Analytical Estimation of Carrier Phase Recovery
Approaches in Long-Haul High-Speed Optical
Communication Systems
Tianhua Xu
1. University College London, London, WC1E7JE, United Kingdom
2. Tianjin University, Tianjin, 300072, China
3. Royal Institute of Technology, Stockholm, SE-16440, Sweden
4. Acreo Swedish ICT AB, Electrum 236, Kista, SE-16440, Sweden
Email: xutianhua@tju.edu.cn, tianhua.xu@ucl.ac.uk
Abstract
The analytical study on the carrier phase estimation (CPE) approaches, involving a one-tap
normalized least-mean-square (NLMS) algorithm, a block-wise average (BWA) algorithm,
and a Viterbi-Viterbi (VV) algorithm has been investigated in the long-haul high-speed
n-level phase shift keying (n-PSK) coherent optical fiber communication systems. The
close-form predictions for the bit-error-rate (BER) performance have been derived and
analyzed by considering both the intrinsic laser phase noise and the equalization enhanced
phase noise (EEPN).
Key Words
Coherent optical detection, optical fiber communication, carrier phase estimation, laser phase
noise, equalization enhanced phase noise
1. Introduction
The performance of long-haul high-speed optical fiber communication systems can be
significantly degraded by the transmission system impairments, such as chromatic dispersion
(CD), polarization mode dispersion (PMD), laser phase noise (PN) and fiber nonlinearities
(FNLs) [1-8]. Using coherent optical detection and digital signal processing (DSP), the
powerful equalization and mitigation of the communication system impairments can be
implemented in the electrical domain, which has become one of the most promising
techniques for the next-generation optical fiber communication networks to achieve a
performance very close to the Shannon capacity limit [9-16], with an entire capture of the
amplitude and phase of the optical signals [17-30]. To compensate the phase noise from the
laser sources, some feed-forward and feed-back carrier phase recovery (CPR) algorithms have
been proposed to estimate the phase of optical carriers [31-40]. Among these carrier phase
estimation (CPE) approaches, the one-tap normalized least-mean-square (NLMS) algorithm,
the block-wise average (BWA) algorithm, and the Viterbi-Viterbi (VV) algorithm have been
validated for compensating the laser phase noise effectively, and are also regarded as
promising DSP algorithms in the real-time high-speed coherent optical fiber transmission
systems [35-40].
In the electronic dispersion compensation (EDC) based optical fiber communication systems,
a phenomenon of equalization enhanced phase noise (EEPN) will be generated due to the
interactions between the electronic dispersion equalization module and the laser phase noise
[41-50]. The performance of long-haul optical fiber communication systems will be degraded
seriously by this EEPN effect, with the increment of fiber dispersion, laser linewidths,
modulation formats, and symbol rates [43-47]. The effects of equalization enhanced phase
noise have been studied in detail in the single-channel, the wavelength division multiplexing
(WDM), the orthogonal frequency division multiplexing (OFDM), the dispersion
pre-distorted, and the multi-mode optical transmission systems [43-59]. Meanwhile, some
investigations have been carried out to study the performance of EEPN in the carrier phase
estimation in the long-haul high-speed optical transmission systems [60-64]. Considering the
impact of equalization enhanced phase noise, the traditional analysis of the carrier phase
estimation algorithms is not suitable any longer for the design and the optimization of the
long-haul high-speed optical fiber networks. Correspondingly, it will also be interesting and
useful to investigate the bit-error-rate (BER) performance in the one-tap normalized
least-mean-square, the block-wise average, and the Viterbi-Viterbi carrier phase estimation
algorithms, when the influence of equalization enhanced phase noise is taken into account.
In this paper, the theoretical assessment on the carrier phase estimation in long-haul coherent
optical fiber communication systems using the one-tap normalized least-mean-square
algorithm, the block-wise average algorithm, and the Viterbi-Viterbi algorithm is presented
and discussed in detail. The close-form expressions for the estimated carrier phase in the
one-tap normalized least-mean-square, the block-wise average, and the Viterbi-Viterbi
algorithms has been derived, and the BER performance such as the BER floors, in the three
carrier phase estimation methods has been predicted. For different phase noise variance (or
effective phase noise variance considering equalization enhanced phase noise), the
performance of the one-tap normalized least-mean-square, the block-wise average, and the
Viterbi-Viterbi carrier phase recovery algorithms have been compared analytically.
2. Analysis of laser phase noise and equalization enhanced phase noise
2.1 Analysis of intrinsic laser phase noise
In coherent optical communication systems, the variance of the intrinsic phase noise from the
transmitter (Tx) laser and the local oscillator (LO) laser can be described using the following
equation, see e.g. in Ref [1,2]
 Tx2 _ LO  2 f Tx  f LO   TS
(1)
where f Tx and f LO are the 3-dB linewidths (assuming a Lorentzian distribution) of the
transmitter laser and the LO laser respectively, and TS is the symbol period of the coherent
transmission system. It can be found that the variance of the laser phase noise decreases with
the increment of the signal symbol rate RS  1 TS .
2.2 Analysis of equalization enhanced phase noise
Considering the interplay between the electronic dispersion compensation module and the LO
laser phase noise, the noise variance of the equalization enhanced phase noise in the long-haul
high-speed optical fiber communication systems can be expressed as follows, see e.g. in Ref
[43,47,50]
2
 EEPN
_ LO 
2 D  L  f LO
2c

(2)
TS
where f LO is the central frequency of LO laser, which is equal to the central frequency of
the transmitter laser f Tx in the homodyne optical communication systems, D is the
chromatic dispersion coefficient of the transmission fiber, L is the length of the transmission
fiber, RS is the signal symbol rate of the communication system, and   c f Tx  c f LO
is the central wavelength of the optical carrier wave.
When the equalization enhanced phase noise is taken into account in the carrier phase
estimation, the total noise variance (effective phase noise variance) in the long-haul
high-speed optical fiber transmission systems can be calculated and described as the
following expression, see e.g. in Ref [32,47,49]
2
 T2   EEPN
  Tx2 _ LO

3
2 D  L  f LO
2c

TS
 2TS f Tx  f LO 
(3)
Analytical assessment of carrier phase estimation algorithms
3.1 Analysis of one-tap normalized LMS carrier phase estimation
Using the one-tap normalized least-mean-square algorithm, the transfer function of the carrier
phase estimation in the coherent optical communication systems can be expressed as the
following equations [36,37],
y k   wNLMS k xk 
wNLMS k  1  wNLMS k  
ek   d k   y k 
(4)

xk 
2
ek x  k 
(5)
(6)
where x(k) is the complex input symbol, k is the index of the symbol, y(k) is the complex
output symbol, wNLMS(k) is the tap weight of the one-tap normalized least-mean-square
equalizer, d(k) is the desired output symbol after the carrier phase estimation, e(k) is the
estimation error between the output symbol and the desired output symbol, and μ is the step
size of the one-tap normalized LMS algorithm.
It has been verified that the one-tap normalized LMS carrier phase estimation behaves similar
to the ideal differential carrier phase recovery, and the estimated phase in the one-tap
normalized LMS carrier phase estimation can expressed as [65,66]
 NLMS k    k  1
(7)
The reported BER floor in the carrier phase estimation for the quadrature phase shift keying
(QPSK) optical transmission systems can be approximately described as follows, see e.g. in
Ref [36,65,66]
NLMS _ QPSK
BER floor

 
1
erfc
2
 4 2 T




(8)
Therefore, the BER floor in the one-tap normalized least-mean-square carrier phase
estimation for the n-level phase shift keying (n-PSK) optical fiber communication systems
can be derived accordingly, and expressed using the following equation:
NLMS
BER floor

  
1

erfc

log 2 n
n
2

T 

(9)
where  T is the total phase noise variance in the long-haul high-speed n-PSK optical
transmission systems.
2
3.2 Analysis of block-wise average carrier phase estimation
As an n-th power carrier phase estimation approach, the block-wise average algorithm
calculates the n-th power of the received symbols to remove the information of the modulated
phase in the n-PSK coherent transmission systems, and the computed phase (n-th power) are
summed and averaged over a certain block (the length of the block is called block size). The
averaged phase value is then divided over n, and the final result is regarded as the estimated
phase for the received symbols within the entire block. For the n-PSK coherent optical
communication systems, the estimated carrier phase for each process block using the
block-wise average algorithm can be expressed as, see e.g. in Ref [36,38,39]

 BWA k   arg
1
n
q  N BWA

 x  p 
 
n
 p 1 q 1  N BWA 
q  k N BWA 
(10)
(11)
where k is the index of the received symbol, NBWA is the block size in the block-wise average
algorithm, and x  means the closest integer lager than x.
The BER floor in the block-wise average carrier phase estimation in the n-PSK coherent
optical communication systems can be derived using the Taylor series expansion, and can be
approximately described using the following equation - see e.g. [28,36]:
BER BWA
floor 

2
BWA
 p 
N BWA


1


  erfc

N BWA log 2 n p 1


n
2

p
BWA



(12)

 T2 2 p  13  3 p  12  2 N BWA  p 3  3N BWA  p 2  N BWA  1
6N
2
BWA
(13)
where  T2 is the total phase noise variance in the long-haul high-speed n-PSK optical
transmission systems.
3.3 Analysis of Viterbi-Viterbi carrier phase estimation
As another n-th power carrier phase estimation approach, the Viterbi-Viterbi algorithm also
calculates the n-th power of the received symbols to remove the information of the modulated
phase. The computed phase are also summed and averaged over the processing block (with a
certain block length). Compared to the block-wise average algorithm, the Viterbi-Viterbi
algorithm only treats the extracted phase as the estimated phase for the central symbol in each
processing block. The estimated carrier phase in the Viterbi-Viterbi algorithm in the n-PSK
coherent optical transmission systems can be described using the following equation, see e.g.
in Ref [36,40]

VV k   arg
1
n
 NVV 1 2

 x k  q 

q   NVV 1
n
2
(14)

where NVV is the block size in the Viterbi-Viterbi algorithm, and should be an odd value of e.g.
1,3,5,7…
Using the Taylor expansion, the BER floor in the Viterbi-Viterbi carrier phase estimation for
the n-PSK coherent optical communication systems can be assessed analytically, and can be
expressed approximately using the following equation, see e.g. in Ref [28,36]
BER VV
floor



1

erfc

2
log 2 n
N VV
1

n

 T

6 N VV








(15)
where  T2 is the variance of the total phase noise in the long-haul high-speed n-PSK optical
fiber transmission systems.
As an example, the BER floors versus different noise variances in the above three carrier
phase estimation algorithms for the long-haul high-speed 8-PSK optical fiber communication
system have been compared and shown in Fig. 1. Here a block size of 11 is used in both the
block-wise average and the Viterbi-Viterbi carrier phase estimation algorithms, since the
additive noise in the transmission channels such as the amplified spontaneous emission (ASE)
noise should be taken into consideration in practical optical communication systems. In this
case, it can be found in Fig.1 that for small phase noise variance (or effective phase noise
variance), the Viterbi-Viterbi carrier phase recovery algorithm outperforms the one-tap
normalized LMS and the block-wise average algorithms, while for the large phase noise
variance (or effective phase noise variance), the one-tap normalized LMS algorithm will show
a better performance than the other two algorithms in the carrier phase estimation for the
8-PSK optical transmission system.
Figure 1. BER floors versus different phase noise variances in the three carrier phase
estimation algorithms in the 8-PSK optical fiber communication systems. Block sizes of the
BWA and the VV algorithms are both 11.
4
Conclusions
The theoretical evaluation of the carrier phase estimation in the long-haul high-speed coherent
optical fiber communication systems, using the one-tap normalized least-mean-square
algorithm, the block-wise average algorithm, and the Viterbi-Viterbi algorithm, has been
investigated and described in detail, both considering the intrinsic laser phase noise and the
equalization enhanced phase noise. The close-form expressions for estimated carrier phase in
the one-tap normalized least-mean-square, the block-wise average, and the Viterbi-Viterbi
algorithms have been derived, and the BER performance such as the BER floors, in the three
carrier phase estimation methods has been predicted analytically.
Acknowledgements
This work is supported in part by European Commission Research Council
FP7-PEOPLE-2012-IAPP (project GRIFFON, No. 324391), in part by UK Engineering and
Physical Sciences Research Council (project UNLOC EP/J017582/1), and in part by Swedish
Research Council Vetenskapsradet (No. 0379801).
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