BLIND PHASE RECOVERY IN QAM COMMUNICATION SYSTEMS USING
CHARACTERISTIC FUNCTION
Ehsan Hassani Sadi, Hamidreza Amindavar
Amirkabir University of Technology, Department of Electrical Engineering, Tehran, Iran
[email protected], [email protected]
ABSTRACT
In this paper, we present a novel non-data aided method for
phase recovery in both square and cross quadrature amplitude
modulation (QAM) communication systems, based on characteristic function. The proposed method is independent of
noise distribution added to the rotated signal. After estimating
the gain multiplied to the received signal and compensating its
effect, we will estimate the phase offset. The key innovation
lies in the use of characteristic function rather than the traditional higher order statistics. We use characteristic function
and its estimate (Empirical Characteristic Function-ECF), in
order to perform phase estimation. The analytical evaluation
of the estimation is provided. Monte Carlo simulation provides feasibility of our new approach.
Index Terms— Characteristic function, phase estimation,
quadrature amplitude modulation, synchronization.
estimator [2]. In [5], a phase histogram-based estimator has
been derived according to a high SNR approximation of the
ML criterion. An estimator based on special phase metric
that exhibits an absolute minimum around the carrier phase
metric, is presented in [6]. It is observed in [7], that the
phase offset produces a cyclic phase rotation of the probability density function (pdf) of the fourth power of the received
signal samples, and a blind phase offset estimator based on
the weighed phase histogram of the signal samples is therein
derived.
The paper is organized as follows. After having introduced the model of the equalized signal in the receiver in section 2, we first describe how to estimate the gain multiplied
to the signal in section 3, and then the estimation of the phase
rotation is described in section 4; it is also described that the
distribution of noise is not necessarily Gaussian. In section 5,
we evaluate the performance of the proposed method by some
simulations. Finally, section 6 concludes the paper.
1. INTRODUCTION
Phase recovery is a problem of paramount importance in synchronous digital communication systems, especially for high
bit rate signalling such as QAM modulation. QAM is particularly attractive for high-throughput-efficiency applications
because of its better performance compared to PSK as the size
of constellation increases. For efficiency reasons, the phase
estimation must be performed in blind manner. The problem
is further complicated for cross constellations, for which the
high SNR corner points used by some simple carrier phase
estimators are not available.
In recent literature, several approaches for blind phase
estimation have been proposed. Many blind phase estimation algorithms have been devised by exploiting higher order
statistics (HOS) of the received signals. A blind phase recovery for square QAM using higher order statistics was
presented in [1]. This method was modified by [2] to work
for general QAM systems (square/non-square). In [3], it is
described that an estimator approximating the maximumlikelihood (ML) estimator for low SNR values, is equivalent
to the fourth-power estimator of [2]. In [4] an estimator based
on eighth-order statistics gives improved performance for
cross QAM systems with respect to the fourth power phase
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
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2. DISCRETE-TIME SIGNAL MODEL
We consider a baseband symmetric QAM communication
system. Let X(k) be the kth transmitted symbol from a
power normalized M -ary constellation. At the receiver side,
after front-end signal processing, a complex low-pass version
of the received signal is available for sampling. We assume
that the received signal is already equalized and frequencysynchronized, and that timing recovery is done. We consider
the following model for the phase rotated signal samples:
Y (k) = Gejθ X(k) + N (k).
(1)
Let Y (k) be the samples of equalized signal at sample rate.
G is the overall gain seen by symbols, θ is an unknown phase
shift to be estimated. We further assume that X is the complex transmitted signal X(k) = a(k)+jb(k), where a(k) and
b(k) are independently and identically distributed (IID) uniform discrete random variables of size M/2 and, N is complex
noise N (k) = u(k) + jv(k), where we assume that u(k) and
v(k) are zero mean and their PDF is symmetric.
ICASSP 2011
N(k)
X(k)
Channel
+
Equalizer
ejT X(k)
+
Y(k)
š
Proposed
Phase
Compensator
X (k)
Decision
Maker
S(k)
(k)
Fig. 1. General block diagram of a receiver after distorting
Channel.
3. GAIN ESTIMATION
In the receiver side, we get the output samples which are usually affected by a narrowband amplifier. In order to fix the
power level of these samples and decrease the signal dynamical range, an automatic gain control loop (AGC) is usually
used. The AGC continuously estimates the average signal
power at the output of the narrowband amplifier and adjusts
its gain so as to have normalized signal power at its output.
However, in practice this gain control cannot be performed
accurately, especially in SONAR scenario, where the environmental noise highly affect the underwater communications.
Therefore, a gain (G), is involved in (1). In order to detect the
equalized signal, we have to first estimate the gain (G) multiplied to the phase rotated signal, and then try to estimate the
phase rotation. Therefore, we need to compensate this gain.
Using (1), we have:
2
2
2
2
E{|Y (k)| } = G E{|X(k)| } + E{|N (k)| }.
(2)
where, E{.} represents the expectation operator. If we know
the noise energy, we can obtain G as:
G=
2
2
E{|Y (k)| } − E{|N (k)| }
2
E{|X(k)| }
.
(3)
Fig. 2. A noisy-rotated 16-QAM constellation.
We present a method to estimate θ using characteristic
function. From (1), we have:
Yr = a(k) cos θ − b(k) sin θ + N r,
Yi = a(k) sin θ + b(k) cos θ + Ni ,
Y = Yr + jYi .
Assuming noise N (k) and the output of equalizer X(k) are
independent, their characteristic functions are multiplied to
yield the characteristic function of Y (k), hence, the characteristic function of Yr equals:
φYr (ω) = E{exp(jωYr (k))}
= E{exp(jωa(k) cos θ) exp(−jωb(k) sin θ)} (6)
×φu (ω).
since X(k) belongs to M -QAM modulation, we have:
M
assuming that noise energy is much less than the signal energy, G can be approximated through the following:
G=
E{exp(jωa(k))} =
2
M
E{exp(jωb(k))} =
2
M
2
m=1
ejωa(m) ,
(7)
M
2
E{|Y (k)| }
2
E{|X(k)| }
.
(4)
4. ESTIMATION OF PHASE ROTATION
The blind carrier phase estimation problem is to find an estimate for θ, only from the knowledge of equalized samples.
As the QAM constellation has quadrant (π/2)symmetry (rotationally invariant to rotations by multiples of 90o ), it follows that the unknown phase recovery has a modulo (mπ/2)
ambiguity. Using appropriate coding schemes (differentially
encoding), this ambiguity can be eliminated. Therefore, without loss of generality, we assume that the unknown phase lies
in the interval [0, π/2).
(5)
2
m=1
ejωb(m) .
So, the characteristic function (CF), i.e. the Fourier transform
of PDF, of Yr can be expressed as follows:
M
M
2
2
2
ejωa(m) cos θ
e−jωb(m) sin θ ] φu (ω).
φYr (ω) = ( )2 [
M
m=1
m=1
(8)
calculating the real part, we have:
⎡M
M
2
2
2 2 ⎣
cos(ωa(m) cos θ)
cos(ωb(m) sin θ)
{φYr (ω)} = ( )
M
m=1
m=1
⎤
M
M
2
2
sin(ωa(m) cos θ)
sin(ωb(m) sin θ)⎦ φu (ω). (9)
+
m=1
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m=1
In order to estimate the unknown phase, we use an ECF based
method. First we estimate φYr (ω) via empirical characteristic
function (ECF) by [8]:
K
1 jωYr (k)
e
.
K
φ̂Yr (ω) =
calculating the real part, we have:
K
1 cos(ωYr (k)).
K
(11)
k=1
We note that the zeros of right side of (9) are approximately
estimated by zeros of (11), and this zero is at ω = ω0 . So,
using ECF, we estimated ω0 . Next step is to estimate θ. Again
by referring to (9), and assuming that φu (ω) has no finite zero,
we have:
M
2
M
cos(ω0 a(m) cos θ)
m=1
2
+
m=1
cos(ω0 b(m) sin θ)
m=1
M
2
M
sin(ω0 a(m) cos θ)
2
+
B(z)
A(z)
(10)
k=1
{φ̂Yr (ω)} =
AGC
sin(ω0 b(m) sin θ) = 0. (12)
m=1
Solving (12) yields θ. Here for both estimating ω0 and θ,
we perform unconstrained optimization by a gradient based
method.
As it is clear, all equations (9), (11) and (12) are independent of noise. Therefore, the proposed method is independent
of any assumption on noise distribution; as a result, it is possible to estimate the unknown phase rotation in presence of
both Gaussian and non-Gaussian noise.
4.1. Phase Noise Distribution
It is a common trend to consider a Normal distribution for
the noise added to the received signal after equalization. In
natural environments, the noise have non-Gaussian and nonstationary behavior. On the other hand, in actual experiments
such as underwater communications like [9], the equalizer is
composed of non-linear structure (Decision Feedback Equalizer). This non-linearity in the equalizer structure makes the
phase error in the equalized signal a non-Gaussian one. Thus,
the algorithms that are optimized for Gaussian noise will degrade in actual experiments. Here, we will show through
Kolmogorov-Smirnov test, that this noise added to the phase
rotated signal after equalizer does not have the Gaussian distribution. In order to show the correctness of this assertion we
have performed the KS test on the phase noise at the output
of the equalizer in [9]. The KS test result, confirms our claim.
As we showed in the previous paragraph, we cannot always assume a Gaussian distribution on the noise in (1), unlike the common assumption in literature. This distribution
can be changed in different scenarios. Therefore, the independence of our new algorithm from noise is a paramount
property.
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Fig. 3. Cascade structure of [9] equalizer in the starting period.
5. SIMULATION AND RESULTS
In order to assess the performance of the proposed approach,
some simulations are performed. Results have been obtained
via Monte Carlo simulations using 500 different runs. We
have used model (1) in our simulation. A static phase offset
is considered by multiplying ejφ to the received signal. The
performance of the algorithm is illustrated through simulations in Figs. 4 and 5, respectively, for 4-QAM and 16-QAM
constellations, θ = π/12(= 15o ) , G = 2. Figs. 4 and 5
show the performance of the algorithm (Root Mean Square
Error) in estimating the phase rotation, as a function of SNR.
Here, we have used correlated noise and a heavy tailed one
(Student t distribution with 3 degrees of freedom) in addition
to Gaussian noise. Fig. 4 is indicative of the advantage of
CF based phase estimator that for SNR>3 dB, the new algorithm determines the unknown phase correctly for any type
of noise, and that the statistical behavior of noise is irrelevant
upon our approach. Fig. 6, shows the performance of our algorithm (RMSE) versus the observed sample size in SNR=10
dB for 4-QAM and 16-QAM constellations. Comparing Figs.
4 and 5, it is clear that noise distribution does not affect the
performance of the algorithm; also, it can be seen that this
new estimator gives about the same RMSE for 4-QAM and
16-QAM constellations with 1500 and 7000 samples at SNR
10 dB. It is apparent that in order to have the same RMSE for
16-QAM, we should have a larger sample size to be analysed
by the algorithm.
6. CONCLUSION
The basic innovation in this paper is to address characteristic function and ECF to the problem of blind phase estimation. We have carried out a theoretical analysis of the approach. First, we compensated the gain multiplied to the rotated signal. Then, we derived the suitable equations based
on characteristic function of data. These equations were used
to estimate the phase offset via a least square optimization
method. Potential improvement, according to its noise independence over existing HOS-based methods is demonstrated.
Monte Carlo simulations have provided the experimental verification of this approach, by analysing its error performance
versus SNR and sample size.
[8] Jun Yu, “Empirical characteristic function estimation and
its applications,” Econometric Reviews, vol. 23, no. 1, pp.
93-123, Dec. 2004.
Student t noise
Gaussian noise
Correlated noise
10
9
[9] J. Labat, O. Macchi, C. Laot, “Adaptive decision feedback equalization: can you skip the training period?,”
IEEE Transactions on Communications, vol. 46, no. 7,
pp. 921-930, Jul. 1998.
8
7
6
5
4
2
1
0
5
10
15
20
SNR (dB)
Fig. 4. Root Mean Square Error versus SNR, 4-QAM,
θ = 15o (Student t distribution has 3 degrees of freedom).
7. REFERENCES
[1] L. Chen, H. Kusaka, M. Kominami, “Blind phase recovery in QAM communication systems using higher order
statistics,” IEEE Signal Processing Letters, vol. 3, no. 5,
pp. 147-149, May 1996.
[2] K. V. Cartwright, “Blind phase recovery in general QAM
communication systems using alternative higher order
statistics,” IEEE Signal Processing Letters, vol. 6, no. 12,
pp. 327-329, Dec. 1999.
[3] E. Serpedin, P. Ciblat, G. B. Giannakis, P. Loubaton,
“Performance analysis of blind carrier phase estimators
for general QAM constellation,” IEEE Signal Processing
Letters, vol. 49, no. 8, pp. 1816-1823, Aug. 2001.
[4] K. V. Cartwright, “Blind phase recovery in cross
QAM communication systems with eight-order statistics,” IEEE Signal Processing Letters, vol. 49, no. 8, pp.
304-306, Dec. 2001.
[5] C. N. georghiades, “Blind carrier phase acquisition for
QAM constellations,” IEEE Transactions on Communications, vol. 45, no. 11, pp. 1477-1486, Nov. 1997.
[6] S. Jarboui, S. Hadda, “Blind phase recovery for general
quadrature amplitude modulation constellations,” IET
Communications, vol. 2, no. 5, pp. 621-629, 2008.
[7] G. Panchi, S. Colonnese, S. Rinauro, G. Scarano, “Gaincontrol-free near-efficient phase acquisition for QAM
constellations,” IEEE Transactions on Signal Processing,
vol. 56, no. 7, pp. 2849-2863, Jul. 2008.
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Root Mean Square Error (deg.)
3
3
Student t noise
Gaussian noise
Correlated noise
2.8
2.6
2.4
2.2
2
1.8
0
2
4
6
8
10
12
SNR (dB)
Fig. 5. Root Mean Square Error versus SNR, 16-QAM,
θ = 15o (Student t distribution has 3 degrees of freedom).
10
Root Mean Square Error (deg.)
Root Mean Square Error (deg.)
11
4-QAM
16-QAM
9
8
7
6
5
4
3
2
1
0
1000
2000
3000
4000
5000
6000
7000
Observed sample size
Fig. 6. Root Mean Square Error versus the observed sample
size in SNR=10 dB, θ = 15o .