Distributed probabilistic data association detector with turbo base-station cooperations in multi-user multi-cell MIMO systems
Yin et al. EURASIP Journal on Wireless Communications and
Distributed probabilistic data association detector with turbo base-station cooperations in multi-user multi-cell MIMO systems
Yufang Yin 1
Gangjun Li 1
Hua Wei 0
0 Chengdu University of Information and Technology , Chengdu , China
1 Chengdu Technological University , Chengdu , China
We address the severe interference condition with unity frequency re-use factor in each cell and propose a novel distributed receiver for reliable detection of the uncoded data transmitted in multi-user multi-cell multiple input, multiple output (MIMO) systems. The new algorithm employs distributed probabilistic data association (PDA) with turbo base-station (BS) cooperations (PDA-TB). From a Bayesian point of view, we modify the original complex PDA detector to incorporate the a priori probability, which makes the complex PDA detector applicable to turbo processing: the soft information is extracted using PDA algorithm at local BS and then used as the a priori probability for further detection at neighboring BSs. Final soft decision is made after the algorithm converges. In this way, the benefit of macro-diversity is achieved and the inter-cell interferences are mitigated. In heavy interference condition, the proposed algorithm outperforms the distributed PDA aided soft combining (PDA-SC) and achieves the near optimal performance. Meanwhile, low complexity is maintained due to the rapid convergence.
PDA; MIMO; Turbo detector; BS cooperations
To design the mobile communication systems, one has to
face two key challenges: fading and interference. Both of
them restrict the system coverage and capacity. Especially
in a multi-user multi-cell multiple input, multiple output
(MIMO) system, the gain of MIMO technique is
determined by the inter-cell co-channel interference (CCI).
Consider a server interference condition with unit
frequency re-use factor where, at the cell boundary areas,
mobile stations (MSs) transmit the signals which can
be detected by the surrounding base stations (BSs). The
conventional non-cooperative BS detects the desired MS
signals independently using local received data. Moreover,
the co-channel users’ signals are considered as
interference and then degrade the system throughput. However,
in the cooperative system, those interference signals are
dealt with specific tools such as virtual MIMO and then
are treated more like the useful information to jointly
2Chengdu University of Information and Technology, Chengdu, China
Full list of author information is available at the end of the article
detect the MS signals. In this way, the number of
cochannel links can be maximized so as to optimize the
In the centralized BS cooperations, the complexity of
the general multi-user multi-cell MIMO detection grows
exponentially with the number of antennas. It raises a
question of scalability and motivates the search for
simplified suboptimal algorithms. To solve this problem, it is
straightforward to distribute, across a network of
interacting BSs, the global uplink task of demodulating all
users’ data symbols . Each BS individually performs
local computations and then passes the processed soft
information to immediate neighbors for further
processing. Distributed BS cooperations were discussed in recent
literatures [2, 5–7]. Among them, Yang  proposed a
simple soft combining method through which the soft
information, calculated in parallel at different BSs, is
statistically aggregated to make the final decision.
Besides the conventional distributed schemes, it is very
natural to try and apply well-known turbo principle ,
which has been employed in turbo equalization [3, 4],
Bayesian data integration [7, 8], and turbo BS
cooperations [5–7, 9]. Turbo principle was originated from turbo
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with turbo BS cooperations is described in Section 3.
Simulation results are presented in Section 4, and finally, the
conclusions are drawn in Section 5.
2 Problem formulation
Consider the uplink in a multi-user multi-cell system,
where the uncoded data is transmitted from Nm mobile
stations (MS) to Nb BSs. Each MS has Nt transmit
antennas, and each BS is equipped with Nr receive antennas. In
the cell edges, assume the signals transmitted by MSs are
in the detectable range (DR) of Nb BSs and then can be
detected by all the BSs. We are interested in this extremely
challenging scenario and discuss the signal model as
The signal received at BSi is
where yi is the Nr × 1 vector of symbols received at BSi,
Hli is the Nr × Nt complex channel matrix between MSl
and BSi, xl is the Nt × 1 vector of symbols transmitted
from MSl, and ni is the complex Gaussian noise with zero
mean and covariance matrix σn2INr . This signal model can
be further formulated as a virtual MIMO system as the
yi = Hix + ni,
decoding, and in the context of cooperative
communication systems, it explores the synergy from distributed
detections on symbols and combines the soft
(probabilistic) detection in a turbo fashion to achieve greater
performance improvement. The implementation of turbo
cooperations separates the joint detections into
commonly two or possibly more connected units (BSs), and
the a posteriori probability (APP) of symbols are
calculated at each unit with the extrinsic information obtained
at other units as the a priori probability. The processes
then iterate among the units until the convergence of
the APPs. The output of the turbo BS cooperation is a
near optimal approximation of true APPs, and Mayer 
illustrated the explicit connection to Turbo-decoding and
To obtain the soft information through local detection,
many low-complexity algorithms have been investigated
as alternative designs for optimal maximum a posteriori
probability (MAP) solution. Among them, probabilistic
data association (PDA) has attracted many attentions and
shown to provide a near-optimal multi-user detection
(MUD) [11, 12] performance in the context of Code
Division Multiple Access (CDMA) . The basic idea of the
PDA is to approximate the Gaussian mixture with a
single Gaussian distribution, a concept seemingly simple yet
very effective in practice. Another appealing advantage
of the PDA is that it is soft in nature and thus
applicable to turbo BS cooperations. Moreover,  indicated
that the larger the number of users, the better
performance achievable due to the Gaussian approximation.
This makes PDA attractive for a MUD scenario with large
number of interferers.
The soft combining in  is based on complex PDA
(CPDA) algorithm [13–15], which matches the full
parameters of the complex Gaussian distribution and
outperforms the approximated CPDA used in our previous work
. However, the original form of CPDA assumes uniform
(or non-informative) prior and then does not incorporate
practical prior knowledge. As a result, empirical
information obtained in other BSs is lost and then the overall
performance is somehow degraded. It implies that further
performance improvement over the PDA-SC algorithm is
possible if the prior information can be carefully
In this paper, we investigate the distributed PDA
detector with turbo BS cooperations. We modify the
structure for original CPDA, where the a priori probability is
appropriately incorporated. This modification considers
the soft information obtained from neighboring BSs and
then makes PDA readily applicable to turbo BS
cooperations. We also provide the performance of the proposed
algorithm by simulations.
The remaining of the paper is organized as follows. The
problem is formulated in Section 2. The distributed PDA
where Hi =
Hi1, · · · , HiNm , x = (x1) , · · · , (xNm )
Assume Hi and σn2 are known.
To mitigate the inter-cell interferences, a natural way is
to perform the joint detection with BS cooperations. Our
objective is to determine xk, k = 1, . . . , Nm × Nt given
received signals yi, i = 1, . . . , Nb. From a Bayesian point of
view, the marginal APP is calculated to make the decision,
xˆk = arg mαamx p xk = αm | y1:Nb
where y1:Nb = y1, · · · , yNb and αm is the m-th element of
the M-ary constellation A.
3 PDA-based turbo BS cooperations
The goal of our soft detector is to obtain p xk|y1:Nb
and then detect the transmitted symbol xk. One way to
reduce the complexity is to perform distributed
detection, using local yi to calculate p(xk|yi) and approximating
p xk|y1:Nb through BS cooperations. This motivates us
to exploit distributed detector with BS cooperations, and
it is very natural to apply well-known turbo principle to
achieve near optimal performance.
p(xk|yi) can be obtained easily if p(yi|xk) is known since
However, to obtain p(yi|xk), marginalization must be
where x−k represents a (NtNm − 1) × 1 vector that
contains all data symbols except xk. Apparently, the optimal
MAP solution of (3) is of complexity exponential
increasing of NtNm and thus computationally infeasible for large
systems. Notice that x−k p(yi|x)p(x−k) is the mixture of
Gaussian distribution; we can further simplify the receiver
by approximating the mixture Gaussian by a single
Gaussian. This motivates us to employ PDA detector at every
BS. In addition, modification is made to allow PDA
detector to accept prior information and then workable in turbo
The proposed algorithm is then named by PDA-TB. It
follows the Bayesian rule and turbo principle that BSi
individually perform local PDA computation with yi, and then
outputs the soft information as the a priori probability for
neighboring BSs to continue their PDA detections until
the APPs calculated at different BSs get convergence.
3.1 The complex PDA detector
3.1.1 NmNt ≤ Nr
Given that (Hi)H Hi is invertible, (2) can be rewritten as
the decorrelated signal model:
y˜i = (Hi)H Hi −1 (Hi)H y,
and it is equivalent to have
y˜i = x + n˜i
= xkek +
where n˜i is a Gaussian noise with zero mean and
covariance = σn2 (Hi)H Hi −1, ek is a column vector with
a 1 in the kth element and zeros elsewhere, and vector
vik = xjej + n˜ i has a multimodal Gaussian mixture
Since the signal model is complex-valued, the complex
formulation of PDA is preferred. The basic idea behind
complex PDA is to approximate vik as a single Gaussian
distribution with the matched mean E(vik), covariance
V (vik), and pseudo-convariance U(vik) ,
V (vik) =
V (xj)ejej + σ 2 (Hi)H Hi −1 ,
V (xj) =
(xj − x¯j)(xj − x¯j)∗p(xj|yi),
(xj − x¯j)(xj − x¯j) p(xj|yi).
w = y˜i − xkek −
R(V (vik) + U(vik)) I(−V (vik) + U(vik))
I(V (vik) + U(vik)) R(V (vik) − U(vik))
where R(∗) and I(∗) represent the real and imaginary part
of a complex variable. The original complex PDA assumes
the uniform or non-informative prior , and then, the
APP of xk is approximated by the likelihood function. To
develop a general complex PDA detector, we incorporate
the prior information p(xik) and calculate the APP of xk by
and the symbol log-likelihood ratio (LLR) ik by
where m = 0, · · · , M − 1 is the index of the elements in
symbol constellation A.
Since (19) is only an approximation to the true APP, the
PDA detector will, after one sweep of update from k = 1 :
NtNm, start another sweep until the APPs converge. In the
end, the detection is performed as
The complexity of PDA at each BS per iteration is on the
order of O[(NmNt)3]. The above procedure is summarized
in Table 1.
Again, the LLR
k2 needs to be updated according to
Table 1 The summary of PDA detector
1) Initialization: p(xk|yi) = 1/M, k = 1, · · · , NtNm.
2) Decorrelated model: Calculate y˜i according to (6)
3) For k = 1 : NtNm
• Data association: Compute E(vik), V(vik) and U(vik) from (10), (11)
• Probability update: Calculate APPs p(xk|yi) according to (20).
4) Convergence testing: If the APPs converge, go to 5). Otherwise, go
back to 3).
5) LLR updating: Update LLR ik according to (21).
6) Detection: Detect xk according to (23).
3.1.2 NmNt > Nr
Given that the total number of transmit antennas is
smaller than the number of receive antennas, (Hi)H Hi is
not invertible any more. In the case of non-decorrelated
model, (2) is rewritten as
yi = hikxk +
where hik is the k-th column of Hi and uik is the
equivalent interference plus noise. Then, PDA algorithm can
be obtained using a similar derivation as in decorrelated
model discussed in Section 3.1.1.
3.2 Turbo BS cooperations
For simple illustration, assume that there are two BSs in
the system: BS1 and BS2. The following method can be
easily extended to the case with Nm ≥ 3. In turbo BS
cooperations, LLR ik is the soft information and exchanged
among BSs iteratively until the APPs p(xk|y1:2) converge.
The detailed procedure is demonstrated in the following.
APP p(xk|y1:2) can be further factorized as
p(xk|y1:2) ∝ p(xk)p(y1:2|xk)
assuming y1 and y2are independent given xk. Initially,
assume p(xk) = 1/M. At BS1, use p(x1) = p(xk) with
data y1 to approximate APP p(xk|y1) and LLR k1. Then
at BS2, p(xk)p(y1|xk) is equivalent to the a priori
probability p(xk2). Given p(xk) = 1/M is a non-informative
prior, we set p(xk2) ∝ p(y1|xk). Specifically, p(xk2 = αm) =
1+ epMx=p−1(1λe1mx)p(λ1p) . In another word, the LLR k1 computed at
BS1 is treated as soft information and set as the a priori
probability p(x2) at BS2. With the local received data y2
and the prior information p(xk2), BS2 performs the PDA
detection and produces the approximated APP p(xk|y1:2).
Before moving to the next step, check first the gap
between p(xk|y1:2) and p(xk|y1). If the difference is small
enough, the algorithm can be stopped and p(xk|y1:2) is
used to make the final decision on xk. Otherwise, this LLR
k2 is passed to BS1 as p(xk1), and then, turbo processing is
continued until the APPs converge. LLR is updated at each
The summary of proposed turbo BS cooperations using
distributed PDA is in Table 2.
3.3 Soft information exchange
Assume there is an idealized backbone connecting all the
BSs and the soft information iks are exchanged among
BSs in a turbo fashion. After the APP converges, the soft
decision is sent to each user’s home BS.
4 Simulation results
For ease of illustration, we assume first Nb = 2, Nt = 1
in the system and a BPSK modulation for the transmitted
data. And it is straightforward to extend those
assumptions to the case with Nb > 2, Nt > 1 and high-order
modulations. Moreover, we also offer the the results for
those general cases with Nb > 2, Nt > 1.
In the simulation, Hi is flat rayleigh fading channels.
The maximum iteration in PDA computation is 5 since
PDA typically converges within 3∼5 iterations .
However, the inner PDA iterations degrade the performance of
Table 2 Distributed PDA detector with turbo BS cooperations
1) Initialize at the 0th iteration
• BS1, set p(xk1) = 1/M and calculate p(xk|y1) and k1 according to
Table 1 using local data y1.
• BS2, set p(xk2) based on k1, calculate p(xk|y1:2) according to Table 1
using local data y2, update k2 by (28).
• if p(xk|y1) − p(xk|y1:2) < ε , go to 3); otherwise go to 2).
2) Iterate until APP p(xk|y1:2) converges. And at the t-th iteration, perform
for i = 1 : 2
• according to Table 1, update p(xk|y1:2) using data yi with prior p(xki)
obtained from sk, s = i.
• update ik according to (30).
3) Detect xk according to (3).
turbo processing  despite of the increasing complexity.
Therefore, we investigate and conclude that in PDA-TB,
the optimal number of inner PDA iteration is 0. In
contrast, the inner PDA iterations are adaptive according to
the convergence condition of CPDA, in both PDA-SC and
PDA-JP. The maximum outer iteration in turbo processing
is 3 based on the empirical data.
4.1 BER comparison
4.1.1 Illustration on the simple cases with Nb = 2, Nt = 1
and BPSK modulation
Figures 1 and 2 show the BER performance of the
proposed PDA-TB algorithm and PDA aided soft
combining (PDA-SC) . “PDA-SC” is the distributed algorithm
where the term “soft combining” means
p(xk|y1:Nb ) ≈
where the a priori probability p(xik) is non-informative. In
those two figures, “PDA” denotes the PDA detection with
local received data y1; “PDA-JP” indicates the joint
processing in a centralized system by employing PDA
algorithm directly on the received data y1:Nb ; “SD-JP” indicates
the sphere decoding with the joint processing. Among
them, PDA-JP and SD-JP are centralized BS cooperations
where a central processing unit collects the received data
y1:Nb and employs PDA or SD respectively to calculate
directly the APP p(xk|y1:2).
Without BS cooperations, “PDA” shows significant
degradation compared with centralized or distributed BS
cooperations. PDA-JP obtains almost the same
performance as SD-JP. “PDA-TB” achieves similar BER
performance as PDA-SC if small number of users (Nm = 4) are
in DR. However, if interference is severe and more users
are in DR, say Nb = 16, PDA-TB is close to the optimal
SD-JP and outperforms PDA-SC by 0.5dB at BER of 10−4.
Even in rand-deficient case showed in Figs. 3 and 4 where
NmNt = 2Nr, PDA-TB outperforms PDA-SC by around
1dB at BER of 10−2.
4.1.2 Illustration on general cases
The proposed algorithm can be extended to the
scenarios with higher-order modulation schemes, arbitrary Nt or
even Nb > 2.
Figure 5 shows the impact of modulation order when
Nm = 4, Nt = 1, Nb = 2, and Nr = 4. As the
modulation scheme changes from BPSK to 4QAM and 8QAM,
the gain between PDA-TB and PDA-SC increases from
almost 0, 0.3, and 1 dB at BER of 10−3, respectively.
Figure 6 demonstrates the impact of the total number
of streams which is equal to Nm × Nt, when Nb = 2 and
Nr = Nm × Nt. As the total number of streams increases
from 4 to 8 and 16, the gain of PDA-TB over PDA-SC
is accordingly from 0, 0.4, and 0.55 dB at BER of 10−4.
The simulation results demonstrate that the PDA-TB can
achieve a slight gain over the PDA-SC when the system
has a fixed stream number.
Figure 7 indicates that PDA-TB achieves about 2.2 dB
gain while incorporating 3 base stations over 2.
However, there is no obvious improvement if PDA-SC is used
to involve 3 base stations rather than 2. Hence, we can
Fig. 1 BER performance with Nm = 4 and Nr = 4
Fig. 2 BER performance with Nm = 16 and Nr = 16
conclude that the diversity gain of the base station
dominates the total gain.
algorithms is the calculation of
4.2 Complexity evaluation
The computational complexity of the proposed PDA-TB
can be evaluated by simply comparing its complexity to
PDA-SC and PDA-JP in a single (one inner and outer)
iteration. The major computational cost of PDA-based
Using the Sherman-Morrison-Woodbury
formulationbased “speed-up” techniques of , the computational
cost of calculating k−1 can be reduced to O(4NmNtNr2)
k−1 and the matrix
Fig. 3 BER performance with Nm = 16 and Nr = 8
Fig. 4 BER performance with Nm = 4 and Nr = 2
real-valued operations (NRO). Moreover, the calculation of
(32) requires O(4MNmNtNr2 + 2MNmNtNr) NRO. In
summary, the computational cost of the proposed PDA-TB is
O(4NmNtNbNr2) + O(4MNmNtNbNr2 + 2MNmNtNbNr)
NRO per iteration.
The computational cost of PDA-SC is the same per
iteration as PDA-TB, but PDA-JP has the complexity of
O(4NmNtNb2N 2) + O(4MNmNtNb2Nr2 + 2MNmNtNbNr)
NRO per iteration.
In addition, Fig. 8 shows that the average iteration
number in turbo processing is rapidly reduced to 1 as
Eb/N0 increases and BER is below 10−4. This implies that
PDA-TB converges even at the first iteration t = 0 and
then indicates that PDA-TB does not require more PDA
Fig. 5 The impact of modulation order
The impact of modulation order
The impact of Nm*Nt
The impact of BS number
Fig. 6 The impact of total number of streams
computation than PDA-SC once BER reaches 10−4 or
below. Meanwhile, the inner PDA iteration in PDA-TB is
0 while that in PDA-SC is set to be adaptive according to
the convergence condition.
Moreover, PDA-TB does not have the step of soft
combining employed by PDA-SC to make final decision.
Instead, PDA-TB uses the output of PDA detector as the
final decision once the algorithm converges. The possible
disadvantage in PDA-TB is that it might introduce the
delay since it is sequential processing; however, PDA-SC
is a parallel algorithm.
We propose a novel distributed PDA detector with turbo
BS cooperations and show that PDA-TB outperforms
existing PDA-SC in severe interference conditions. This
Fig. 7 The impact of BS number
Fig. 8 Average turbo iteration number in PDA-TB
improvement is due to the appropriate manipulation of
the a priori probability. The future direction will be
complexity reduction for the distributed PDA detector.
This work is supported by the Scientific Research Foundation of CUIT
(No.KYTZ201415, KYTZ201501, KYTZ201502), Sichuan Provincial Department of
Science and Technology Innovation and R&D projects in Science and
Technology Support Program (No. 2015RZ0060, 2015GZ0340, 2015GZ0286,
2015GZ0290), and the Scientific Research Foundation for the Returned
Overseas Chinese Scholars, State Education Ministry. We give thanks for the
insightful discussion and help of Li Li. Finally, we give great thanks to the
anonymous reviewers for the their suggestions to improve the quality of this
The work is sponsored by Sichuan Provincial Department of Science and
Technology Innovation, P.R.China, Chengdu Technological University, and
Chengdu University of Information and Technology (Grant No. KYTZ201415,
KYTZ201501, KYTZ201502, 2015GZ0340, 2015GZ0286, 2015GZ0290).
YY and HW contributed to the conception and design of the study. YY and GL
contributed to the acquisition of data and simulation. YY, GL, and HW
contributed to the analysis and interpretation of simulation data. All authors
read and approved the final manuscript.
About the authors
Dr. Yufang Yin obtained her Ph.D. degree in 2007 from EE department, the
University of Texas at San Antonio. From Jan. 2009 to June. 2012, she worked as
the specialist and research scientist in Nokia Networks. From July 2012 to Oct.
2014, she worked as the staff engineer in Spread Spectrum Communication.
Since Nov. 2016, she joined in Chengdu Technological University. Her research
area includes statistical signal processing and applied machine learning.
Gangjun Li received the B.S. degree in Mechatronic Engineering from the
University of Electronic Science and Technology of China in 1987 and the M.S.
degree in Mechanics from the Harbin Institute of Technology in 1990. He
received the Ph.D. degree in Robotics Engineering from the Southwest
Jiaotong University in 2002. Since 2010, he is a professor of Chengdu
Technological University(CDTU). His research interests are in areas of robotics,
simulation, modeling of complex systems and nonlinear control.
Hua Wei obtained his Ph.D. degree from ECS department, University of
Southampton in September, 2005. From Jan. 2006 to Sep. 2014, he worked as
the principle engineer in Spread spectrum Communication for wireless
modem design. Since September 2014, he joined in Chengdu University of
information Technology. His research area includes adaptive equalization,
signal processing and VLSI design for wireless baseband processor.
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
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