BLIND IDENTIFICATION OF DISTRIBUTED ANTENNA SYSTEMS WITH MULTIPLE
CARRIER FREQUENCY OFFSETS
Yuanning Yu, Athina P. Petropulu and H. Vincent Poor+
Electrical & Computer Engineering Department, Drexel University
+School of Engineering and Applied Science, Princeton University
ABSTRACT
In spatially distributed multiuser antenna systems, the received
signal contains multiple carrier-frequency offsets (CFOs) arising
from mismatch between the oscillators of transmitters and receivers.
This results in a time-varying rotation of the data constellation,
which needs to be compensated at the receiver before symbol re-
covery. In this paper, a new approach for blind CFO estimation
and symbol recovery is proposed. The received base-band signal
is over-sampled, and its polyphase components are used to formu-
late a virtual Multiple-Input Multiple-Output (MIMO) problem. By
applying blind MIMO system estimation techniques, the system re-
sponse can be estimated and decoupled versions of the user symbols
can be recovered, each one of which contains a distinct CFO. By
applying a decision feedback Phase Lock Loop (PLL), the CFO can
be mitigated and the transmitted symbols can be recovered. The esti-
mated MIMO system response provides information about the CFOs
that can be used to initialize the PLL, speed up its convergence, and
avoid ambiguities usually linked with PLL.
keywords-Multi-user Systems, Distributed Antenna Systems,
Carrier Frequency Offset, Blind MIMO System Identification
1. BACKGROUND
In both wireless and wireline communication systems, received sig-
nals are often corrupted by carrier-frequency offsets (CFOs), due
to Doppler shift and/or local oscillator drift. The CFO causes a
time-varying rotation of the data symbols, and thus before symbol
recovery, it must be estimated and accurately compensated for by
the receiver. The CFO can be estimated via the use of pilots sym-
bols; however, even a small error in this estimation tends to cause
large data recovery errors. This necessitates transmission of pilot
symbols rather often. In single user systems, or in multiuser sys-
tems where the transmitters are physically connected to the same
oscillator, there is only one CFO that needs to be estimated. This
is typically done via a decision feedback Phase Lock Loop (PLL)
at the receiver. The PLL is a closed-loop feedback control system
that can adaptively track both frequency and phase offsets between
the equalized signals and the reference constellation. However, de-
pending on the constellation used during transmission, the PLL can
have an M -fold symmetric ambiguity, and thus it has limited CFO
acquisition range; e.g., |fk| < 1/8 for 4QAM signals. Moreover,
the PLL require a long convergence time. To solve these problems,
several methods have been proposed [3], [5], [6], [9] [11] that allow
for blind estimation of the CFO and symbols using the second-order
cyclo-stationary statistics of the over-sampled received signal. Blind
CFO estimation has also been studied in the context of orthogonal
This work was supported by the U. S. National Science Foundation under
Grants ANI-03-38807, CNS-06-25637 and CNS-04-35052.
frequency-division multiplexing (OFDM) systems, where the CFO
destroys the orthogonality between the carriers (see [4] and the ref-
erences therein).
In a spatially distributed multiuser antenna system where data
are transmitted simultaneously from multiple antennas, the received
signal contains multiple CFOs, one for each transmit antenna. A
PLL does not work in this case as there is no single frequency to
lock onto. The literature on estimation of multiple CFOs is rather
sparse. In [8], multiple CFOs were estimated by using pilot sym-
bols that were uncorrelated among the different users. To account
for multiple offsets, [10] proposed that multiple nodes transmit the
same copy of the data with an artificial delay at each node. The
resulting system was modeled as a convolutive single-input/single-
output (SISO) system with time-varying system response caused by
the multiple CFOs. A minimum mean-square error (MMSE) deci-
sion feedback equalizer was used to track and equalize the channel
and to recover the input data. Training symbols were required in
order to obtain a channel estimate, which was used to initialize the
equalizer.
In this paper, a new approach to blind CFO estimation and sym-
bol recovery is proposed. The received base-band signal is over-
sampled, and its polyphase components are used to formulate a vir-
tual MIMO problem. By applying blind MIMO system estimation
techniques, the system response can be estimated, and decoupled
versions of the user symbols can be recovered, each one of which
contains a distinct CFO. By applying a PLL, the CFO can be miti-
gated and the transmitted symbols can be recovered. The estimated
MIMO system response provides information about the CFOs that
can be used to initialize the PLL, speed up its convergence, and avoid
ambiguities usually linked with PLLs.
2. SYSTEM MODEL
We consider a distributed antenna system, where K users trans-
mit simultaneously to a base station. Narrow-band transmission is
assumed here, where the channel between any user and the base
station is frequency non-selective. In addition, quasi-static fading
is assumed, i.e., the channel gains remain fixed during the packet
length. The continuous-time base-band received signal y(t) can be
expressed as
y(t) =
akxk(t− τk)e
j2πFkt + w(t) , (1)
where ak represents the effect of channel fading between the k−th
user and the base station and also phase offset; τk is the delay asso-
ciated with the path between the k−th user and the base station; Fk
is the frequency offset of the k−th user and w(t) represents noise;
http://arxiv.org/abs/0704.1925v1
xk(t) denotes the transmitted signal of user k:
xk(t) =
sk(i)p(t− iTs) , (2)
where sk(i) is the i−th symbol of user k; Ts is the symbol period;
and p(t) is a pulse function with support [0, Ts].
Our objective is to obtain an estimate of s(i) =
[s1(i), ..., sK(i)]
T in the form
ŝ(i) = Λ̂P
s(i) , (3)
where P is a column permutation matrix and Λ̂ a constant diago-
nal matrix. These are considered to be trivial ambiguities, and are
typical in any blind problem.
3. FORMATION OF THE MIMO PROBLEM
The received signal y(t) is sampled at rate 1/T = P/Ts, where
the over-sampling factor P ≥ K is an integer. In order to guar-
antee that all the users’ pulses overlap at the sampling times, the
over-sampling period should satisfy: Ts/P ≥ τk, k = 1, ...K.
Or, in other words, the over-sampling factor P is upper bounded
by Ts/min{τ1, ..., τK}.
Let t = iTs +mT, m = 1, . . . , , P, denote the sampling times.
The over-sampled received signal can be expressed as
ym(i) = y(iTs +mT )
j2πfk(i+
xk((i+
)Ts − τk) + w((i+
j2πfk(i+
sk(i)p(
Ts − τk) + w(iTs +
am,k(sk(i)e
j2πfki) + w(i+
), m = 1, . . . , P , (4)
where fk = FkTs is the normalized frequency offset between the
k−th user and the base station, and the typical element of the virtual
MIMO channel matrix A is given by
am,k = ake
Ts − τk
. (5)
Define the following: y(i)
= [y1(i), ..., yP (i)]
A = {am,k}, a tall matrix of dimension P × K;
s̃(i)
= [s1(i)e
j2πf1i, ..., sK(i)e
j2πfK i]T ; and w(i)
[w(i + 1
), ..., w(i + P
)]T . Then, (4) can be written in ma-
trix form as
y(i) = As̃(i) +w(i) . (6)
We could use the training based method of [8] to solve the
MIMO system (4). That method assumes that the pilot symbols of
different users are uncorrelated. The CFOs are obtained by search-
ing for the location of a peak in the cross-correlation between the
Discrete-Time Fourier Transform (DTFT) of a pilot sequence and
that of the received signal.
In the following we show how to estimate CFOs and recover the
transmitted signals in a bind fashion, i.e., without the need for pilot
symbols. The advantage of a blind approach is bandwidth efficiency
since no bandwidth is wasted transmitting pilot symbols.
4. BLIND CHANNEL ESTIMATION AND
COMPENSATION OF THE CFOS
Let us make the following assumptions.
• A1) For each m = 1, . . . , P , wm(.) is a zero-mean Gaus-
sian stationary random processes with variance σ2w, and is
independent of the channel inputs.
• A2) For each k, the sequence sk(i) is a zero
mean with independent and identically distributed
(i.i.d.) elements having nonzero kurtosis; i.e.,
γ4sk = Cum[sk(i), s
k(i), sk(i), s
k(i)] 6= 0. The sequences
sk’s are also mutually independent.
• A3) The over-sampling factor P is no less than K.
Under assumption (A2), it is easy to verify that the rotated input sig-
nals s̃k(.) are also zero mean and i.i.d with nonzero kurtosis. Also,
the s̃k(i)’s are mutually independent for different k’s. Assumption
(A3) guarantees that the virtual MIMO channel matrix A in (6) has
full rank with probability one. If the delays of users are randomly
distributed in the interval [0, Ts/P ), then each row of the channel
matrix can be viewed as having been drawn randomly from a contin-
uous distribution so that the channel matrix has full rank with prob-
ability one.
One can apply any blind source separation algorithm (e.g.,
[1],[2] or [7] ) to obtain
= APΛ . (7)
Subsequently, using a least-squares equalizer we can obtain an
estimate of the de-coupled signals s̃(i), within permutation and
scalar ambiguities as
ˆ̃s(i) = (Â
y(i) = e
jArg{−Λ}
s̃(i) . (8)
Without loss of generality we can assume that the transmitted signal
has unit power. Then, on denoting by θk the k−th diagonal element
of Arg{Λ}, the j−th separated input signal can be expressed as
ˆ̃sk(i) = sk(i)e
j(−θk+2πfki) . (9)
In order to recover the transmitted signals, we still need to mit-
igate the effect of CFO in each decoupled signal. This can be done
via a PLL. By using the decoupled signals as inputs, and the constel-
lation used in transmission as a reference, the PLL can effectively
mitigate the CFO by minimizing the feedback error, which is calcu-
lated based on the distance of the recovered signal and the closest
valid constellation point. However, depending on the constellation
used in transmission, there is a four-fold symmetric ambiguity for
MQAM signals, or M -fold symmetric ambiguity for MPSK signals.
For example, for 4QAM signals, and an initial CFO value of fk = 0,
the effective tracking range for fk is |fk| < 1/8. Moreover, de-
pending on the value of the CFO, the PLL generally needs a long
convergence time, during which the input signals are not correctly
recovered.
Next we will show that by exploring the structure of the virtual
channel matrix, we can obtain an estimate of the CFOs, which can
then be used to initialize the PLL. By doing this, we can prevent
the symmetric ambiguity problem and enlarge the effective tracking
range of the PLL from |fk| < 1/8 to |fk| < 1/2. Also, the conver-
gence time of the PLL can be greatly reduced.
By taking the phase of the estimated channel matrix Â, we ob-
Ψ = Arg
+ φ1 . . .
. . .
2πf1P
+ φ1 . . .
2πfKP
where φk = Arg{ak} + θk, which accounts for both the phase of
ak and the estimated phase ambiguity in (9).
By applying linear fitting on the j−th column of Ψ we obtain
the least squares estimate of fj as
f̂j =
p=1 pΨp,j)− (
p=1 p)(
p=1 Ψp,j)
p2)− (
. (11)
We can write f̂j = fj + ǫj where ǫj represents estimation error.
On noting that the de-coupled signals ˆ̃sj(i) in (9) are shuffled in
the same manner as the estimated CFOs in (11), we can use the esti-
mated CFOs to compensate for the effects of CFO in the decoupled
signals (9) and thereby obtain estimates of the input signals as
ŝ(i) = e
jArg{−Λ}
s(i) . (12)
Due to errors in the channel estimates, we can only compensate
for most, but not all, of the CFO effects in (9) and so we can write
ŝk(i) = sk(i)e
j(−θk−2πǫki) . (13)
By subsequently applying a PLL to ŝj(i), we can further mitigate
the effects of the residual CFO ǫk. For 4QAM signals, as long as
|ǫk| < 1/8, the residual CFO can be effectively removed by the PLL.
The initial CFO estimator (11) can prevent the symmetric ambiguity
of the PLL, and can also greatly reduce the convergence time of the
PLL. From (10), we can see that the CFO estimator will achieve full
acquisition range for the normalized CFO.
5. SIMULATION RESULTS
In this section, we verify the validity of the proposed method via sim-
ulations, under the following assumptions. The channel coefficients
ak, k = 1, . . . ,K are zero-mean Gaussian random variables. The
waveform p(·) is a Hamming window. The delays, τk, k = 1, . . . ,K
are uniformly distributed in the range [0, Ts/P ). The input signals
are 4QAM signals.
The blind source separation algorithm used here
is the JADE method, which was downloaded from
http://www.tsi.enst.fr/c̃ardoso/guidesepsou.html.
First we show results for a two-user systems with f1 =
−0.1552, f2 = 0.4335, a1 = 0.3173 − 0.6483i, a2 = 0.1625 +
0.5867i, with SNR = 20dB, and N = 1024. In Fig. 1 we show
the polyphase outputs y1, y2. Due to the mixing and the CFOs no
obvious constellation is visible. In Fig. 2, we show the de-coupled
signals ˆ̃sk, k = 1, 2 right after JADE. Although still rotated by the
CFOs, two signals s̃k, k = 1, 2 are clearly separated. In Fig. 3, we
show the recovered input signals ŝk, k = 1, 2, where we can see
that after compensating for the effect of CFOs the constellations are
recovered.
Next we show estimation results averaged over 300 indepen-
dent channel runs, and 20 Monte-Carlo runs for each channel. For
each channel case, the coefficients ak, k = 1, 2 were generated
randomly, and the continuous CFOs where chosen randomly in the
range [− 1
). The delays, τk, k = 1, ..., K where chosen uni-
formly in the range of [0, Ts/P ). The transmitted signal was 4QAM.
The performance of both the pilot-based method and the pro-
posed method at different data lengths and with SNR set to 30dB
is shown in Fig. 4. For the pilot-based method, each user trans-
mits a pilot sequence of length 32, and the pilots are random se-
quences uncorrelated between different users. Fig. 4 shows the
mean-square error (MSE) for the CFO estimate (11) for different
values of the over-sampling factor P . The MSE is calculated usings
[(f̂k − fk)]
2 = 1
[(F̂k − Fk)Ts]
2. We can see that
by increasing P we can improve the estimation accuracy. Fig. 5
shows the Bit Error Rate (BER) for different values of P . For both
blind and training methods, the BER is calculated based on the re-
covered signals after the PLL. As expected, the BER performance
also improves by increasing P . The proposed method appears to
work well even for short data length.
Next we show the performance of both methods at various noise
levels. We use packet length N = 1, 024. In Fig. 6 we show the
MSE of the blind CFO estimator (11) as well as that of the training
based method. We can see that by increasing P we improve estima-
tion accuracy. In Fig. 7, we show the BER performance after the
PLL for both blind and training based methods. We see that the pro-
posed blind method has almost the same performance as the training
based method for SNR values lower than 20dB, while the training
based method can achieve better BER performance for higher values
of SNR.
6. CONCLUSION
In this paper we have proposed a novel blind approach for identifica-
tion of a distributed multiuser antenna system with multiple CFOs.
By over-sampling of the received base-band signal, the MISO prob-
lem is converted into a MIMO one. Blind MIMO system estimation
then yields the system response, and MIMO input recovery yields
the decoupled transmitted signals, each one containing a CFO. By
exploring the structure of the MIMO system response we obtain a
coarse estimate of the CFOs, which is then combined with a deci-
sion feedback PLL to compensate for the CFOs in the decoupled
transmitted signals. The proposed blind method has full acquisition
range for normalized CFOs.
7. REFERENCES
[1] T. Acar, Y. Yu and A. P. Petropulu, “Blind MIMO system estimation
based on PARAFAC decomposition of higher order output tensors”
IEEE Trans. Signal Process., Vol. 54, No. 11, pp. 4156 - 4168, Nov.
[2] J. F. Cardoso and A. Souloumiac, “Blind beamforming for non-
Gaussian signals,” IEE Proceedings: Radar and Signal Process., Vol.
140, No. 6, pp. 362-370 Dec. 1993.
[3] P. Ciblat, P. Loubaton, E. Serpedin and G. B. Giannakis, “Perfor-
mance analysis of blind carrier frequency offset estimators for non-
circular transmissions through frequency-selective channels,” IEEE
Trans. Signal Process., Vol. 50, No. 1, pp. 130 - 140, Jan. 2002.
[4] T. Roman, S. Visuri and V. Koivunen, “Blind frequency synchroniza-
tion in OFDM via diagonality criterion,” IEEE Trans. Signal Pro-
cess., Vol. 54, No. 8, pp. 3125 - 3135, Aug. 2006.
[5] M. Ghogho, A. Swami and T. Durrani, “On blind carrier recovery
in time-selective fading channels,” in Proc. 33rd Asilomar Conf. Sig-
nals, Systems and Computers, Vol. 1, pp. 243-247, Pacific Grove, CA,
Oct. 1999,
http://www.tsi.enst.fr/~cardoso/guidesepsou.html
[6] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing
recovery in flat-fading channels: A cyclostationary approach,” IEEE
Trans. Commun., Vol. 46, No. 3, pp. 400 - 411, Mar. 1998.
[7] L. de Lathauwer, B. de Moor and J. Vandewalle, “Independent com-
ponent analysis and (simultaneous) third-order tensor diagonaliza-
tion,” IEEE Trans. Signal Process., Vol. 49, No. 10, pp. 2262-2271,
Oct. 2001.
[8] F. Prihoda, E. Garbarine and A. P. Petropulu, “Resolving wireless
collisions in random access networks,” Proc. 2006 Asilomar Conf.
Signals, Systems and Computers, Pacific Grove, CA, Oct. 2006.
[9] K. E. Scott and E. B. Olasz, “Simultaneous clock phase and frequency
offset estimation,” IEEE Trans. Commun., Vol. 43, No. 7, pp. 2263 -
2270, July 1995.
[10] D. Veronesi and D. L. Goeckel, “Multiple frequency offset compen-
sation in cooperative wireless systems,” Proc. IEEE Global Telecom-
mun. Conf., San Francisco, CA, Nov. 2006.
[11] Y. Wang, P. Ciblat, E. Serpedin and P. Loubaton, “Performance anal-
ysis of a class of nondata-aided frequency offset and symbol timing
estimators for flat-fading channels,” IEEE Trans. Signal Process.,
Vol. 50, No. 9, pp. 2295- 2305, Sept. 2002.
−5 0 5
received signals y
−5 0 5
received signals y
Fig. 1. Received mixing signal y
−4 −2 0 2 4
decoupled signals 1 
−5 0 5
decoupled signals 2
Fig. 2. De-coupled inputs ˆ̃s
−4 −3 −2 −1 0 1 2 3 4
recovered input signal 1
−4 −3 −2 −1 0 1 2 3 4
recovered input signal 2
Fig. 3. Recovered input signals ŝ with P = 2
0 500 1000 1500 2000 2500 3000 3500 4000 4500
packet length N
MSE of CFOs at SNR=30dB
P=2, bind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 4. MSE of CFOs vs N for K=2, with SNR=30dB, 4QAM
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Packet length N
BER Comparison for SNR 30dB
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 5. BER vs N for K=2, with SNR=30dB, 4QAM
0 5 10 15 20 25 30
SNR dB
MSE of CFOs at N=1024
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 6. MSE of CFOs vs SNR for K=2, 4QAM
0 5 10 15 20 25 30
SNR dB
BER Comparison at N=1024
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 7. BER vs SNR for K=2, 4QAM, T=1024
	 Background
	 System Model
	 Formation of the MIMO Problem
	 Blind channel estimation and compensation of the CFOs
	 Simulation Results
	 Conclusion
	 References

  In spatially distributed multiuser antenna systems, the received signal
contains multiple carrier-frequency offsets (CFOs) arising from mismatch
between the oscillators of transmitters and receivers. This results in a
time-varying rotation of the data constellation, which needs to be compensated
at the receiver before symbol recovery. In this paper, a new approach for blind
CFO estimation and symbol recovery is proposed. The received base-band signal
is over-sampled, and its polyphase components are used to formulate a virtual
Multiple-Input Multiple-Output (MIMO) problem. By applying blind MIMO system
estimation techniques, the system response can be estimated and decoupled
versions of the user symbols can be recovered, each one of which contains a
distinct CFO. By applying a decision feedback Phase Lock Loop (PLL), the CFO
can be mitigated and the transmitted symbols can be recovered. The estimated
MIMO system response provides information about the CFOs that can be used to
initialize the PLL, speed up its convergence, and avoid ambiguities usually
linked with PLL.

BLIND IDENTIFICATION OF DISTRIBUTED ANTENNA SYSTEMS WITH MULTIPLE
CARRIER FREQUENCY OFFSETS
Yuanning Yu, Athina P. Petropulu and H. Vincent Poor+
Electrical & Computer Engineering Department, Drexel University
+School of Engineering and Applied Science, Princeton University
ABSTRACT
In spatially distributed multiuser antenna systems, the received
signal contains multiple carrier-frequency offsets (CFOs) arising
from mismatch between the oscillators of transmitters and receivers.
This results in a time-varying rotation of the data constellation,
which needs to be compensated at the receiver before symbol re-
covery. In this paper, a new approach for blind CFO estimation
and symbol recovery is proposed. The received base-band signal
is over-sampled, and its polyphase components are used to formu-
late a virtual Multiple-Input Multiple-Output (MIMO) problem. By
applying blind MIMO system estimation techniques, the system re-
sponse can be estimated and decoupled versions of the user symbols
can be recovered, each one of which contains a distinct CFO. By
applying a decision feedback Phase Lock Loop (PLL), the CFO can
be mitigated and the transmitted symbols can be recovered. The esti-
mated MIMO system response provides information about the CFOs
that can be used to initialize the PLL, speed up its convergence, and
avoid ambiguities usually linked with PLL.
keywords-Multi-user Systems, Distributed Antenna Systems,
Carrier Frequency Offset, Blind MIMO System Identification
1. BACKGROUND
In both wireless and wireline communication systems, received sig-
nals are often corrupted by carrier-frequency offsets (CFOs), due
to Doppler shift and/or local oscillator drift. The CFO causes a
time-varying rotation of the data symbols, and thus before symbol
recovery, it must be estimated and accurately compensated for by
the receiver. The CFO can be estimated via the use of pilots sym-
bols; however, even a small error in this estimation tends to cause
large data recovery errors. This necessitates transmission of pilot
symbols rather often. In single user systems, or in multiuser sys-
tems where the transmitters are physically connected to the same
oscillator, there is only one CFO that needs to be estimated. This
is typically done via a decision feedback Phase Lock Loop (PLL)
at the receiver. The PLL is a closed-loop feedback control system
that can adaptively track both frequency and phase offsets between
the equalized signals and the reference constellation. However, de-
pending on the constellation used during transmission, the PLL can
have an M -fold symmetric ambiguity, and thus it has limited CFO
acquisition range; e.g., |fk| < 1/8 for 4QAM signals. Moreover,
the PLL require a long convergence time. To solve these problems,
several methods have been proposed [3], [5], [6], [9] [11] that allow
for blind estimation of the CFO and symbols using the second-order
cyclo-stationary statistics of the over-sampled received signal. Blind
CFO estimation has also been studied in the context of orthogonal
This work was supported by the U. S. National Science Foundation under
Grants ANI-03-38807, CNS-06-25637 and CNS-04-35052.
frequency-division multiplexing (OFDM) systems, where the CFO
destroys the orthogonality between the carriers (see [4] and the ref-
erences therein).
In a spatially distributed multiuser antenna system where data
are transmitted simultaneously from multiple antennas, the received
signal contains multiple CFOs, one for each transmit antenna. A
PLL does not work in this case as there is no single frequency to
lock onto. The literature on estimation of multiple CFOs is rather
sparse. In [8], multiple CFOs were estimated by using pilot sym-
bols that were uncorrelated among the different users. To account
for multiple offsets, [10] proposed that multiple nodes transmit the
same copy of the data with an artificial delay at each node. The
resulting system was modeled as a convolutive single-input/single-
output (SISO) system with time-varying system response caused by
the multiple CFOs. A minimum mean-square error (MMSE) deci-
sion feedback equalizer was used to track and equalize the channel
and to recover the input data. Training symbols were required in
order to obtain a channel estimate, which was used to initialize the
equalizer.
In this paper, a new approach to blind CFO estimation and sym-
bol recovery is proposed. The received base-band signal is over-
sampled, and its polyphase components are used to formulate a vir-
tual MIMO problem. By applying blind MIMO system estimation
techniques, the system response can be estimated, and decoupled
versions of the user symbols can be recovered, each one of which
contains a distinct CFO. By applying a PLL, the CFO can be miti-
gated and the transmitted symbols can be recovered. The estimated
MIMO system response provides information about the CFOs that
can be used to initialize the PLL, speed up its convergence, and avoid
ambiguities usually linked with PLLs.
2. SYSTEM MODEL
We consider a distributed antenna system, where K users trans-
mit simultaneously to a base station. Narrow-band transmission is
assumed here, where the channel between any user and the base
station is frequency non-selective. In addition, quasi-static fading
is assumed, i.e., the channel gains remain fixed during the packet
length. The continuous-time base-band received signal y(t) can be
expressed as
y(t) =
akxk(t− τk)e
j2πFkt + w(t) , (1)
where ak represents the effect of channel fading between the k−th
user and the base station and also phase offset; τk is the delay asso-
ciated with the path between the k−th user and the base station; Fk
is the frequency offset of the k−th user and w(t) represents noise;
http://arxiv.org/abs/0704.1925v1
xk(t) denotes the transmitted signal of user k:
xk(t) =
sk(i)p(t− iTs) , (2)
where sk(i) is the i−th symbol of user k; Ts is the symbol period;
and p(t) is a pulse function with support [0, Ts].
Our objective is to obtain an estimate of s(i) =
[s1(i), ..., sK(i)]
T in the form
ŝ(i) = Λ̂P
s(i) , (3)
where P is a column permutation matrix and Λ̂ a constant diago-
nal matrix. These are considered to be trivial ambiguities, and are
typical in any blind problem.
3. FORMATION OF THE MIMO PROBLEM
The received signal y(t) is sampled at rate 1/T = P/Ts, where
the over-sampling factor P ≥ K is an integer. In order to guar-
antee that all the users’ pulses overlap at the sampling times, the
over-sampling period should satisfy: Ts/P ≥ τk, k = 1, ...K.
Or, in other words, the over-sampling factor P is upper bounded
by Ts/min{τ1, ..., τK}.
Let t = iTs +mT, m = 1, . . . , , P, denote the sampling times.
The over-sampled received signal can be expressed as
ym(i) = y(iTs +mT )
j2πfk(i+
xk((i+
)Ts − τk) + w((i+
j2πfk(i+
sk(i)p(
Ts − τk) + w(iTs +
am,k(sk(i)e
j2πfki) + w(i+
), m = 1, . . . , P , (4)
where fk = FkTs is the normalized frequency offset between the
k−th user and the base station, and the typical element of the virtual
MIMO channel matrix A is given by
am,k = ake
Ts − τk
. (5)
Define the following: y(i)
= [y1(i), ..., yP (i)]
A = {am,k}, a tall matrix of dimension P × K;
s̃(i)
= [s1(i)e
j2πf1i, ..., sK(i)e
j2πfK i]T ; and w(i)
[w(i + 1
), ..., w(i + P
)]T . Then, (4) can be written in ma-
trix form as
y(i) = As̃(i) +w(i) . (6)
We could use the training based method of [8] to solve the
MIMO system (4). That method assumes that the pilot symbols of
different users are uncorrelated. The CFOs are obtained by search-
ing for the location of a peak in the cross-correlation between the
Discrete-Time Fourier Transform (DTFT) of a pilot sequence and
that of the received signal.
In the following we show how to estimate CFOs and recover the
transmitted signals in a bind fashion, i.e., without the need for pilot
symbols. The advantage of a blind approach is bandwidth efficiency
since no bandwidth is wasted transmitting pilot symbols.
4. BLIND CHANNEL ESTIMATION AND
COMPENSATION OF THE CFOS
Let us make the following assumptions.
• A1) For each m = 1, . . . , P , wm(.) is a zero-mean Gaus-
sian stationary random processes with variance σ2w, and is
independent of the channel inputs.
• A2) For each k, the sequence sk(i) is a zero
mean with independent and identically distributed
(i.i.d.) elements having nonzero kurtosis; i.e.,
γ4sk = Cum[sk(i), s
k(i), sk(i), s
k(i)] 6= 0. The sequences
sk’s are also mutually independent.
• A3) The over-sampling factor P is no less than K.
Under assumption (A2), it is easy to verify that the rotated input sig-
nals s̃k(.) are also zero mean and i.i.d with nonzero kurtosis. Also,
the s̃k(i)’s are mutually independent for different k’s. Assumption
(A3) guarantees that the virtual MIMO channel matrix A in (6) has
full rank with probability one. If the delays of users are randomly
distributed in the interval [0, Ts/P ), then each row of the channel
matrix can be viewed as having been drawn randomly from a contin-
uous distribution so that the channel matrix has full rank with prob-
ability one.
One can apply any blind source separation algorithm (e.g.,
[1],[2] or [7] ) to obtain
= APΛ . (7)
Subsequently, using a least-squares equalizer we can obtain an
estimate of the de-coupled signals s̃(i), within permutation and
scalar ambiguities as
ˆ̃s(i) = (Â
y(i) = e
jArg{−Λ}
s̃(i) . (8)
Without loss of generality we can assume that the transmitted signal
has unit power. Then, on denoting by θk the k−th diagonal element
of Arg{Λ}, the j−th separated input signal can be expressed as
ˆ̃sk(i) = sk(i)e
j(−θk+2πfki) . (9)
In order to recover the transmitted signals, we still need to mit-
igate the effect of CFO in each decoupled signal. This can be done
via a PLL. By using the decoupled signals as inputs, and the constel-
lation used in transmission as a reference, the PLL can effectively
mitigate the CFO by minimizing the feedback error, which is calcu-
lated based on the distance of the recovered signal and the closest
valid constellation point. However, depending on the constellation
used in transmission, there is a four-fold symmetric ambiguity for
MQAM signals, or M -fold symmetric ambiguity for MPSK signals.
For example, for 4QAM signals, and an initial CFO value of fk = 0,
the effective tracking range for fk is |fk| < 1/8. Moreover, de-
pending on the value of the CFO, the PLL generally needs a long
convergence time, during which the input signals are not correctly
recovered.
Next we will show that by exploring the structure of the virtual
channel matrix, we can obtain an estimate of the CFOs, which can
then be used to initialize the PLL. By doing this, we can prevent
the symmetric ambiguity problem and enlarge the effective tracking
range of the PLL from |fk| < 1/8 to |fk| < 1/2. Also, the conver-
gence time of the PLL can be greatly reduced.
By taking the phase of the estimated channel matrix Â, we ob-
Ψ = Arg
+ φ1 . . .
. . .
2πf1P
+ φ1 . . .
2πfKP
where φk = Arg{ak} + θk, which accounts for both the phase of
ak and the estimated phase ambiguity in (9).
By applying linear fitting on the j−th column of Ψ we obtain
the least squares estimate of fj as
f̂j =
p=1 pΨp,j)− (
p=1 p)(
p=1 Ψp,j)
p2)− (
. (11)
We can write f̂j = fj + ǫj where ǫj represents estimation error.
On noting that the de-coupled signals ˆ̃sj(i) in (9) are shuffled in
the same manner as the estimated CFOs in (11), we can use the esti-
mated CFOs to compensate for the effects of CFO in the decoupled
signals (9) and thereby obtain estimates of the input signals as
ŝ(i) = e
jArg{−Λ}
s(i) . (12)
Due to errors in the channel estimates, we can only compensate
for most, but not all, of the CFO effects in (9) and so we can write
ŝk(i) = sk(i)e
j(−θk−2πǫki) . (13)
By subsequently applying a PLL to ŝj(i), we can further mitigate
the effects of the residual CFO ǫk. For 4QAM signals, as long as
|ǫk| < 1/8, the residual CFO can be effectively removed by the PLL.
The initial CFO estimator (11) can prevent the symmetric ambiguity
of the PLL, and can also greatly reduce the convergence time of the
PLL. From (10), we can see that the CFO estimator will achieve full
acquisition range for the normalized CFO.
5. SIMULATION RESULTS
In this section, we verify the validity of the proposed method via sim-
ulations, under the following assumptions. The channel coefficients
ak, k = 1, . . . ,K are zero-mean Gaussian random variables. The
waveform p(·) is a Hamming window. The delays, τk, k = 1, . . . ,K
are uniformly distributed in the range [0, Ts/P ). The input signals
are 4QAM signals.
The blind source separation algorithm used here
is the JADE method, which was downloaded from
http://www.tsi.enst.fr/c̃ardoso/guidesepsou.html.
First we show results for a two-user systems with f1 =
−0.1552, f2 = 0.4335, a1 = 0.3173 − 0.6483i, a2 = 0.1625 +
0.5867i, with SNR = 20dB, and N = 1024. In Fig. 1 we show
the polyphase outputs y1, y2. Due to the mixing and the CFOs no
obvious constellation is visible. In Fig. 2, we show the de-coupled
signals ˆ̃sk, k = 1, 2 right after JADE. Although still rotated by the
CFOs, two signals s̃k, k = 1, 2 are clearly separated. In Fig. 3, we
show the recovered input signals ŝk, k = 1, 2, where we can see
that after compensating for the effect of CFOs the constellations are
recovered.
Next we show estimation results averaged over 300 indepen-
dent channel runs, and 20 Monte-Carlo runs for each channel. For
each channel case, the coefficients ak, k = 1, 2 were generated
randomly, and the continuous CFOs where chosen randomly in the
range [− 1
). The delays, τk, k = 1, ..., K where chosen uni-
formly in the range of [0, Ts/P ). The transmitted signal was 4QAM.
The performance of both the pilot-based method and the pro-
posed method at different data lengths and with SNR set to 30dB
is shown in Fig. 4. For the pilot-based method, each user trans-
mits a pilot sequence of length 32, and the pilots are random se-
quences uncorrelated between different users. Fig. 4 shows the
mean-square error (MSE) for the CFO estimate (11) for different
values of the over-sampling factor P . The MSE is calculated usings
[(f̂k − fk)]
2 = 1
[(F̂k − Fk)Ts]
2. We can see that
by increasing P we can improve the estimation accuracy. Fig. 5
shows the Bit Error Rate (BER) for different values of P . For both
blind and training methods, the BER is calculated based on the re-
covered signals after the PLL. As expected, the BER performance
also improves by increasing P . The proposed method appears to
work well even for short data length.
Next we show the performance of both methods at various noise
levels. We use packet length N = 1, 024. In Fig. 6 we show the
MSE of the blind CFO estimator (11) as well as that of the training
based method. We can see that by increasing P we improve estima-
tion accuracy. In Fig. 7, we show the BER performance after the
PLL for both blind and training based methods. We see that the pro-
posed blind method has almost the same performance as the training
based method for SNR values lower than 20dB, while the training
based method can achieve better BER performance for higher values
of SNR.
6. CONCLUSION
In this paper we have proposed a novel blind approach for identifica-
tion of a distributed multiuser antenna system with multiple CFOs.
By over-sampling of the received base-band signal, the MISO prob-
lem is converted into a MIMO one. Blind MIMO system estimation
then yields the system response, and MIMO input recovery yields
the decoupled transmitted signals, each one containing a CFO. By
exploring the structure of the MIMO system response we obtain a
coarse estimate of the CFOs, which is then combined with a deci-
sion feedback PLL to compensate for the CFOs in the decoupled
transmitted signals. The proposed blind method has full acquisition
range for normalized CFOs.
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−5 0 5
received signals y
−5 0 5
received signals y
Fig. 1. Received mixing signal y
−4 −2 0 2 4
decoupled signals 1 
−5 0 5
decoupled signals 2
Fig. 2. De-coupled inputs ˆ̃s
−4 −3 −2 −1 0 1 2 3 4
recovered input signal 1
−4 −3 −2 −1 0 1 2 3 4
recovered input signal 2
Fig. 3. Recovered input signals ŝ with P = 2
0 500 1000 1500 2000 2500 3000 3500 4000 4500
packet length N
MSE of CFOs at SNR=30dB
P=2, bind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 4. MSE of CFOs vs N for K=2, with SNR=30dB, 4QAM
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Packet length N
BER Comparison for SNR 30dB
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 5. BER vs N for K=2, with SNR=30dB, 4QAM
0 5 10 15 20 25 30
SNR dB
MSE of CFOs at N=1024
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 6. MSE of CFOs vs SNR for K=2, 4QAM
0 5 10 15 20 25 30
SNR dB
BER Comparison at N=1024
P=2, blind
P=2, pilots
P=4, blind
P=4, pilots
Fig. 7. BER vs SNR for K=2, 4QAM, T=1024
	 Background
	 System Model
	 Formation of the MIMO Problem
	 Blind channel estimation and compensation of the CFOs
	 Simulation Results
	 Conclusion
	 References
