We derive joint probability density functions (JPDFs) for two adjacent data from directdetection optical receivers in dense wavelengthdivision multiplexing systems. We show that the decision using two data per bit can increase the receiver sensitivity compared with the conventional decision. Our JPDFs can be used for softwaredefined optical receivers enhancing the receiver sensitivities for intensitymodulated channels.
I. INTRODUCTION
Recently, softwaredefined networks that can provide practical ways of handling multistandard environments have been investigated extensively
[1
,
2]
. In a similar way, softwaredefined optical receivers are under development in optical fiber communication systems to satisfy many different kinds of modulation formats and baud rates
[3
,
4]
. Although coherent optical systems have become more practical than during their early stages, currently deployed optical transmission systems are mostly the intensitymodulated and directdetection (IM/DD) type.
Conventional optical receivers for IM/DD systems use a single data per single bit for the decision
[5]
and require clock recovery circuits. If we could use multiple data for the decision, we do not need the clock recovery circuits
[4]
while enhancing the receiver sensitivities. However, there are no analyses, to our knowledge, about the correlations between two adjacent data from an optical receiver.
In this paper, we propose to use correlations of two adjacent data for the decision in directdetection optical receivers. Using the receiver eigenmodes
[6

12]
, we derive the joint characteristic function (JCF) for two successive data from an optical receiver and evaluate the corresponding joint probability density function (JPDF)
[13]
. The receiver eigenmodes can describe accurately the effects of the amplified spontaneous emission (ASE), received optical waveforms, and shapes of optical and electrical filters within the receiver
[7]
. Recently, receiver eigenmode contributions have been analyzed as a function of time for the optical receiver output
[12]
. It has been found that, in conventional dense WDM systems
[14

17]
, where the channel spacing is comparable to the bit rate, the lowestorder (0th) receiver eigenmode contributes dominantly. We will use this fact to find the correlations of two adjacent data and the threshold line for the decision to get higher sensitivities than conventional receivers.
II. JOINT CHARACTERISTIC FUNCTION
Before the derivation of the JPDF, we derive the JCF first. We will consider two adjacent samples at times
t
_{1}
and
t
_{2}
. Considering only the polarization component parallel to the received optical signal, we can find the voltage at
t
_{1}
, denoted as
y
_{1}
, as
[7]
where
S_{m}
(
t
_{1}
) and
N_{m}
(
t
_{1}
) are complex numbers that represent the signal and the noise amplitudes, respectively, for the
m
th receiver eigenmode.
λ_{m}
is the
m
th eigenvalue and
k
is a proportional constant. For all
m
, the real and the imaginary parts of
N_{m}
(
t
_{1}
) are mutually independent zero mean Gaussian random variables having an identical variance of
In conventional dense WDM systems, the lowestorder receiver eigenmode contributes dominantly
[12]
and we can approximate
We normalize
y
_{1}
such that it becomes a power ratio such that
where the denominator is the noise power per receiver eigenmode per polarization. Similarly, we have the normalized voltage at
t
_{2}
as
N
_{0}
(
t
_{1}
) and
N
_{0}
(
t
_{2}
) are related by a correlation function as
[12]
The correlation function,
C
(
t
_{1}
−
t
_{2}
), relates the
0
th order eigenmode amplitudes at different times, which is given by
where
ø
_{0}
(
ω
) is the
0
th order eigenfunction. Let’s denote the real part of
N
_{0}
(
t
_{1}
) and
N
_{0}
(
t
_{2}
) as
N
_{1r}
and
N
_{2r}
, respectively, and the imaginary part of
N
_{0}
(
t
_{1}
) and
N
_{0}
(
t
_{2}
) as
N
_{1i}
and
N
_{2i}
, respectively. The covariance matrix for
N
_{1r}
,
N
_{1i}
,
N
_{2r}
, and
N
_{2i}
can be found as
[13]
where
C_{r}
and
C_{i}
are real and imaginary parts of
C
(
t
_{1}
−
t
_{2}
), respectively. The inverse matrix of Σ is
where
For the Gaussian vector,
X
=
(N
_{1r}
,
N
_{1i}
,
N
_{2r}
,
N
_{2i}
)
T
, where
T
means the transpose, its JPDF is
[13]
Thus the JPDF for
N
_{1r}
,
N
_{1i}
,
N
_{2r}
, and
N
_{2i}
can be written as
The JCF for
and
can be written as
where
E
{⋅} is the ensemble average. The integrations can be done exactly using the Gaussian integration formula,
where
α
and
β
are constants with Re α > 0 . Thus we find
where
S
_{1r}
and
S
_{1i}
are real and imaginary parts of
S
_{0}
(
t
_{1}
), respectively. Also,
S
_{2r}
and
S
_{2i}
are real and imaginary parts of
S
_{0}
(
t
_{2}
), respectively. If we include the other polarization, where only the ASE components are present, we obtain the JCF exactly as follows:
III. JOINT PROBABILITY DENSITY FUNCTION
The JPDF of
and
can be found from the twodimensional Fourier transform
When the signal is absent, the JCF becomes
which gives the JPDF of
and
in an exact form
where
I
_{1}
is the modified Bessel function. We have used the following relation (
x
,
M
> 0) :
If the two data are independent, we find from (22)
This JPDF is just a product of each sample’s PDF. To find (24), we have increased
G
satisfying the
G
< 1 condition. When the signal is present, the JPDF can be obtained numerically. Note that the integration over ξ
_{1}
can be done exactly to yield
where
The remaining integral can be done using a fastFouriertransform algorithm.
We may use the asymptotic form for the Bessel function,
, and evaluate the integral of (25) using the method of steepest descents [18], which gives
The complicated expression of (27) is valid when the correlation is low (
G
≈1). We perform numerical evaluations of (25) to find BER throughout.
IV. RESULTS AND DISCUSSION
For our analysis, we use a Gaussian optical receiver
[11
,
12]
, where both optical and electrical filters are Gaussian. We assume
A
_{1}
=
A
_{2}
. We choose the 3dB bandwidth of the optical filter two times larger that of the electrical filter. The 3dB bandwidth of the electrical filter is 0.7 times the bit rate. Then the correlation function (6) becomes
C
(τ) = exp[−(0.7π)
^{2}
2
q
/{(1−
q
^{2}
)ln2}(τ/
T
)
^{2}
]
[12]
. We have
q
= 0.268 according to the bandwidth ratio between the optical and the electrical filters.
T
is the bit period. The two sampling points are
T
/4 apart and we obtain
G
= 0.396. We assume no phase changes of the signal between two sampling points, which gives
Figure 1
shows two JPDFs evaluated numerically along the
=
line for
A
_{1}
= 40.65 and for
A
_{1}
= 0 simultaneously. The former corresponds to the mark and the latter corresponds to the space. The JPDF for the space has been scaled down by the factor of 10 compared with the JPDF for the mark.
Joint probability density functions (JPDFs) evaluated numerically along the = line for the mark (A_{1,2}= 40.65) and for the space (A_{1,2} = 0). The JPDF for the space has been scaled down by the factor of 10 compared with the JPDF for the mark. The two sampling points are T/4 apart, where T is the bit period. A Gaussian optical receiver is used with G = 0.396.
In
Fig. 2
, we show the foregoing JPDFs in a 3dimensional way. It has been obtained by adding the JPDFs’ functional values to
after multiplying a constant 4×103. Only the points, where at least one of the JPDFs is larger than 1010, are shown. The JPDF for the space has been scaled down by the factor of 10 here also. Since
A
_{1}
=
A
_{2}
, the JPDFs, including (22), are symmetric with respect to the =
=
line. Note that
f
(
,
) = 0 when at least one of
and
is zero. Along the
=
line, the width of each JPDF increases as
increases. In conventional optical receivers, where a threshold voltage is used, we have
BER
= (2
A
_{1}
/π)
^{1/4}
exp(−
A
_{1}
/4)/4
[11]
, which is 2.2×10
^{5}
. In our case, there is a threshold line as is shown in
Fig. 2
where both JPDFs have the same value. The area of each JPDF beyond that line contributes to the BER
[5]
which is evaluated numerically as 6.8×10
^{6}
. This corresponds to about 0.5 dB enhancement in
A
_{1}
compared with the conventional BER of 2.2×10
^{5}
.
A
_{1}
and
A
_{2}
are actually signaltonoise ratios
[11]
. Thus we can have higher system margins using the correlation between the data from the directdetection optical receiver.
Plot of the JPDFs of Fig. 1 in the (, ) plane. We have added the JPDF’s functional value to after multiplying a constant 4×10^{3}. We show only the points where at least one of JPDFs is larger than 10^{10}. The JPDF for the space has been scaled down by the factor of 10.
If the two sampling points are
T
/2 apart with all other parameters fixed, G increases to 0.866. The JPDFs in this case are shown in
Fig. 3
, which become more circularly symmetric owing to the reduced correlation. The BER is 2.6×10
^{7}
and the enhancement in
A
_{1}
increases to about 1.6 dB. If the two sampling points are 3
T
/4 apart with all other parameters fixed,
G
increases to 0.989. The BER is 1.3×10
^{8}
and the enhancement in
A
_{1}
is about 2.4 dB. If the received signal has a different phase between two sampled points, this will decrease
H
in (18). In this case, our numerical analyses show that the BER decreases as the phase difference increases. Thus our BER values can be reduced introducing the phase change within a bit.
Plot of the JPDFs in (, ) plane, where the two sampling points are T/2 apart with all other parameters the same as Fig. 2. G = 0.866. The JPDFs become more circularly symmetric than those of Fig. 2 owing to the reduced correlation.
In order to use the threshold line for the decision, we need analogtodigital converters (ADCs) and digital signal processing (DSP) circuits instead of conventional Dflipflop type decision. The speed of ADC and DSP circuits has been increased remarkably up to coherent 200 Gb/s per channel during recent years
[19]
. Realizations of softwaredefined optical receivers using these devices are now technically feasible and our JPDFs can be used to upgrade their capabilities for IM/DD channels.
V. CONCLUSION
We have derived the JPDFs for two voltage data from directdetection optical receivers in dense WDM systems. We have shown that, with our JPDFs, we can reduce the BER and enhance the receiver sensitivities by 0.5 dB ~ 2.4 dB. Our decision method can be used for softwaredefined optical receivers supporting both coherent and intensitymodulated channels simultaneously.
Acknowledgements
The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2012 and this research was also supported partly by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2013R1A1A2012918).
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