XXXV SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SÃO PEDRO, SP

Tensor-Based Multiuser Detection for Uplink

DS-CDMA Systems with Cooperative Diversity
Antonio Augusto Teixeira Peixoto and Carlos Alexandre Rolim Fernandes

Abstract— In the present paper, a semi-blind receiver for model called PARATUCK was proposed. An overview of
a multiuser uplink DS-CDMA (Direct-Sequence Code-Division some of these tensor models can be found in [8].
Multiple-Access) system based on relay aided cooperative com- There are other examples of tensor decompositions in wire-
munications is proposed. For the received signal, a quadrilinear
Parallel Factor (PARAFAC) tensor decomposition is adopted, less cooperative communications like in [9], where a receiver
such that the proposed receiver can semi-blindly estimate the was proposed for a two-way AF relaying system with multiple
transmitted symbols, channel gains and spatial signatures of all antennas at the relay nodes adopting tensor based estimation.
users. The estimation is done by fitting the tensor model using In [10], an unified multiuser receiver based on a trilinear tensor
the Alternating Least Squares (ALS) algorithm. With computa- model was proposed for uplink multiuser cooperative diversity
tional simulations, we provide the performance evaluation of the
proposed receiver for various scenarios. systems employing an antenna array at the destination node.
There are also recent works, as [11], where a two-hop MIMO
Keywords— Semi-blind receiver, DS-CDMA, Cooperative com- relaying system was proposed adopting two tensor approaches
munications, PARAFAC, Tensor model, Alternating least squares.
(PARAFAC and PARATUCK), and in [12], where a one-way
two-hop MIMO AF cooperative scheme was employed with
I. I NTRODUCTION a nested tensor approach. In [13], receivers were based on a
trilinear decomposition on a cooperative scenario exploiting
Cooperative diversity is a new way for granting better spreading diversity at the relays. In [14], a similar scenario
data rates, capacity, fading mitigation, spatial diversity and was proposed without spreading, but with different time-
coverage in wireless networks [1], so that, its promising slots for each relay transmission. [15] presented a new tensor
characteristics have put it into research interest lately. The decomposition called nested Tucker decomposition (NTD),
basic idea behind it is making the network nodes help each applied to an one-way two-hop MIMO relay communication
other, allowing an improvement in the throughput without system.
increasing the power at the transmitter, similarly to multiple- In contrast to the works earlier mentioned, which are
input multiple-output (MIMO) systems. There are some coop- based on trilinear tensor models, we move to a quadrilinear
erative protocols available, like the amplify-and-foward (AF) PARAFAC decomposition in this paper. Indeed, we propose
and the decode-and-foward (DF) [2]. In means of simplicity, a semi-blind multiuser receiver able to jointly estimate the
the AF protocol is of good choice because the relay node will channel gains, antenna responses and transmitted symbols,
just amplify the user’s signals and fowards it to the destination. exploiting the uniqueness properties of a fourth order tensor.
Latency and complexity are then keep small on this protocol. More specifically, we are considering a cooperative AF relay
An important mathematical tool used in this work is tensor aided scenario where direct-sequence spreading is used at the
based models. An advantage of using tensors in comparison to relays, thus, taking advantage of cooperative and spreading
matrices is the fact that tensors allows us the use of multidi- diversity.
mensional data, allowing a better understanding and precision This work extends [5] by considering a cooperative link
for a multidimensional perspective. Due to its powerful signal with R relays. Moreover, in comparisson to [10] and [14], our
processing capabilities, tensors can be found applied to many work admits spreading at the relays by using orthogonal codes,
fields, for example, in chemometrics and others [3]. and, in contrast to [13], the proposed system considers the
The Parallel Factor (PARAFAC) decomposition [3], [4] relays transmitting in different time-slots instead of all relays
was first used in wireless communications systems in [5], transmitting simultaneously to the base station. An advantage
where a blind receiver was proposed for a DS-CDMA system of the proposed work, with respect to the previous ones, is its
and a tensor was used to model the received signal as a greater flexibility on the choice of some system parameters.
multidimensional variable. After, many other works using By choosing the system parameters, such as the number of
tensor decompostions in telecommunications were developed. relays or the spreading code length, we get the models from
Wireless MIMO systems had also been proposed with tensor [5], [10], [13] and [14]. It is also worth mentioning that the
approaches, which led to the development of new tensor proposed receiver explores spatial and cooperative diversities.
models, as in [6], where a constrained factor decomposition The present work is structured as follows. Section II lays out
was proposed, and in [7], where a new constrained tensor the adopted system model, including the cooperative scenario
and environment assumptions. Section III shows the quadri-
Antonio Augusto Teixeira Peixoto and Carlos Alexandre Rolim Fernandes,
Computer Engineering, Federal University of Ceará, campus de Sobral, CE, linear tensor model used, Section IV presents the proposed
Brazil, E-mails: augusto.peixoto@outlook.com, alexandre ufc@yahoo.com.br. receiver, Section V shows the simulations results and Section

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XXXV SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SÃO PEDRO, SP

VI summarizes the conclusions.

The notation used in this paper is presented here. Scalars
are denoted by italic Roman letters (a,b,...), vectors as lower-
case boldface letters (a,b,...), matrices as upper-case boldface
letters (A,B,...) and tensors as calligraphic letters (A,B,...). To
retrieve the element (i,j) of A, we use [Ai,j ].AT and A† stands
for the transpose and the pseudo-inverse of A respectively. The
operator diagj [A] is the diagonal matrix formed by the j-th
row of A. The operator ◦ denotes the outer product of two
vectors and  denotes the Khatri-Rao product between A ∈
CI×R and B ∈ CJ×R , resulting in A  B = [a1 ⊗ b1 , ..., aR ⊗ Fig. 1. System model - Uplink for multiuser cooperative scenario.
BR ] ∈ CIJ×R .
II. S YSTEM M ODEL
(RD)
The system model considered in this work is a DS-CDMA user, vk,r,n,p is the corresponding noise of the RD link, gr,m
uplink with M users transmitting to a base station with the is the amplification factor applied by the r-th relay of the m-
help of relay-aided links using the AF protocol. The links th user and cp,m is the p-th chip of the spreading code of the
between a given user and one relay are called source-relay m-th user. Substituting (1) into (2), we get:
(SR) and the ones between a relay and the base station are
M
called relay-destination (RD). The base station has an uniform (RD)
X (RD) (SRD)
linear array of K antennas. Each of the M users will transmit xk,r,n,p = hk,r,m h(SR)
r,m gr,m sn,m cp,m + vk,r,n,p , (3)
m=1
to its R associated AF relays. The R relays of a given user use
direct-sequence spreading on the user signal, with a spreading M
(SRD) (RD) (RD)
X
(SR)
code of length P, where the same code is used by all relays vk,r,n,p = hk,r,m gr,m vr,m,n cp,m + vk,r,n,p . (4)
of a given user. Also, the relays and users are single antenna m=1
devices operating in half-duplex mode.
(SRD)
It is assumed perfect synchronization at the symbol level The term vk,r,n,p is the total noise component through the
to avoid intersymbol interference, frequency-flat fading is source-relay-destination (SRD) link, from an user to the base
considered and all channels are independent. We consider that station.
each user communicates with its R associated relays and that Regarding the propagation scenario adopted in the system
each relay fowards the signal using a different time-slot. We model, let us also consider the following assumption. All links
also assume that an user and its relays are all located inside a are subject to multipath propagation and all possible scatters
cluster, such that, the signal received at a relay located within are located far away from the base station, so that all the
the cluster of the m-th user contains no significant interference signals transmitted by the relays arrive at the destination with
from the other users, as Fig. 1 shows. This assumption was also approximatively the same angle of arrival. The angle spread is
made in [10] and in [14]. An interpretation of this assumption small compared to the spatial resolution of the antenna array
is that a user and its relays are located in a cell, while the at the base station. This is truly valid when the user and its
other users and their associated relays are located in other relays are close to each other and the base station experiences
cells, modeled as co-channel interferers. no scattering around its antennas. This is very common in
The signal received by the r-th relay of the m-th user is suburban areas where the base station is placed on the top
given by: of a tall building or in a tower [16]. The channel coefficient
(RD)
u(SR) (SR) (SR)
r,m,n = hr,m sn,m + vr,m,n , (1) hk,r,m may be defined as:
(SR)
where hr,m is the channel coefficient between the m-th user L(RD)
r,m
and its r-th relay, sn,m is the n-th symbol of the m-th user (RD)
X (RD)
(SR) hk,r,m = ak (θm )βl,r,m , (5)
and vr,m,n is the additive white gaussian noise (AWGN) l=1
component. All the data symbols sn,m are independent and
identically distributed, with 1 ≤ m ≤ M, and uniformly where θm is the mean angle of arrival of the m-th scattering
distributed over a Quadrature Amplitude Modulation (QAM) cluster, ak (θm ) is the response of the k-th antenna of the m-th
or a Phase-Shift Keying (PSK) alphabet. scattering cluster, defined as ak (θm ) = exp(jθm ), where θm
The signal received at the k-th antenna of the base station, is an uniform random variable with zero mean and variance
(RD)
trough the r-th time slot, on the n-th symbol period and p-th of 2π, βl,r,m is the fading envelope of the l-th path between
chip of the spreading code, on the RD link is given by: the r-th relay of the m-th user and the base station. Lr,m is
M the total number of multipaths. (5) can be approximated as
(RD) (RD) (RD) follows:
X
xk,r,n,p = hk,r,m gr,m u(SR)
r,m,n cp,m + vk,r,n,p , (2)
(RD) (RD)
m=1 hk,r,m ≈ ak (θm )γr,m , (6)
(RD)
where hk,r,m is the channel coefficient between the k-th PL(RD)
(RD) (RD) r,m (RD)
receive antenna and the r-th relay associated with the m-th where γr,m is defined as γr,m = l=1 βl,r,m . Now, by

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XXXV SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SÃO PEDRO, SP

substituting (6) into (3), we get: where  denotes the Khatri-Rao product (column-wise Kro-
M necker product) [4]. There are also other unfolded matrices,
(RD) (SD) as, for instance:
X
(RD) (SR)
xk,r,n,p = ak (θm )γr,m hr,m gr,m sn,m cp,m + vk,r,n,p
m=1
(7) Y2 = (C  A  H)ST , (14)
and again, substituting (6) into (4), we get: Y3 = (S  C  A)H , T
(15)
M
(SRD)
X
(RD) (SR) (RD) Y4 = (H  S  C)AT , (16)
vk,r,n,p = ak (θm )γr,m gr,m vr,m,n cp,m + vk,r,n,p . (8)
m=1 with Y2 ∈ CP KR×N , Y3 ∈ CN P K×R and Y4 ∈ CRN P ×K .
The transmission rate for each user is given by 1/(R+1), thus,
the total transmission rate on the system is M/(R+1). B. Uniqueness Properties
One of the most important properties of the tensor model
III. P ROPOSED T ENSOR M ODEL obtained in (10) and (12) is its essential uniqueness under
The model above described for the RD links can be viewed certain conditions [17], [18]. The uniqueness property of the
as a four-way array with its dimensions directly related to quadrilinear PARAFAC decomposition by Kruskal’s condition
space (receive antennas at the base station), cooperative slots described in [17], [18], and in [19], is given as follows:
(cooperative channels), time (symbols) and spreading codes
κA + κH + κS + κC ≥ 2M + 3, (17)
(chip). In this section, we model the received signal as a 4-
th order tensor using a PARAFAC decomposition as shown where κA is the Kruskal rank of the matrix A, (similarly to
in [8] and in [17]. Let Y be a M-component, quadrilinear H, S and C). The Kruskal rank of a matrix corresponds to
PARAFAC model, so that Y ∈ CK×R×N ×P is a 4-th order the greatest integer κ, such that every set of κ columns of the
tensor collecting the baseband RD data signals at the base matrix is linearly independent. If the condition (17) is satisfied,
station: the factor matrices A, H, S and C are essentially unique, hence,
(RD)
[Y]k,r,n,p = xk,r,n,p (9) each factor matrix can be determined up to column scaling
and permutation. This uniqueness properties of the PARAFAC
for k = 1,...,K, r = 1,...,R, n = 1,...,N and p = 1,...,P. decomposition
0 0
means that any other set of matrices (A , H , C
0

In order to simplify the presentation, we omit the AWGN 0

and S ) that satisfies (11) is0 related with the original matrix
terms and assume that the channel is constant for N symbol 0 0
set (A, H, C0 and S) by A = AΠ∆A , H = HΠ∆H , C =
periods throughout the rest of this section. A typical element
CΠ∆C and S = SΠ∆S , where Π ∈ CM ×M is a permutation
of Y, denoted by yk,r,n,p = [Yk,r,n,p ] is given by:
matrix and ∆A , ∆H , ∆C and ∆S are diagonal matrices that
M
X meet ∆A ∆H ∆C ∆S = I.
yk,r,n,p = ak (θm )hr,m sn,m cp,m . (10) Now, let us assume that A, H, C and S are all full κ-rank (a
m=1 matrix is said to have full κ-rank if its κ-rank is equal to the
The channel coefficient hr,m is defined as: minimum between the number of rows and columns), where
the κ-rank denotes the Kruskal rank of a matrix, thus (17)
(RD) (SR)
hr,m = γr,m hr,m gr,m . (11) becomes:
(10) corresponds to a PARAFAC decomposition with spatial, min(K, M ) + min(R, M ) + min(N, M ) + min(P, M ) ≥ 2M + 3.
cooperative slots, time and code indices, in other words, a (18)
quadrilinear data tensor. The data tensor Y can be expressed Given that a matrix whose columns are drawn independently
as: from an absolutely continuous distribution has full rank with
M
X probability one [5], then matrix H has full κ-rank with
Y= A.,m ◦ H.,m ◦ S.,m ◦ C.,m , (12) probability one. Also, the matrix A is full κ-rank because we
m=1 model it as a Vandermonde matrix with distinc generators, as
where ◦ denotes the outer product, A ∈ CK×M is the antenna the user signals arrive at the base station array with different
array response matrix with [A]k,m = ak (θm ), H ∈ CR×M is angles of arrival. The symbols matrix S is full κ-rank with
the channel matrix with [H]r,m = hr,m , S ∈ CN ×M is the high probability if N is sufficiently large in comparison to the
symbol matrix with [S]n,m = sn,m and C ∈ CP ×M is the modulation cardinality and the number of users. At last, for
spreading codes matrix with [C]p,m = cp,m . In (12), we have the matrix C, full κ-rank is possible if a certain length of
the PARAFAC decomposition of the data tensor Y as a sum spreading codes is used.
of M rank-1 components. With the assumptions above, we can determine some pa-
rameters of the adopted system, for example, the number of
A. Unfolding Matrices users that the proposed receiver can handle and the minimum
We can also rewrite (12) in an unfolding matricial form. Let acceptable parameters (number of antennas at base station,
Y1 ∈ CKRN ×P be defined as the tensor Y ∈ CK×R×N ×P length of the spreading codes, number of relays or the data
unfolded into a matrix, as follows: block length) that matches a target number of user channels
to be detected. Hence, we will have flexibility when choosing
Y1 = (A  H  S)CT , (13) K, R, N and P, which is the one of the main reasons for

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XXXV SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SÃO PEDRO, SP

Algorithm 1 ALS FITTING 0

10

1)Initialization : Set i = 0; Initialize Â(i=0) and −1

10
Ĥ(i=0) ;
2)i = i + 1; −2
10
T
3)Ŝ(i) = (C  Â(i−1)  Ĥ(i−1) )† Ỹ2 ;

SER
−3
10
T
4)Ĥ(i) = (Ŝ(i)  C  Â(i−1) )† Ỹ3 ; −4
T 10
5)Â(i) = (Ĥ(i)  Ŝ(i)  C)† Ỹ4 ; R=1
R=2

6)Repeat steps 2 − 5 until convergence; −5

10 R=3
R=4
R=5
−6
10
0 5 10 15 20 25 30
SNR (dB)

considering the tensor approach. It provides different tradeoffs

for our system based on the parameters. Indeed, we have from
Fig. 2. SER versus SNR performance of the proposed receiver for a different
(18): number of relays.
• If P ≥ M, N ≥ M (typical DS-CDMA scenario), then,
min(K,M) + min(R,M) ≥ 3. For example, let K = 2 and
R = 1, we satisfy (18) and at least 1 relay and 2 antennas is assumed that the first row of S is known and the scaling
are sufficient for M users. ambiguity is removed by dividing the first row of Ŝ by the
• If P ≥ M, N ≥ M and K ≤ M, then, R can be 0 if K = first row of S. After obtaining the scaling matrix of  and Ŝ,
3, giving us the model described in [3], a noncooperative we can find the scaling matrix ∆H of Ĥ with ∆A ∆S ∆H = I.
DS-CDMA uplink.
• If K ≥ M, N ≥ M, P ≤ M and R ≤ M, then for R = 2, P V. S IMULATION R ESULTS
= 1 chip is sufficient for M users, therefore we get [14] This section presents computer simulations results for per-
(the same can be achieved if R = 3, thus P can be zero). formance evaluation purposes with the following scenario. The
• If K ≥ M, P ≥ M, R ≤ M and N ≤ M, then, for R =
wireless links have frequency-flat Rayleigh fading with path
1, N = 2 symbols are enough to guarantee uniqueness. It loss expoent equal to 3, the base station antenna array is
means that a short block length is sufficient for detection. composed by K antennas, 16-QAM modulation is used and
Based on the assumptions above, we can conclude that the Hadamard codes are considered for spreading sequences. The
proposed tensor model gives us flexibility about many param- symbol error rate (SER) curves are shown as a function of the
eters and diversity tradeoff. signal-to-noise ratio (SNR) of the RD link. The mean results
were obtained by 10000 independent Monte Carlo samples.
IV. R ECEIVER A LGORITHM The AF relays have variable gains and the source power Ps
Assuming that there is no channel information at the and the relay power Pr were considered as unitary.
receiver or transmitter, the receiver algorithm presented in Figure 2 shows the SER versus SNR for the proposed
this section is based on the ALS (Alternating Least Squares) technique with P = 8 chips, a datablock of N = 16 symbols, K
method, which consists in fitting the quadriliear model to the = 2 receive antennas and M = 4 users. Then we have curves
received data tensor [20]. The idea behind the ALS procedure for various values of R (number of relays on the cluster). From
is very simple: each time, update one of the factor matrices by Fig. 2, we can observe a better performance when we increase
using the least squares estimation technique with the previous the number of relays on the system. This happens because
estimations of the other factor matrices. Each factor matrix when the number of relays is augmented, the model turns to
is estimated, in an alternate way, always using the previous a more cooperative scenario, exploiting cooperative diversity
estimations of the other factor matrices. This procedure is and resulting in better link quality.
repeated until convergence. The unfolding matrices in (13)- Now, we compare the SER of the proposed receiver with
(16) are used to estimate A, H and S, where we assume the ones of the: Zero Forcing (ZF) receiver, that works under
knowledge of the spreading codes (matrix C) at the receiver. complete knowledge of A, H and C, the semi-blind DS-
The Quadrilinear ALS algorithm is shown in Algorithm 1. CDMA receiver proposed in [5] (non-cooperative DS-CDMA),
The measured error at the end of the i-th iteration is given by the receiver proposed in [10] using AF (same cenario of the
e(i) = kỸ1 −(Â(i)  Ĥ(i)  Ŝ(i) )CT kF , where k.kF denotes the present work, but without spreading codes) and the receiver
Frobenius norm, Ỹ1 , Ỹ2 , Ỹ3 and Ỹ4 are the noisy unfolding shown in [13], where the relays transmit at the same time.
matrices and Â(i) , Ĥ(i) and Ŝ(i) ) are the estimates of the factor For Figure 3, we set N = 16, P = 4, M = 4, K = 3 and
matrices at the i-th iteration. The convergence of the algorithm R = 1 for both the ZF and the proposed receiver. For the
is obtained when |e(i) − e(i − 1)| < 10−6 . receiver proposed in [10], only one relay is used and we set
After obtaining the estimation of A, H and S, it is necessary K = 3, N = 16 and M = 2. For [13], we set N = 16, P
to remove the scaling ambiguity. The scaling ambiguity of  = 2, M = 4, K = 3 and R = 1. For the receiver of [5], we
is removed by considering that the first row of A is known, set P = 4, K = 3, N = 16 and M = 4. These simulations
which is possible because A is a vandermonde matrix. The parameters were chosen to give us the same or similar spectral
same can be done to remove the scaling ambiguity from Ŝ. It efficiency for all the receivers. The direct link between user

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XXXV SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇÕES E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SÃO PEDRO, SP

0
10 system, like the number of relays, number of antennas at the
base station, spreading codes or data block length. Thus, we
−1
10
are able to cover lots of pratical scenarios. The results showed
us that the proposed receiver performs well in comparison to
SER

the receivers described in [5], [10], [13] and the ZF receiver.

−2
10
Non−coopetive CDMA [5]
Cooperative CDMA [13]
This work may be extendend by using another algorithm
Cooperative non−CDMA [10]
ZF
instead of the ALS, as in [14]. Also, the frequency-flat fading
−3
Proposed could be changed to frequency-selective fading.
10
0 5 10 15 20 25 30 R EFERENCES
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that allows some flexibility in choosing the parameters of the

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