Accurate Digital Frequency Offset Estimator
for Coherent PolMux QAM Transmission Systems
Mehrez Selmi(1) , Yves Jaouën(1) , Philippe Ciblat(1)
(1)
Institut Telecom/Telecom ParisTech, CNRS UMR LTCI, 46 rue Barrault 75634 Paris CEDEX 13 France,
B mehrez.selmi@telecom-paristech.fr
Abstract An accurate blind frequency offset estimator adapted to QAM modulated signal is proposed. For coherent
100Gbit/s QAM PolMux transmission, frequency offset can be recovered with an accuracy of a few kHz.
Introduction property, it is still possible to build FO estimate based
M-ary quadrature amplitude modulation (M-QAM) for- on yp4 (k) in QAM context.
mats combined with coherent detection and digital sig- Indeed, by following the approach described in 7 , one
nal processing (DSP) are promising candidates for the can remark that yp4 (k) can be decomposed as
implementation of next generation optical transmission
yp4 (k) = Ap e2jπ4(φ0,p +kφ1 ) + ep (k) (2)
systems. However, those modulation formats are more
sensitive to signal distortions and phase errors than where Ap = E[s4p (k)] is a constant amplitude and
QPSK. These phase errors may correspond to con- where ep (k) a zero mean process that can be viewed
stant phase offset, frequency offset (FO) and laser as a noise process. The most important thing is to re-
phase noise 1 . Several FO estimators have been al- mark that yp4 (k) is actually a constant-amplitude com-
ready presented for QPSK based optical transmissions. plex exponential with frequency 4φ1 disturbed by a zero
These algorithms rely either on the phase difference mean additive noise. Therefore one can deduce a
between two adjacent receive samples 2,3 or the max- FO estimate based on the maximization of the peri-
imisation of the discrete-frequency spectrum of the odogram of yp4 (k) as proposed below
fourth-power receive samples 4 . Let N be the number
of available independent receive samples. The Mean 1 X
φ̂1,N = arg max fp (φ) (3)
Square Error (MSE) on the FO decreases as 1/N for 4 φ∈[−1/2,1/2)
p∈{X,Y }
the first kind of algorithms, and as 1/N 2 for the second
kind of algorithms. As M-QAM is more sensitive to FO, with ˛ N −1 ˛2
˛1 X 4
˛ ˛
designing more accurate estimators is still required. fp (φ) = ˛ yp (k)e−2jπkφ ˛
(4)
˛N
˛
We here propose a new non-data-aided FO estima- k=0
˛
tor for any QAM format in PolMux context. We espe-
and with N the number of available samples.
cially show that its MSE decreases as 1/N 3 . Note that
Unlike what is usually done in optical communica-
this algorithm can be adapted to PSK formats as well.
tions, we propose to compute the maximisation of peri-
odogram into two steps as follows
Frequency estimator description
1. a coarse step which detects the maximum magni-
The proposed estimator is carried out after the com-
tude peak which should be located at around the
pensation of group velocity dispersion (GVD) and polar-
sought frequency . This is carried out by a N -Fast
isation dispersion (PMD). As a consequence, assuming
Fourier Transform (N -FFT). This step is been al-
a perfect compensation, the receive signal on polarisa-
ready implemented for QPSK format 4 . One can
tion p (with p ∈ {X, Y }) takes the following form
easily check that MSE associated with this step is
yp (k) = sp (k)e2jπ(φ0,p +kφ1 ) + np (k) (1) of order of magnitude 1/N 2 .
2. a fine step which inspects the cost function around
where {sp (k)} are independent sequences of QAM the peak detected by the coarse step. This step
symbols, np (k) is the additive channel noise. The term is implemented by a gradient-descent algorithm.
φ0,p corresponds to the constant phase while φ1 = MSE associated with this step 7 is of order of mag-
∆f Ts is the discrete-time FO to be estimated where nitude 1/N 3 .
∆f is the continuous-time FO expressed in Hertz and
Finally, to the best of our knowledge, our proposition
Ts is the symbol period. Eq. (1) holds if ∆f 1/Ts .
of treating both polarisation ways jointly is new.
If sp (k) is a QPSK modulated data stream, it has
been remarked for a long time that s4p (k) is constant Numerical results
and independent of the data stream. Consequently The performance of the algorithm is evaluated by us-
it is possible 5 to build FO estimate based on yp4 (k). ing Monte-Carlo simulations. A 100Gbit/s transmis-
When sp (k) is QAM modulated, s4p (k) is not constant sion is achieved by multiplexing both polarisations
anymore. But a QAM modulated signal is fourth-order with 16-QAM modulated signals which corresponds
non-circular 6 , i.e., E[s4p (k)] 6= 0. Based on this nice to 12.5Gbaud transmission per polarisation. The
linewidths of lasers are set to zero. The polarisation
dependent effects (PDE) are simulated using the con-
catenation of random birefringence matrices 8 .
At the receiver, the continuous-time signal is sam-
pled at symbol rate. The linear PDE is compensated
using a 5 taps FIR MIMO filters calculated by means of
CMA algorithm 9 . Afterwards, FO is estimated by using
one of the four following methods: i) coarse step based
on an unique periodogram associated with polarisation
X (this algorithm is usually carried out in QPSK con-
text), ii) coarse step based on the sum of both pe-
riodograms associated with polarisations X and Y, iii)
coarse and fine step based on an unique periodogram
associated with polarisation X, and iv) coarse and fine
step based on the sum of both periodograms associ- Fig. 2: BER vs. OSNR
ated with polarisations X and Y. At each Monte-Carlo
trial, FO is randomly located between 0 and 3.12GHz.
The number of Monte-Carlo runs is fixed to 100.
In Fig. 1, MSE of FO (defined as E[|φ1 − φ̂1,N |2 ])
is plotted versus OSNR for N = 256 and N = 4096.
One observe that the outliers effect 10 at low SNR is
stronger for the methods based on one polarisation
than for those based on both polarisations. We also
confirm that the convergence speed is faster with fine
step than with coarse step. Notice that MSE for FO esti-
mation by using Leven algorithm 2 is around 10−5 when
N = 4096 whereas the proposed method achieves a
MSE less than 10−12 for the same value of N .
Fig. 3: BER vs. frequency offset ∆f
Additional simulations (not plotted in the paper due
to the space limitation) show that the proposed FO es-
timate tolerates more amounts of GVD and PMD. Nev-
ertheless if the estimation algorithm is applied before
CMA algorithm, we have observed a failure probability
of about 5%.
Conclusion
We proposed an accurate non-data aided FO estimator
when QAM modulated signals are considered. Conse-
quently we showed that frequency offset can be prop-
erly mitigated which implies that coherent 16-QAM may
remain an attractive candidate for 100Gbit/s transmis-
sion.
Fig. 1: MSE vs. OSNR for a) N = 256, b) N = 4096
Acknowledgement
This work has been funded by Networks of Future Labs
In Fig. 2, Bit Error Rate (BER) is analysed with re-
located at Institut TELECOM.
spect to OSNR when N = 512. BER of 10−4 is ob-
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