art alfalmeida Space time spreading

Space– time spreading MIMO-CDMA downlink systems using

constrained tensor modeling

Andre´ L.F. de Almeida

a

_{, 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 19 October 2007 Received in revised form 1 February 2008 Accepted 31 March 2008 Available online 12 April 2008

Keywords: Allocation matrices Blind detection Constrained tensor model Downlink transmission MIMO-CDMA system Multicode transmission Space–time spreading

a b s t r a c t

This paper considers the downlink of a multiple-input multiple-output (MIMO) code-division multiple-access (CDMA) system based on space–time spreading (STS) using multiuser/multicode transmission. A constrained tensor model is proposed for the STS transmitted signal. The new STS model generalizes existing ones by allowing multiuser spatial multiplexing with different code reuse/multiplexing patterns for each user. This model is parametrized by two allocation matrices with variable structure, which allow to control, respectively, the allocation of multiple data streams and spreading codes to the transmit antennas. A methodology is proposed to design these allocation matrices leading to finite sets of feasible STS schemes with guaranteed blind symbol recovery. The construction of feasible STS schemes for 2, 3 and 4 transmit antennas is presented as an illustration of this methodology. A blind receiver based on the alternating least squares algorithm is used for symbol recovery. The bit-error-rate performance of this receiver is tested and compared to some existing STS-based downlink transceivers.

&_{2008 Elsevier B.V. All rights reserved.}

1. Introduction

It is well known that the use of multiple antennas at both the transmitter and receiver is promising since it potentially provides an increased spectral efficiency compared to traditional systems employing multiple antennas at the receiver only[1]. In the context of current and upcoming wireless communication standards, the integration of multiple-antenna and code-division multi-ple access (CDMA) technologies has been the subject of several studies[2].

Spatial multiplexing schemes for multiple-input multi-ple-output (MIMO)-CDMA systems were addressed in [2–4]. In [2], a BLAST-based CDMA system performing direct-sequence spreading on the symbols to be transmitted from each antenna is proposed, where each data stream is

associated with a spreading sequence. In[3], the multiple transmit antennas are organized in groups and a unique spreading code is used by all the antennas of the same group. The separation of the different groups at the receiver is done by using layered space–time processing. Focusing on the downlink reception,[4]also considers the spatial reuse of the spreading codes and proposes a chip-level equalizer at the receiver to handle the loss of code orthogonality.

Based on space–time coding principles, a space–time spreading (STS) scheme is proposed in[5]for providing maximum diversity gain without using extra spreading codes. Ref. [5] does not consider the multiuser inter-ference problem. In [6], a multilayered STS downlink model is proposed based on CDMA codes. The data stream is divided into multiple streams, or ‘‘layers’’, each one being spread by a unique STS sequence. In[7,8], a two-dimensional STS scheme based on Walsh–Hadamard spreading matrices is proposed for the downlink of a multiuser MIMO system.

Existing STS techniques are generally used in conjunc-tion with nonlinear multiuser detecconjunc-tion receivers based

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Signal Processing

0165-1684/$ - see front matter&2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.03.022

_{Corresponding author. Tel.: +33 492 942 736; fax: +33 492 942 896.}

E-mail addresses:lima@i3s.unice.fr (A.L.F. de Almeida),

on successive interference canceling or less-complex linear zero forcing (ZF) or minimum mean square error (MMSE) receivers. Also, performance is usually evaluated assuming perfect channel knowledge at the receiver, which is an optimistic assumption in practice.

In a seminal paper[9], the problem of uplink multiuser detection/separation is linked to a parallel factor (PARAF-AC) tensor model[10]. It is shown that the received signal sample associated with a receive antenna, a symbol and a chip, can be interpreted as an element of a three-way array, or third-order tensor. Using this interpretation,[9] shows that the use of tensor modeling allows the receiver to fully exploit three forms of diversity (time, space and code) for blindly recovering the transmitted symbols/ channel/codes. Generalizations of[9]to frequency-selec-tive channels were proposed in subsequent works[11–15], under different assumptions concerning the multipath propagation structure. However, all these works are limited to single-antenna transmissions.

Multiple-antenna transmissions based on tensor mod-eling are proposed in[16–19]. The approach of[16]relies on temporal-only spreading of the data streams and enjoys built-in blind symbol recovery. Since there is no spatial spreading of the data streams across the transmit antennas, no spatial spreading diversity is obtained in [16]. In [17], a generalized tensor model is proposed for MIMO-CDMA systems with blind detection. However, this modeling approach only considers spatial multiplexing and does not cover STS. The common characteristic of the tensor modeling approaches of[16,17]is that the number of data streams is restricted to be equal to the number of transmit antennas. The approach of[18]performs full STS and allows the number of data streams to be different from the number of transmit antennas, as opposed to [16,17]. A ‘‘constrained’’ STS model with variable structure was proposed in [20] by incorporating two constraint matricesinto the model which jointly control the alloca-tion of data streams and spreading codes to the transmit antennas.

In this paper, we present a STS model for the downlink of a MIMO-CDMA system with multiuser/multicode transmission. From the viewpoint of tensor modeling, the proposed STS model is a generalization of the uplink model of[20]to the case of multiuser spatial multiplexing (i.e. several user transmissions are multiplexed across the same set of transmit antennas). Another distinguishing feature of this work compared to previous ones is the design of the allocation matrices. We propose a complete parametrization of the two allocation matrices along with new design constraints. These design constraints allow to easily derive finite sets of feasible STS schemes with guaranteed blind symbol recovery for a given number of data streams, spreading codes and transmit antennas. These STS schemes go from full code reuse to full code multiplexing and have different levels of spectral effi-ciency. Design examples are presented for 2, 3 and 4 transmit antennas. The performance of different STS schemes is tested using a blind receiver combining multiuser interference elimination/despreading and an alternating least squares (ALS) algorithm for symbol recovery.

This paper is organized in the following manner. Section 2 describes the proposed STS MIMO-CDMA downlink model. In Section 3, a set of design constraints for the two allocation matrices is detailed. Some design examples are also illustrated in this section. The receiver algorithm is presented in Section 4. Simulation results for performance evaluation are provided in Section 5 and the paper is concluded in Section 6.

2. STS MIMO-CDMA downlink system

We consider the downlink of a single cell synchronous MIMO-CDMA wireless communication system with Q

active users and spreading factor P. The base-station and theq-th user are equipped with Ktransmit and M

receive antennas, respectively. For each user, a high-rate serial input stream is split into R fixed low-rate streams that are transmitted usingJorthogonal spreading codes with RpK and JpK. This type of scheme is

known as multicode transmission1_{[21,22]. After}

spread-ing, theRdata streams of all theQusers are then added together at each one of the Ktransmit antennas. Fig. 1 depicts the block-diagram of the proposed system, considering the i-th user in transmission and the q-th user in reception. The design of the joint stream-code allocation block at the transmitter and the two signal processing blocks at the receiver will be described in the next sections.

The wireless channel is assumed to be flat-fading and constant during a time-slot of durationNsymbol periods.2

The data streams of all users are transmitted with equal powers, which are normalized so that the average of the total transmitted power is independent of the total number RQ of data streams and of the number M of transmit antennas. We assume that the receiver has full knowledge of the associated spreading codes. The spreading codes are designed from user-specific OVSF (orthogonal variable spreading factor) codes as suggested in the high-speed downlink packet access (HSDPA) standard[23].

2.1. Transmitted signal model

Let hðqÞ_m

k be the channel response between the k-th transmit antenna and them-th receive antenna of theq-th user,q¼1;. . .;Q,sðqÞ_n;r¼:sðqÞ_{ððr1ÞNþnÞbe then-th symbol}

of ther-th data stream transmitted to theq-th user, and

cðqÞ_p;jbe thep-th element of thej-th spreading code. Let us define uðqÞ_n;p;k¼:uðqÞ_k ððn1ÞPþpÞ as the ðn;p;kÞ-th sample of the q-th transmitted signal tensor UðqÞ2CNPK_.

1_{Multiple short codes for a high-rate user should be orthogonal to}

reduce the so-called ‘‘intercode interference’’. We point out that a random scrambling long pseudo-noise (PN) code common to all parallel short code users can be applied after spreading without affecting the orthogonality property between the users.

2_{The flat-fading assumption is considered here for simplicity}

reasons. However, in the case of delay spread channels with path delays of the order of a few chips, we suggest to use a guard interval (also called ‘‘guard chips’’), with duration exceeding the maximum delay spread. The guard interval can be implemented by appending trailing zeros to each transmission burst (refer to[9]for further details).

We propose the following tensor model for the multiuser transmitted signal that is broadcast to theQusers:

un;p;k¼

XQ

i¼1

uðiÞ_n;p;k¼X

Q

i¼1

XR

r¼1

XJ

j¼1 sðiÞ

n;rcðiÞp;jcr;kfj;k, (1)

where cr;kand fj;k are typical elements of matricesW2 CRK and _U2CJK. These matrices, which are common for all the users, can be viewed as stream-to-antenna

and code-to-antenna allocation matrices, respectively. They determine the reuse of users’ data streams and spreading codes across the transmit antennas. At this point, we define the basic structure of these matrices:

The columns of W and U are canonical vectors3

associated with the following canonical bases, respec-tively:

EðRÞ¼ feðRÞ₁ ;. . .;eðRÞ_R g; EðJÞ¼ feðJÞ₁;. . .;eðJÞ_J g,

WandUare both full rank matrices, which means that each canonical vector appears at least once inWandU.

As each column vector_Wkand_Ukis a canonical vector of the bases EðRÞ _{and E}ðJÞ_{, respectively, we havekWrk}2_¼a

r

andkUjk2¼b_j, wherear40 and_b

j40 denote the number

of times the canonical vectorseðRÞr andeðJÞj appear inWand U, respectively. It follows thatWWTandUUTare diagonal matrices given by

WWT¼diagða1;. . .;aRÞ; UUT¼diagðb1;. . .;bJÞ, (2)

and we have

traceðWWTÞ ¼X

R

r¼1 ar¼K,

traceðUUTÞ ¼X

J

j¼1 b_j¼K.

The signaluðiÞ_n;p;kassociated with thei-th user, defined in (1), can be rewritten in the following manner:

uðiÞ_n;p;k¼ X

R

r¼1 sðiÞ

n;rcr;k

! XJ

j¼1 cðiÞ_p;jf_j;k 0

@

1

A

¼aðiÞ_n;kbðiÞ_p;k, (4) whereaðiÞ_n;k¼ ½AðiÞn;kandbðiÞp;k¼ ½B

ðiÞ

p;kare typical elements

of the matrices

AðiÞ¼:SðiÞW2CNK,

BðiÞ_¼:_CðiÞ

U2CPK_; i¼1;. . .;Q. (5)

A matrix collectingNsymbolsPchips transmitted from thek-th transmit antenna to thei-th user is defined as

UðiÞ_k¼

uðiÞ_1;1;k uðiÞ_1;P;k

.. .

.. .

uðiÞ_N;1;k uðiÞ_N;P;k

2

6 6 6 4

3

7 7 7 5

2CNP. (6)

This matrix can be factored in the following manner:

UðiÞ_k¼AðiÞ_kB_kðiÞT¼SðiÞWkUTkCðiÞT

or, equivalently,

UðiÞ_k¼SðiÞ_CkCðiÞT_{, (7)}

where

Wk¼ ½c_1;k c_R;kT2CR,

Uk¼ ½f1;k fJ;kT2CJ (8)

and

Ck¼WkUTk2CRJ (9)

determines the coupling between data streams and spreading codes at thek-th transmit antenna.

From (4) and (7), the overall (multiuser) transmitted signal tensor obtained by summing over the Q user

spreading codes

K M

H

data streams

From other users

From other users Joint allocation of

data streams and

spreading codes MUI

elimination

Blind detection Rich-scattering

medium

Estimated data streams

Transmission by the base-station

Reception by the q-th user

• • •

• • •

• • •

• • •

• • •

• • •

n,(i)1

(i)

(i) (i)

n,1 (q)

n,R

p,1 p, j

^

n,R

(q) ΨΦ

ΨΦT

1 1

Fig. 1.Block-diagram of the proposed downlink MIMO-CDMA system.

3_{A canonical vector e}ðNÞ

n 2RN is a unitary vector containing an

contributions is given by

u_n;p;k¼X

Q

i¼1

uðiÞ_n;p;k¼X

Q

i¼1 aðiÞ_n;kbðiÞ_p;k

¼X

Q

i¼1

SðiÞ_nCkCðiÞT_p , (10)

where

SðqÞ_n ¼ ½sðqÞ_n;1 sðqÞ_n;R 2C1R_,

CðqÞ_p ¼ ½cðqÞ

p;1 cðqÞp;J 2C1J.

Model (10) leads to the downlink transmission block-diagram shown inFig. 2.

We also define a stream-to-code allocation matrix

as follows:

C¼X

K

k¼1

Ck¼WUT2CRJ_{. (11)}

2.2. Received signal model

In absence of noise, the discrete-time baseband version of the signal received by them-th antenna of theq-th user can be represented by a tensorXðqÞ2CNPMwith typical element xðqÞn;p;m¼:xðqÞmððn1ÞPþpÞ, q¼1;. . .;Q. Using (1),

we arrive at the following tensor model for the received signal:

xðqÞ n;p;m¼

XK

k¼1

hðqÞ_m;kun;p;k

¼X K k¼1 XQ i¼1 XR r¼1 XJ j¼1 sðiÞ

n;rcðiÞp;jh ðqÞ

m;kcr;kfj;k (12)

or equivalently xðqÞ n;p;m¼ XQ i¼1 XK k¼1 XR r¼1 sðiÞ

n;rcr;k

!

hðqÞ_m;k X J

j¼1 cðiÞ

p;jfj;k

0 @ 1 A ¼X Q i¼1 XK k¼1

aðiÞ_n;khðqÞ_m;kbðiÞ_p;k

!

, (13)

whereaðiÞ_n;kandbðiÞ_p;kare defined in (4).

Let us defineXðqÞ_mas a matrix collectingNsymbolsP

chips of the signal received at them-th transmit antenna

of theq-th user:

XðqÞ_m¼

xðqÞ_1;1;m xðqÞ_1;P;m

.. .

.. .

xðqÞ_N;1;m xðiÞ_N;P;m

2 6 6 6 4 3 7 7 7 5

2CNP. (14)

Note that XðqÞ_m is the m-th slice of the received signal tensorXðqÞ2CNPM_{obtained by slicing it along its third} dimension. This matrix-slice can be factored as

XðqÞ_m¼X

Q

i¼1

AðiÞdiagðhðqÞ_m;1;. . .;hðqÞ_m;KÞBðiÞT

¼X

Q

i¼1

SðiÞWDmðHðqÞÞUTCðiÞT,

where diagðhðqÞ_m;1;. . .;hðqÞ_m;KÞforms a diagonal matrix holding

hðqÞ_m;1;. . .;h_m;KðqÞ on its main diagonal, whileDmðHðqÞÞforms a

diagonal matrix from them-th row of HðqÞ2CMK_{. This} matrix-slice can be rewritten in a block factorized form as

XðqÞ_m¼SHðqÞ_mCT, (15)

with

S¼ ½Sð1Þ;. . .;SðQÞ 2CNRQ_,

C¼ ½Cð1Þ;. . .;CðQÞ 2CPJQ_{, (16)}

being the multiuser symbol and code matrices, and

HðqÞ_m ¼

WDmðHðqÞÞUT

. . .

WDmðHðqÞÞUT

2 6 6 6 4 3 7 7 7 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Qtimes

¼WDmðHðqÞXÞUT, (17) being the downlink channel associated with the q-th receiver, where

W¼IQW2CRQKQ, U¼IQU2CJQKQ, X¼1T_QIK2CKKQ

are block-constraint matrices of the transmission model,

1Q being the ‘‘all-ones’’ vector of dimension Q. Taking

these definitions into account, we obtain the following compact writing for the m-th sliceXðqÞ

m of the received

signal tensor:

XðqÞm¼SWDmðHðqÞXÞUTC T

. (18)

Although fW;U;Xg are three constraint matrices of model (18), it is worth noting that W and U are ‘‘configurable’’ matrices with variable canonical structure characterizing the STS transmission scheme. For this reason, W can be interpreted as a stream-to-antenna allocation matrix, whileUas a code-to-antenna allocation matrix. On the other hand,_Xis a ‘‘fixed’’ constraint matrix that only depends on the numbersKof transmit antennas andQof users. The design of these matrices is the subject of the next section.

S_n • (1) C_p • (1) un,p,k un,p,k (1) un,p,k (Q) C_p • (Q) S n• Γk Γk (Q) • • • • • •

Fig. 2.Transmission block-diagram of the proposed downlink MIMO-CDMA system.

3. Design of the allocation matrices

In this section, we present a complete parametrization of the allocation matrices _Wand _U. Using this parame-trization, a set of four design constraints are proposed which allow to easily derive a codebook of feasible STS schemes with guaranteed blind symbol recovery for a given number of data streams, spreading codes and transmit antennas. Then, design examples are presented for 2;3 and 4 transmit antennas.

Recall that the same allocation pattern of data streams and spreading codes is assumed for theQusers, since we haveW¼IQWandU¼IQU. Therefore, we will focus

on the design of the allocation matrices_Wand_U(instead ofWandU) without loss of generality. The use of different per-user allocation patterns is a straightforward general-ization of the proposed tensor model given in (1).

Let us partition theKtransmit antennas intoRsubsets ofKrantennas each,r¼1;. . .;R, withRpK, each antenna

of the r-th subset being associated with the r-th data stream. So, the r-th data stream is spread across Kr

antennas. TheseKr antennas are then partitioned intoJ_r

subsets associated with different spreading codes, thejr

-th code being reusedg_r;j

r times,jr¼1;. . .;Jr. The stream-to-antenna allocation matrix W is partitioned into R

blocks, the r-th block containing Kr columns. The structure of this matrix is given by

W¼

½1 1 |fflfflfflfflfflffl{zfflfflfflfflfflffl}

K1

½1 1 |fflfflfflfflfflffl{zfflfflfflfflfflffl}

K2

. . .

½1 1 |fflfflfflfflfflffl{zfflfflfflfflfflffl}

KR 2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

3

7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

¼ ½W1 Wr WR, (19)

with Wr of dimension RKr, and the r-th row of W

given by

Wr¼ ½0 0 |fflfflffl{zfflfflffl}

Kr1

1 1 |fflfflfflffl{zfflfflfflffl}

Kr

0 0 |fflfflfflffl{zfflfflfflffl}

KKr

ð1KÞ

and

Kr¼

Xr

i¼1

K_i; KR¼K.

The general structure ofWcan be formulated with the aid of canonical vectors in the following manner:

W¼ ½1T_K1eðRÞ₁ ;. . .;1T_KReðRÞ_R .

Similarly, the code-to-antenna allocation matrix U is partitioned intoRblocks, ther-th block being partitioned intoJ_rsub-blocks. This matrix has the following structure

U¼ ½U1 Ur UR, (20)

where

;

(21)

0_J

r1Kr and0ðJJrÞKr are zero matrices of dimensionJr1 Kr andðJJrÞ Kr, respectively, where

J_r¼X

r

i¼1

J_i; J_R¼J

and

ðUrÞ_J

r1þjr¼ ½0|fflfflfflffl{zfflfflfflffl} 0

gr;jr1

1 1 |fflfflfflffl{zfflfflfflffl}

gr;jr

0 0 |fflfflfflffl{zfflfflfflffl}

Krgr;jr

ð1KrÞ

denotes the j_r-th row of the nonzero row-block of _Ur, where

gr;jr¼ Xjr

l¼1

gr;l; r¼1;. . .;R,

withgr;Jr ¼Kr.

Using the canonical vectors, the general structure ofUcan be explicited as

U¼ ½1T_g

1;1e ðJÞ

1;. . .;1Tg1;j1e ðJÞ J1;1

T g2;1e

ðJÞ J1þ1;. . .;1

T gR;JRe

ðJÞ J .

We have to notice that the r-th row of _W defines the allocation of the r-th data stream to Kr transmit

antennas. Similarly, the j_r-th row of the nonzero row-block of _Ur defines the allocation of the j_r-th code tog_r;j

r antennas that transmit ther-th data stream. When c_r;kf_j

r;k¼1, it means that the jr-th spreading code is used by the k-th transmit antenna to transmit the r-th data stream. The element g_r;j

r of C defined in (11), and given by

gr;jr¼ XK

k¼1

cr;kfjr;k (22)

denotes the reuse factor of the j_r-th spreading code. It corresponds to the number of transmit antennas trans-mitting the r-th data stream using the j_r-th spreading code,j_r¼1;. . .;J_r, whereJ_r is the number of spreading codes used to transmit ther-th data stream by means ofKr

transmit antennas, so that we have

XR

r¼1

XJr

jr¼1 g_r;j

3.1. Design constraints (relations linking J_r, Krandgr;jr)

Note that W and U are parametrized by the set of parametersfJr;Kr;gr;jrg,r¼1;. . .;R. Recall that

J_r: number of codes allocated to ther-th data stream;

Kr: spatial reuse factor of (i.e. number of antennas allocated to) ther-th data stream;

g_r;j

r: spatial reuse factor of thejr-th code allocated to ther-th data stream.

In the following, we provide some design constraints for the allocation matricesWand Uwith the two following objectives:

1. To guarantee the uniqueness of the symbol matrixS

thus allowing a blind symbol recovery.

2. To simplify the transmission design by eliminating redundant STS schemes from the set of possible choices covered by the general design framework.

Constraint 1. The spatial reuse factor of the j_r-th code cannot exceed the spatial reuse factor of the r-th data stream, which leads to

gr;jrpKr. (23)

This constraint implies thattransmit antennas transmitting different data streams always use different spreading codes. This design constraint is related to the blind recovery of the transmitted symbols. To be specific, the reuse of the same spreading code by any two different data streams would induce a rotational indeterminacy affecting the corresponding columns of the symbol matrix S, i.e. this matrix would be identifiable only up to a post-multi-plication by a nonsingular transformation matrix (refer to [20]for further details).

Constraint 2. The code reuse factor does not exceed the spatial reuse factor of ther-th data stream:

J_rpK_r, (24)

which implies that only one spreading code is used at each transmit antenna, i.e. we do not allow multiple spreading codes associated with the same data stream to be multiplexed at the same transmit antenna. This constraint is merely a choice that simplifies the design of the allocation matrices by reducing the set of possible transmission schemes. In other words, transmission structures where two different codes are used by the same data stream at the same transmit antenna are not considered in our design.

From (24), we get

J¼X

R

r¼1 JrpX

R

r¼1 Kr¼K.

SinceRpJ(R¼Jonly occurs ifJ_r¼1 forr¼1;. . .;R), we obtain the following inequalities relating the number of data streams, spreading codes and transmit antennas:

RpJpK. (25)

Constraint 3. The spatial reuse factors of the R data streams are chosen in a decreasing order:

KrXK_r0; 8ror0; r;r0_{¼1;. . .;R. (26)}

This constraint eliminates alternative (redundant) designs that are just permutation of the rows ofWin (19).

Constraint 4. The spatial reuse factors of the J₁;. . .;J_R

spreading codes for theR data streams are chosen in a decreasing order:

maxðgr;1;. . .;gr;JrÞXmaxðgr0;1;. . .;gr0;Jr0Þ,

8ror0; r;r0_{¼1;. . .;R}

and

g_r;iXg_r;j; 8ioj; i;j¼1;. . .;J_r. (27)

This constraint is composed of two inequalities. The first one eliminates redundant designs which are just permu-tations of the blocksU1;. . .;URin (20). On the other hand, the second inequality avoids row permutations within each blockUr,r¼1;. . .;Rin (21). It should be noted that for given stream-to-antenna and code-to-antenna tion patterns, there are different stream-to-code alloca-tion patterns which are unimportant to the context of this work. The goal of the two inequalities in (27) is to fix the stream-to-code allocation, thus eliminating redundant choices forU.

Remark 1. In practice,WandUcan be designed by taking users’ requirements such as transmit diversity gain and data-rate into account. These requirements can be used to provide additional guidelines for choosingJ_r,Kr andgr;jr. Performance-oriented optimization of the allocation ma-trices based on a priori knowledge about the channel could resort to limited feedback precoding methods [24,25]. This research topic is beyond the scope of this work and is under investigation.

3.2. Spectral efficiency

Taking the above design requirement and spreading matrix structure into account, the rate of the proposed STS model can be calculated by using the simple formula: SF¼ RQ

P

log₂ðmÞ ðbits per channel useÞ, (28) wherem is the modulation cardinality. In this work, we make the assumption PXJQ, which means that the spreading code matrix C2CPJQ defined in (16) is full column-rank (the reason for this assumption will be clarified in Section 4). In order to facilitate the comparison of different STS schemes in terms of spectral efficiency, we assumeP¼JQwithout loss of generality. In this case (28) is equivalent to

SF¼ R

J log2ðmÞ. (29)

Recall that the two allocation matricesW2CRK _andU2 CJK are full row-rank, i.e. rankð_WÞ ¼R and rankð_UÞ ¼J. Therefore, the spectral efficiency SF in (29) can be calculated from the ranks of W and U, i.e. SF¼ ðrankðWÞ=rankðUÞÞlog₂ðmÞ.

3.3. Design examples

For a fixed numberKof transmit antennas, there is a finite setSKðW;UÞof feasible transmit schemes satisfying

the design constraints given in the previous section. Each element of this set is given by a different matrix pair ðW;UÞ. Here, we present feasible sets of STS schemes forK¼2, 3 and 4 transmit antennas, which is generally the case in practical MIMO-CDMA downlink settings. The allocation matrices W and U can then be derived from a set ofgenerating arrays as shown in the examples below.

K¼2: The set S2ðW;UÞ of feasible schemes can be

derived from the following generating arrays:

g₂¼ ½1 1; G11¼I2,

and we have

S2ðW;UÞ ¼ fðg2;g2Þ

|fflfflfflffl{zfflfflfflffl}

J¼1

;ðg2;G11Þ

|fflfflfflfflffl{zfflfflfflfflffl}

J¼2

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

R¼1

;ðG11;G11Þ

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

R¼J¼2

g.

Therefore, three transmit schemes are possible forK¼2. The first one, where W¼U¼g₂, indicates that a single data stream is transmitted over both transmit antennas using the same spreading codeðR¼J¼1Þ. This is a full transmit spatial diversity scheme with full code reuse. From (9) we have

C1¼C2¼1.

The transmitted signal matrices are given by (7)

UðqÞ₁¼UðqÞ₂¼sðqÞ_cðqÞT_.

In the second scheme we have W¼g2 and U¼G11,

meaning that a single data stream is now associated with two different spreading codes at the first and second transmit antennas. In this case, transmit spatial diversity with full code multiplexing takes place, and we have

C1¼ ½1 0; C2¼ ½0 1.

The transmitted signal matrices are given by

UðqÞ₁¼sðqÞ_CðqÞT

1 ; U

ðqÞ 2¼sðqÞC

ðqÞT 2 .

Finally, the third schemeW¼U¼G11indicates that two

different data streams are transmitted from both antennas using different spreading codes, which corresponds to a full spatial multiplexing scheme with full code multi-plexing, i.e. different codes are used at each antenna. We have

C1¼ 1 0

0 0

; C2¼ 0 0

0 1

.

In this case, the transmitted signal matrices are given by

UðqÞ₁¼SðqÞ₁CðqÞT₁ ; UðqÞ₂¼SðqÞ₂CðqÞT₂ .

This scheme is similar to that of [2], although the modeling approach used here is different. Note that, for each one of the STS schemes mentioned above, the corresponding spectral efficiency can be calculated using (29), for a fixed modulation cardinality. Note that the first

and third schemes ofS2ðW;UÞhave SF¼log2ðmÞwhile the

second one has SF¼log2ðmÞ=2.

K¼3: Consider the three generating arrays as follows:

g3¼ ½1 1 1; G21¼

1 1 0

0 0 1

,

G111¼

1 0 0

0 1 0

0 0 1

2

6 4

3

7 5.

Using the proposed design for the constraint matrices, we get the following feasible setS3ðW;UÞfrom these

generat-ing arrays:

S3ðW;UÞ

¼ ðg3;g3Þ

|fflfflfflffl{zfflfflfflffl}

J¼1

;ðg3;G21Þ

|fflfflfflfflffl{zfflfflfflfflffl}

J¼2

;ðg3;G111Þ

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

J¼3

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

R¼1

;

8 > > > > > <

> > > > > :

ðG21;G21Þ

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

J¼2

;ðG21;G111Þ

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

J¼3

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

R¼2

;ðG111;G111Þ

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

R¼J¼3

9 > > > > > =

> > > > > ;

yielding a total of six transmit schemes. The first scheme

W¼U¼g3corresponds to a full transmit diversity scheme

with full code reuse. The last one_W¼_U¼G111is a spatial

multiplexing system using different spreading codes at each transmit antenna. Between these two solutions, we have four intermediary schemes which are different combinations of transmit diversity and spatial multiplex-ing with different code reuse patterns. For instance, the second and third schemes transmit a single data stream while the fourth and fifth schemes transmit two data streams using two and three spreading codes, respectively. More explicitly, for the fifth scheme (R¼2 andJ¼3), the transmitted signal matrices are

UðqÞ₁¼SðqÞ₁CðqÞT₁ ; UðqÞ₂¼SðqÞ₁CðqÞT₂ ,

UðqÞ₃¼SðqÞ₂CðqÞT₃ ,

which shows that the first data streamðSðqÞ₁Þis transmitted by the two first antennas (k¼1 and 2) using two different codes (CðqÞ₁ andCðqÞ₂), while the second data streamðSðqÞ₂Þis transmitted by the third antennaðk¼3Þ using the third codeðCðqÞ₃Þ. It is worth noting that three levels of spectral efficiency are presented in the setS3ðW;UÞ. For instance,

assuming QPSK modulationðm¼2Þ, the first, fourth and sixth STS schemes (i.e. those with R¼J) have the same spectral efficiency SF¼2 bits per channel use. The second, third and fifth STS schemes have SF¼1₂; 1

3and23bits per

channel use, respectively.

g4¼ ½1 1 1 1; G31¼

1 1 1 0

0 0 0 1

,

G22¼

1 1 0 0

0 0 1 1

; G211¼

1 1 0 0

0 0 1 0

0 0 0 1

2 6 4 3 7 5,

G1111¼

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2 6 6 6 4 3 7 7 7 5 .

Using the proposed design constraints, the feasible set

S4ðW;UÞis composed of 14 different transmit schemes:

We remark thatðG31;G31ÞandðG22;G22Þare two schemes

having the same number of transmitted data streams and spreading codes but differing in their allocation pattern to the transmit antennas. Let us consider now two schemesðG31;G211ÞandðG22;G211Þ. Both transmit two

data streams over four transmit antennas using three spreading codes.

Let us comment scheme ðG31;G211Þ. It transmits the

first data stream over three transmit antennas using two different spreading codes (one is reused at two transmit antennas) and the second data stream is transmitted by a single antenna using a single spreading code. On the other hand, the schemeðG22;G211Þtransmits both data streams

by assigning two transmit antennas to each one. The first data stream is associated with a single spreading code while the second one uses two different spreading codes. We can distinguish six spectral efficiency levels in

S4ðW;UÞ, which are SF¼1; 14; 13; 12; 23and34bits per channel

use (assuming QPSK modulation), and we have four schemes with SF¼1; two schemes with SF¼1₂; and one scheme with SF¼1

4; 13or34.

It is worth noting thatnot allthe pairwise combina-tions of generating arrays is a feasible transmit scheme ensuring blind symbol recovery. For instance, the scheme

ðW;UÞ ¼ ðG22;G31Þis not feasible. In this scheme, the first

spreading code is reused by the twodifferentdata streams, inducing a coupling between different data streams. It is shown in [20] that such a coupling leaves rotational freedom within the antenna subset associated with the second data stream, thus affecting uniqueness ofS(only

the first data stream can be uniquely recovered). There-fore, care must be taken when deriving the MIMO-CDMA transmit schemes from the generating arrays.

4. Receiver algorithm

In this paper, we are interested to blindly detect the transmitted data streams of each user separately without the knowledge about (ora prioriestimate of) the MIMO downlink channel. We consider a simple blind receiver composed of two processing stages. The first one

attempts to mitigate multiuser interference using code-matched filtering/despreading. The second stage works on the resulting signal tensor after the code-matched filtering/despreading operation, and performs blind symbol recovery separately for each user, by means of the alternating LS algorithm. We assume that the q-th user has perfect knowledge of its code matrixCðqÞ.

Requirement 1. Multiuser interference elimination by each user requires

CðqÞTCðq0Þ_¼ IJ if q¼q 0_;

0 if qaq0; (

fq;q0_{g ¼1;. . .;Q.}

Note that this condition is equivalent to CTC¼IJQ,

which arrives whenPXJQ.

Remark 2. If Requirement 1 is not satisfied, multiuser detection can alternatively be used at the receiver without despreading, as in [9,20]. This is the case when ICI is present and the code matrixCis unknown and nonorthogonal. The price to pay when multiuser detection is used is an increased receiver complexity.

After the despreading operation by each user, we have the following interference-free model:

YðqÞ₁

.. .

YðqÞ_M

2 6 6 6 4 3 7 7 7 5 ¼

XðqÞ₁

.. .

XðqÞ_M

2 6 6 6 4 3 7 7 7 5

FðqÞ; FðqÞ¼ ½CðqÞ₂CPJ_.

From (18), the resulting (single-user) received signal model can be written, for theq-th user, as

S4ðW;UÞ ¼ ðg4;g4Þ

|fflfflfflffl{zfflfflfflffl}

J¼1

;ðg₄;G31Þ;ðg4;G22Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

J¼2

;ðg₄;G211Þ

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

J¼3

;ðg₄;G1111Þ

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} J¼4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} R¼1 8 > > > > > < > > > > > :

ðG31;G31Þ;ðG22;G22Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

J¼2

;ðG31;G211Þ;ðG22;G211Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

J¼3

;ðG31;G1111Þ;ðG22;G1111Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

J¼4

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

R¼2

ðG211;G211Þ

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

J¼3

;ðG211;G1111Þ

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

J¼4

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

R¼3

;ðG1111;G1111Þ

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} R¼J¼4 9 > > > > > = > > > > > ; .

YðqÞ_m¼SðqÞWDmðHðqÞÞUT2CNJ. (30)

We can obtain the two other following slice representa-tions for theq-th user interference-free signal:

YðqÞ

n¼UDnðSðqÞWÞHðqÞT2CJM,

YðqÞ_j ¼HðqÞDjðUÞWTSðqÞT2CMN. (31)

By concatenating column-wise each set of slicesfYðqÞ_jgand

fYðqÞ_ng, we obtain

YðqÞ₁ ¼

YðqÞ₁

.. .

YðqÞ_J

2

6 6 6 4

3

7 7 7 5

¼

HðqÞ_D

1ðUÞ

.. .

HðqÞDJðUÞ

2

6 6 6 4

3

7 7 7 5 WT

|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

Z1ðHðqÞÞ2CMJR

SðqÞT¼ ðU}HðqÞÞWTSðqÞT,

(32)

YðqÞ₂ ¼

YðqÞ₁

.. .

YðqÞ_N

2

6 6 6 4

3

7 7 7 5

¼

UD1ðSðqÞWÞ

.. .

UDNðSðqÞWÞ

2

6 6 4

3

7 7 5

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Z2ðSðqÞÞ2CJNK

HðqÞT_{¼ ðS}ðqÞ

W}UÞHðqÞT_.

(33)

Requirement 2. Assuming JX2, the identifiability of the symbol and channel matrices SðqÞ andHðqÞ in the LS sense requires thatZ1ðHðqÞÞandZ2ðSðqÞÞbe full column-rank,and we must have MJXR and JNXK.

Remark 3. The full rank assumption is reasonable in practice, if: (i) the entries of the channel matrix are assumed to be independently and randomly drawn from an absolutely continuous distribution, and if (ii) ‘‘persistence of excitation’’ of the transmitted symbols holds. The sufficient condition of[9], which involves the concept ofk-rank, does not apply to model (30), since the

k-rank of W and U are always equal to one, except for allocation schemes corresponding to a full spatial multi-plexingðW¼IKÞand/or a full code multiplexingðU¼IKÞ.

The derivation of a tighter sufficient condition for blind symbol recovery in our case is under investigation.

The second stage of the receiver is concerned with blind symbol recovery based on the interference-free received signal. The alternating least squares (ALS) algorithm, which is the classical solution for estimating the compo-nent matrices of a tensor model, is considered here. In our case, by using the knowledge of_Wand_U, this algorithm consists in alternated estimations of the symbolðSðqÞÞand channelðHðqÞÞmatrices. At each step of this algorithm, one component matrix is estimated in the LS sense, while the other is fixed to its value obtained at the previous step. At the beginning of the algorithm, _H^

ð0Þ is randomly

initi-alized. At thet-th iteration we have

^

SðqÞT_ðtÞ ¼ ½Z1ðH^ ðqÞ ðt1ÞÞyY~

ðqÞ

1 ; H^

ðqÞT ðtÞ ¼ ½Z2ðS^

ðqÞ ðtÞÞyY~

ðqÞ

2 ,

where ðÞy denotes the matrix pseudo-inverse, _Y~ðqÞ

j ¼

YðqÞ_j þVðqÞ_j ,j¼1;2, is the noisy version ofYðqÞ_j , the additive white gaussian noise matrix being represented byVðqÞ_j , and

fYðqÞ₁ ;Z1ðHðqÞðtÞÞg and fY ðqÞ 2 ;Z2ðS^

ðqÞ

ðtÞÞg are defined in (32) and

(33), respectively. Convergence at the t-th iteration is decided when the error between the received signal tensor and its reconstructed version from the estimated component matrices does not change significantly be-tween iterations t and tþ1. The inherent scaling ambiguity factor over each column ofSðqÞ is eliminated by assuming that the first transmitted symbol of each data stream is equal to one, i.e.SðqÞ₁ ¼ ½1 1 1.

5. Simulation results

The bit-error-rate (BER) performance of the ALS-based blind receiver is evaluated from computer simulations, for several transmit configurations. A total of 1000 Monte Carlo runs is considered for performance evaluation. The signal-to-noise ratio (SNR) at the q-th receiver is defined as

SNRðqÞ_dB¼10 log10

kYðqÞ₁ k2_F

kVðqÞ₁ k2_F !

,

where VðqÞ₁ 2CMJN _{is an additive white Gaussian noise} matrix. At each run, the channel coefficients are drawn from an i.i.d. Gaussian generator. Unless otherwise stated, the transmitted symbols are drawn from a pseudo-random QPSK sequence at each run. We recall that user-specific OVSF spreading codes [23] with appropriate lengths P are used throughout the simulations. The number N of symbols and the spreading factor P are shown on the top of each figure. All the plotted BER results are averaged over all the data streams of all the users.

Performance of different STS schemes: In the first experiment, we consider a MIMO-CDMA downlink system with Q ¼4 users, K¼4 transmit antennas and M¼2 receive antennas. We consider three STS schemes with different stream-to-antenna allocation matrices fWðiÞg,

i¼1;2;3:

Wð1Þ¼G1111; Wð2Þ¼G211; Wð3Þ¼G22.

The code-to-antenna allocation matrix is the same for all the schemes and is given by

U¼G1111.

The parameters fJ_rg and fKrg, r¼1;. . .;R, characterizing these allocation matrices are summarized inTable 1. In scheme 1, there is no reuse of data streams and spreading codes across the transmit antennas. In scheme 2, the first data stream is spread twice across two antennas, while in scheme 3 both data streams are spread in the spatial domain across two antennas. According to Fig. 3, the performance improves when going from schemes 1 to 3.

Table 1

Transmit parametersfJrgandfKrgused in the STS schemes of Fig. 3

Scheme 1 Scheme 2 Scheme 3

fJ1;K1g f1;1g f2;2g f2;2g

fJ2;K2g f1;1g f1;1g f2;2g

fJ3;K3g f1;1g f1;1g –

The performance gain of scheme 2 over scheme 1 is certainly due to the spatial spreading of the first data stream contributing to an improved average performance. Scheme 3 offers the best performance by spatially spreading each data stream across two antennas. How-ever, it is important to note that such a performance improvement obtained by decreasing the number of data streams (and, consequently, by increasing the transmit diversity) comes at the expense of a reduction in spectral efficiency. For instance, the spectral efficiency is reduced by a factor of two when going from schemes 1–3. Recalling that QPSK modulation is used, we have SF¼2, 1.5 and 1 for schemes 1, 2 and 3, respectively.

In the second experiment, we consider Q¼4 users,

K¼6 transmit antennas and M¼2 receive antennas. Note that this is a more challenging system configuration, where the number of receive antennas is much smaller than the number of transmit antennas/users. For this system configuration, we simulate four STS schemes with different pairs of allocation matrices f_WðiÞ;UðiÞg,

i¼1;. . .;4:

fWð1Þ;Uð1Þg ¼ fG2211;G2211g,

fWð2Þ;Uð2Þg ¼ fG2211;G111111g,

f_Wð3Þ;Uð3Þg ¼ fG222;G2211g,

f_Wð4Þ;Uð4Þg ¼ fG222;G111111g.

The parametersfJ_rgandfKrg,r¼1;. . .;R, associated with

these allocation matrices are given inTable 2. We should note that the four schemes differ in the number of per-user data streams and spreading codes allocated for transmission. For schemes 1 and 2 the number of data streams per user isR¼4. On the other hand, for schemes

3 and 4 we have R¼3. Concerning the number of spreading codes/spreading factor, we haveJ¼4 andP¼

16 for schemes 1 and 3, andJ¼6 andP¼32 for schemes 2 and 4. The results are displayed in Fig. 4. Note that scheme 3 outperforms scheme 1. This performance gain can be justified as follows. FromTable 2, we can see that scheme 1 transmits four data streams over six antennas. Note that only the first and second data streams are spread across two transmit antennas ðK1¼K2¼2Þ, the

third and fourth data streams being transmitted by a single antenna. On the other hand, scheme 3 transmits three data streams over six antennas, i.e. all the three data streams are spread across two antennas ðK1¼ K2¼K3¼2Þ, thus benefiting from an average transmit

diversity gain of order two. The performance gain of scheme 3 w.r.t. scheme 1 is also related to the receiver capability in separating the transmitted signals, which is facilitated when fewer data streams are transmitted. The same comments are valid when comparing schemes 2 and 4. The difference is that these schemes useJ¼6 codes for transmission, while schemes 1 and 3 useJ¼4 codes only. In fact, the number of used codes explains the better 0 3 6 9 12 15 18 21 24

10−5

10−4

10−3 10−2

10−1

100

SNR (dB)

BER

Q = 4, K = 4, M = 2, N = 5, P = 16

(R, J) = (4, 4) (R, J) = (3, 4) (R, J) = (2, 4)

Fig. 3.Performance of different STS schemes forQ¼4,K¼4 andM¼2.

Table 2

Transmit parametersfJrgandfKrgused in the STS schemes of Fig. 4

Scheme 1 Scheme 2 Scheme 3 Scheme 4

fJ1;K1g f1;2g f2;2g f1;2g f2;2g

fJ2;K2g f1;2g f2;2g f1;2g f2;2g

fJ3;K3g f1;1g f1;1g f2;2g f2;2g

fJ4;K4g f1;1g f1;1g – –

performance of scheme 2 w.r.t. scheme 1, and of scheme 4 w.r.t. scheme 3 due to a higher code diversity for a fixed number of data streams (R¼4 or 3).

Comparison with the ZF-MUD receiver: In order to provide a performance reference for the proposed STS model using the ALS algorithm, we have simulated the Zero Forcing MUltiuser Detection (ZF-MUD) receiver, where all the data streams of all the users are jointly detected. The ZF-MUD receiver assumes perfect knowl-edge of all user channels, in contrast to the ALS-based receiver which performs a channel-blind detection. Using our notation, the ZF-MUD receiver consists in a single-step estimation of the symbol matrix as follows:

ST¼

HðqÞXD1ðCUÞW T

.. .

HðqÞXDPðCUÞW T

2

6 6 6 6 4

3

7 7 7 7 5

y XðqÞ₁

.. .

XðqÞ_P

2

6 6 6 6 4

3

7 7 7 7 5

¼ ½ðCU HðqÞXÞWTy_XðqÞ

1 .

Two STS configurations are chosen, in the first one we have ðR;J;KÞ ¼ ð3;4;4Þ with allocation matrices

fWð1Þ;Uð1Þg ¼ fG211;G1111g. In the second one, we have

ðR;J;KÞ ¼ ð2;3;3Þ with allocation matrices fWð2Þ;Uð2Þg ¼ fG21;G111g. Note that these system configurations are

similar w.r.t. the transmission of the first and second data streams (fJ₁;K1g ¼ f2;2g and fJ2;K2g ¼ f1;1g for both

schemes). The difference is that the first system config-uration has an additional transmit antenna which is used to transmit a third data stream. According toFig. 5, there is a performance gap between ALS-based receiver and the

ZF-MUD receiver in both configurations. Considering, for instance, the first configuration, we can observe a SNR gap around 5 dB between both receivers for a BER¼102. This is an expected result, since the proposed algorithm works with no knowledge about the channel while ZF-MUD assumes perfect channel knowledge. Note also that the second configuration outperforms the first one. In fact, the second configuration transmits less data streams than the first one, thus improving the signal separation performance for both receivers.

Comparison with competing STS schemes: In [5], a transmit diversity STS scheme was proposed for the downlink of a DS-CDMA system. For two transmit antennas and one receive antenna, it achieves the same spatial diversity gain as the Alamouti code[26]with the advantage that no extra spreading codes are required. Here, we compare this classical scheme with the proposed one considering a single-user transmissionðQ ¼1Þwith

R¼2,M¼K¼2 andP¼2. Orthogonal Walsh–Hadamard codes were used in both schemes. It should be noted that perfect channel knowledge is assumed for the STS scheme of[5], which leads to the best performance this scheme can achieve. In contrast, the proposed STS scheme uses ALS-based blind detection. The results are depicted in Fig. 6. Both schemes provide very close performance. Note that the gap is around 1 dB for a BER¼102_.

We now compare the proposed STS scheme with the multilayered STS scheme of[6]. In this scheme, the serial input stream is divided into multiple data streams, or ‘‘layers’’, each one being spread by a unique STS sequence. At the receiver, a linear MUD receiver is used to recover all the layers under the assumption of perfect channel knowledge. Recall that the proposed STS scheme assumes

0 5 10 15 20 25

10−6

10−5

10−4

10−3

10−2 10−1

100

SNR (dB)

BER

Q = 4, K = 6, M = 2, N = 10

(R,J) = (4,4), P = 16 (R,J) = (4,6), P = 32 (R,J) = (3,4), P = 16 (R,J) = (3,6), P = 32

a channel-blind detection. The fixed system parameters are K¼4, Q¼J¼4 and R¼1. In order to keep the spectral efficiency at the same value, we choose P¼16 and 16-QAM for the proposed STS scheme, while P¼8 and 4-QAM are used in the STS scheme of[6]. The results are shown inFig. 7 consideringM¼2 receive antennas. We can see that the proposed scheme significantly

outperforms the multilayered STS scheme for SNR values higher than 15 dB.

6. Conclusion

In this paper, we have presented a new constrained tensor model for the downlink of a MIMO-CDMA system

0 3 6 9

10−5

10−4 10−3

10−2

10−1 100

SNR (dB)

BER

Q = 1, K = 2, M = 2, N = 5, P = 2

Proposed scheme (R,J) = (1,2) STS scheme of [5]

Fig. 6.Performance comparison between the proposed STS scheme and the STS scheme of [5].

0 3 6 9 12 15 18

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BER

Q = 2, M = 3, N = 10, P = 8

ALS, (R,J,K) = (3,4,4) ALS, (R,J,K) = (2,3,3) ZF−MUD, (R,J,K) = (3,4,4) ZF−MUD, (R,J,K) = (2,3,3)

Fig. 5.ALS-based receiver (channel-blind) vs. ZF-MUD receiver (channel knowledge). A.L.F. de Almeida et al. / Signal Processing 88 (2008) 2403–2416

with multiuser spatial multiplexing. We have proposed a complete parametrization for the stream and code alloca-tion matrices along with new design constraints. Based on these design constraints, a codebook of feasible STS schemes with guaranteed blind symbol recovery can be derived for a given number of data streams, spreading codes and transmit antennas. We have presented some design examples for 2, 3 and 4 transmit antennas. Bit-error-rate performance of some STS schemes has been evaluated using a two-stage receiver. Assuming that the spreading codes are known at the receiver, multiuser interferences are first eliminated and then the users’ symbols and channels are jointly and blindly estimated using the ALS algorithm. We have also compared our STS schemes with some competing schemes. Practical per-spectives of this work include variable per-user STS structures (i.e. users with different allocation matrices). We are currently investigating performance-oriented de-signs for the allocation matrices to cope with the user selection problem.

References

[1] G.J. Foschini, M.J. Gans, On limits of wireless communications when using multiple antennas, Wireless Pers. Comm. 6 (3) (1998) 311–335.

[2] H. Huang, H. Viswanathan, G.J. Foschini, Multiple antennas in cellular CDMA systems: transmission, detection, and spectral efficiency, IEEE Trans. Wireless Comm. 1 (3) (2002) 383–392. [3] S. Sfar, R.D. Murch, K.B. Letaief, Layered spacetime multiuser

detection over wireless uplink systems, IEEE Trans. Wireless Comm. 2 (4) (2003) 653–668.

[4] L. Mailaender, Linear MIMO equalization for CDMA downlink signals with code reuse, IEEE Trans. Wireless Comm. 4 (5) (2005) 2423–2434.

[5] B. Hochwald, T.L. Marzetta, C.B. Papadias, A transmitter diversity scheme for wideband CDMA systems based on space–time spread-ing, IEEE J. Selected Areas Comm. 19 (1) (2001) 48–60.

[6] B.K. Ng, E. Sousa, Space–time spreading multilayered CDMA system, in: Proceedings of the GLOBECOM, vol. 3, 2000, pp. 1854–1858. [7] R. Doostnejad, T.J. Lim, E. Sousa, Space–time spreading codes for a

multiuser MIMO system, in: Proceedings of the 36th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA, 2002, pp. 1374–1378.

[8] R. Doostnejad, T.J. Lim, E. Sousa, Space–time multiplexing for MIMO multiuser downlink channels, IEEE Trans. Wireless Comm. 5 (7) (2006) 1726–1734.

[9] N.D. Sidiropoulos, G.B. Giannakis, R. Bro, Blind PARAFAC receivers for DS-CDMA systems, IEEE Trans. Signal Process. 48 (3) (2000) 810–822.

[10] R.A. Harshman, Foundations of the PARAFAC procedure: model and conditions for an ‘‘explanatory’’ multi-mode factor analysis, UCLA Working Papers in Phonetics, vol. 16, 1970, pp. 1–84.

[11] 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.

[12] 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.

[13] A.L.F. de Almeida, G. Favier, J.C.M. Mota, PARAFAC models for wireless communication systems, in: Proceedings of the Interna-tional Conference on Physics in Signal and Image Processing (PSIP), Toulouse, France, 2005.

[14] A.L.F. de Almeida, G. Favier, J.C.M. Mota, PARAFAC-based unified tensor modeling for wireless communication systems with applica-tion to blind multiuser equalizaapplica-tion, Signal Processing 87 (2) (2007) 337–351.

[15] D. Nion, L. De Lathauwer, A block factor analysis based receiver for blind multi-user access in wireless communications, in: Proceed-ings of the ICASSP, Toulouse, France, 2006.

[16] N.D. Sidiropoulos, R. Budampati, Khatri–Rao space–time codes, IEEE Trans. Signal Process. 50 (10) (2002) 2377–2388.

[17] A. de Baynast, L. De Lathauwer, B. Aazhang, Blind PARAFAC receivers for multiple access-multiple antenna systems, in: Proceedings of the VTC Fall, Orlando, USA, 2003.

[18] A.L.F. de Almeida, G. Favier, J.C.M. Mota, Space–time multiplexing codes: a tensor modeling approach, in: IEEE International

Work-0 5 10 15 20 25

10−5 10−4 10−3

10−2

10−1

100

SNR (dB)

BER

K = 4, M = 2, N = 10

Proposed scheme STS scheme of [6]

shop on Signal Processing Advances in Wireless Communications (SPAWC), Cannes, France, 2006.

[19] A.L.F. de Almeida, G. Favier, J.C.M. Mota, Tensor-based space–time multiplexing codes for MIMO-OFDM systems with blind detection, in: IEEE International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC), Helsinki, Finland, 2006. [20] A.L.F. de Almeida, G. Favier, J.C.M. Mota, Constrained tensor

modeling approach to blind multiple-antenna CDMA schemes, IEEE Trans. Signal Process., 2008, to appear.

[21] I. Chih-Lin, R.D. Gitlin, Multi-code CDMA wireless personal communication networks, in: Proceedings of the International Conference on Communications, ICC, Seattle, WA, 1995, pp. 1060–1064.

[22] S.J. Lee, H.W. Lee, D.K. Sung, Capacities of single-code and multi-code DS-CDMA systems accommodating multiclass services, IEEE Trans. Vehic. Technol. 48 (2) (1999) 376–384.

[23] Third generation partnership project, Universal mobile telecommu-nications system (UMTS); spreading and modulation (FDD). Technical Report ETSI TS 25.213 V6.2.0.

[24] R.W. Heath, D.J. Love, Multimode antenna selection for spatial multiplexing systems with linear receivers, IEEE Trans. Signal Process. 53 (8) (2005) 3042–3056.

[25] D.J. Love, R.W. Heath, Multimode precoding for MIMO wireless systems, IEEE Trans. Signal Process. 53 (10) (2005) 3674–3687. [26] S. Alamouti, A simple transmit diversity technique for wireless

commu-nications, IEEE J. Selected Areas Comm. 16 (8) (1998) 1451–1458. A.L.F. de Almeida et al. / Signal Processing 88 (2008) 2403–2416