Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system S. Zhang1, P. Y. Kam1, J. Chen2, and C. Yu1,2,* 1
Department of Electrical & Computer Engineering, National University of Singapore, 117576, Singapore. 2 A*STAR Institute for Infocomm Research (I2R), 138632, Singapore. *Corresponding author:
[email protected]
Abstract: A novel decision-aided maximum likelihood (DA ML) technique is proposed to estimate the carrier phase in coherent optical phase-shiftkeying system. The DA ML scheme is a totally linear computational algorithm which is feasible for on-line processing in the real systems. The simulation results show that the DA ML receiver can outperform the conventional Mth power scheme, especially when the nonlinear phase noise is dominant. 2008 Optical Society of America OCIS codes: (060.2330) Fiber optics communications; (060.1660) Coherent communications .
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18.
A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. 23, 115– 130 (2005). E. Ip and J. M. Kahn, “Feedforward Carrier Recovery for Coherent Optical Communications,” J. Lightwave Technol. 25, 2675–2692 (2007). M. Nazarathy, X. Liu, L. Christen, Y. K. Lize, and A. Willner, “Self-coherent multisymbol detection of optical differential phase-shift-keying,” J. Lightwave Technol. 26, 1921–1934 (2008). R. No´e, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett. 17, 887–889 (2005). L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne phase-shift-keying systems: past challenges and future opportunities,” J. Lightwave. Technol. 24, 4876–4884 (2006). D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. 24, 12–21 (2006). J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15, 1351–1353 (1990). H. Kim and A. H. Gnauck, “Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise,” IEEE Photon. Technol. Lett. 15, 320–322 (2003). S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Receiver sensitivity improvement using decision-aided maximum likelihood phase estimation in coherent optical DQPSK system,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2008), paper CThJJ2. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, New York, 2002). K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave Technol. 26, 1817–1822 (2008). K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer, New York, 2005). P. Y. Kam, “Maximum-likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. COM-34, 522–527 (1986). S. Zhang, P. Y. Kam, J. Chen, and C. Yu, “Adaptive decision-aided maximum likelihood phase estimation in coherent optical DQPSK system,” in Proceedings of Opto-Electronics and Communications Conference (2008), paper TuA-4. X. Wei, X. Liu, and C. Xu, “Numerical simulation of the SPM penalty in a 10-Gb/s RZ-DPSK system,” IEEE Photon. Technol. Lett. 15, 1636–1638 (2003). K.-P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave. Technol. 22, 779–783 (2004). E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” J. Lightwave Technol. 25, 2033–2043 (2007). X. Liu, X. Wei, R. E. Slusher, and C. J. McKinstrie, “Improving transmission performance in differential phase shift-keyed systems by use of lumped nonlinear phase-shift compensation,” Opt. Lett. 27, 1351–1353 (2002).
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19. K. Kikuchi, “Electronic post-compensation for nonlinear phase fluctuations in a 1000-km 20-Gbit/s optical quadrature phase-shift keying transmission system using the digital coherent receiver,” Opt. Express 16, 889-896 (2008).
1. Introduction Spectrum efficiency (SE) has become an increasingly important factor in the design of an optical transmission system, especially when the data rate of a single channel is beyond 40Gb/s or even 100Gb/s. For example, signals with higher SE would lead to less crosschannel-talk to neighboring channels in a 100GHz- or even 50GHz-channel-spacing, wavelength-division-multiplexing (WDM) system since the required bandwidth of the signal is reduced. The SE of a system employing on-off keying (OOK) or binary differential phaseshift keying (DPSK) cannot exceed 1 bit/s/Hz per polarization [1]. Recent research has focused on M-ary phase-shift keying (MPSK) or even quadrature amplitude modulation (QAM) with coherent detection, which can increase the SE by log2M [2]. However, one of the major challenges in coherent detection is to overcome the carrier phase noise when using a local oscillator (LO) to beat with the received signals to retrieve modulated phase information. Phase noise can result from lasers, which will cause power penalty to the receiver sensitivity. A self-coherent multisymbol detection of optical differential MPSK is introduced to improve the system performance; however, higher analogto-digital conversion resolution and more digital signal-processing power are required as compared to a digital coherent receiver [3]. Further, differential encoding is also necessary in this scheme. As for the coherent receiver, initially, an optical phase-locked loop (PLL) is an option to track the carrier phase with respect to the LO carrier in homodyne detection. However, an optical PLL operating at optical wavelengths in combination with distributed feedback (DFB) lasers is quite difficult to be implemented because the product of laser linewidth and loop delay is too large [4]. Another option is to use electrical PLL to track the carrier phase after downconverting the optical signal to an intermediate frequency (IF) electrical signal in a heterodyne detection receiver. Compared to heterodyne detection, homodyne detection offers better sensitivity and requires smaller receiver bandwidth [5]. On the other hand, coherent receivers employing high-speed analog-to-digital converters (ADCs) and high-speed baseband digital signal processing (DSP) units are becoming increasingly attractive rather than using an optical PLL for demodulation. A conventional Mth power scheme is proposed in [4,6] to raise the received MPSK signals to the Mth power to estimate the phase reference in conjunction with a coherent optical receiver. However, this scheme requires nonlinear operations, such as Mth power and arctan(·), and the ±2π/M phase ambiguity, which will incur a large latency to the system. Such nonlinear operations may limit the further potential for real-time processing of the scheme. In addition, nonlinear phase noises always exist in long-haul systems due to the G-M effect [7], which severely affect the performance of a phase-modulated optical system [8]. The results in [9] show that such Mth power phase estimation techniques may not effectively deal with nonlinear phase noise. In this paper, we propose a computationally efficient, decision-aided, maximum likelihood (DA ML) phase estimation technique to recover the carrier phase. A real-time structure of the DA ML receiver, to our knowledge, is first proposed to estimate the carrier phase reference online in optical coherent detection. The DA ML scheme is a totally linear computational algorithm which is feasible for on-line processing in the real systems. The simulation results show that the DA ML receiver can outperform the conventional Mth power scheme, especially when the nonlinear phase noise is dominant. This paper is organized as follows. The general received signal model in a coherent optical MPSK transmission system is derived in Section 2. Section 3 provides a detailed derivation of DA ML phase estimation and the real-time DA ML receiver structure for optical MPSK signals. In Section 4, their performances are numerically examined in both a linear phase noise and a nonlinear phase noise system. Finally, Section 5 concludes the paper.
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2. System setup and signal model Figure 1 shows a typical long-haul transmission system with coherent detection, where optical amplifiers (OAs) are periodically employed to compensate for the fiber transmission loss Lf. The gain of OAs is designed to restore the signal to the original level, i.e. G=1/Lf, while amplified spontaneous emission (ASE) noises will be introduced by the OAs along the transmission. LO-ASE beat noise is the dominant noise source at the receiver due to the accumulated ASE noises from each amplifier and the high power of the LO laser. The received signal after NA-span transmission is assumed to be (1) Er (t ) = Pr e j (θ s (t ) +φs (t )) + nɶ x (t ) e jωct x + nɶ y (t )e jωct y
where Pr and θ s (t ) are the power and the phase of the transmitted signals, φs (t ) is the phase noise associated with the source laser, ωc is the carrier frequency of the signal, and x and y are the two orthogonal polarization states of the signal. In this paper, non-return to zero (NRZ) pulse shape is assumed. Both nɶ x (t ) and nɶ y (t ) are the low-pass representation of the ASE noise. The spectral density of the received ASE noise in each polarization is given by [10] S ASE = N A nsp (G − 1)hν
(2)
where nsp and hν are the spontaneous emission factor of the OAs and a photon energy.
Fig.1. A typical long-haul transmission system with a coherent receiver. ADC: Analog-to Digital Converters
In coherent receivers, a LO laser is required to beat with the received signals to retrieve the signal amplitude and phase information. The state of the polarizations (SOP) of Er (t ) and the LO lasers are assumed to be the same by using either polarization-diversity receiver or polarization controller (PC) [6]. ELO (t ) can be described by
j (ωLO t + φLO (t )) ELO (t ) = PLO e x
(3)
where PLO is the power of the LO laser. φ LO ( t ) is the phase noise of the LO, and ω LO is the carrier frequency of the LO laser. In a coherent quadrature receiver as shown in Fig. 1, the transfer matrix of a 2 x 4 90°optical hybrid is given as
1 1 1 1 −1 (4) S= 2 1 j 1 − j Thus, the photocurrents from the upper balanced detectors are the in-phase components of the received signals: #102401 - $15.00 USD
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2 2 1 1 R Er (t ) + ELO (t ) − R Er (t ) − ELO (t ) + ish1 4 4 = R Re Er (t ) ⋅ E *LO (t ) + ish1
iI (t ) =
{
}
= R Pr PLO ⋅ cos(ω IF t + θ s (t ) + φs (t ) − φLO (t ))
{
}
+ R PLO Re nɶ x (t )e j (ωIF t −φLO (t )) + ish1
(5)
where R is the photodetector responsivity and ω IF = ωc − ωLO . And the photocurrents from the lower balanced receiver correspond to the quardrature-phase components of the received signals: 2 2 1 1 iQ (t ) = R Er (t ) + jELO (t ) − R Er (t ) − jELO (t ) + ish 2 4 4 = R Pr PLO ⋅ sin(ωIF t + θ s (t ) + φs (t ) − φLO (t ))
{
}
+ R PLO Re nɶ x (t )e j (ωIF t −φLO (t ) −π / 2) + ish 2
(6)
where ish1 and ish 2 are the shot noises with the two-side power density N sh = eRPLO (7) where e is the electron charge. The two-sided power densities of the LO-ASE beat noises, i.e.,
{
}
{
}
R PLO Re nɶ x (t )e j (ωIF t −φLO (t )) and R PLO Re nɶ x (t )e j (ωIF t −φLO (t ) −π / 2) , are
N LO− ASE = R 2 PLO S ASE ( f )
(8)
The two laser phase noises can be combined into one term as θ (t ) = φs (t ) − φ (t ) , which results from the total 3-dB laser linewidth ∆υ of the LO and transmitter laser. The power LO
jθ ( t )
spectrum of e has a Lorentzian line-shape. The variance of the phase deviation induced in a symbol interval T is given by
σ 2 = 2π∆υT (9) ɶ Let n (t ) be the total noise, i.e., shot noises and LO-ASE noises from two branches. Its power density function is the sum of Eq. (7) and Eq. (8), where thermal noises are neglected due to large LO power. As a result, the input signal after the optical hybrid has the form r (t ) = iI (t ) + j ⋅ iQ (t ) + nɶ (t ) (10) In our paper, we employ a homodyne receiver with DSP carrier phase estimation, ω i.e., IF = 0 , which promises a 3-dB advantage over a heterodyne receiver. This is due to the fact that the bandwidth of the low-pass filter required for heterodyne detection is Rs, whereas one half of this bandwidth, i.e., Rs/2, is enough for the homodyne case [11]. Here, Rs is the symbol rate of the system. The signal power divided by the total noise power in the same band, Rs/2 in the homodyne receiver, gives the SNR per symbol at the detector. The SNR per symbol is related to the number of photons per symbol through SNR sym =
R 2 Pr PLO 2 ⋅ R PLO S ASE ( f ) Rs / 2 2
≈
Ns N A nsp
(11)
since Pr = GN s hν Rs when LO-ASE beat noise predominates in the total noise, and preamplifier with gain G is considered in the coherent receiver. Here, Ns is the average
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number of photons per symbol. The results comply with the claimed ASE-limited receiver sensitivity for quardrature receivers in [12]. The SNR per bit (SNRb) equals (1/ log 2 M ) ⋅ SNR sym , for M bits encoded in one symbol. Eq. (11) shows that the performance of a coherent quardrature receiver with optical preamplifier is degraded by 1/nsp, which conforms to the results in [11]. 3. Decision-aided maximum likelihood detection in optical MPSK system
3.1 Principle of decision-aided maximum likelihood (DA ML) phase estimation Traditionally, carrier phase should be recovered before correctly demodulating the received signals. The DA ML phase estimation is derived to retrieve a phase reference according to the received signals and feedback decisions. Following the received signal in Section 2, the sampled signal can be written as follows, by sampling the recovered signal in Eq. (10) at t =kT : r (k ) = Es m(k )e jθ ( k ) + nɶ (k )
(12)
where k denotes kth sample over time interval [kT, (k+1)T), data
symbol
and
Es
is
the
symbol
energy,
m( k ) ∈ {Ci = exp [ jφi ( k ) ]}
i.e.,
2
R Pr P
LO
.
{2π i / M , i = 0,1,…, M − 1} is the phase modulation, M denotes M-ary PSK,
Here,
is the φi (k ) ∈
θ ( k ) is the phase
noise introduced from the laser phase noise or during fiber transmission, {nɶ ( k )} , samples of nɶ (t ) at t =kT, is a sequence of complex, white, Gaussian random variables with E[ nɶ ( k )] = 0 and E[ nɶ (k ) 2 ] = N 0 , and its power density is given in Eq. (7) and Eq. (8). The conditional probability density function (pdf) of the received signal r ( k ) is given by p ( r ( k ) | θ (k ), m(k )) =
1
π N0
exp( −
r (k ) − m(k )e
jθ ( k )
N0
2
(13)
).
It is noted that no statistical assumption about the phase noise θ ( k ) is made in (13). Since m(k) and θ ( k ) are independent, the maximum a posteriori probability (MAP) estimate of the transmitted symbol and the carrier phase can be developed into maximum likelihood (ML) estimate (optimal over linear channel) when assuming all symbols are equiprobable. In ML algorithm, the phase reference estimate θˆ( k ) at time t=kT is computed over the immediate past L received signals, i.e., r (l ), k − L ≤ l ≤ k − 1 , where L is called the memory length. The likelihood function Λ (θ , k ) is given by the joint pdf p ( r ( k − L ), ⋯ , r ( k − 1) | θ ) . Symbol timing is assumed to be known, and θ is assumed to be time invariant at least over an interval longer than LT. It is reasonable that the phase noise changes slowly compared to the symbol rate in a slow-varying phase noise system. Therefore, the selection of memory length L will affect the performance of our DA ML method, which will be discussed in Section 4. Note that if there is no intersymbol interference (ISI), r (l ) and r ( j ) are independent because of the independence of nɶ (l ) and nɶ ( j ) when i ≠ j . The pdf p ( r (l ) | θ ) is equal to M −1 ∑ i=0 p (r (l ) | θ , m(l ) = Ci ) p (m(l ) = Ci ) . In order to make the derivation explicit, it is possible to arrange
the
constellation
{
points
such
that
Ci
= −C− i
in
MPSK
case,
i.e.,
}
m(k ) ∈ Ci : − M / 2 ≤ i ≤ M / 2, i ≠ 0 . By means of this arrangement of the constellation points, the log-likelihood function L(θ , k ) = ln Λ (θ , k ) can now be expressed as
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M / 2
k −1
L(θ , k ) =
∑ ln ∑ exp(−S ) cosh q (l,θ ) + c i
i =1
l =k −L
i
(14)
where Si =| Ci |2 / N 0 , qi (l ,θ ) = (2 / N 0 ) Re r (l )Ci*e− jθ , and c is a constant independent of θ . It
is now easy to derive the likelihood equation ( ∂L(θ , k ) / ∂θ = 0 ) that leads to the following equation that defines the ML estimate θˆ of θ : k −1
cos θˆ(k )
∑
∑
M /2 i =1
∑
l =k − L
k −1
= sin θˆ(k )
exp(− Si ) sinh qi (l , θˆ(k )) Im r (l )Ci* M /2 ˆ exp(− S ) cosh q (l , θ (k ))
∑
l =k − L
∑
M /2 i =1
i
i =1
i
exp(− Si ) sinh qi (l ,θˆ(k )) Re r (l )Ci* M /2 exp(− S ) cosh q (l , θˆ(k ))
∑
i =1
i
(15)
i
Although Eq. (15) is highly nonlinear and it is hard to get an explicit solution for θˆ(k ) from this equation, approximations can be made in order to make it implementable. Here, high SNR is of most interest in practice. When considering the high SNR limit, decision-feedback provides a nearly optimum implementation of the ML estimator. In the case of binary PSK (M=2), Eq. (15) reduces to k −1 Im r (l )mˆ * (l ) θˆ(k ) = arctan l =kk−−1L Re r (l )mˆ * (l ) l =k − L
∑
(16)
∑
where the Ci* at time t=lT in Eq.(15) is replaced by the receiver decision mˆ (l ) in decisionaided phase estimation Eq. (16). In the case of signal constellations with M >2, for each fixed l in Eq. (16), each of the summations over L is dominated in magnitude by the term whose index corresponds to the decision. As pointed out in [13], the decision-feedback approximation is more accurate for BPSK (M =2) than for M-ary PSK with M >2. In order to totally eliminate the nonlinear operation and phase ambiguity in the phase estimation algorithm, we introduce a complex vector V as k −1
V (k ) ≡
∑ r (l )mˆ (l ) *
(17)
l =k − L
The quantities {Re V (k ),Im V (k )} are the in-phase and quardrature outputs of the receiver, respectively. V(k) is called the signal phase reference. In the form Eq. (17), it is clear that the π-radian ambiguity of the arctan(·) function is resolved by the signs of Re V ( k ) and Im V ( k ) if an initial training data sequence is sent to start up the receiver.
In an M-ary PSK constellation, the phase reference can be generated by Eq. (17) due to its circular symmetry. The decision statistics are given by qi (k ) = arg max Re r (k )V * (k )e − jφi ( k )
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(18)
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The receiver computes qi (k) for each possible value φi of the data symbol m(k), and decides − jφ ( k ) * is maximized. Therefore, the receiver is totally linear, consisting of the linear structure in Eq. (17) for phase reference V(k) instead of θˆ(k ) , and the linear computations for the decision statistics in Eq. (18). Fig. 2 summarizes the receiver structure and the signal processing operations involved in the DA ML algorithm. As previously stated, the receiver needs a known data training sequence of length L symbols to start it up, and the subsequent decisions will be fed back to form the signal phase reference V(k). We call this the DA ML receiver.
that m(k)=Ci if
Re r ( k )V ( k )e
i
Phase Reference Estimator
r (k )
V (k ) =
k −1
∑ r (l )mˆ (l ) *
l=k −L
V (k )
mˆ (k ) mˆ (k )
Decision Set mˆ (k ) = Cd if Cd = max Re[r (k )Ci*V * (k )], i = ±1,...,± M / 2
Fig.2. DA ML receiver structure.
This DA ML receiver has been shown in [13] to achieve coherent detection performance if the carrier phase is a constant and the memory length L is sufficiently long. The performance of the DA ML algorithm will be examined in an optical MPSK system in the next section. 3.2 Realization of real-time BPSK/QPSK DA ML receiver In order to operate the DA ML receiver on-line, we introduce a simple way to implement the decision statistics Eq. (18) for different M-ary PSK. As for binary PSK (BPSK), the decision statistic can be reduced to qi (k ) = sgn Re r (k )V * (k ) (19)
(
)
due to e − jφi ( k ) ∈ {±1} , which can be easily implemented using a slicer with the real part of r (k )V * (k ) as input. In case of QPSK modulation, the decision statistics in Eq. (18) can be equivalently expressed as qi (k ) ≡ arg max Re 2e jπ / 4 r (k )V * (k )e − jφi ( k ) e− jπ / 4 * = arg max Re µ (k )m′ (k ) . (20)
{
}
Here µ (k ) ≡ 2e jπ / 4 r (k )V * (k ) , m′(k ) ≡ e jφi ( k ) e jπ / 4 ∈ (±1 ± j ) / 2 are rotated symbols in a 45º-tilted quadrature PSK (QPSK) constellation. The scale factor of 2 in Eq. (20) does not affect the decision rule, since the decision statistics take index i when the real part is maximum. The scaling will promise a simplification to our receiver, as illustrated in following steps. It is easy to show that the decision in Eq. (20) over the tilted constellation reduces to decisions on the signs of the real and imaginary parts of µ (k ) :
mˆ ′(k ) =
sgn(Re[ µ (k )]) + j sgn(Im[ µ (k )])
. (21) 2 The decision on variables Re[ µ (k )] and Im[ µ (k )] can be implemented using two signdecision devices or two slicers. Although mˆ ′(k ) is a complex rotated signal which is different
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from the previous m(k), the gray bit mapping of the QPSK signals is preserved after the decision: (22) 2 mˆ ′(k ) = ±1 ± j ∈ 2e jπ / 4 {1, j , −1, − j} ↔ {00,10,11, 01} As for M-ary PSK (M>4), the decision rule cannot be easily simplified into taking the sign of the decision statistics. An alternative is to use high-speed ADCs to implement the decision statistics in Eq. (18) in DSP processors, which also make the real-time implementation of M-ary PSK (M>4) harder compared to BPSK and QPSK signals. r ′(k ) = r ( k )(1 + j )
r (k )
CM
CCI
slicer 1,0
1 1
Conj(·)
+
+
1,0
r ′ (k ) *
D
D
r ′* (k − 1)
D r ′* (k − 2)
D
D
D
D
CCI D
Decision Feedback Loop
CCI Phase Reference Recovery
Fig. 3 The structure of real-time QPSK DA ML receiver (L =2): D: Time delay. The input complex signal is formed by its real and imaginary parts.
Here, we use the analog devices to implement our real-time DA ML receiver for BPSK/QPSK modulated signals. By considering the QPSK simplified decision statistics in Eq. (21), decision variable µ (k ) should be fed into the slicer to decode the received signals, so a four-quadrant complex multiplexer (CM) is required to perform multiplications of the form: µ (k ) = 2e jπ / 4 r (k ) ⋅ 2V * (k )
(
≡ Re 2e jπ / 4 r (k ) + j Im 2e jπ / 4 r (k )
(
⋅ Re 2V * (k ) + j Im 2V * (k )
)
) (23)
where the term r ′(k ) = 2 e jπ / 4 r ( k ) = (1 + j ) r ( k ) ≡ (1 + j ) ⋅ ( Re [ r ( k ) ] + j Im [ r ( k ) ]) can be realized using complex-controlled inverters (CCIs), whose control bit inputs are fixed to {1,1}. The transfer function of a CCI is given by re yout (t ) = (2b0 − 1) xinre − (2b1 − 1) xinim , im yout (t ) = (2b1 − 1) xinre + (2b0 − 1) xinim
where {xinre , xinim } are the real and re im { yout , yout } are the counterparts of
(24)
imaginary parts of input signals to the CCI and its output signals, {b0 , b1} are the control bits (binary
inverted) to realize the multiplication between ±1 ± j and xinre + jxinim [3]. Since CCI is a specialized complex multiplier with ±1 ± j , its implementation is considerably simpler than that of a general four-quadrant CM. Furthermore, the decision-feedback phase reference V(k)
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is generated by the decision-feedback loop, which consists of 2L time delay (T) of the slicer output. Since *
k −1 k −1 2V (k ) = 2 r (l )mˆ * (l ) = 2r * (l )mˆ ′(l )e − jπ / 4 l = k − L l = k − L
∑
*
k −1
=
∑
∑ [r ′(k )] mˆ ′(l ) , *
(25)
l =k −L
phase reference V*(k) can be obtained by two adders and L CCIs. V*(k) times the rotated received signal r ′(k ) = r (k )e jπ / 4 2 to generate the decision variable µ (k ) , which is the input signal to the slicer to decode the received signal r(k). From the previous implementation, the scale factor of 2 is split into two factors of 2 so that the r ′(k ) can be obtained through a simple CCI, not the four-quadrant CM, to reduce the complexity. As for BPSK modulated signals, the difference is that the received r(k) does not require a 45º rotation and the slicer just decodes the signals based on the real part of r (k )V * (k ) . The real-time QPSK DA ML receiver with memory length L=2 is illustrated in Fig. 3. Note that the QPSK DA ML receiver with memory length L comprises 2L+1 CCIs, 4 L time delays, two adders, one CM and one slicer. A similar structure for differentially encoded MPSK signals is shown in [3] in conjunction with several pairs of optical delay interferometers (ODIs) with respective time delays and phase shifters. However, such a receiver structure with a window length D suffers from D-1 pairs of complex ODIs, and it requires data be differentially encoded. In our receiver structure, there is no need for data to be differentially encoded, which promises performance approaching to the quantum-limited one. 4. Numerical results and discussion
To examine the performance of our DA ML phase estimation technique, we performed Monte Carlo simulations to obtain BERs versus OSNR in two cases: a linear phase noise system and a nonlinear phase noise system. The conventional Mth power scheme [4,6] is also studied to compare with our results.
Fig. 4. Simulated BER performances of 40-Gb/s QPSK in a linear optical phase noise channel with two different schemes: DA ML and Mth power (L= 5, 10, σ=0.02).
In an optical linear phase channel, as illustrated in Section 2 Eq. (9), the phase noise difference in two adjacent symbol intervals, i.e., θ (t +T ) −θ (t ) , obeys a Gaussian distribution with mean zero and variance σ2 determined by the linewidth of the transmitter laser and the #102401 - $15.00 USD
(C) 2009 OSA
Received 6 Oct 2008; revised 19 Dec 2008; accepted 21 Dec 2008; published 21 Dec 2008
19 January 2009 / Vol. 17, No. 2 / OPTICS EXPRESS 711
LO. Simulated BER performance of a coherent 40Gb/s QPSK system as a function of SNRb, obtained with different memory lengths L, is shown in Fig. 4 when σ=0.02, corresponding to the overall linewidth ∆υ =1.27 MHz. With forward error control (FEC) coding implemented to achieve highly reliable communication in an optical transmission system, 1x10-4 BER level is a reasonable reference to compare the performances of the different receivers. When σ=0.02, the phase noise leads to a small SNR penalty of Mth power compared to DA ML at 1x10-4 BER level with memory length L=5. Also DA ML has a slight performance improvement over Mth power though the advantage is not sufficiently obvious. Note that the performances of both DA ML and Mth power improve with the increase of memory length L. If the laser linewidth increases to ∆υ =7.8 MHz, i.e., σ=0.05, the results in Fig. 5 indicate that the performances of DA ML and Mth power with memory length L=5 remain comparable. While a larger memory length is expected to improve the system performance, on the contrary, DA ML with L=10 has lost its advantage over the case with L=5, due to the fact that fast changing phase noise becomes less correlated over the long memory length compared to the case of short memory length. As for the conventional Mth power with L=10, its performance is improved; however, its performance can also drift away with large L [6]. From the above results, we can say that the faster the phase noise varies the shorter the memory length L to be used. One alternative is to use an adaptive phase estimation technique, which has been shown to be able to achieve the optimum system performance by adaptively adjusting the phase estimate [14]. Although the performances are quite comparable for DA ML and Mth power in a linear phase noise system, of greater practical interest is the performance in a nonlinear phase noise dominant channel, which is the predominant phase noise in a long-haul transmission system.
Fig. 5. Simulated BER performances of 40-Gb/s QPSK in linear optical phase noise channel with two different schemes: DA ML and Mth power (L= 5, 10, σ=0.05).
In a multi-span optical communication system with erbium-doped fiber amplifiers (EDFAs), the performance of optical phase-modulated systems is severely limited by the nonlinear phase noise converted by ASE noise through the fiber Kerr nonlinearity, known as the Gordon-Mollenauer effect [7]. Studies in [8] and [15] have shown that nonlinear phase noise should be taken into account when evaluating the performance of phase modulation systems. Following the nonlinear phase noise channel model in [16], the modeled system comprises a 40-Gb/s single channel QPSK modulated optical signal over NA 100-km equallyspaced amplifiers to fully compensate for the attenuation of optical power along the fiber at the end of each span. In our simulation, we have span number NA = 22, fiber nonlinear coefficient γ=2W-1km-1, fiber loss α= 0.2 dB/km, amplifier gain G= 20 dB, noise figure = 6 dB, bandwidth of optical filters = 0.32 nm, and optical wavelength λ0= 1553 nm. Here, the #102401 - $15.00 USD
(C) 2009 OSA
Received 6 Oct 2008; revised 19 Dec 2008; accepted 21 Dec 2008; published 21 Dec 2008
19 January 2009 / Vol. 17, No. 2 / OPTICS EXPRESS 712
dispersion of fibers is fully compensated for by the dispersion compensation fiber so that we can investigate the nonlinear phase effect on the three phase estimation techniques. Besides, provided that sampling is performed above the Nyquist rate, linear channel impairments, such as chromatic dispersion and polarization-mode dispersion, may be compensated for using a linear equalizer with low power penalty [17]. Many techniques have been proposed to compensate for nonlinear phase noise at the receiver end using either electronic circuits [16] or a nonlinear optical component [18]. It was shown that the use of the nonlinear phase noise compensation before the Mth power can greatly improve the system performance in a nonlinear-phase-noise system [19]. This technique can also be applied to our DA ML receiver, though it will increase the complexity of both the Mth-power and DA ML receivers. In this paper, these nonlinear phase compensators are not employed so that we can compare the tolerance of Mth power and DA ML receivers to the nonlinear phase noise directly.
Fig. 6. Simulated nonlinear phase effect of QPSK signals in a 22-span nonlinear optical channel (NA =22 and L=5).
As shown in Fig. 6, the receiver sensitivity of DA ML with L=5 has improved by about 1 dB over Mth power scheme at the BER level of 1x10-4. This result shows that DA ML is capable of effectively mitigating the nonlinear phase noise effect. Note that no laser phase noise is considered in the nonlinear-phase-noise system (the so called “nonlinear-phase-noisedominant system”). Besides, the performance of the system improves as signal power increases; with power exceeding the optimum one, phase noise becomes too severe and degrades system performance. This is due to the fact that the variance of total phase noise decreases with the increased signal power before it reaches the optimum; with further increase in the signal energy beyond the optimum value, the variance of the total phase noise increases. It is found that the optimum performance occurs at Pin ≈0 dBm when the nonlinear phase shift is almost 1 radian, which agrees with the results in [7]. The BER performances of the two schemes at the optimum input power are simulated in Fig. 7 (a) with different numbers of spans NA. The optimum incident power decreases with the number of increased fiber spans, since more ASE noises will be converted into nonlinear phase noise in longer spans. From the two BER curves at the optimum input power, we see that DA ML always gives a better performance in the nonlinear phase noise system. On the other hand, the Mth power scheme is not the most favorable technique in a long-haul transmission system. This advantage can also be illustrated in the Q-factor improvement, where the Q-factor is obtained from the formula BER = erfc(Q / 2 ) / 2 . At the optimum point, the DA ML receiver outperforms the Mth power by about 0.7 dB in Q-factor, as shown in Fig. 6. For different span numbers (20~30), DA ML consistently has 0.5~1 dB improvement of Qfactor over the Mth power scheme, as shown in Fig. 7 (b). With the increase of number of fiber spans, Q-factor improvement also increases accordingly. #102401 - $15.00 USD
(C) 2009 OSA
Received 6 Oct 2008; revised 19 Dec 2008; accepted 21 Dec 2008; published 21 Dec 2008
19 January 2009 / Vol. 17, No. 2 / OPTICS EXPRESS 713
The simulation results in a nonlinear phase noise system show that the DA ML receiver is much more effective for mitigating the nonlinear phase noise effect than the conventional Mth power scheme. This can be explained as follows: for the DA ML receiver, the receiver obtains the phase reference V(k) for demodulating the current symbol from the past L received signals, which carry a good knowledge about the channel characteristics. As a result, the past decision errors can cause performance degradation to the DA ML receiver. However, of interest is the high SNR case, where the performance will not be severely affected by the past decision errors according to the simulation results and analysis in [13]. This leads to the improvement of DA ML over Mth power in a nonlinear-phase-noise-dominant system based on the assumption made in Section 2 that the phase noise θ is constant over L symbol durations, since the nonlinear phase noise changes slowly due to the nearly constant envelope of the Mary PSK compared to the laser phase noise. On the other hand, an SNR loss will occur in estimating the carrier phase using the Mth power scheme. For instance, compared to the DA ML estimator, non-decision-aided Mth power scheme suffers a squaring loss for BPSK (M =2), which arises from noise x noise and (noise)2-(noise)2 terms [13]. It is observed that the improvement of DA ML over the entire signal-power range is due to the fact that nonlinear phase noise exists in the presence of the fiber Kerr effect and optical amplifier ASE noises.
(a)
(b)
Fig. 7. Performances at the optimum input power versus no. of spans (NA) when L=5: (a) BER performance and optimum input power. (b) Q-factor improvement over Mth power.
From the numerical results, the DA ML carrier phase estimation shows a better receiver sensitivity improvement than Mth power for an optical QPSK signal in a nonlinear optical channel when the nonlinear phase noise is dominant. In a linear channel, although the performances of DA ML and Mth power are almost the same, Mth power requires nonlinear computations, such as an arg(·) and raising each received signal sample to the fourth power, as shown in [4,6], which will incur a large latency in the system and leads to phase ambiguity in estimating the block phase noise. In contrast, DA ML is a linear and efficient algorithm, and there is no need to deal with the ±2π/M-radian phase ambiguity, which enables the DA ML receiver to be implemented in real-time. 5. Conclusion
We have presented the general signal model in a coherent detection receiver with a 2 x 4 90° optical hybrid. With the derived general signal model, we propose a DA ML phase estimation method to recover the carrier phase. The proposed real-time DA ML receiver is applied in coherent detection for BPSK and QPSK modulated system. The numerical results show that the performances of DA ML and conventional Mth power are comparable in a linear phase noise system, whereas DA ML outperforms Mth power in a nonlinear phase noise system. In addition, our DA ML receiver requires only linear computations, which gives less latency to
#102401 - $15.00 USD
(C) 2009 OSA
Received 6 Oct 2008; revised 19 Dec 2008; accepted 21 Dec 2008; published 21 Dec 2008
19 January 2009 / Vol. 17, No. 2 / OPTICS EXPRESS 714
the system compared to the Mth power scheme, and which means our DA ML receiver is feasible for on-line processing in the real systems.
#102401 - $15.00 USD
(C) 2009 OSA
Received 6 Oct 2008; revised 19 Dec 2008; accepted 21 Dec 2008; published 21 Dec 2008
19 January 2009 / Vol. 17, No. 2 / OPTICS EXPRESS 715
