes an investigation of the relationship be-
tween QP and ECP for signal-dependent-noise-dominated sys-
tems, which is first derived for NRZ signals with a simplified
model and is then extended to return-to-zero (RZ) modulation
formats. A model for RZ signals is derived based on Gaussian
pulse shapes that differs significantly from the NRZ theoret-
ical analysis. While the model developed in [8] also applied to
RZ signals, it was specific to linearly chirped pulses and did
not extend the results to finding the
pendent noise. After the model development, computer simula-
tion is then applied to evaluate the viability of these models for
signal distortions caused by uncompensated chromatic disper-
sion and by spectral filtering due to passage through multiple
optical filters that might represent a train of optical add–drop
multiplexers (OADMs) in an optical network. Three different
RZ pulse formats are studied, and we find significant agreement
between the model predictions of QP and the results obtained
from the simulation experiments, confirming the validity of the
analysis over certain penalty ranges. Finally, experimental re-
sults are presented for both NRZ and RZ signals with distor-
tion caused by uncompensated dispersion and compared with
the models.
The remainder of this paper is divided into three sections. In
Section II, the analytical expressions relating QP to ECP are de-
rived for NRZ and RZ signal formats. The expressions are com-
pared for different RZ formats, and a boundary for an extreme
case is also determined. In Section III, the simulation experi-
ments are described and the results presented in comparison to
the analytical predictions. The laboratory experimental results
arealsogiveninthissection.Theworkissummarized,andsome
conclusions drawn in Section IV.
than that
or PP with signal-de-
0733-8724/$20.00 © 2005 IEEE

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2032JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 6, JUNE 2005
Fig. 1.
diagram and eye-closure definition.
Example illustration of a noiseless distorted signal electrical eye
II. ANALYTICAL DERIVATION OF PENALTY EXPRESSIONS
We beginbynoting thedefinitionsfor eye closureand
eye-closure (or eye-opening) parameter is the difference in the
absenceofnoisebetweenthe“ones”valueandthe“zeros”value
atthebit center,where thisdifferenceisusually maximum.This
difference is defined to be between the innermost rails of the
signal electrical eye diagram and is given simply as
. The
Eye Closure (EC)
(1)
where
eye center, and
rail. As defined in this paper, the rails in an eye diagram rep-
resent the distinct signal trajectories and voltage levels at the
center ofa noiseless eye,due to distortionand bit patterndepen-
dencies. An example of a noiseless distorted eye is illustrated
schematically in Fig. 1.
Within the framework of the Gaussian approximation for
signal power distribution near the optimal decision level, the
effective
factor is defined as
is the voltage level of the minimum “ones” rail at the
is the voltage level of the maximum “zeros”
(2)
where
tion at the eye center,
“zeros” distribution, and
deviations of the “ones” and “zeros” distributions, respectively.
These effective signal parameters are not necessarily the actual
histogram values of the entire “ones” and “zeros” distributions
but instead represent the values that fulfill the relationship be-
tween effective
factor and bit-error rate (BER) at the optimal
decision level [14]
is the effective average voltage of the “ones” distribu-
is the effective average voltage of the
andare the effective standard
BER
It has been previously demonstrated that the statistics of the
rails closest to the decision threshold level generally dominate
theBERandthus
fordistortedsignals[15].Withthisassump-
tion, we can create a simple model of the signal distribution as
if all bits have the means and standard deviations of these in-
nermost rails. The numerator of
as being equivalent to the eye closure (EC) or
.
can therefore be regarded
and
TheQPsandECPsaredefinedastheratiosofthesequantities
in the absence of a distorting impairment to the quantities in the
presenceofthedistortion.Thus,QPisexpressedingeneralform
as
QP
(4)
where the subscript “ ” represents the undistorted state of the
signal, and the subscript “ ” represents the distorted state of the
signal. An assumption underlying (4) is that the average signal
powers are the same for the undistorted and distorted signal
states. The ECP is written generally as
ECP(5)
where it is again assumed here that the two states have equal
average signal powers. QP and ECP then have the general rela-
tionship of
QP
ECP(6)
Note that if the noise is independent of the signal such as
for thermal detector noise, the noise variances will be identical
in the undistorted and distorted states. Then, the QP simply re-
ducestotheECPin(6),andQP
dependent noise, as mentioned previously. This should be true
for any modulation format.
Now let us consider NRZ signals with signal-dependent
noise. We follow an approach previously used to calculate the
PP for distorted signals with SDN [11], [12] and apply it to
calculate the QP. The approach treats signal distortion effects
as if they were caused by a finite extinction ratio. The QP is
defined as the ratio of the
extinction ratios. We define a finite-voltage extinction ratio
corresponding to a distorted signal as
ECPforthecaseofsignal-in-
values with infinite and finite
(7)
where
. For the infinite extinction ratio
. For an NRZ signal with equal probability of “ones” and
“zeros,” the average voltage can be reasonably approximated as
, and thus
(8)
Thus, for infinite extinction ratio (undistorted state) and finite
extinction ratio (distorted state), we can write straightforward
expressions as
(9)
(10)
and
(11)

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DOWNIE: RELATIONSHIP OF QP TO ECP FOR NRZ AND RZ SIGNALS WITH SIGNAL-DEPENDENT NOISE2033
For SDN such as signal–spontaneous beat noise caused by
optical preamplification, it is well known that the variance of
the noise is proportional to the optical power, and thus it is
alsoessentiallyproportionaltotheoutputvoltage.Using(4)and
(7)–(11), we can then write the
penalty as
QP
(12)
where
is a proportionality constant. This reduces to
QP
(13)
Plugging (9)–(11) into the expression for ECP in (5), we find
ECP
(14)
and simplification produces the expression
QP
ECP
ECP
ECP
ECP
ECP
(15)
as the relationship between ECP and QP for NRZ signals in
the presence of signal-dependent noise. In principle, this rela-
tionship strictly holds for the penalty caused by finite extinction
ratio but is an approximation for NRZ signal distortion caused
by other means, as was pointed out in [12] for the related PP
expression. The validity of the expression for general distortion
primarily rests on the assumption that the eye is closed sym-
metrically from the top and bottom. It is also noted that (15)
here can be related to [12, eq.(2)] by recognizing that for SDN,
the PP
QP in linear units and the ECP in (15) is the same
as the PP for signal-independent noise (SIN). The relationship
between PP and QP also allows us to equate any results for
QP expressed in 20log() format to the PP, or equivalently, the
optical-signal-to-noise ratio (OSNR) penalty, expressed in the
conventional 10log() format. Equation (15) will be presented in
graphical form along with an analogous expression for RZ for-
mats later in the paper.
For RZ modulation formats, a slightly different approach is
taken to estimating the QP as a function of ECP for signals with
SDN. In this case, we treat RZ signal pulses as Gaussian func-
tions and make the assumption that the pulse in the distorted
state is still Gaussian shaped, but with wider pulsewidth and
lowerpeakpowerthanintheundistortedstate.Thisis,ofcourse,
a somewhat oversimplified approximation to general signal dis-
tortion caused by impairments such as dispersion and spectral
filtering.Underthisassumptionandthatofequalpulseenergies,
we can express the functional forms of the undistorted and dis-
torted RZ voltage pulses as
(16)
and
(17)
where
pulse types, and
Now we can calculate QP and ECP using (4) and (5) for a
general RZ modulation format. To begin, the pulse values of the
“ones” bits at the maximum eye-opening position are taken as
andrepresent the
is a constant.
pulsewidths of the two
(18)
and, similarly
(19)
assuming that the pulse is centered at
“zeros” bits are approximated as
. The values of the
(20)
and
(21)
where
of the expressions for
the maximum “zeros” rail at the eye-opening position results
from the bit sequence 101, i.e., where the zero is surrounded by
“ones”pulses.Inthiscase,wemightreasonablyexpectthevalue
ofthe“zero”tobetheadditionoftheGaussianpulsevaluesfrom
both sides. By substituting (18)–(21) into (5), we calculate ECP
as
represents a bit period. The coefficient of 2 in front
and reflects the assumption that
ECP
(22)
Similarly, the
penalty can be expressed from (6) as
QP
ECP
(23)
and finally
QP
ECP
(24)
In (22) and (24),
torted pulsewidth, while
eter whose value dictates the eye-closure and
is a constant that describes the undis-
is the distorted pulsewidth param-
penalties.

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2034 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 23, NO. 6, JUNE 2005
Fig. 2.
modulation formats in an optical back-to-back condition with discrete points
representing the best Gaussian fits to each pulse shape.
Noiseless electricaleye diagrams of(a)CSRZ,(b) RZ50,and (c)RZ33
Before continuing, we consider the extreme RZ case that
sets a lower boundary to the function of QP as a function of
ECP. Consider an RZ format with a pulsewidth narrow enough
(short enough duty cycle) such that the “zero”’s value at the
eye-opening location for both the undistorted and distorted
states of the signal is approximately zero. That is, we assume
that
(25)
Fig. 3.Model results for QP as a function of ECP.
implying that the signal distortion is manifested through the
“one”’s value without appreciably changing the “zero”’s In this
case, the ECP is given simply as
ECP
(26)
and the QP then becomes
QP
ECP ECP(27)
Combining (26) and (27), we then see that in this extreme
case the QP is related to the ECP as
QP
ECP(28)
or expressed in units of decibel
QP dB
ECP dB(29)
In this case, a QP of 0.5 dB results from each decibel of ECP
if both penalties in decibel units are calculated with the same
coefficient in front of the logarithm. For general RZ signals, the
relationship between QP and ECP as expressed by (22) and (24)
ensures a QP dependence larger than the extreme case, or lower
bound.
To gain an understanding of the RZ model results given in
(22) and (24), we consider three different types of RZ mod-
ulation formats. The three formats are carrier-suppressed RZ
(CSRZ), 50% duty cycle RZ (RZ50), and 33% duty cycle RZ
(RZ33). To determine the Gaussian parameter
scribes these formats, a 10-Gb/s signal of each type was gener-
ated within a optical system simulation tool. The electrical eye
time-domain data for each was captured in an optical back-to-
back condition with no noise, and a Gaussian function as de-
scribed by (16) was fit to the eye diagram. The electrical filters
used in the receiver in the simulations were fifth-order Bessel
functions with bandwidths of 8 GHz for the CSRZ and RZ50
signals and 11 GHz for the RZ33 signal. The three RZ format
eye diagrams with accompanying data points representing the
best Gaussian fit are shown in Fig. 2. For a 10-Gb/s signal bit
that best de-

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DOWNIE: RELATIONSHIP OF QP TO ECP FOR NRZ AND RZ SIGNALS WITH SIGNAL-DEPENDENT NOISE2035
Fig. 4. Simulation experiment setup to measure ? results as a function of uncompensated fiber dispersion.
rate, the
CSRZ, RZ50, and RZ33 were 40, 34, and 27 ps, respectively.
The relationships between QP and ECP for each of the three
RZ formats can be determined according to the simple model
described previously by (22) and (24), as parameterized by the
distorted pulsewidth
. As discussed previously, the values of
were determined by fitting Gaussian functions to the undis-
torted pulse shapes. The parameter
, and ECP and QP were calculated. The QP is shown
as a function of the ECP in Fig. 3 for the three RZ formats. Also
shown in the figure is the QP function for the NRZ format as
given in (15), and the lower bound for the RZ formats described
by(29).TheresultsshowthatallthreeRZformatshaveasignif-
icantly weaker dependence of QP on ECP than the NRZ format.
Consistent with this, RZ formats with a larger duty cycle show
a stronger dependence on ECP than shorter duty cycle formats.
Thus,foragiveneye-closurevalue,CSRZhasagreaterQPthan
RZ50, which in turn has a greater QP than RZ33. We also note
that the QP function for RZ33 is very close to the theoretical
lower bound for smaller values of ECP less than 2 dB or so. The
data in Fig. 3 are shown in units of decibels in 20log() format
for both QP and ECP.
pulsewidth values found by the fitting process for
was then varied with
III. SIMULATION AND LABORATORY
EXPERIMENTS AND RESULTS
A. Simulations and Results
In order to explore the validity of the model results shown
in Fig. 3, a series of simulation experiments were conducted
with an opticalfibercommunicationsystem modelingtool.Two
different impairments were studied that cause signal distortion:
1) chromatic dispersion and 2) spectral filtering from passage
through multiple optical filters. For each impairment type, sim-
ulations were performed with all three RZ modulation formats
as well as NRZ.
data was collected for the systems with three
different nominal OSNR values by use of the
functioninthemodelingtool.ECPdatawerecollectedbytrans-
mission of the signals through identical systems without noise,
and the electrical eye data was analyzed to calculate the ECP. A
schematic diagram of the optical system set up in thesimulation
environment to model the performance as a function of disper-
sion is shown in Fig. 4.
As shown in the figure, the transmitter consisted of a contin-
uous-wave(CW) laseratwavelength 1550nm, a pseudorandom
bit sequence (PRBS) generator, and two Mach–Zehnder (MZ)
modulators for RZ signals (one modulator for NRZ signals).
The first MZ modulator was modulated in NRZ format with
the PRBS pattern at a bit rate of 10 Gb/s, while the second MZ
device was sinusoidally driven and carved out the RZ pulses of
interest. Three different RZ pulse types were studied: CSRZ,
calculation
RZ50, and RZ33. The RZ signal was propagated through a
piece of fiber to produce a given state of dispersion. The fiber
had a dispersion parameter
fects were precluded by low channel launch powers ( 0 dBm)
and the short fiber lengths required ( 14 km). The signal was
then passed through a variable optical attenuator (VOA) and
erbium-doped fiber amplifier (EDFA) with the VOA set to
control the OSNR of the signal. The receiver was represented
by an optical filter and optical-to-electrical (O/E) conversion
with a given electrical filter bandwidth. The optical filter was a
fourth-order Butterworth filter with 40-GHz 3-dB bandwidth.
Theelectrical filtersusedwere thesameasdescribedpreviously
for the RZ signals, and the electrical filter for NRZ signals was
also a fifth-order Bessel filter with 8.0-GHz bandwidth. The
length of the PRBS used in the
Signal OSNR values of 20, 23, and 26 dB were investigated.
Uncompensated accumulated fiber dispersion values of 0 to
900 ps/nm were simulated in the experiments by varying the
length of the fiber for RZ signals, and up to 1400 ps/nm for
NRZ signals. The simulation tool used is a general optical
system modeling tool that employs a conventional split-step
Fourier method for signal propagation [1] and optical and
electrical filtering applied in the frequency domain.
The
calculation used in the modeling tool essentially de-
termines the means and standard deviations of the signal power
for the “ones” and “zeros” in the bit sequence and implicitly
assumes Gaussian noise statistics. A pattern dependence fea-
turewasemployedinthecalculationsthateffectivelyrecognizes
the rail structure of the eye and calculates a
weighted average of the
values determined from the different
rails. This approach using patterning effects more closely re-
flects the measurement of
in a laboratory experiment where
is determined from BER measurements in the Gaussian tails
of the signal distribution and is also generally consistent with
our modeling assumption that the innermost rails dominate the
BER and
values at the optimal decision threshold. We will
discuss implications of this assumption subsequently.
The setup used to calculate the ECP was identical to that
shown in Fig. 4, except that the EDFA was removed. The signal
was noiseless in this case. The definitions of the optical filter
and the electrical filter in the O/E converter for ECP data were
the same as those used in the
The PRBS length used when generating the ECP data with the
electrical eye probe was 512 b. The ECP was calculated with
reference to the eye of a back-to-back system and normalized to
eliminateanycontribution from simplesignal loss.Thisensures
the ECP calculated was due to the signal distortion alone.
The second signal distortion impairment studied here was
spectral filtering produced by passage of the signal through a
series of optical filters that might represent a series of OADM
100 ps/nm/km. Nonlinear ef-
measurements was 4096 b.
value that is a
probe when measuringdata.