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PARAFAC-based channel estimation and data recovery in nonlinear

MIMO spread spectrum communication systems

Carlos A.R. Fernandes

a,1

, Ge´rard Favier

b,

, Jo

ao C.M. Mota

~

c,2 aComputer Engineering, Campus Sobral, Federal University of Ceara´, Prac

-a Senador Figueira, rua Anahid Andrade, S/N, Brazil

bI3S Laboratory, University of Nice-Sophia Antipolis/CNRS, Les Algorithmes/Euclide B-2000 route des Lucioles, BP 121, 06903 Sophia-Antipolis Cedex, France cDepartamento de Engenharia de Teleinforma´tica, Federal University of Ceara´, Campus do Pici, 60.755-640, 6007 Fortaleza, Brazil

a r t i c l e

i n f o

Article history:

Received 3 July 2009 Received in revised form 21 June 2010

Accepted 16 July 2010 Available online 23 July 2010

Keywords:

PARAFAC decomposition Channel estimation Data recovery MIMO Volterra Nonlinear channel

Direct sequence spread spectrum Radio over fiber

a b s t r a c t

In this paper, a new tensorial modeling is first proposed for nonlinear multiple-input multiple-output (MIMO) direct sequence spread spectrum communication systems. The channel is modeled as an instantaneous MIMO Volterra system. Then, a direct data approach for joint blind channel estimation and data recovery is developed using the parallel factor (PARAFAC) decomposition of a third-order tensor composed of received signals, exploiting space, time and code diversities. A blind channel estimation method based on the PARAFAC decomposition of a fifth-order tensor composed of covariances of the received signals is also proposed, considering phase shift keying (PSK) modulated transmitted signals. The proposed estimation algorithms are evaluated by simulating a nonlinear uplink MIMO radio over fiber (ROF) communication system.

&2010 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, parallel factor (PARAFAC) based methods are proposed for channel estimation and data recovery in nonlinear multiple-input multiple-output (MIMO) direct sequence spread spectrum communication systems mod-eled by an instantaneous MIMO Volterra model. We consider that the outputs of the MIMO Volterra system represent observations at the receiver obtained through an antenna array, while the inputs represent sources transmitting at the same time and frequency band, which may correspond to a single user with multiple transmit antennas or multiple users with a single transmit antenna each.

This kind of model has several applications in communication systems as, for instance, the modeling of MIMO satellite communication channels with nonlinear power amplifiers (PA). Due to power limitation, the satellite station usually employs a PA that is driven at or near saturation in order to obtain a more efficient transmission[1,2]. At saturation, the PA exhibits a non-linear characteristic, resulting in the introduction of nonlinear bandlimited distortions [3]. MIMO Volterra models are also used for modeling uplink channels in radio over fiber (ROF) communication systems[4,5]. The ROF links have found a new important application with their introduction in microcellular wireless networks. In such systems, the uplink transmission is carried out from a mobile station towards a radio access point, the transmitted signals being converted in optical frequencies by a laser diode and then retransmitted through optical fibers. Important nonlinear distortions are introduced by the electrical–optical (E/O) conversion[4,6,7]. When the length of the optical fiber is short (few kilometers) and the Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/sigpro

Signal Processing

0165-1684/$ - see front matter&2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.07.010

Corresponding author. Tel.: +33 492942736; fax: +33 492942896.

E-mail addresses:alexandrefernandes@ufc.br (C.A. Fernandes), favier@i3s.unice.fr (G. Favier), mota@gtel.ufc.br (J.C.M. Mota).

1Tel./fax: +55 88 36132829.

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radio frequency has an order of GHz, the dispersion of the fiber is negligible[8]. In this case, the nonlinear distortion due to the E/O conversion process becomes preponderant. Up to several Mbps, the ROF channel can be considered as a memoryless link [6]. Thus, in a MIMO system, the channel is composed of a wireless link, modeled as a linear instantaneous mixing system, followed by E/O conversions, modeled as memoryless nonlinearities. The overall channel can then be viewed as a memoryless MIMO Wiener system, corresponding to a particular case of MIMO Volterra system.

In recent years, some works have been done for channel estimation and data recovery in multiuser non-linear ROF systems, specially by X.N. Fernando and S.Z. Pinter. In the multiple-input single-output (MISO) case, i.e. considering multiple users and a single antenna at the reception, several authors have considered ROF links in the context of code division multiple access (CDMA) systems. An estimation technique for a ROF uplink channel was presented in [9,4] for a multiuser CDMA system using pseudo-noise spreading codes. The same authors have proposed a receiver for the ROF downlink channel in a CDMA environment using Walsh codes[10]. It should be highlighted that all these techniques are supervised, i.e. they assume that the transmitted signals are known during a training period. A blind zero forcing receiver for a multiuser nonlinear channel in a CDMA system was proposed in[11]. However, the channel model considered in [11] does not correspond to a nonlinear mixture of sources, contrarily to the one used in the present paper.

In chip-synchronous linear direct sequence spread spectrum systems, the use of code-matched filters is a usual solution for separating the sources. In this case, orthogonal or quasi-orthogonal spreading codes are required. However, in a nonlinear spread spectrum system, the nonlinearities induce a loss of orthogonality of the spread signals, leading to the introduction of nonlinear multiple-access interference (MAI). In this paper, we propose two tensor approaches for estimating the nonlinear channel and the transmitted symbols, by using complex-valued spreading codes. The techniques hereafter developed are based on tensor decompositions that result from the fact that the signals received by an antenna array can be viewed as 3-D variables, with indices corresponding to time, chip and space.

The main contributions of this paper are divided into three parts. First, a new tensor modeling is proposed for nonlinear MIMO spread spectrum communication sys-tems. Then, a semi-blind receiver is proposed based on the PARAFAC decomposition of a third-order tensor composed of received signals with space, time and code diversities. This tensor-based approach allows joint estimation of the channel coefficients and transmitted signals. The PARAF-AC decomposition is computed by applying a modified version of the alternating least squares (ALS) algorithm that uses some pilot symbols, i.e. a short training sequence, and takes the structure of one of the factor matrices into account. In this paper, the techniques that use a tensor composed of received signals will be called direct-data techniques. Channel estimation and data recovery based on tensors composed of received signals have been addressed by several authors in the case of linear CDMA channels [12–15]. In fact, the PARAFAC analysis carried out in Section 3 can be viewed as an extension of [12] to nonlinear channels. In the case of nonlinear channels, blind PARAFAC-based receivers for single-user Volterra channels of a system were developed in[16,17]. Moreover, a direct-data based blind identifica-tion method using the PARAFAC decomposiidentifica-tion was recently proposed for single-user Wiener–Hammerstein channels[18], in the context of a time division multiple access (TDMA) system.

In a third part, a blind technique for memoryless MIMO Volterra channel estimation in spread spectrum systems is developed, considering phase shift keying (PSK) modulated transmitted signals. It relies on the PARAFAC decomposition of a fifth-order tensor composed of covariances of the received signals, the PARAFAC model being estimated by means of the ALS algorithm. In the present paper, this method will be denoted covariance technique.

A great advantage of these tensor-based approaches is that they allow to work with weak uniqueness conditions compared with previous works[11,19,20], which require that the number of channel outputs be greater than the number of nonlinear terms of the Volterra series. The proposed tensorial techniques for nonlinear spread spec-trum communication systems provide a great flexibility on the number of antennas and spreading factor, leading to an interesting tradeoff between complexity and

Nomenclature

x scalar variable—lower-case letter x vector—boldface lower-case letter X matrix—boldface capital letter X high-order tensor—calligraphic letters ½Xi,j (i,j)th element of the matrixX

½Xi

1,i2,...,iNði1,i2,. . .,iNÞth element of the tensorX

diag[x] diagonal matrix built from the vectorx

diagi[X] diagonal matrix built from theith row of the matrixX

Khatri–Rao (columnwise Kronecker) product

rX rank of the matrixX kX k-rank of the matrixX

JXJF Frobenius norm of the matrixX

Xy Moore–Penrose pseudo-inverse of the matrix

X

R number of receive antennas N number of symbol periods

P spreading gain

2K+1 channel nonlinearity order

Q number of sources (real plus virtual)

(3)

capacity. Unlike our previous work in [21], we use spreading codes to induce a new diversity on the received signals. That allows to improve the channel estimation accuracy and to relax the uniqueness conditions, as well as to propose direct-data based techniques that perform joint channel estimation and symbol recovery.

It should be highlighted that this paper does not intend to propose new algorithms for fitting the PARAFAC model. Many algorithms have been developed for this purpose, most of them for third-order tensors. See [22] for a comparison of several algorithms. We have decided to use only the ALS algorithm as it is the standard algorithm for the PARAFAC model, with the following properties: easy to implement, monotonic convergence and very simple to extend to tensors of any order [22]. For the direct-data approach, we also use a modified version of the ALS algorithm that provides a significant performance im-provement.

Input backoff and signal predistortion are other popular approaches used to combat nonlinearities in communication systems. However, the backoff approach reduces the power efficiency, as the transmitter uses only a small portion of its allowed input range. Our approach allows a higher utilization of the input range, increasing the power efficiency. That provides a higher coverage and a higher signal strength for users located far away from the receive antennas. Moreover, our approach of compen-sating the nonlinear distortions at the receiver provides some advantages over predistortion schemes that try to compensate nonlinear distortions at the transmitter. The first one is that it allows the global optimization of the problem, i.e. the joint compensation of the distortions due to the linear and nonlinear subsystems that constitute the Volterra channel. Another advantage is that, in a uplink transmission, most part of the signal processing is done at the base station (BS), no modification in the portable units being then necessary to accommodate the nonlinearity compensation. Moreover, the Volterra representation of the channel has the advantage of taking into account other possible channel nonlinearities, contrarily to pre-distortion schemes that generally compensates the non-linear distortions of a single nonnon-linear block.

This paper is organized as follows. Section 2 presents the channel model used in this work. In Section 3, we describe a direct-data method for joint channel estimation and data recovery of memoryless MIMO Volterra chan-nels. In Section 4, a covariance approach for estimating memoryless MIMO Volterra channels is proposed. In Section 5, we provide a brief comparison between the uniqueness conditions of the proposed covariance and direct-data tensor-based techniques. The performance of these techniques is evaluated by means of computer simulations in Section 6, and Section 7 draws some conclusions.

2. MIMO Volterra model of a nonlinear direct sequence spread spectrum system

The discrete-time equivalent baseband model of the nonlinear MIMO communication channel is assumed to be

a chip-synchronous memoryless MIMO Volterra model:

yr,n,p¼

XK

k¼0

XT

t1¼1

X

T

tkþ1¼tk

XT

tkþ2¼1

X

T

t2kþ1¼t2k

hðrÞ

2kþ1ðt1,. . .,t2kþ1Þ

Y

kþ1

i¼1

uti,n,p

Y 2kþ1

i¼kþ2

u

ti,n,pþur,n,p, ð1Þ

where yr,n,pð1rrrR,1rnrN,1rprPÞ is the signal received by therth antenna at thepth chip period of the nth symbol period, i.e. received at the [(n1)P+p]th chip period, R is the number of receive antennas, N is the number of symbol periods,Pis the length of the spreading code (spreading gain), (2K+1) is the nonlinearity order of the model, hðrÞ

2kþ1ðt1,. . .,t2kþ1Þ are the (2k+1)th-order kernel coefficients of therth sub-channel,ut,n,pð1rtrTÞ

is the spread signal transmitted by thetth source (input) at thepth chip period of thenth symbol period,Tis the number of sources andur,n,pis the additive white Gaussian noise (AWGN).

The equivalent baseband Volterra model (1) includes only the odd-order kernels with one more non-conjugated term than conjugated terms because the other nonlinear products of input signals correspond to spectral compo-nents lying outside the channel bandwidth, and can therefore be eliminated by bandpass filtering[23,24].

The spread signalut,n,pis obtained by multiplying the information signal st(n) by the spreading code ct(p), leading to

ut,n,p¼stðnÞctðpÞ ð2Þ

for 1rtrT, 1rnrNand 1rprP, wherect(p) is thepth element of the spreading code andst(n) thenth informa-tion symbol of thetth source. The signalst(n) is assumed to be stationary and independent from stuðnÞ, for tatu. Substituting (2) into (1), we get

yr,n,p¼

XK

k¼0

XT

t1¼1

X

T

tkþ1¼tk

XT

tkþ2¼1

X

T

t2kþ1¼t2k

hðrÞ

2kþ1ðt1,. . .,t2kþ1Þ

Y

kþ1

i¼1

stiðnÞ

Y 2kþ1

i¼kþ2

s

tiðnÞ

Y

kþ1

i¼1

ctiðpÞ

Y 2kþ1

i¼kþ2

c

tiðpÞþur,n,p, ð3Þ

Eq. (3) can be expressed as follows:

yr,n,p¼

XQ

q¼1

hr,q~sn,qc~p,qþur,n,p, ð4Þ

where

~

sn,q¼

Y

kþ1

i¼1

stiðnÞ

Y 2kþ1

i¼kþ2

s

tiðnÞ ð5Þ

contains products of information signals,

~

cp,q¼

Y

kþ1

i¼1

ctiðpÞ

Y 2kþ1

i¼kþ2

c

tiðpÞ ð6Þ

contains products of spreading codes and

hr,q¼hð2rkÞþ1ðt1,. . .,t2kþ1Þ are the Volterra kernel coeffi-cients of therth sub-channel, the indexqdepending on the set of indices ft1,t2,. . .,t2kþ1g. In fact, the index q corresponds to a certain ordering of the indices

(4)

the indicesqandt1,t2,. . .,t2kþ1, see[21]. The number of parameters of each sub-channel is given by

Q¼PKk¼0CT,kCT,kþ1, withCT,k¼ ðTþk1Þ!=ðT1Þ!k!. Note

thats~n,qð1rqrQÞcorresponds to a product of informa-tion signals that can be viewed as a source, T of them being real sources and (QT) virtual sources. In the sequel, it is assumed thatK,TandQare known.

From (4), we conclude that the received signals contain linear and nonlinear MAI, and that the channel nonlinea-rities induce the loss of orthogonality of the spread signals, that is, the ‘‘virtual codes’’c~p,qare not orthogonal, even if the spreading codes ct(p) are. In this case, usual solutions for linear channels, like code-matched filters, are unable to cancel the nonlinear interferences.

If the information signalsst(n) are PSK modulated and the spreading codesct(p) have an unitary modulus, then the transmitted signalsut,n,pare constant modulus. In this case, the nonlinear terms corresponding toti¼tj, for all i2 f1,. . .,kþ1gand j2 fkþ2,. . .,2kþ1g, can be omitted from (1) due to the fact that the termjuti,n,pj

2reduces to a multiplicative constant absorbed by the associated chan-nel coefficient, and the memoryless MIMO Volterra model (1) can be rewritten as

yr,n,p¼

XK

k¼0

XT

t1¼1

X

T

tkþ1¼tk

XT

tkþ2¼1

X

T

t2kþ1¼t2k

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

tkþ2,...,t2kþ1at1,...,tkþ1

~

2rkÞþ1ðt1,. . .,t2kþ1Þ

Y

kþ1

i¼1

uti,n,p

Y 2kþ1

i¼kþ2

u

ti,n,pþur,n,p: ð7Þ

For instance, for a linear-cubic channel (2K+1=3), we have

yr,n,p¼

XT

t1¼1 ~

1rÞðt1Þst1ðnÞct1ðpÞþ

XT

t1¼1

XT

t2¼t1

XT

t3¼1 |ffl{zffl}

t3at1,t2

~

3rÞðt1,t2,t3Þ

st1ðnÞst2ðnÞst3ðnÞct1ðpÞct2ðpÞct3ðpÞþur,n,p: ð8Þ

Eqs. (7) and (8) can also be written as (4), with

Q¼TþT2ðT1Þ=2. The estimation methods developed in Section 3 can be used with PSK or quadrature amplitude modulation (QAM) input signals, while the ones devel-oped in Section 4 are applicable only with PSK signals.

3. Direct-data approach for joint MIMO Volterra channel estimation and data recovery

In this section, a method for joint channel estimation and symbol recovery is developed using the PARAFAC decomposition of a third-order tensor composed of received signals, with space, time and code diversities. The PARAFAC decomposition of this tensor allows a joint estimation of the channel, spreading codes and trans-mitted signals, using only one known pilot symbol. The technique developed in the present section can be used with PSK or QAM modulations.

3.1. Third-order received signal tensor

For simplifying the development, we consider the noiseless case. Let Y2

C

RNP be the third-order tensor composed of received signalsyr,n,pfor 1rrrR, 1rnrN and 1rprP, with ½Yr,n,p¼yr,n,p. Eq. (4) represents the

scalar writing of the PARAFAC decomposition ofY with rank rQ and matrix factors H2

C

RQ, C~2

C

PQ

and ~

S2

C

NQ

, where

H¼ ½h1. . .hRT ð9Þ

is the channel matrix, withhr¼ ½hr,1hr,2. . .hr,QT2

C

Q1,

~

S¼ ½s~

1 s~NT ð10Þ

is the nonlinear symbol matrix, withs~n¼ ½~sn,1 s~n,QT 2

C

Q1,s~

n,qbeing defined in (5), and

~

C¼ ½c~1 c~PT

ð11Þ

is the nonlinear code matrix, with c~p¼ ½c~p,1 c~p,QT 2

C

Q1,c~p,qbeing defined in (6).

For instance, for T=2, K=1 and constant modulus transmitted signals, the matrices H, S~ and C~ are, respectively, given by

11Þð1Þ11Þð2Þ31Þð1,1,2Þ31Þð2,2,1Þ

^ ^ ^ ^

1RÞð1Þ hð1RÞð2Þ hð3RÞð1,1,2Þ hð3RÞð2,2,1Þ

0 B B @ 1 C C

A, ð12Þ

~

s1ð1Þ s2ð1Þ s21ð1Þs2ð1Þ s22ð1Þs1ð1Þ

^ ^ ^ ^

s1ðNÞ s2ðNÞ s21ðNÞs2ðNÞ s22ðNÞs1ðNÞ 0

B @

1

C

A ð13Þ

and

~

c1ð1Þ c2ð1Þ c21ð1Þc2ð1Þ c22ð1Þc1ð1Þ

^ ^ ^ ^

c1ðPÞ c2ðPÞ c21ðPÞc2ðPÞ c22ðPÞc1ðPÞ 0

B @

1

C

A: ð14Þ

The matrix slices of the tensorYare given by

Yr,,¼S~diagr½HC~ T

2

C

NP, ð15Þ

Y,n,¼C~diagn½S~HT2

C

PR,

ð16Þ

Y,,p¼Hdiagp½C~S~ T

2

C

RN, ð17Þ

where diagi½ denotes the diagonal matrix formed with

the ith row of the matrix argument. By columnwise stacking these matrix slices, we get the following unfolded matrix representations of the tensorY:

Y½

Y1,, ^

YR,, 2

6 4

3

7

52

C

NRP,

Y½

Y,1, ^

Y,N, 2

6 4

3

7

52

C

PNR,

ð18Þ

Y½3¼

Y,,1 ^

Y,,P

2 6 6 6 4 3 7 7 7 52

C

RPN, ð19Þ

which leads to

Y½1¼ ðHS~ÞC~

T

(5)

Y½2¼ ðS~C~ÞHT, ð21Þ

Y½3¼ ðC~HÞS~

T

, ð22Þ

wheredenotes the Khatri–Rao (columnwise Kronecker) product.

3.2. Uniqueness condition

The essential uniqueness of the PARAFAC decomposi-tion of Y is assured when the Kruskal’s condition is satisfied[25]:

kHþkS~þkC~Z2Qþ2, ð23Þ

wherekAdenotes the k-rank of matrixA, i.e. the greatest integerkAsuch that every set ofkAcolumns ofAis linearly independent. The essential uniqueness means that if another set of matrices Hu, S~uand C~usatisfies (20)–(22), thenHu¼H

PL

1,S~u

¼S~

PL

2andC~u

¼C~

PL

3, where

L

1,

L

2

and

L

3are diagonal matrices such that

L

1

L

2

L

3¼IQ, and

P

is a permutation matrix of orderQ. Assuming that the matrices H, S~ and C~ are full k-rank, condition (23) becomes

minðR,QÞþminðN,QÞþminðP,QÞZ2Qþ2: ð24Þ

In particular, if we chooseNZQ, we get

minðR,QÞþminðP,QÞZQþ2: ð25Þ

The flexibility on the choice of R, N and Pprovided by Kruskal’s condition is one of the main advantages of this PARAFAC-based approach. It leads to an interesting tradeoff between complexity (number of receive antennas R) and capacity (spreading factor P). In particular, note that it is possible to chooseRoQ andPoQ. However,P andRcannot both be chosen much smaller thanQ.

An important property of the above described tensor modeling is that the spreading codes do not need to be orthogonal. The only constraint on these codes is that the matrixC~ be full k-rank.

3.3. ALS algorithm with direct decision and block initialization (ALS-DD-BI)

In this section, we present an estimation method based on the ALS algorithm for computing the PARAFAC decomposition of the tensorY. In the following develop-ments, the spreading codes are assumed to be known at the receiver, implying that the matrixC~ is also known, as calculated using (6). So, if Kruskal’s condition (23) is satisfied, we have

P

¼

L

3¼IQ and

L

L

11. Therefore,

^

H¼H

L

1andS^~¼S~

L

1

1 . This means that the permutation ambiguity is eliminated. Moreover, due to the structure of the matrix S~, the scaling ambiguity matrix

L

1 can be determined by using one known pilot symbol for each sourceðstðnÞ,t¼1,: :,TÞ, i.e. by assuming that the first row

ofS~ is known:

L

1¼diag s~^1,1 ~

s1,1 ,. . .,s~1,Q

^ ~

s1,Q

" #

" #

, ð26Þ

where diag½ denotes the diagonal matrix formed from the vector argument, ands^~1

,qdenotes an estimated value

of s~1,q. The scaling ambiguity can also be cancelled by using an automatic gain control and a phase-locked loop, or a differential modulation. However, solving the ambiguity via differential coding may lead to a significant performance loss. It is important to notice that the extension of the proposed estimation algorithm to the case whereC~ is unknown is straightforward.

The standard ALS (alternating least squares) algorithm is obtained by alternately minimizing the two following conditional least squares (LS) cost functions:

J1¼JY½3ðC~H^ ðit1Þ

ÞS~TJ2F, ð27Þ

J2¼JY½2ð ^ ~

SðitÞC~ÞHT

J2F, ð28Þ

with respect toS~andH, whereJJFdenotes the Frobenius norm, andY½3andY½2 are noisy versions ofY[3]andY[2],

respectively. The performance of the ALS algorithm can be improved by taking into account the structure ofS~and the fact that the transmitted symbols belong to a finite alphabet. The proposed estimation algorithm consists in a modified version of the ALS algorithm including direct

decisions to construct the matrixS~^ðitÞ, with the use of a short training sequence, i.e. a sequence of known pilot symbols, to obtain an initial estimateH^ð0Þ

of the channel matrix. This algorithm will be called ALS-DD-BI algorithm.

Let us denote byS~^ð

itÞ

L 2

C

NT

the matrix composed of

theT first columns ofS~^ð

itÞ

, i.e. the matrix containing the

linear part ofS^~ðitÞthat corresponds to the real sources, and

byS^~ð

itÞ

NL 2

C

NðQTÞthe matrix composed of the (QT) last

columns ofS^~ð

itÞ

, i.e. the matrix containing the nonlinear

part ofS^~ðitÞ that corresponds to the virtual sources. That

givesS^~ðitÞ¼ ½S~^ðitÞ

L j

^ ~

itÞ

NL.

Moreover, let us define S^~ð

itÞ

L,DD2

C

NT

as the matrix

composed of the elements of S~^ðLitÞ after their projection

onto the alphabet of symbols, i.e after a decision device.

Finally, letS~^ðitÞ

NL,DD2

C

NðQTÞbe the nonlinear part of the

information signal matrix reconstructed fromS^~ðitÞ

L,DD, with

^ ~

itÞ

DD¼ ½

^ ~

itÞ

L,DDj

^ ~

itÞ

NL,DD.

Let us denote byS~02

C

NtQ the matrix composed of the Nt first rows of S~ and by Y

½2,02

C

PNtR the corre-sponding unfolded matrix of the tensorY, whereNtis the length of the training sequence. The initial estimation ofH

is obtained from (28) as

^

Hð0Þ¼ ½ðS~0C~ÞyY½2,0T, ð29Þ

where ðÞy denotes the Moore–Penrose matrix pseudo-inverse. Note that a necessary identifiability condition for calculating this initialization isrðS~

0C~Þ¼Q, which implies NtPZQ. The ALS-DD-BI algorithm is summarized in

(6)

^ ~

itÞ

denote, respectively, the estimates of the matricesH

andS~ at iterationit.

The classical ALS algorithm corresponds to steps 1, 5 and 6 of the ALS-DD-BI algorithm, which can be viewed as a generalization of the method proposed in [12] to nonlinear channels. Indeed, in [12], the factor matrices

H,S~andC~contain only the elements corresponding to the linear part of the model (4).

4. Covariance approach for MIMO Volterra channel estimation

In this section, a method is proposed for estimating the considered MIMO Volterra channel using the PARAFAC decomposition of a tensor composed of channel output covariances. It is assumed that the information signals st(n)ð1rtrTÞbelong to a PSK constellation and that the spreading codes ct(p) have an unitary modulus, which leads to transmitted signalsut,n,pwith constant modulus.

4.1. Covariance matrices of the received signals

Eq. (4) can be expressed in vector form as

yðn,pÞ ¼Hdiagp½C~s~nþvðn,pÞ, ð30Þ

where yðn,pÞ ¼ ½y1,n,p. . .yR,n,pT2

C

R1 and vðn,pÞ ¼ ½u1,n,p

. . .uR,n,pT2

C

R1. Let us define the spatio-temporal covar-iance matrices of the received signals as

Ryðd,p1,p2Þ ¼E½yðnþd,p1ÞyHðn,p2Þ 2CRR

¼Hdiagp1½

~

CR~sðdÞdiagp2½

~

CHH

þ

s

2I

RdðdÞdðp1p2Þ, ð31Þ

where

E

½is the mathematical expectation operator,ðÞH

denotes the Hermitian transpose of a matrix, D is the number of delaysð0rdrD1Þtaken into account and

Rsð~dÞ ¼

E

½s~ðnþdÞs~HðnÞ 2

C

QQ: ð32Þ

Assuming that the noise variance

s

2is known, the noise covariance matrix can be ignored in (31) as it can be subtracted from Ry(0,p,p). If the noise variance is not known, the proposed estimation method can be applied

without using the covariance matrices Ry(0,p,p), for p=1,y,P. Therefore, from now on and without loss of generality, the noise term will be omitted.

4.2. Precoding scheme

In telecommunication systems, the transmitted signals are often assumed to be independent and identically distributed (i.i.d.). When the transmitted signals are i.i.d. and PSK modulated, the covariance matrixRs~ð0Þis non-null and diagonal[21], and the matricesRs~ðdÞare null for

d40. By exploiting some properties of PSK signals, a

precoding scheme that introduces time correlation in such a way that the matricesRsð~ dÞare non-null and diagonal was developed in[21].

This precoding scheme consists in modeling the transmitted signals as discrete time Markov chains (DTMC), the states of the DTMC being given by the PSK symbolsap¼ fAteE2pðp1Þ=Pg, forp=1,2,y,P, whereAtis the

amplitude of the signal of thetth source,Pis the number of points of the PSK constellation andEis the imaginary

unit ðE¼ ffiffiffiffiffiffiffi1

p

Þ. Some conditions on the transition prob-ability matrices (TPMs) of the Markov chains have been established to introduce temporal correlation and satisfy statistical constraints that lead to non-null and diagonal matricesR~sðdÞ. These matrices are entirely characterized by the configuration of TPMs used by the sources. Several configurations of TPMs can be used for ensuring these statistical constraints. For instance, configuration A proposed in[21]uses 8-PSK signals and corresponds to a code rate of 1/3. For further details about this coding scheme, see[21].

Thus, if the matricesRsð~dÞ,d=0,y,D1, are diagonal, Eq. (31) can be rewritten as

Ryðd,p1,p2Þ ¼Hdiagp1½

~

Cdiagdþ1½Zdiagp2½

~

CHH, ð33Þ

where the rows of the matrix Z2

C

DQ contain the diagonal elements of Rsð~ dÞ for 0rdrD1, i.e.

zdþ1,q¼ ½Zdþ1,q¼ ½Rsð~ dÞq,q.

As we will see later, the use of this precoding scheme is not mandatory for the techniques proposed in this section, as the channel estimation algorithms may work with D=1, i.e. using only the covariance matrices

Ryð0,p1,p2Þ, for 1rp1,p2rP. However, the precoding has the advantage of introducing some redundancy in the transmitted signals, which can be exploited to induce a supplementary dimension to the tensor.

4.3. Fifth-order tensor of covariances

Let us defineR2

C

DRRPP

as the fifth-order tensor composed of the covariance matrices Ryðd,p1,p2Þ, for 0rdrD1 and 1rp1,p2rP, constructed in such a way that the (r1,r2)th element of the matrix Ry(d,p1,p2)

corresponds to the (d+1,r1,r2,p1,p2)th element of R, i.e. ½Ryðd,p1,p2Þr1,r2¼ ½Rdþ1,r1,r2,p1,p2¼rdþ1,r1,r2,p1,p2. From (33),

the tensorRcan be written elementwise as

rdþ1,r1,r2,p1,p2¼

XQ

q¼1

hr1,qc~p1,qzdþ1,qc~p2,qh

r2,q: ð34Þ

Table 1

ALS-DD-BI algorithm—direct-data tensor.

Initialization(it=0):

CalculateH^ð0Þ

using (29). Iterations(it=it+1):

(1) CalculateS~^ðitÞ¼ ½ðC~H^ðit1Þ

ÞyY ½3T

(2) Eliminate the scaling ambiguity inS^~ðitÞ

L by using (26).

(3) ConstructS^~ðitÞ

L,DDby projecting

^ ~

SðLitÞonto the alphabet of symbols.

(4) Reconstruct the nonlinear part of the symbol matrixS^~ðitÞ

NL,DDfrom

^ ~ SðLit,DDÞ .

(5) CalculateH^ðitÞ

¼ ½ðS~^ðDDitÞC~ÞyY½2T

(6) IfJ ^

HðitÞH^ðit1ÞJ2F

JH^ðit1ÞJ2F

4e or J

^ ~

SðitÞS~^ðit1ÞJ2F

JS^~ðit1ÞJ2

F

(7)

Eq. (34) corresponds to the PARAFAC decomposition ofR

with rank rQ and matrix factors H2

C

RQ, C~ 2

C

PQ, Z2

C

DQ,C~

2

C

PQ andH2

C

RQ.

Unfolded matrix representations of the tensor R are obtained by columnwise stacking all the matrix slices of a given type. The channel estimation algorithms presented in the next section are based on the following unfolded matrix representations ofR:

R½1¼ ðHC~ZC~

ÞHH2

C

RP2DR, ð35Þ

R½2¼ ðC~ZC~

HÞHT

2

C

RP2DR,

ð36Þ

R½3¼ ðC~ZC~

ÞðHHÞT

2

C

P2DR2: ð37Þ

These matrices are constructed so that the element

½Rðdþ1Þ,r1,r2,p1,p2 of the tensor is placed at the position

(ilin,icol), withilinandicoldefined as

ilin¼ ðr11ÞP2Dþðp11ÞPDþdPþp2, icol¼r2, ð38Þ

ilin¼ ðp11ÞRPDþdRPþðp21ÞRþr2, icol¼r1, ð39Þ

ilin¼ ðp11ÞPDþdPþp2, icol¼ ðr21ÞRþr1 ð40Þ

for the matricesR[1],R[2]andR[3], respectively.

The essential uniqueness of the PARAFAC decomposi-tion of the tensorRis assured by Kruskal’s condition[25]:

2kHþ2kCþkZZ2Qþ4: ð41Þ

If this condition is satisfied, then the matrix factors

H,H,C~,C~

and Z are unique up to column scaling and permutation ambiguities,which means that two sets of matrices fH,H,C~,C~

,Zg and fHu,H00,C~u,C~00,Zug satisfying (34), are such asHu¼H

PL

a,H00¼H

PL

b,C~u¼C~

PL

c,C~00¼

~

C

PL

d and Zu¼Z

PL

e, where

L

a,

L

b,

L

c,

L

d and

L

e are diagonal matrices with

L

a

L

b

L

c

L

d

L

e¼IQ and

P

is a

permutation matrix of order Q. Assuming that the matrices H, C~ and Z are full k-rank, condition (41) becomes

minðR,QÞþminðP,QÞþmin2ðD,QÞZQþ2: ð42Þ

The matrixZcontains the information about the time correlation introduced by the precoding scheme. It can be assumed to be known, as shown in [21]. It should be mentioned that the precoding scheme provides full k-rank matrices Z. Moreover, as in Section 3, the matrix C~ containing the code products can also be assumed to be known at the receiver, the spreading codes being chosen such thatC~ is full k-rank. Thus, if Kruskal’s condition (42) is satisfied, we have C~u¼C~, C~00¼C~

, Zu¼Z and, hence,

P

¼IQ,

L

L

L

e¼IQ and

L

L

a1¼

L

1, where

L

is

a QQ diagonal matrix, which gives Hu¼H

L

and

H00¼H

L

1. The scaling ambiguity does not represent an effective problem, as it can be canceled by using an automatic gain control and a phase-locked loop, or a differential modulation. Another possible solution con-sists in using a few pilot symbols to eliminate this ambiguity.

The uniqueness condition (42) is weaker than that associated with other channel estimation methods

[21,11,19,20]. As in Section 3, it is possible to choose

RoQ andPoQ. This flexibility on the choice ofRandP provided by Kruskal’s condition is one of the main advantages of the PARAFAC-based approach, allowing an interesting tradeoff between complexity and capacity.

4.4. Two-steps ALS algorithm

In this section, we present a blind algorithm to compute the PARAFAC decomposition of the tensorR. It is worth mentioning that this algorithm does not require the precoding if the number of used covariance matrices is set to one (D=1).

The proposed channel estimation method is a two-step version of the ALS algorithm, based on an alternate minimization of the two following conditional LS cost functions:

J1¼JR^½1ðH^ ðit1Þ

a C~ZC~

ÞHTbJ2F, ð43Þ

J2¼JR^½2ðC~ZC~

H^ðbitÞÞH T

aJ2F, ð44Þ

where the matricesH^ðitÞ

a andH^

ðitÞ

b denote, respectively, the

estimates ofHandH*

at theitth iteration,R^½1andR^½2are the sample estimates of R[1] and R[2], respectively,

calculated using the channel outputs measured duringN symbol periods. The corresponding ALS algorithm is described in Table 2, where

e

is an arbitrary small positive constant and H^ðitÞ

ab ¼0:5½H^

ðitÞ

a þðH^

ðitÞ

b Þ. Three

channel estimates are obtained: H^ðitÞ

a , ðH^

ðitÞ

b Þ and H^

ðitÞ

ab,

the final channel estimate being chosen as the one providing the smallest value of the cost function (44).

As with the direct-data approach, if the matrix C~ is unknown at the receiver, the ALS algorithm can be used to jointly estimate the matricesHandC~.

5. Comparison of the uniqueness conditions of the covariance and direct-data approaches

This section provides a brief comparison of the uniqueness conditions of the proposed PARAFAC-based estimation methods. Table 3 shows the uniqueness conditions of the methods developed in Sections 3 and

Table 2

Two-steps ALS algorithm—covariance approach.

Initialization(it=0): ^

Hða0Þ-RQrandom matrix Iterations(it=it+1):

(1)H^ðitÞ

b ¼ ½ðH^

ðit1Þ

a C~ZC~

ÞyR^

½1T

(2)H^ðitÞ

a ¼ ½ðC~ZC~

H^ðbitÞÞyR^½2T

(3) IfJ ^ HðabitÞH^

ðit1Þ

ab J2F

JH^ðabit1ÞJ2F

4e, return to step 1.

Table 3

Uniqueness conditions for the proposed estimation methods.

Approach T=2 T=3

Direct-data RþPZ6 RþPZ14

(8)

4, for K=1 (third-order nonlinearity) with T=2 and 3, which corresponds to Q=4 and 12, respectively. It is assumed that all the factor matrices are full k-rank. Moreover, for simplifying the comparison, we make some realistic assumptions concerning the number of delays (D), symbol periods (N), receive antennas (R) and spreading factor (P). We consider that D,NZ4 and 2rR,Pr4 when T=2. For T=3, we consider that D=8,

NZ12 and 2rR,Pr12. Note that the direct-data approach provides a uniqueness condition more relaxed than the one of the covariance approach in terms of constraints on the number of receive antennas and spreading gain. We recall that the ALS-DD-BI method also requiresNtPZQ.

The expressions of Table 3 are based on Kruskal’s condition, which is the classical condition for essential uniqueness of the PARAFAC decomposition. In [26], a generic condition was derived for the uniqueness of the PARAFAC decomposition of third- and fourth-order ten-sors. This condition assumes that one of the tensor dimensions is higher than the tensor rank. Considering the assumptions made in the last paragraph, the unique-ness condition derived in [26]for the PARAFAC decom-position of the data tensor defined in Section 3 is given by

PðPþ1ÞRðRþ1ÞZ2QðQ1Þ: ð45Þ

This condition is more relaxed than the one inTable 3for the data tensor.

6. Simulation results

In this section, the proposed channel estimation and data recovery methods are evaluated by means of simulations. A linear-cubic MIMO Volterra model of an uplink channel in a MIMO radio over fiber communication system[4,5]is considered for the simulations. The RT wireless link, corresponding toR receive andT transmit antennas, is characterized by a frequency flat Rayleigh fading. The electrical–optical (E/O) conversion in each antenna is modeled by the following polynomial

f1xþf3jxj2x, with f1=1 and f3=0.35 [4,9]. The results

were obtained via Monte Carlo simulations with at least 100 independent data and noise realizations, and the spreading codes are generated from a truncated fast Fourier transform (FFT) matrix.

The data recovery performance is evaluated by means of the bit-error-rate (BER) and the channel estimation by means of the normalized mean square error (NMSE) of the estimated channel parameters, defined as

NMSE¼10log10 1

NR

XNR

l¼1

JHH^lJ2F JHJ2F

!

, ð46Þ

where NR is the number of Monte Carlo simulations,H^

l

represents the channel matrix estimated at thelth Monte Carlo simulation. The signal-to-noise-ratio (SNR) is re-lated to theEB/N0(energy per bit to noise power spectral

density) ratio in the following waySNR¼2Eb=N0for 4-PSK signals and SNR¼Eb=N0 for the signals generated using configuration A of the precoding scheme of[21].

6.1. Direct-data approach

In this section, we compare the following algorithms:

Standard ALS with random initialization;

ALS-DD-BI: ALS with block initialization and direct decision inside the loop;

ALS-BI: ALS with block initialization and without DD inside the loop;

ALS-DD: ALS with random initialization and direct decision inside the loop.

The performance is also compared with the estimator (29) with Nt=N=32 (supervised estimation). All the simula-tions concerning the direct-data based estimation meth-ods were obtained with 4-PSK transmitted signals.

Fig. 1shows the NMSE versus SNR provided by the ALS, ALS-DD-BI and ALS-DD methods forP=3,R=3,T=2,Nt=4 and N=32. The NMSE results obtained with the ALS-BI algorithm are not shown because they are quite similar to the results obtained with the ALS algorithm. From this figure, it can be concluded that the NMSE provided by the ALS-DD-BI algorithm is significantly smaller than the one obtained with the ALS algorithm. Discarding the

information contained in S~^ðitÞ

NL makes the ALS-DD-BI

algorithm suboptimal. Even so, it performs better than the ALS algorithm because it exploits, inside the loop, the finite-alphabet property of one of the factor matrices, which is not the case for the standard ALS algorithm.

The performance of the ALS-DD technique with the use of only one pilot symbol is different from that of the ALS-DD-BI algorithm when the SNR is equal to 30 dB. This is due to the fact that the ALS-DD-BI and ALS-DD algorithms are not monotonically convergent, contrarily to the ALS algo-rithm, which may lead to convergence problems. However, as it can be viewed inFig. 1, the use of a block-initialization eliminates this problem. This means that the use of direct decisions in the loop improves the steady-state error in most of the cases, but it may sometimes lead to a bad convergence. Moreover, this figure indicates that, when random initialization is used, DD inside the loop induces

0 5 10 15 20 25 30 −50

−40 −30 −20 −10 0 10

SNR (dB)

NMSE (dB)

ALS − N=32 ALS−DD−BI − N=32 ALS−DD − N=32 Supervised − N=32 ALS − N=8

(9)

more convergence errors at high SNRs than at low SNRs. This means that the noise induces a good convergence.

Fig. 1also shows the NMSE of the ALS algorithm with N=8, which allows to draw the remarkable conclusion that the proposed ALS method is able to blindly estimate the channel using the channel outputs measured during only eight symbol periods.

Fig. 2 shows the number of iterations needed to achieve the convergence versus SNR for the ALS, ALS-DD-BI, ALS-DD and ALS-BI algorithms with Nt=4, P=3, R=3,T=2 andN=32. Note that the ALS-DD-BI technique converges more quickly than the ALS algorithm in most of the cases, needing approximately 2 iterations to achieve the convergence when the SNR is higher than 15 dB. However, at low SNRs, decision errors slow down the convergence of the ALS-DD-BI algorithm. Note also that the ALS-DD is a bit slower than the ALS-DD-BI due to its random initialization. Moreover, as expected, the ALS-BI algorithm converges more fastly than the ALS algorithm. So, the BI does not affect the steady-state error of the ALS algorithm, it modifies only its convergence time.

Fig. 3shows the BER versus SNR provided by the ALS and ALS-DD-BI algorithms withN=32,Nt=4,P=3,R=3 and T=2. It is also shown the BER provided by the zero forcing (ZF) receiver assuming an exact channel knowledge:

^ ~

S¼ ½ðC~HÞyY½3T: ð47Þ

Note that (47) corresponds to the first step of the ALS algorithm with a known channel. The ALS-DD-BI algorithm provides a BER smaller than that of the ALS algorithm and close to that of the ZF receiver with a known channel. We do not show the BERs provided by the ALS-DD and ALS-BI algorithms because they are quite similar to the ones of the ALS-DD-BI and ALS, respectively, the only exception being the performance of the ALS-DD forSNR=30 dB, as expected from the results ofFig. 1.

In chip-synchronous linear channels, the use of ortho-gonal codes is sufficient to separate the sources by using a code-matched filter. Thus, in order to compare the proposed technique with a classical approach considered

in the case of linear channels, we also calculated the BER provided by a receiver that is composed of two stages: the first one is composed of a code-matched filter and the second one is constituted with a maximum ratio combin-ing (MRC) spatial filter. In this case, the spreadcombin-ing codes are orthogonal and generated from a FFT matrix of orderP=3. InFig. 3this technique is denoted by ‘‘Code-matched’’. We can conclude that the loss of orthogonality induced by the channel nonlinearities has an important impact over the BER of the code-matched receiver. Indeed, this BER is significantly higher than the ones obtained with the ALS-DD-BI algorithm and the ZF receiver with known channel.

6.2. Covariance approach

In this section, the proposed covariance based method for estimating memoryless MIMO Volterra channels is evaluated. The information signals are 8-PSK modulated and generated using the precoding scheme.Fig. 4shows 0 5 10 15 20 25 30

0 10 20 30 40 50

SNR (dB)

Number of Iterations

ALS ALS−DD−BI ALS−BI ALS−DD

Fig. 2.Number of iterations needed to achieve the convergence versus SNR for the direct-data based receivers.

0 5 10 15 20

10−4 10−3 10−2 10−1 100

SNR (dB)

BER

ALS ALS−DD−BI Known−Channel Code−matched

Fig. 3.BER versus SNR provided by the direct-data based receivers, code-matched receiver and ZF with known-channel.

0 5 10 15 20 25 30

−25 −20 −15 −10 −5 0

SNR (dB)

NMSE (dB)

ALS, 3x2, N=256 ALS, 3x2, N=8 ALS, 2x2, N=256 ALS, 2x2, N=8

(10)

the NMSE versus SNR provided by the ALS algorithm for D=4, P=3, T=2, two values of N (8 and 256) and two values of R (2 and 3), denoted by 32 and 22, respectively. It can be viewed that the channel estimates using three receive antennas are 1–2 dB more accurate than the ones obtained with two receive antennas. Moreover, it can be concluded that the proposed covariance based method performs relatively well using onlyN=8 symbols to estimate the covariances.

Fig. 5 shows the BER versus SNR provided by the following minimum mean square error (MMSE) receiver based on (22):

^

WMMSE¼R~sð0ÞðC~H^ÞH½ðC~H^ÞR~sð0ÞðC~H^ÞHþ

s

2IRP12CQRP

ð48Þ

using ALS based channel estimates, forD=4,P=3,R=3,T=2 and two values ofN(8 and 256). For comparison, the BER provided by the MMSE receiver assuming an exact knowledge of the channel is also plotted, as well as the BER obtained with the code-matched receiver above described using N=256. This figure shows that, for N=256, the BER provided by the ALS algorithm is very close to that of the MMSE receiver calculated with the exact channel. Moreover, as inFig. 4, the proposed method performs relatively well forN=8. It can also be viewed that the code-matched receiver performs significantly worst than the proposed ALS method forN=256.

Fig. 6shows the influence of the spreading gainPand the number D of covariance delays on the channel estimation accuracy. The results were obtained with the ALS algorithm, forP=1, 2 and 3, withN=256,SNR=20 dB, R=3 andT=2. WhenP=3, we can see that the ALS gives similar performances for all the tested values ofD. Indeed, forP=R=3, the uniqueness condition (42) becomesDZ0. In this case, the use of the precoding scheme is not very attractive. However, it can be concluded fromFig. 6that, forP=2 and 1, the ALS does not work well forD=1. Indeed, in these cases, condition (42) becomes DZ2 and DZ4, respectively. The use of the precoding scheme is therefore mandatory in such cases.

6.3. Comparison between direct-data and covariance approaches

In this subsection, we compare the direct-data and covariance ALS methods, using 4-PSK transmitted signals and input signals generated with the precoding scheme, respectively. Figs. 7 and 8 show the NMSE and the BER versus SNR, forD=4,P=3,R=3,T=2 and two values ofN(8 and 128). The BER associated with the covariance method was calculated using the MMSE receiver (48).

It can be concluded fromFig. 7that, for low SNRs, the NMSE provided by the covariance-based ALS method is smaller than the one obtained with the direct-data ALS method, while the direct-data method performs better for high SNRs. That is due to the fact that the noise is the main source of performance degradation for the direct-data method. On the other hand, for the covariance approach, the errors on the covariance estimation are the

1 2 3 4 5

−25 −20 −15 −10 −5 0 5

Number of delays (D)

NMSE (dB)

P=3 P=2 P=1

Fig. 6.NMSE versus the numberDof covariance delays provided by the covariance based receiver.

0 5 10 15 20 25 30

−30 −25 −20 −15 −10 −5 0 5

SNR (dB)

NMSE (dB)

ALS − direct−data − N=8 ALS − covariance − N=8 ALS − direct−data − N=128 ALS − covariance − N=128

Fig. 7.NMSE versus SNR provided by the direct-data and covariance based receivers using the ALS algorithm.

0 5 10 15 20

10−6 10−4 10−2 100

SNR (dB)

BER

ALS − N=256 Known−Channel Code−matched − N=256 ALS − N=8

(11)

main source of performance degradation. That is con-firmed by noting that, for N=8, the NMSE of the covariance method does not increase a lot fromSNR=20 to 30 dB.

FromFig. 8, we can see that the covariance ALS method provides lower BERs than the direct-data method. How-ever, as previously mentioned, these two methods do not use the same kind of transmitted signals. The transmis-sion rate provided by the signals generated using the precoding scheme is twice smaller than the one of a 4-PSK signal.

Thus, we can conclude that the use of the direct-data approach is more interesting when we have to use small blocks of data (small values ofN) and the SNR is high. On the other hand, if a high value ofNcan be used and the SNR is low, then the covariance approach is more attractive.

7. Conclusion

In nonlinear chip-synchronous direct sequence spread spectrum communication systems, usual solutions for linear channels, like code-matched filters, are unable to cancel nonlinear interferences. In this paper, the problem of channel estimation and data recovery in nonlinear MIMO spread spectrum communication systems has been addressed. Exploiting space, time and code diversities of the received signals, original methods based on the PARAFAC decomposition of two types of tensors have been proposed. The main advantage of these tensor-based approaches is that they provide techniques with different characteristics and they allow a great flexibility on the choice of some design parameters.

A method for joint channel estimation and symbol recovery based on the PARAFAC decomposition of a third-order tensor composed of received signals has been first presented. Then, assuming that the transmitted signals are PSK modulated, a MIMO Volterra channel estimation method based on the PARAFAC decomposition of a

fifth-order tensor composed of covariances of the received signals has been proposed.

These estimation methods have been applied to an uplink channel of a nonlinear MIMO ROF communication system. The simulation results have illustrated the good performance of the proposed algorithms. The main advantage of the direct-data approach is that it allows joint blind channel estimation and data recovery using a small number of channel output measurements. More-over, the ALS-DD-BI algorithm provides a significant performance improvement with respect to the standard ALS algorithm. On the other hand, the main advantage of the covariance approach is that it leads to weaker uniqueness conditions and it is more robust to noise than the direct-data techniques.

In a future work, we will propose channel estimation methods to the case of nonlinear spread spectrum systems modeled as Volterra channels with memory, using other tensor decompositions[15]. Another approach to be considered will consist in taking the finite alphabet constraint and the special structure of the nonlinear symbol matrix into account into the LS optimization, in order to improve the ALS performance while maintaining its monotone convergence[27].

Acknowledgement

This work was partially supported by the CAPES/Brazil agency.

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Imagem

Fig. 1 shows the NMSE versus SNR provided by the ALS, ALS-DD-BI and ALS-DD methods for P=3, R =3, T =2, N t =4 and N=32
Fig. 1 also shows the NMSE of the ALS algorithm with N=8, which allows to draw the remarkable conclusion that the proposed ALS method is able to blindly estimate the channel using the channel outputs measured during only eight symbol periods.
Fig. 5 shows the BER versus SNR provided by the following minimum mean square error (MMSE) receiver based on (22):
Fig. 8. BER versus SNR provided by the direct-data and covariance based receivers using the ALS algorithm.

Referências

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