art alfalmeida constrained tucker 3

Fast communication

Constrained Tucker-3 model for blind beamforming

Andre´ L.F. de Almeida

a,b,1

_{, Ge´rard Favier}

a,

_{, Joa˜o C.M. Mota}

b

a_{I3S Laboratory, University of Nice-Sophia Antipolis (UNSA), CNRS, France}

b_{Wireless Telecom Research Group, Federal University of Ceara´, Fortaleza, Brazil}

a r t i c l e

i n f o

Article history:

Received 31 October 2007 Accepted 20 November 2008 Available online 13 December 2008

Keywords: Blind beamforming Equalization Multipath propagation Tensor modeling Wireless communications

a b s t r a c t

This paper presents a constrained Tucker-3 model for blind beamforming. The constrained structure of this model is parameterized by two constraint matrices that capture the spatial and temporal structure of the channel. We first present a generalized Tucker-3 model that considers a multipath propagation scenario with large delay spread and different number of paths per source. This tensor model generalizes the models of [N.D. Sidiropoulos, G.Z. Dimic, Blind multiuser detection in WCDMA systems with large delay spread, IEEE Signal Process. Lett. 8 (3) (2001) 87–89] and [A. de Baynast, L. De Lathauwer, De´tection autodidacte pour des syste`mes a` acce`s multiple base´e sur l’analyse PARAFAC, in: Proceedings of XIX GRETSI Symposium on Signal and Image Processing, Paris, France, 2003] by considering a more general multipath propagation scenario. From this general model, two particular cases can be derived by adjusting the constrained structure of the Tucker-3 core tensor. Identifiability of the proposed tensor model is studied and some simulation results are presented to illustrate the application of this model to blind beamforming.

&2008 Elsevier B.V. All rights reserved.

1. Introduction

In the deterministic blind beamforming problem, the spatial/temporal structure of the wireless channel and/or the temporal structure of the source signals are exploited

for reconstructing the source signals [1,2]. Classical

solutions are based on algebraic (subspace

decomposi-tion) techniques working on blocks of data [3]. It was

shown in [4], that the problem of deterministic blind

beamforming can be addressed using a tensor modeling approach by means of the parallel factor (PARAFAC)

decomposition[5]. The tensor model of[4] is restricted

to the case of small delay spread and multiple paths per

source. In [1], a large delay spread channel model was

considered but the authors restricted the tensor model to

the case of a single resolvable multipath per source. The

approach of[2]also works under the same assumptions as

[1] but proposes a different tensor model. The case of

large delay spread with multiple paths per source was first

addressed in[6], and later in[7]and [8]under different

tensor modeling approaches. Specifically,[6]and[7]use a

constrained PARAFAC model while[8] proposes a

block-component tensor model. However, these models are restricted to the case where all the sources have the same number of multipaths.

In this paper, we present a constrained Tucker-3 model

[9] for the blind beamforming problem. The associated

Tucker-3 core is factored as a function of constraint matrices, the structure of which depends on the structure assumed for the multipath channel. We first present the general model assuming large delay spread and multiple paths per source. Then, two particular models are derived by assuming (i) far-field reflections without angular spread and (ii) local scattering with small delay spread. The identifiability of the proposed tensor model is studied in the general case. Our modeling approach not only Contents lists available atScienceDirect

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

Signal Processing

0165-1684/$ - see front matter&2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.11.016

_{Corresponding author. Tel.: +33 492 942 736; fax: +33 492 942 896.} E-mail addresses:lima@i3s.unice.fr (A.L.F. de Almeida),

favier@i3s.unice.fr (G. Favier),mota@gtel.ufc.br (J.C.M. Mota).

1_{Supported by a postdoctoral fellowship under CAPES/COFECUB N.}

generalizes existing tensor models by considering a more general multipath propagation scenario but also provides an unification of these models using a common multi-linear algebraic notation.

2. Tensor signal model for blind beamforming

Let us considerQsource signals propagating through a

wireless channel characterized by multipath propagation (seeFig. 1). The receiver is equipped with a uniform linear

array of M half-wavelength-spaced omnidirectional

an-tennas. We assume that all the sources contribute with a

small number of dominant paths. Theq-th source signal

reaches the receiver via LðqÞ distinct propagation paths.

The channel impulse response has finite duration and

is assumed to be zero outside the interval ½0;KTÞ, where

T is the symbol period. We suppose that the baseband

received signal at each antenna is sampled at a rate equal

toPtimes the symbol rate. At the receiver, the received

samples are collected duringNsymbol periods. We adopt

a parametric structure for the multipath channel, under the assumption of no cluster angular spread. Let

aðqÞ

m;lðqÞ¼

:

amð

y

ðqÞ lðqÞÞ

be them-th antenna response for the angle

y

ðqÞ

lðqÞassociated

with thelðqÞ-th path of theq-th source,

hðqÞ

lðqÞ

;k;p¼

b

ðqÞ

lðqÞgð

qÞ_ðk1þp1

P

t

ðqÞ

lðqÞÞ

be a delayed pulse-shape response scaled by the

corre-sponding attenuation coefficient

b

ðqÞ

lðqÞ, where

t

ðqÞ

lðqÞ denotes the propagation delay normalized by the symbol period.

LetsðnqÞbe then-th symbol transmitted by theq-th source.

Let us denote by xðqÞ

m;n;p¼xðmqÞðnþ ðp1Þ=PÞ the p-th

received signal sample associated with then-th symbol

period,m-th antenna andq-th source. In absence of noise,

this signal can be expressed as

xðqÞ

m;n;p¼

XLð

qÞ

lðqÞ_¼1 aðqÞ

m;lðqÞ

XK

k¼1 sð_nqÞ_khðqÞ

lðqÞ_;k;p. (1)

The total signal received at them-th antenna is then given by

xm;n;p¼

XQ

q¼1 xðqÞ

m;n;p.

xm;n;pis an element of the third-order tensor

X

2

C

MNP

collecting the received signal samples over theMreceive

antennas,Nsymbol periods andPoversamples/symbol.

3. Constrained Tucker-3 model

Let us define aðqÞ

lðqÞ¼ ½a

ðqÞ

1;lðqÞa

ðqÞ

2;lðqÞ a

ðqÞ

M;lðqÞT2

C

M

as the array response vector and let

AðqÞ¼ ½a₁ðqÞ að_LqðqÞÞ

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} LðqÞ_paths

2

C

MLðqÞ (2)

be a matrix concatenating the array responses associated

with the LðqÞ _{paths of the q-th source. Let h}ðqÞ

lðqÞ_;k¼

½hðqÞ

lðqÞ_;k;1h

ðqÞ

lðqÞ_;k;2 h

ðqÞ

lðqÞ_;k;P

T₂

_C

P _{be a vector collecting the}

pulse-shape waveform samples, scaled by the complex

attenuation coefficient of thelðqÞ-th path, within a symbol

period. Now, let us define the matrices concatenating the vectorshðqÞ

lðqÞ_;kof theq-th source, respectively overKsymbol

periods, and then for theLðqÞ_paths

HðqÞ lðqÞ¼ ½h

ðqÞ

lðqÞ_;1 h

ðqÞ

lðqÞ_;K 2

C

PK

and

HðqÞ_{¼ ½H}ðqÞ

1 H

ðqÞ

LðqÞ

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} LðqÞ_paths

2

C

PKLðqÞ .

LetSðqÞ₂

_C

NK _{be theq-th source symbol matrix defined}

as ½SðqÞ

n;k¼:snk which has a Toeplitz structure. Let

Xð₁qÞ2

C

NMP_,XðqÞ 2 2

C

PNM

and Xð₃qÞ2

C

MPN _{be the three}

unfolded matrix representations of the received signal tensorxðqÞ

m;n;pbuilt as

½XðqÞ

1 ðn1ÞMþm;p¼ ½Xð qÞ

2 ðp1ÞNþn;m

¼ ½Xð₃qÞ_ðm1ÞPþp;n¼xðmqÞ;n;p.

XðqÞ

i¼1;2;3admit the following factorizations[7]:

XðqÞ

1 ¼ ðAð

qÞ

W

ðqÞSðqÞ

U

ðqÞ_ÞHðqÞT_{, (3)}

Xð₂qÞ¼ ðSðqÞ

U

ðqÞ_HðqÞ_ÞðAðqÞ

W

ðqÞÞT, (4)

Xð₃qÞ¼ ðHðqÞAðqÞ

W

ðqÞÞðSðqÞ

U

ðqÞÞT, (5) where

W

ðqÞ¼ILðqÞ1T_K;

U

ðqÞ¼1T_LðqÞIK (6)

are two full row-rank constraint matrices, whereand

denote, respectively, the Kronecker and the Khatri–Rao

products. Note that both

W

ðqÞ and

U

ðqÞ_{are linked to the}

channel model parametersLðqÞ_{andK. Using the following}

property of the Khatri–Rao product:

ACBD¼ ðABÞðCDÞ (7)

for arbitrary compatible matrices A2

C

IR, B2

C

JS,

C2

C

RP

,D2

C

SP

, we can rewrite (3) as

XðqÞ

1 ¼ ðA

ðqÞ_SðqÞ_ÞGðqÞ

1 H

ðqÞT_,

where

GðqÞ

1 ¼

W

ðqÞ

U

ðqÞ. (8)

Substituting (6) into (8), we get

GðqÞ

1 ¼ ðILðqÞ1T_KÞ ð1T_LðqÞIKÞ ¼ ðILðqÞIKÞ ¼IKLðqÞ, (9)

L(1) paths

L(Q) paths Source 1

Source Q

M

Receiver

which allows us to rewrite the overall received signal (summed over all the sources) in the following form:

X1¼

XQ

q¼1

Xð₁qÞ¼X

Q

q¼1

ðAðqÞSðqÞÞHðqÞT_{, (10)}

or equivalently

X1¼ ½Að1ÞSð1Þ;. . .;AðQÞSðQÞ

Hð1ÞT

. . .

HðQÞT

2

6 6 4

3

7 7

5¼ ðAj jSÞH

T_,

(11)

where

A¼ ½Að1Þ

;. . ._;AðQÞ ₂

_C

ML

, (12)

S¼ ½Sð1Þ

;. . .;SðQÞ ₂

_C

NKQ

, (13)

H¼ ½Hð1Þ_{;. . .;H}ðQÞ ₂

_C

PKL

(14)

are block matrices, L¼PQ_q¼1LðqÞ_{, and j j denotes a}

‘‘block-wise Kronecker product’’, i.e. a column-wise

con-catenation of Q Kronecker product blocks. Similarly, by

taking property (7) into account in (4) and (5), we can obtain two other equivalent Tucker-3 matrix representa-tions

X2¼ ðSj jHÞG2AT; X3¼ ðHj jAÞG3ST. (15) The model given by (11) and (15) follows a constrained

Tucker-3 decomposition [9,10] with Tucker-3 core

ma-trices given by

G1¼diagðGð11Þ;. . .;G

ðQÞ

1 Þ ¼IKL, (16)

G2¼diagðGð21Þ;. . .;G

ðQÞ

2 Þ 2

C

K2LL_{, (17)}

G3¼diagðGð31Þ;. . .;G

ðQÞ

3 Þ 2

C

KLKQ

, (18)

whereL¼PQ_q¼1ðLðqÞ_Þ2_{, whileG}ðqÞ

2 andG

ðqÞ

3 are factored in

terms of the constraint matrices as

GðqÞ

2 ¼ ð

U

ðqÞ_I

KLðqÞÞ

W

ð

qÞT₂

_C

K2_LðqÞ_LðqÞ

, (19)

GðqÞ

3 ¼ ðIKLðqÞ

W

ð

qÞ_Þ

U

ðqÞT₂

_C

KðLðqÞ_Þ2

K

. (20)

3.1. Case 1: far-field reflections without angular spread

In some practical propagation scenarii, multipath reflections occur in remote objects that are located in the far-field of the receiver antenna array. We assume that most energy is concentrated in a single spatially

resol-vable path in the direction of the transmitter[3]. In[1,2],

two different tensor modelings for the received signal were proposed under the assumption of far-field reflection without angular spread. Using this assumption, we show that a constrained Tucker-3 formulation is also possible.

Let us assume that theLðqÞ_{reflections relative to theq-th}

source signal are located in the far-field of the antenna

array, so that theLðqÞ_{spatial signatures can be considered}

identical, i.e., aðqÞ

1 ¼ ¼a

ðqÞ

LðqÞ¼aðqÞ, q¼1;. . .;Q:

Conse-quently, the array response matrix A defined in (12) is

now constituted ofQblocks ofLðqÞ_{identical columns each,}

and can be explicited as

A¼ ½að1Þ₁T

Lð1Þ;. . .;aðQÞ1T_LðQÞ ¼A

!

(21) withA¼ ½að1Þ_{;. . .;a}ðQÞ_and

!

¼

1T_Lð1Þ

._. .

1T_LðQÞ

2

6 6 6 4

3

7 7 7 5

. (22)

Substituting (21) into (11) and (15) yields

X1¼ ðA

!

j jSÞHT; X2¼ ðSj jHÞG2ðA

!

ÞT,

X3¼ ðHj jA

!

ÞG3ST. (23)

Let us consider the following property of the block-wise Kronecker product:

ACj jBD¼ ðAj jBÞdiagðCð1Þ_Dð1Þ_{;. . .;C}ðQÞ_DðQÞ_{Þ, (24)}

for arbitrary block-matricesA,B,CandDcomposed ofQ

blocks, with C and D block-diagonal matrices with

compatible dimensions. Using this property in each of the three unfolded representations (23), we get the following equivalent constrained Tucker-3 representa-tions:

X1¼ ðAj jSÞG1HT; X2¼ ðSj jHÞG2AT,

X3¼ ðHj jAÞG3ST, (25)

where G1¼diagðGð11Þ;. . .;G

ðQÞ

1 Þ, G2¼G2!T and G3¼ diagðGð₃1Þ;. . .;Gð₃QÞÞ are the associated constrained core

matrices, withGðqÞ

1 ¼1 T

LðqÞIK andGð₃qÞ¼ ðIKLðqÞ1T_LðqÞÞGð

qÞ

3 .

3.2. Case 2: local scattering with small delay spread

Now, consider that all scattering objects are close to the receiver antenna array so that the relative propagation delays are much smaller than the symbol period, i.e. maxð

t

ðqÞ

lðqÞÞ5T, q¼1;. . .;Q. It is assumed that multipath delays are different. This is also known as the incoherent multipath assumption with small delay spread, which was

addressed in[4] using tensor modeling. Since temporal

dispersion at the symbol-level is no longer present

ðK¼1Þ, we can drop the dependence of both the

pulse-shape matrix and symbol matrix on the indexk. These

matrices can be rewritten as

H¼ ½hð₁1Þ;. . .;hð_L1ð1ÞÞ

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Lð1Þ_paths

;. . .hð₁QÞ;. . .;hð_LQðQÞÞ

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} LðQÞ_paths

2

C

PL,

S¼ ½sð1Þ_{;. . .;s}ðQÞ ₂

_C

NQ

, (26)

wheresðqÞ_{¼ ½s}ðqÞ

1 ;s

ðqÞ

2 ;. . .;s

ðqÞ

N T2

C

N

. Using (6), (8), (19) and

(20) withK¼1, we get the following constraint matrices:

W

ðqÞ¼ILðqÞ and

U

ðqÞ¼1T_LðqÞ, which gives G1¼G2¼IL and G3¼diagðGð31Þ G

ðQÞ

3 Þ with G

ðqÞ

3 ¼ ðILðqÞI_LðqÞÞ1_LðqÞ. Table 1 summarizes the constrained Tucker-3 models presented in this paper for each scenario.

4. Identifiability

The goal of blind beamforming is to reconstruct theQ

to channel knowledge or training sequences. Now, the identifiability of the blind beamforming problem is studied for the constrained Tucker-3 model (11) and (15).

Theorem 1. Identifiability ofH,AandSin the least square

(LS)sense requires thatðAj jSÞ,ðSj jHÞandðHj jAÞbe

full column-rank. This implies that

MNXKL; NPXK2L; PMXKL. (27)

Proof. We are interested in estimating H, A and S by alternately minimizing three LS cost functions con-structed from (11) and (15). Clearly, this requires that

ðAj jSÞ,ðSj jHÞG2 andðHj jAÞG3be left-invertible, i.e. full column-rank. Recall that the two constraint matrices

W

ðqÞ and

_U

ðqÞ _{are full row-rank by definition, implying}

rankð

W

ðqÞ_{Þ ¼L}ðqÞ _{and rankð}

U

ðqÞ_{Þ ¼K. Therefore, we can}

easily conclude from (19) and (20) that Gð₂qÞ2

C

K2LðqÞLðqÞ

and Gð₃qÞ2

C

KðLðqÞÞ2K are full column-rank, implying that

the block-diagonal core matrices G2¼diagðGð21Þ;. . .;G

ðQÞ

2 Þ and G3¼diagðGð31Þ;. . .;Gð

QÞ

3 Þ be also full column-rank.

Therefore, it follows that ðSj jHÞG2and ðHj jAÞG3 will

be left-invertible provided thatSj jHandHj jAare full

column-rank, which leads to (27). &

Theorem 2. IfA,SandHare full-rank,then any alternative

set fA˜;S˜;H˜g yielding the same tensor model is linked

to the set fA;S;Hg by: A˜¼ATa, S˜¼STs, H˜ ¼HTh;

where Ta¼diagðTða1Þ Tð

QÞ

a Þ 2

C

LL

, Ts¼diagðTðs1Þ Tð QÞ

s Þ

2

C

KQKQ, Th¼diagðTðh1Þ T

ðQÞ

h Þ 2

C

KLKL _{are non-singular}

block-diagonal matrices satisfying the following constraint:

ðTðqÞ

a Tð qÞ

s Þ1¼Tð qÞT

h ; q¼1;. . .;Q. (28) Proof. The non-singularity of Ta;Ts and Th comes from

the full-rank assumption forA,SandH. Theorem 2 can be

proved by following the same reasoning as in [11].

Inserting TaTa1¼IL, TsT

1

s ¼IKQ and ThTh1¼IKL in (11)

and using property (24), we get

X1¼ ðATaTa1j jSTsTs1ÞThTTThHT

¼ ðA˜j jS˜Þ½diagððTð1Þ

a Þ1 ðTsð1ÞÞ1;. . .;ðTð QÞ

a Þ1

ðTðQÞ

s Þ1ÞT

T

h H˜

T

¼ ðA˜j jS˜ÞG˜1H˜T, (29) whereG˜1¼diagððTða1ÞÞ

1 _ðTð1Þ

s Þ

1

;. . ._;ðTðQÞ

a Þ

1 _ðTðQÞ

s Þ

1_ÞTT

h :

Comparing (29) with (16), we deduce that we must have

˜

Gð₁qÞ¼ ððTðqÞ

a Þ1 ðTð qÞ

s Þ1ÞðT

ðqÞ

h Þ

T_¼I

KLðqÞ; q¼1;. . .;Q, which implies that

ðTðqÞ

a TðsqÞÞ1¼ ðTð qÞ

h ÞT; q¼1;. . .;Q: & (30)

This constrained Tucker-3 model has restricted

rota-tional ambiguities in the sense that the onlyadmissible

transformation matrices are those that satisfy (28). This is in contrast to the standard (unconstrained) Tucker-3

model [9]. For the model without angular spread (25)

the transformational ambiguity across sets ofLð1Þ

;. . ._;LðQÞ

columns disappears, i.e. Ta¼diagð

d

ða1Þ; ;

d

ðQÞ

a Þ. For

the small delay spread model, the transformational

ambiguity across sets of Kcolumns disappears, i.e.Ts¼

diagð

d

ðs1Þ; ;

d

ðQÞ

s Þ.

Remark. With the general modelðK41_Þ, the recovery of

the Q source signals requires the elimination of the

transformational ambiguities Tð1Þ

s ;. . .;TðsQÞ. This is done

by enforcing the Toeplitz structure ofSð1Þ_{;. . .;S}ðQÞ_{using a}

subspace method[3]. The inherent scaling factors on the

source symbols can be eliminated by using differential

modulation[12].

5. Simulation results

We illustrate the performance of a tensor-based blind beamforming receiver exploiting the proposed constrained Tucker-3 model with the known Tucker-3 core structure. The receiver combines the alternating least squares (ALS) algorithm for minimizing the three LS cost functions constructed from (11) and (15), with the

subspace method of [3] and finite-alphabet projection.

The algorithm starts with a random initialization _A^

ð0Þ

and_S^

ð0Þ.

At thei-th iteration, the updating steps are

ðiÞH^T_ðiÞ¼ ½ð

^

Aði1Þj jS^ði1ÞÞyX1,

ðiiÞA^T_ðiÞ¼ ½ð

^

Sði1Þj jH^ðiÞÞG2yX2,

ðiiiÞS^T_ðiÞ¼ ½ð

^

HðiÞj jA^ðiÞÞG3yX3,

where y denotes the matrix pseudo-inverse. Using the

estimate ofS^ðiÞ¼ ½S^ðð1iÞÞ; ;

^ SðQÞ

ðiÞ, a subspace method[3]is

used at each iteration of the algorithm to find an estimate Table 1

Unification of constrained Tucker-3 models for blind beamforming.

Factor matrices:fA;S;Hg, core matrices:fG1;G2;G3g

unfolded matrices of the received signal:

X1¼ ðAj jSÞG1HT,X2¼ ðSj jHÞG2AT,X3¼ ðHj jAÞG3ST

General case Case 1: no angular spread

Case 2: small delay spread

A¼A¼ ½Að1Þ

AðQÞ_{A¼ ½a}ð1Þ _aðQÞ

A¼A¼ ½Að1Þ

AðQÞ AðqÞ_2CMLðqÞ

aðqÞ₂CM _AðqÞ_2CMLðqÞ

S¼ ½Sð1Þ _SðQÞ _{S¼ ½S}ð1Þ _SðQÞ _{S¼ ½s}ð1Þ _sðQÞ

SðqÞ_2CNK

SðqÞ_2CNK

sðqÞ₂CN

H¼ ½Hð1Þ

HðQÞ

H¼ ½Hð1Þ

HðQÞ

H¼ ½Hð1Þ

HðQÞ

HðqÞ_2CPKLðqÞ

HðqÞ_2CPKLðqÞ

HðqÞ_2CPLðqÞ

Gi¼diagðGði1Þ;. . .G

ðQÞ

i Þ,i¼1;2;3

GðqÞ 1 ¼G

ðqÞ

1 G

ðqÞ 1 ¼1

T

LðqÞIK G1ðqÞ¼ILðqÞ

GðqÞ 2 ¼G

ðqÞ

2 G

ðqÞ 2 ¼G

ðqÞ

21LðqÞ G₂ðqÞ¼I_LðqÞ

GðqÞ 3 ¼G

ðqÞ

3 G

ðqÞ

3 ¼ ðIKLðqÞ1T_LðqÞÞG

ðqÞ

3 G

ðqÞ

3 ¼ ðILðqÞI_LðqÞÞ1_LðqÞ

WithGðqÞ 1,G

ðqÞ 2 andG

ðqÞ

of the source symbol vectors _s^

1;. . .;s^Q, followed by a

finite-alphabet projection. Then, the block-Toeplitz sym-bol matrix is reconstructed and the above updating steps

are repeated until convergence (for further details see[7]).

The performance is evaluated in terms of the

bit-error-rate (BER) versus signal-to-noise ratio (SNR) over 1000

Monte Carlo runs. At each run the fading gains are generated from an i.i.d. Rayleigh generator while the user symbols are generated from an i.i.d. distribution and are

modulated using BPSK. We considerQ¼2,K¼3,Lð1Þ_¼2,

Lð2Þ_{¼3, N¼50, P¼12 and M¼ f2;3g. The}

angles-of-arrival and the time-delays are described in Table 2.

Fig. 2shows that the performance for source 2 is better than that for source 1. Indeed, the signal from source 2

is received via Lð2Þ¼3 paths (against Lð1Þ¼2 paths for

source 1), thus achieving a higher path diversity gain. Performance is also improved for both sources, when the number of receive antennas is increased. The figure also shows the performance of the non-blind MMSE receiver

which assumes perfect knowledge ofAandH, for source 2

with M¼3. The performance loss of the proposed

receiver w.r.t. the MMSE one is 3 dB for a BER of 102_.

6. Conclusion

In this paper, we have presented a constrained Tucker-3 model for blind beamforming in the case of convolutive channels with multiple paths per source. We have shown that the associated Tucker-3 core can be factored as a function of two constraint matrices depending on the number of paths and the delay spread. Two particular propagation scenarii have been derived by specifying the Tucker-3 core tensor in each case by means of a unified modeling. A necessary and sufficient identifiability con-dition has been derived. A blind receiver based on this tensor model and the use of the ALS algorithm has also been proposed and illustrated by some simulation results.

References

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[3] A.-J. van der Veen, Algebraic methods for deterministic blind beamforming, Proc. IEEE 86 (10) (1998) 1987–2008.

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Table 2

Multipath parameters for the simulated scenario.

Angles-of-arrival Time-delays

Source 1 ð50

;20

Þ ð0;TÞ

Source 2 ð0

;30 ;50

Þ ð0;0:2T;TÞ

0 5 10 15 20 25

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BER

M=2 or 3, N=50, P=12

source 1, M=2 source 2, M=2 source 1, M=3 source 2, M=3 source 2, M=3 (MMSE)

source 1: L(1)₌₂

source 2: L(2)=3

Fig. 2. Receiver performance for two sources with different multipath