IEEE SIGNAL PROCESSING LETTERS, VOL. 23, NO. 10, OCTOBER 2016 1349
A BP–MF–EP Based Iterative Receiver for Joint
Phase Noise Estimation, Equalization, and Decoding
Wei Wang, Zhongyong Wang, Chuanzong Zhang, Qinghua Guo, Peng Sun, and Xingye Wang
Abstract—In this letter, with combined belief propagation (BP),
mean field (MF), and expectation propagation (EP), an iterative
receiver is designed for joint phase noise estimation, equalization,
and decoding in a coded communication system. The presence of
the phase noise results in a nonlinear observation model. Conven-
tionally, the nonlinear model is directly linearized by using the
first-order Taylor approximation, e.g., in the state-of-the-art soft-
input extended Kalman smoothing approach (Soft-in EKS). In this
letter, MF is used to handle the factor due to the nonlinear model,
and a second-order Taylor approximation is used to achieve Gaus-
sian approximation to the MF messages, which is crucial to the
low-complexity implementation of the receiver with BP and EP. It
turns out that our approximation is more effective than the direct
linearization in the Soft-in EKS, leading to a significant perfor-
mance improvement with similar complexity as demonstrated by
simulation results.
Index Terms—Iterative receiver, message passing, phase noise
estimation.
I. INTRODUCTION
L
OCAL oscillators, which provide a reference signal for
time and frequency synchronization, are one of the key
modules in a communication system. The instability of oscilla-
tors results in phase noise, which may severely affect the system
performance [1].
Various Bayesian and non-Bayesian approaches have been
proposed to solve the phase noise problem. Bhatti and
Moeneclaey modeled the phase noise with a discrete cosine
transform (DCT) expansion [2], where the DCT coefficients
can be easily estimated. However, the DCT method is a non-
Bayesian one, and it does not make use of the time dependence
Manuscript received May 31, 2016; revised July 19, 2016; accepted July 20,
2016. Date of publication July 22, 2016; date of current version August 24, 2016.
This work was supported by the National Natural Science Foundation of China
under Grant NSFC 61172086 and Grant NSFC 61571402, and by Australian
Research Council’s DECRA under Grant DE120101266. The associate editor
coordinating the review of this manuscript and approving it for publication was
Prof. Mohammad M. Mansour.
W. Wang,Z.Wang, and P. Sun are with the School of Information Engineering,
Zhengzhou University, Zhengzhou 450001, China (e-mail: iewwang@zzu.edu.
cn; iezywang@zzu.edu.cn; iepengsun@gmail.com).
C. Zhang is with the School of Information Engineering, Zhengzhou Uni-
versity, Zhengzhou 450001, China, and also with the Department of Electronic
Systems, Aalborg University, Aalborg 9220, Denmark (e-mail: ieczzhang@
gmail.com).
Q. Guo is with the School of Electrical, Computer and Telecommunications
Engineering, University of Wollongong, Wollongong, NSW 2522, Australia,
and also with the School of Electrical, Electronic and Computer Engineer-
ing, University of Western Australia, Crawley, WA 6009, Australia (e-mail:
qguo@uow.edu.au).
X. Wang is with the School of Information Engineering, North China Univer-
sity of Water Resources and Electric Power, Zhengzhou 450000, China (e-mail:
wangxingye6507@sina.com).
Color versions of one or more of the figures in this letter are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LSP.2016.2593917
of the phase noise process. In Bayesian methods such as par-
ticle filter [3], Tikhonov parametric estimation [4], and ex-
tended Kalman smoothing (EKS) [5], phase noise is modeled
as a Wiener process. The particle filtering method [3] needs
to sample a posteriori probability density function (PDF) of
continuous-valued phase noise variables, where a larger num-
ber of particles yields better performance at the cost of higher
complexity. The Tikhonov parametrization method [4] is an it-
erative method to deal with strong phase noise for additive white
Gaussian noise (AWGN) channels. The intractable integral op-
eration associated with continuous variables is circumvented
by constraining the PDF to the Tikhonov distribution (also
known as the von Mises distribution [6]). However, the work
in [4] was focused on AWGN channel, and a straightforward
extension to intersymbol interference (ISI) channels (which is
allowed by incorporating a maximum a posteriori equalizer)
will lead to complexity growing exponentially with the channel
memory length. In the soft-input EKS (Soft-in EKS) method
1
proposed in [5], the nonlinear observation model is directly lin-
earized by using the first-order Taylor expansion. Soft-in EKS
has been used in single-input single-output and multiple-input
multiple-output systems [5], [8]–[10].
Recently, the message passing techniques, including the be-
lief propagation (BP) [11] and the variational message passing
(VMP) [12], have been widely used for iterative receivers de-
sign. BP is effective for discrete probability models and linear
Gaussian models. The BP-based equalizer in [13] has a linear
complexity, which is much lower than that of the equalizer in
[14]. The VMP method, also referred to as a mean filed (MF),
is especially suitable for handling variables with exponential
distributions. Recently, a unified message passing framework
has been proposed in [15], where BP and MF are merged to
keep the virtues of BP and MF while avoiding their drawbacks.
It has been applied to joint channel estimation and decoding
in orthogonal frequency division multiplexing [16], [17] and
single carrier frequency domain equalization [18]. The expec-
tation propagation (EP)[19] has been used to achieve Gaussian
approximation to non-Gaussian messages, and it has been em-
ployed for channel estimation in [16]. In addition, combined
EP and BP has been applied to signal detection in flat-fading
channels or ISI channels, e.g., in [7] and [20].
In this letter, with combined BP, MF, and EP, we propose an
iterative approach to joint phase noise estimation, equalization,
and decoding for a coded system over ISI channels. BP and
EP are used to deal with the linear model for the phase noise
1
The EKS method in [5] was proposed for AWGN channels. It can be extended
to the case of ISI channels, e.g., by incorporating the BP–EP algorithm [7] to
handle ISI channels.
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