Blind Channel Estimation for STBC Systems Using Higher-Order Statistics

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 2, FEBRUARY 2011 495

Blind Channel Estimation for STBC SystemsUsing Higher-Order Statistics

Vincent Choqueuse, Member, IEEE , Ali Mansour, Senior Member, IEEE , Gilles Burel, Senior Member, IEEE ,Ludovic Collin, and Kof Yao, Member, IEEE

Abstract —This paper describes a new blind channel estimationalgorithm for Space-Time Block Coded (STBC) systems. Theproposed method exploits the statistical independence of sourcesbefore space-time encoding. The channel matrix is estimatedby minimizing a kurtosis-based cost function after Zero-Forcingequalization. In contrast to subspace or Second-Order Statistics(SOS) approaches, the proposed method is more general since itcan be employed for the general class of linear STBCs includingSpatial Multiplexing, Orthogonal, quasi-Orthogonal and Non-Orthogonal STBCs. Furthermore, unlike other approaches, themethod does not require any modi cation of the transmitterand, consequently, is well-suited for non-cooperative context. Nu-merical examples corroborate the perfo rmance of the proposedalgorithm.

Index Terms —MIMO, space time cod ing, channel estimation,independent component analysis, higher -order statistics.

I. INTRODUCTIO N

S PACE-TIME Block Coding is a set of practical signaldesign techniques aimed at appro aching the information

theoretic capacity limit of Multiple- Input Multiple-Output(MIMO) channels. Since the pioneer work of Alamouti [1],space-time coding has been a fast gro wing eld of research.In the last decade, numerous codin g schemes have beenproposed. These include orthogonal (O STBCs) [1]–[3], quasi-orthogonal (QOSTBCs) [4], [5] and non-orthogonal STBCs(NOSTBCs) [6]. At the receiver side, the decoding is achievedby a space-time equalizer. Most space-time equalizers requireChannel State Information (CSI). This information is usuallyobtained through training based techniques at the expense of the bandwidth ef ciency. On the other hand, the differentialschemes proposed in [7]–[10], which do not require CSI, incura penalty in performance of at least 3dB as compared to thecoherent Maximum-Likelihood (ML) receiver. The drawbacksof training-based approaches and differential schemes havemotivated an increasing interest in the development of blind

Manuscript received October 23, 2009; revised March 2, 2010 and October1, 2010; accepted October 18, 2010. The associate editor coordinating thereview of this paper and approving it for publication was L. Deneire.

V. Choqueuse was with the Lab-STICC, UEB, Université de Brest; UMRCNRS 3192, CS 93837, 29238 Brest cedex 3, France. He is now with LBMS,EA 4325, same address (e-mail: [email protected]).

A. Mansour is with the Department of Electrical and Computer Engineer-ing, Curtin University, Perth, Australia (e-mail: [email protected]).

G. Burel, L. Collin, and K. Yao are with the Lab-STICC, UEB, Universitéde Brest; UMR CNRS 3192, CS 93837, 29238 Brest cedex 3, France (e-mail:{Gilles.Burel, Ludovic.Collin, Kof -Clement.Yao}@univ-brest.fr).

The authors would like to thank Janet Leschaeve for her help in checkingEnglish usage, as well as the associate editor and the anonymous reviewersfor their useful comments.

Digital Object Identi er 10.1109/TWC.2010.112310.091576

channel estimation algorithms for STBC systems. Develop-ment of blind receivers also has applications in militarycommunication system when the transmitted symbols have tobe estimated in a blind fashion.

Blind channel estimation algorithms based on Maximum-Likelihood (ML) have been proposed in [11], [12]. Despitetheir high performances, the computational costs of the ML-based methods become prohibitive for high-order modulations.In the case of BPSK or QPSK constellations, the blind-MLdetection can be simpli ed to a Boolean Quadratic Program(BQP) [13]. For more general settings, iterative procedurecan be employe d to avoid the computational complexity of the ML approach . These include the Cyclic ML [12] and theExpectation-Max imisation (EM) [14], [15] algorithms. How-ever, these iterati ve methods require a careful initialization of the channel and/ or symbols. In particular, a poor initializationcan strongly affe ct the Symbol-Error Rate (SER) performance.

To avoid the drawbacks of ML-based channel estimationalgorithms, sever al authors have investigated the use of sub-space [16], [17] or Second-Order Statistics (SOS) [18]–[21]approaches. How ever, excluding some speci c low-rate codes,these approache s fail to extract the channel in a full-blindcontext [16]–[22 ]. Several approach have been proposed inliterature to solve this problem, including the transmission of ashort training sequence [16], [17] or the use of precoders [18],[20], [21]. However, these semi-blind methods cannot beemployed in a non-cooperative scenario since they requiremodi cation of the transmitter.

One solution to avoid the limitations of SOS and subspacealgorithms is to exploit Higher-Order Statistics (HOS). Thisapproach is usually called Independent Component Analysis(ICA) [23]. ICA was originally developed for non-codedsystems. Recently several authors have investigated its ex-tension to STBC communications [24]–[31]. Nevertheless,these extensions have several limitations and drawbacks. Inparticular, the algorithms [24]–[29] are limited to a sub-classof Orthogonal STBCs and their extension to the general classof STBCs is far from trivial. On the other hand, the methods[30], [31] do not take into account the speci c structure of theSTBC.

Despite this rich literature, none of the previous algorithmsis able to estimate the channel matrix for general STBCs with-out modi cation of the transmitter (pilot sequence, precoding).In this paper, an original algorithm is proposed which is well-suited to the general class of linear STBCs whatever the code-rate and/or the modulation. The channel matrix is estimated

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496 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 2, FEBRUARY 2011

by minimizing a kurtosis-based cost function. In contrast withclassical ICA algorithms, the cost-function is computed fromthe Zero-Forcing (ZF) space-time equalized symbols. Ourproposed method has low-complexity and does not requireany channel initialization, the use of pilot sequence and/orknowledge of modulation. Furthermore, our method does notrequire any modi cation of the transmitter and, consequently,can be employed in a non-cooperative scenario.

This paper is organized as follows. The signal models andthe assumptions are presented in section II. The kurtosis-basedcost function is described in section III and the minimizationalgorithm is described in section IV. The set of the remainingambiguities after channel estimation is provided in section V.Finally, the simulation results are presented in section VI.

II . S IGNAL MODELS AND ASSUMPTIONS

Hereinafter, bold upper case letters denote matrices, e.g.,X; bold lower case letters stand for column vectors, e.g., x,and lower case letters represent scalars. Superscripts (⋅)

T and

(⋅)H

denote transpose and Hermitian, respectively. Symbol = √−1 is the imaginary unit, (⋅)∗ corresponds to the

complex conjugate and the operatorsℜ(⋅) and

ℑ (⋅) denotethe real and imaginary parts, respecti vely. The symbol

⊗is

the Kronecker product, trace (⋅) is the trace function andℰ [⋅]is the expectation operator. The × matrices I and 0

correspond to the identity and zero ma trices, respectively. Theunit vector , e( ) , is an -dimensional row vector with "1" inits ℎ component and zero elsewhere i.e.

ℎ

↓

[ ]e( ) = 0

⋅ ⋅ ⋅

0 1 0

⋅ ⋅ ⋅

0(1)

The elementary matrix , E ( ) , is an × matrix which is "1"in the ℎ row and ℎ column and ze ro elsewhere i.e.

E ( ) =

ℎ

↓⎡⎢⎢⎢⎢⎢

⎣

⎤⎥⎥⎥⎥⎥

⎦

...0

⋅ ⋅ ⋅ 0 1 0 . . . ← ℎ

0...

(2)

A. Transmitted signal model

Let us consider a linear STBC that transmits symbolsduring time slots through antennae. The space-time blockencoder generates an × block matrix from a block of symbols s = [ 1 , ⋅ ⋅ ⋅, ]T . The block matrix, (s), can beexpressed under the general form [32]

(s) = ∑=1(Aℜ( ) + A + ℑ ( )) (3)

where the × matrices A are the space-time codingmatrices.

B. Signal model of received samples

Let us consider a receiver composed of antennae. Letus also assume a quasi-static frequency- at channel modelledby an × complex matrix H . The ℎ received block,denoted by the × matrix Y , is given by [32]

Y = H (s ) + B (4)

where the × matrix B = [b (1) , ⋅ ⋅ ⋅,b ( )] refers to theadditive noise and b ( ) is a -dimensional column vector.

The aim of this study is to estimate H from the received datablocks, Y , under the following assumptions:

· AS1) the × channel matrix, H , is of full-columnrank. Furthermore, the number of receiver antennae isstrictly greater than the number of transmitters, i.e. >

.· AS2) the noise vector is both spatially and temporally

white with a variance of 2 per complex dimension. Inparticular, it implies that:

ℰ [B BH ] = 2 I (5)

· AS3) the t ransmitted symbols, s , are non gaussian,independent and identically distributed (i.i.d).

· AS4) the average transmit power on each antenna isnormalized to unity which also implies that:

ℰ [ (s ) H (s )] = I . (6)

· AS5) the sp ace-time code is known at the receiver side.

Assumptions AS1), AS2) and AS3) are widely used andAS4) is respecte d for most STBCs 1 . Moreover in many sce-

narios, the space -time code is usually assumed to be known,otherwise, it can be estimated with a blind STBC recognitionalgorithm [33]–[ 36]. It should be noted that condition AS5)also implies that , , and A are known at the receiverside.

III. C HANNEL ESTIMATION STRATEGY

In this section, a new blind channel estimation strategybased on HOS is proposed. The method is composed of twosteps which are detailed in the subsections III-A and III-B,respectively.

A. Step 1: Data Whitening

In the preprocessing step, the channel is estimated up to aunitary matrix through the use of SOS. By using assumptionsAS1), AS2), AS3) and AS4), the × covariance matrixof the noiseless transmitted signals R = ℰ [Y YH ] − 2 Ican be expressed as

R = Hℰ [ (s ) H (s )]H H

= HH H . (7)

Under assumption AS1), the rank of the symmetric matrix Ris equal to . Therefore, R can be decomposed as follows:

1It should be noted that if ℰ [ (s ) H (s )] = I , the scaling factorcan be absorbed into the channel matrix H without loss of generality

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CHOQUEUSE et al. : BLIND CHANNEL ESTIMATION FOR STBC SYSTEMS USING HIGHER-ORDER STATISTICS 497

R = UΛUH , where U is an × matrix satisfying UH U =I and Λ is an × diagonal matrix containing real entries.From (7), it follows that the channel matrix H can be expressedas

H =1√ UΛ

12 W H (8)

where W is an × full rank unitary matrix.

After the preprocessing step, the determination of the matrixH reduces to the determination of the × unitary matrixW . To determine W , let us de ne the × whitened datablock, X , as

X = √ Λ− 12 UH Y . (9)

B. Determining the unitary matrix W

1) The Zero-Forcing receiver: In this paragraph, a linearZero-Forcing (ZF) decoder is expressed in terms of the ×whitened data block X . Using (4) and (8), it can be shownthat

X = W H (s ) + √ Λ− 12 UH B

N

(10)

where the × matrix N is a multidimensional zero-meanGaussian signal. Let us de ne, s , the 2 real-valued columnvector obtained by concatenating the r eal and imaginary partof s i.e.

s ≜

⎡⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎣

ℜ( ( )1 )

...

ℜ( ( ) )

ℑ

( ( )1 )

...

ℑ ( ( ) )

⎤⎥⎥⎥⎥

⎥⎥⎥⎥⎥⎦

. (11)

Let us also introduce the vectorization operator, vec {.}, ob-tained by stacking all columns of a matrix on top of eachother. Using the property of the vec {.}operator in equations(10) and (3) [37], it can be shown that

x = WG s + n (12)

where the 2 -dimensional column vectors x and n , the2 × 2 matrix W and the 2 × 2 matrix G arerespectively de ned by

x ≜ ℜ(vec{XH})

ℑ (vec{XH})(13)

n ≜ ℜ(vec{NH})

ℑ (vec{NH})(14)

G ≜ ℜ(vec{AH1 }) ⋅ ⋅ ⋅ ℜ(vec{AH

2 })

ℑ (vec{AH1 }) ⋅ ⋅ ⋅ ℑ(vec{AH

2 })(15)

W ≜ ℜ(W T )⊗I −ℑ (W T )⊗I

ℑ (W T )⊗Iℜ(W T )⊗I

. (16)

As W is a unitary matrix, it is demonstrated in appendix Athat W is orthogonal i.e. W T W = I2 . If the unitary matrixW is known at the receiver side, the transmitted symbols canbe recovered with a linear Zero-Forcing (ZF) equalizer. The

ZF equalizer computes an inverse matrix to compensate thecombined effects of the channel and space-time coding i.e.

⎡⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎣

ℜ ( ˆ( )1 )

...ℜ ( ˆ

( ) )ℑ (

ˆ( )1 )

..

.ℑ ( ˆ

( ) )

⎤⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎦

= G † [ ℜ (W ) ⊗ I ℑ (W ) ⊗ I−ℑ (W ) ⊗ I ℜ (W ) ⊗ I ]

W T

x

(17)

where the 2 ×2 matrix G † denotes the pseudo-inverse of G (G †G = I2 ) and ˆ

( ) is the ℎ estimated symbol of theℎ block. Using (17), ˆ

( ) can be expressed as

ˆ( ) = [e( ) e( ) ]G †W T x . (18)

In a blind context, the unitary matrix W is unknown at thereceiver side. To estimate W , this study exploits the statisticalindependence of the equalized symbols. More precisely, theunitary matrix W is estimated by maximizing the statisticalindependence of the Zero-Forcing equalized symbols, ˆ

( ) .2) Kurtosis-ba sed cost function: A simple approach to

maximize the sta tistical independence of ˆ( ) is to maximize

the nongaussiani ty of ˆ( ) [23]. One measure of nongaussian-

ity of a random variable is the (unnormalized) Kurtosis,[ ], which is de ned as

[ ] ≜

ℰ [∣∣4] −2(ℰ [∣∣

2])2 −ℰ [ ]ℰ [ ∗ ∗]. (19)

It follows that t he unitary matrix W can be estimated bymaximizing the f unction =1 ∣ [ ˆ

( ) ]∣where∣.∣denotes the

absolute value. I t should be noted that in the most practical

cases, the sign o f the kurtosis is assumed to be known andthe same for all the transmitted symbols. In particular, it isshown in referen ce [38] that the cumulant of most of the digitalmodulation (ASK , PSK and QAM) are negative. Therefore, anestimate of W , denoted ˆW , can be obtained as follows

ˆW :⎧⎨⎩

minW (W ) = ∑=1 ˆ

( )

subject to WW H = I

(20)

where (W ) is a real-valued cost function which dependson the × complex-valued matrix W . It should benoted that criterion (20) has already appeared in literature for

simpler channel estimation problems. In particular, it has beenemployed for classical ICA problems, where (s ) = s [39]–[42]. In our study, an extension to STBC systems is obtainedby applying criterion (20) on the Zero-Forcing space-timeequalized symbols ˆ

( ) in (18).

IV. A LGORITHM IMPLEMENTATION

In this section, the focus is on the minimization of the real-valued cost function : ℂ × →ℝ under the unitaryconstraint WW H = I . As no closed form solution exists,a Steepest-Descent (SD) approach is employed. To performa descent step, SD algorithm requires the computation of the gradient. The gradient expression has been provided inseveral studies for classical ICA problems [42], [43], however,

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its expression is no longer valid for STBC systems. In thesubsection IV.IV-A, the gradient expression is establishedfor STBC systems. Then, two constrained-minimization SDalgorithms are described in subsection IV.IV-B.

A. Expression of the gradient in the Euclidean space

In the Euclidean space, the gradient of the cost function

(W

) is the × matrix ΓW which is de

ned as [44]ΓW = (W )

W∗(21)

where:

(W )W∗

≜12

(W )

ℜ(W )+ (W )

ℑ (W ). (22)

Let us denote, , the element on the ℎ row and the ℎ

column of matrix W . Using the × elementary matrixE ( )

, ΓW can be expressed as

ΓW =

∑ =1

∑=1

12

E ( ) (W )

ℜ

(

)+ (W )

ℑ

(

).(23)

From (20), it follows that

ΓW = ∑=1 ∑=1 ∑=1

12

E ( ) ( [ ˆ( ) ]

ℜ ( )+

[ ˆ( ) ]

ℑ ( ))(24)

where ˆ is given by (18). By inter changing the order of derivative and expectation [45], the d erivative of the (unnor-malized) Kurtosis with respect to a com plex element is givenby

[ ˆ]= 2ℰ ∣

ˆ∣2

ˆ ˆ

∗

+

ˆ

∗ ˆ− 4ℰ [∣ ˆ∣

2

]ℰ ˆ∗

+ ˆ∗

ˆ− 2 ℰ [ ˆ

2 ]ℰ ˆ∗ ˆ

∗

+ ℰ [ ˆ2∗]ℰ ˆ .(25)

From (18), the × matrix ΓW can be expressed as

ΓW = ∑=1 ∑=1 ∑=1

E ( )(ℰ ∣ ˆ( )∣2

ˆ( ) q (2) + ˆ

∗( ) q (1) x

− 2ℰ ∣ ˆ( )∣2ℰ ˆ

( ) q (2) + ˆ∗( ) q (1) x

− ℰ [

ˆ2( ) ]ℰ

ˆ∗( ) q (2) x − ℰ [

ˆ2∗( ) ]ℰ

ˆ( ) q (1) x )

(26)where the 2 -dimensional column vector x is de ned in(13) and where the 2 -dimensional row vectors q (1)

andq (2)

are given respectively by

q (1) = [ e( ) e( ) ]G † E ( )⊗ I E ( )

⊗ I− E ( )

⊗ I E ( )⊗ I

(27)

q (2) = [ e( ) − e( ) ]G † E ( )⊗ I E ( )

⊗ I− E ( )

⊗ I E ( )⊗ I

.(28)

Remark 1: In practice, the signals are assumed to be er-godic; that means that the expectation operator

ℰ [⋅]in (26)

can be approximated by a time-average.

B. Constrained minimization algorithm

Several SD algorithms for the minimization of a real-valuedcost function under the unitary constraint have been proposedin literature. In this subsection, two algorithms are described.

For constrained-minimization, classical approaches solvethe optimization problem on the Euclidean space by usinggradient-based algorithms [23], [42], [46]. At each iterationstep, an update of W is performed in the direction of thenegative gradient. Then, a symmetric orthogonalizationis applied to restore the unitary constraint of W . Thistwo-step approach is described in the algorithm 1 for a xed step size 2 . The major drawback of the Euclidean SDis that it can lead to undesired suboptimal solutions [47], [48].

Algorithm. 1 Channel estimation for STBC systems usingclassical SD algorithm

1: compute R2: perform the eigenvalue decomposition R = UΛUH

3: compute the whitened data X with (9)4: initialize W randomly5: repeat6: set ← (W )7: compute t he gradient ΓW in the Euclidean space with

(26)8: update W ←W − ΓW

9: update W ←W (W H W )− 1/ 2

10: until − (W ) < where is a threshold11: compute ˆH with (8).

Recently, maj or improvements have been obtained by tak-ing into account the geometrical aspect of the optimizationproblem. Nonge odesic and geodesic approaches have beenproposed in [48] , [49]. Coupled with Armijo step size rule[50], these algorithms always converge to a local minimumif it is not initialized at a stationary point. In the followingequations, the geodesic SD algorithm [48] is chosen since ithas lower computational complexity than the nongeodesic one.The geodesic SD algorithm moves towards the SD gradientdirection,

∇W , in the Riemannian space. This direction canbe expressed as [48]

∇W = Γ W W H −WΓHW (29)

where ΓW is the gradient in the Euclidean space (see (26)).Then, the update rule is given by

W ←exp ( ∇W ) W (30)

where exp (⋅) =∞

=0 (⋅) / ! is the matrix exponential andcorresponds to the step size. Using the Armijo step size rule,the algorithm almost always converges to a local minimum.The geodesic SD algorithm with the Armijo step size rule isdescribed in the algorithm 2.

Figures 1 and 2 illustrate the convergence of algorithm 2 fora STBC system. The STBC system employs Alamouti coding

2As discussed in [47], line search optimization is not well-adapted forEuclidean SD with the projection method.

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Algorithm. 2 Channel estimation for STBC systems usinggeodesic SD algorithm with Armijo step size rule

1: compute R2: perform the eigenvalue decomposition R = UΛUH

3: compute the whitened data X with (9)4: initialize W randomly and set = 15: repeat6: compute the gradient ΓW in the Euclidean space with

(26)7: compute the direction

∇W in the Riemannian spacewith (29)

8: compute⟨∇W ,∇W⟩= 1

2ℜtrace(∇W∇

HW )

9: initialize Σ = exp(− ∇W ) and Υ = ΣΣ .10: while (W ) − (ΥW ) ≥ ⟨∇W ,∇W⟩

do11: set Σ = Υ , Υ = ΣΣ and ←212: end while13: while (W ) − (Σ W ) < 2⟨∇W ,∇W⟩

do14: set Σ = exp(− ∇W ) and ←1

215: end while16: update W

←Σ W

17: until⟨∇

W ,∇W⟩< where is a threshold

18: compute ˆH with (8).

0 1 2 3 4 5 6 7 8 9−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

Number of iterations

C o s t

f u n c t i o n

K [

s 1 (t )]

K [

s 2 (t )]

Cost Function: J (W )

Fig. 1: Cost function (W ) versus iteration.

( = = = 2 ) and QPSK modulation. The number of

transmitted blocks, the number of receiver antennae and theSignal to Noise Ratio (SNR) are equal to = 512 , = 4and 20dB, respectively, and the threshold is xed at = 10 − 5 .Figure 1 displays [ ˆ1], [ ˆ2 ] and (W ) with respect to theiteration number. The gure shows that the cost function isminimized after 9 iterations. The kurtosis [ ˆ1] and [ ˆ2]converge to −1 which is the kurtosis of QPSK modulation[38]. Figure 2 shows the constellation of the symbols ˆ1and ˆ2 in the complex plane before and after convergence.After convergence, it should be noted that the constellationof the equalized symbols is phase-rotated as compared to theQPSK constellation. However, as opposed to the classical ICAmodel, the phase rotation ambiguities of

ˆ1 and

ˆ2 are not

independent. The effect of the STBC structure on the channelambiguities is studied in the following section.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Q u a

d r a

t u r e

In−Phase

(a) After Whitening:ˆ

1

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Q u a

d r a

t u r e

In−Phase

(b) After Whitening: ˆ

2

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Q u a

d r a

t u r e

In−Phase

(c) Iteration 9:ˆ

1

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Q u a

d r a

t u r e

In−Phase

(d) Iteration 9:ˆ

2

Fig. 2: Constellation of ˆ1 and ˆ2

V. REMAINING AMBIGUITIES

For the classi cal ICA model, it is well known that thechannel can be estimated up to a permutation and phaserotation ambigui ties [23], [51]. For STBC systems, the pro-posed method re duces the number of channel ambiguities byexploiting the sp atio-temporal redundancy of the transmittedsymbols in (18).

Theorem 1: Let us consider three matrices M , P and Dwhere M is an

×unitary matrix, P is an

×permutation

matrix and D is an × diagonal matrix with entries of unitmodulus ( DD∗ = I ). If these matrices satisfy

M (s ) = (PDs ) (31)

for any s , then HM H is also a solution of the blind channelestimation problem.

Proof: From (4) and (31), one gets:

Y = H (s ) + B

= HM H M (s ) + B

= HM H (PDs ) + B . (32)

As the elements of s are i.i.d, the elements of the vectorPDs are also i.i.d. Therefore HM H is a solution of the blindchannel estimation problem.

Let us express the set Θ, which contains all the matrices Msatisfying (31), with respect to the coding matrices. Condition(31) can be described in a vector form as

ℜ(vec{ H (s )M H})

ℑ (vec{ H (s )M H})

k1

= ℜ(vec{ H (PDs )})

ℑ (vec{ H (PDs )})

k 2

.

(33)The 2 -dimensional column vector vec { H (s )M H}can be

expressed asvec{ H (s )M H}= ( M∗

⊗I ) vec{ H (s )} (34)

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TABLE I: Set of ambiguity matrices for different STBCs using = 2 , 3, 4 transmit antennae. The matrices M 1-M 10 arede ned in (40)-(49).

Number of Number of Design transmit symbols Code rate Set of ambiguity matrices

approach antennae per block / after channel estimation

Spatial Multiplex. Θ = {PD }Alamouti [1] 2 2 1 Θ = {M 1 ( ), M 2 ( )}OSTBC [2] 3 3 1/2 Θ = {± I3 }

OSTBC [32] 3 3 3/4 Θ = {± I3

}OSTBC [3] 4 3 3/4 Θ = {± I4 }OSTBC [2] 4 4 1/2 Θ = {± I4 , ± M 4 (0) , ± M 5 (0) , ± M 6 (0) }

QOSTBC [4] 4 4 1 Θ = {M 3 ( ), M4 ( ), M 5 ( ), M 6 ( )}NOSTBC [52] 4 4 1 Θ = {M 7 ( 1 , 2 ), M 8 ( 1 , 2 ), M 9 ( 1 , 2 ), M10 ( 1 , 2 )}

Asℜ(M∗) = ℜ(M ) and

ℑ (M∗) = −ℑ (M ), k1 can bewritten in a linear form as

k1 = ℜ(M )⊗Iℑ (M )⊗I

−ℑ (M )⊗Iℜ(M )⊗I G s . (35)

The right term in (33) can also be expressed into a linear formas

k2 = G ℜ(vec{PDs })

ℑ (vec{PDs })

= GPℜ(D) −P

ℑ (D)Pℑ (D) P

ℜ(D) s . (36)

Using (35) and (36), (33) can be simp li ed as

M T Gs = GPℜ(D) −P

ℑ (D)P

ℑ

(D) P

ℜ

(D) s (37)

where M T is an 2 ×2 matrix wi th real elements, whichis de ned as

M T = ℜ(M )⊗Iℑ (M )⊗I

−ℑ (M )⊗Iℜ(M )⊗I . (38)

As (37) must be satis ed for any s , one obtains

G †M T G =Pℜ(D) −P

ℑ (D)Pℑ (D) P

ℜ(D) . (39)

Finally, the following result is obtainedTheorem 2: For any STBC , the set Θ of ambiguity

matrices is the one containing all the × matrices Msatisfying (39) where P is a × permutation matrixand D is a × diagonal matrix with entries of unit modulus.

It should be noted that the condition (39) depends onthe matrix G which only depends on the STBC (see (15)).Unfortunately, it appears to be dif cult to nd the exactrelationship between G and the matrices M , P and D. Toprovide a clear relationship between these matrices, we haveperformed several Monte-Carlo simulations with the RayleighMIMO channel. Table I provides the set Θ of ambiguitymatrices for several STBCs using = {2, 3, 4} transmit

antennae. In Table I, matrices M 1-M 10 are equal to

M 1( ) = 00 − (40)

M 2( ) = 0

− − 0 (41)

M 3( ) = M 1( )0

20 2 M 1(− ) (42)

M 4( ) = M 2( ) 0 20 2 M 2(− ) (43)

M 5( ) =0 2 M 1( )

−M 1(− ) 0 2(44)

M 6( ) =0 2 M 2( )

−M 2(− ) 0 2(45)

M 7( 1 , 2) = M 1( 1) 0 20 2 M 1( 2 ) (46)

M 8( 1 , 2) = M 1( 1) 0 2

0 2 M 2( 2 )(47)

M 9( 1 , 2) = M 2( 1) 0 20 2 M 1( 2 ) (48)

M 10 ( 1 , 2) = M 2( 1) 0 20 2 M 2( 2 ) . (49)

Let us emphasize the differences between Table I and thetables reported in [17], [18], [21], [22]. Tables reported in [17],[18], [21], [22] focus on the blind channel-identi ability con-dition for subspace and SOS approaches. Without modi cationof the transmitter (precoding, pilot sequence), they show thatsubspace and SOS methods are unable to estimate the channelfor

≥1-rate STBCs and some speci c low-rate STBCs. Unlike

subspace and SOS approaches, the proposed method can beapplied to the whole class of linear STBCs without anymodi cation of the transmitter or the use of a pilot sequence.Moreover, unlike the general subspace methods [16], [17], itdoes not introduce additional ambiguities to those associatedto the blind channel estimation problem. For example for the34 -rate OSTBC using = 3 antennae, the proposed methodcan estimate the channel up to a sign whereas the subspacemethod introduces an unknown phase rotation [16], [17].

VI. S IMULATION RESULTS

Monte-Carlo simulations were run to assess the perfor-mances of the algorithms 1 and 2. Let us denote by Hand ˆH the original and estimated channel, respectively. After

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TABLE II: Spatial Multiplexing: Average Computation timesfor each algorithm.

Algorithm

SNR -10dB 0dB 10dB

Classical SD 0.07 s 0.11 s 0.09 sGeodesic SD 0.34 s 0.16 s 0.09 sJADE 0.004 s 0.004 s 0.004 s

channel estimation, the remaining ambiguity is removed bypost-multiplying ˆH with ˆM where

ˆM = arg minM∈Θ∥

H −ˆHM∥

2 (50)

and where the set of ambiguity matrices, Θ, depends on theSTBC (see Table I). After ambiguity removal, the estimatedchannel is denoted as ˆH = ˆH ˆM . Performances of theproposed blind algorithms were quanti ed through:

· the Normalized Mean Square Error (NMSE), which isde ned as:

= ∥

H

−ˆH

∥

2

∥H∥

2

. (51)

· the average Symbol Error Rate (S ER) obtained after MLdecoding.

Each simulation was carried out unde r the following condi-tions: i) a Rayleigh distributed chann el i.e. each element of H follows an i.i.d. circular Gaussian distribution with zero-mean and unit-variance, ii) a QPSK modulation, iii) 512transmitted blocks, iv) a temporally a nd spatially zero-meanwhite Gaussian additive noise with variance 2 (which isunknown at the receiver side), v) a thres hold equal to = 10 − 5

and vi) a receiver satisfying assumpti on AS1). Performancesof the algorithms 1 and 2 were evalua ted for several Signal-to-Noise Ratios (SNRs) where the SN R was de ned as [53]

= 10 log10 ( / 2). (52)

For each SNR, two thousand Monte-Carlo simulations wereperformed to approximate the NMSE and SER. As there isno guarantee that the algorithms 1 and 2 will nd the globalminimum, performances of the proposed methods were alsoevaluated with multistart initialization. Multistart initializationprocedure runs an algorithm several times with new randomstarting points and selects the estimated unitary matrix W

which minimizes the cost-function (W ). In the followingsubsections, performances are presented for 3 different STBCsystems.

A. Spatial Multiplexing

In this subsection, we consider the case of a SpatialMultiplexing system using = 2 transmit antennae. Thetransmitted blocks are given by

(s) = 1

2. (53)

For Spatial Multiplexing, the channel estimation problem re-duces to the classical ICA problem. After channel estimation,the set of ambiguity matrices is given by Θ = {PD}where

−10 −5 0 5 1010

−3

10−2

10−1

10 0

SNR (dB)

N M S E

Proposed Method: Classical SDProposed Method: Classical SD (5 starts)Proposed Method: Geodesic SDProposed Method: Geodesic SD (5 starts)JADE

Fig. 3: Spatial Multiplexing: NMSE.

−10 −5 0 5 1010

−2

10−1

10 0

SNR (dB)

S E R

Co herent receiverPro posed Method: Classical SDPro posed Method: Classical SD (5 starts)Pro posed Method: Geodesic SDPro posed Method: Geodesic SD (5 starts)JADE

Fig. 4: Spatial Multiplexing: Symbol Error Rate.

P and D are permutation and phase matrices. Figures 3 and4 present the performances of the algorithms 1 and 2 for areceiver composed of = 3 antennae. These two algorithmsare compared with JADE [54]. Figure 3 displays the channelNMSE versus SNR. In this simulation, algorithms 1 and 2always match or outperform the JADE algorithm, dependingon the SNR. Figure 3 also indicates that the multistart initial-

ization does not seem to improve the performances. Figure 4presents the SER versus the SNR. The SER is compared to theone obtained with the coherent ML receiver (perfect CSI). Itshould be noted that the blind channel-estimation algorithmsachieve near-optimal performances at high SNR since theirSERs approach the ones of the coherent ML receiver. Acomparison of the average computation times is shown inTable II for simulations implemented on a 2.6 GHz IntelPentium processor using Matlab. For multistart initializationapproaches, the computation times must be multiplied by thenumber of random starts. Table II shows that the classicalSD is less computationally demanding than the geodesic SDat low-SNR, but their computational complexities are similarat high-SNR. Table II also suggests that the JADE algorithmis signi cantly less complex than the proposed algorithms.

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−10 −5 0 5 1010

−4

10−3

10−2

10−1

10 0

10 1

SNR (dB)

N M S E

Proposed Method: Classical SDProposed Method: Classical SD (5 starts)Proposed Method: Geodesic SDProposed Method: Geodesic SD (5 starts)

Fig. 5: Alamouti Coding: NMSE.

−10 −5 0 5 1010

−4

10−3

10−2

10−1

10 0

SNR (dB)

S E R

Coherent receiverProposed Method: Classical SDProposed Method: Classical SD (5 s tarts)Proposed Method: Geodesic SDProposed Method: Geodesic SD (5 s tarts)

Fig. 6: Alamouti Coding: Sym bol Error Rate.

TABLE III: Alamouti Coding: Average Computation times foreach algorithm.

Algorithm

SNR -10dB 0dB 10dB

Classical SD 0.10 s 0.16 s 0.12 sGeodesic SD 0.51 s 0.21 s 0.10 s

However, one should note that JADE is limited to Spatial-

Multiplexing systems and cannot be employed for more gen-eral settings.

B. Alamouti Coding

In this subsection, we consider a STBC system using theAlamouti Code. This Orthogonal code is de ned by [1]

(s) = 1 − ∗

22

∗

1(54)

For Alamouti coding, the direct use of an ICA algorithm,like JADE, is irrelevant since the transmitted symbols betweenconsecutive time instances are not independent. Furthermore,it is demonstrated in [16]–[18], [22] that subspace and SOSapproaches cannot estimate the channel matrix when the

transmitter employs Alamouti Coding. Regarding the proposedmethods, Table I shows that the set of ambiguity matrices afterchannel estimation is Θ = {M1( ), M 2( )}. Figure 5 displaysthe NMSE versus SNR for a receiver composed of = 3antennae. Without multistart initialization, the geodesic SDclearly outperforms the classical SD since the latter exhibitsan error oor at SNR greater than 4dB. This error oor isdue to the fact that the Euclidean SD can lead to undesiredsuboptimal solutions even at high SNR [47], [48]. It shouldbe observed that the multistart initialization strategy removesthe error oor and improves the NMSE performances of thetwo proposed algorithms. Figure 6 compares the SER withthe one obtained with a coherent ML receiver. As previouslydiscussed, without multistart initialization, the performancesof the Euclidean SD lead to an error oor at SNR greaterthan 4dB. However, it should be observed that algorithms 1and 2 achieve near-optimal performance when a multistart ini-tizalization is used. A comparison of the average computationtimes is shown in Table III. It should be noted that classicalSD is less computationally demanding than the geodesic SD

at low-SNR, but this trend is reversed at high SNR.

C. 34 -rate OSTB C using 3 antennae

In this subsect ion, we consider the case of a 34 -rate OSTBC

using 3 antennae . This OSTBC is de ned by [32]

(s) = ⎡⎣

1 0 2 − 30 1

∗

3∗

2

− ∗

2 − 3 1 0⎤⎦

(55)

For this low-rat e code, the channel can be estimated withsubspace and S OS approaches. The remaining ambiguity

reduces to a sign for the SOS approach [18] and to a rotationfor the subsp ace method [16], [17]. In the following g-ures, performanc es of the proposed algorithms are comparedwith the SOS-ba sed method [18] for a receiver composed of

= 5 antennae. Figure 7 displays the NMSE versus SNR.Without multiple-start initialization, it should be noted thatthe algorithms 1 and 2 exhibit an error oor at SNR greaterthan 6dB. For the Euclidean SD algorithm, the error ooris due to the fact that this approach can lead to undesiredsuboptimal solutions [47], [48]. For the Geodesic SD one,even if it always converges to a local minimum [47], theerror oor is due to the fact that the local minimum does

not necessarily coincide with the global one. As previouslyobserved, multistart initialization removes the error oor andimproves the performances of the two proposed algorithms. Inparticular, Figure 7 shows that for > 0, algorithms 1 and2 with multistart initialization outperform the SOS method.Figure 8 shows that method [18] and the proposed multistartalgorithms both achieve near-optimal SER performances for

> 0. A comparison of the average computation timesis presented in Table IV. It is shown that the SOS method isless computationally demanding than the proposed algorithms.Therefore, for SOS-identi able OSTBCs, it seems that theclosed-form SOS algorithm is de nitely preferable since thisalgorithm does not suffer from convergence problems and it isless computationally expensive than the proposed approaches.However, it should be emphasized that the SOS method is

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−10 −5 0 5 10

10−4

10−3

10−2

10−1

10 0

SNR (dB)

N M S E

Proposed Method: Classical SDProposed Method: Classical SD (5 starts)Proposed Method: Geodesic SDProposed Method: Geodesic SD (5 starts)SOS Method

Fig. 7: 3/ 4-rate OSTBC: NMSE.

−10 −5 0 5 1010

−6

10−5

10−4

10−3

10−2

10−1

10 0

SNR (dB)

S E R

Coherent receiverProposed Method: Classical SDProposed Method: Classical SD (5 st arts)Proposed Method: Geodesic SDProposed Method: Geodesic SD (5 st arts)SOS Method

Fig. 8: 3/ 4-rate OSTBC: Sym bol Error Rate.

TABLE IV: 34 -rate OSTBC: Average Computation times for

each algorithm.

AlgorithmSNR -10dB 0dB 10dB

Classical SD 1.32 s 0.85 s 0.78 sGeodesic SD 3.33 s 0.62 s 0.51 sSOS method 0.48 s 0.48 s 0.48 s

limited to a subclass of OSTBCs [22], whereas the proposedalgorithms can be applied to the whole class of linear STBCs.

VII. C ONCLUSION

This paper proposed an original blind channel estimationalgorithm for space-time block coding communications. Themethod is based on the minimization of a kurtosis-basedcost function after Zero-Forcing equalization. The proposedmethod can be applied to the whole class of linear STBCs,whatever the code-rate and the modulation. This paper alsopresented the set of the remaining channel ambiguities forseveral STBCs using 2, 3 or 4 transmit antennae. The goodperformances of the proposed algorithm were demonstratedthrough computer simulations for different STBCs. In par-ticular, simulations have shown that the proposed method

matches or outperforms the JADE algorithm [54] for SpatialMultiplexing and matches the performances of the closed-formSOS approach [18] for identi able OSTBCs.

APPENDIX

Let us consider the 2 ×2 matrix W de ned in (12).As (A⊗

B)T = AT⊗

BT , one gets:

W T W = ℜ(W )⊗Iℑ (W )⊗I

−ℑ (W )⊗Iℜ(W )⊗I

× ℜ(W T )⊗I −ℑ (W T )⊗I

ℑ (W T )⊗Iℜ(W T )⊗I

(56)

From the mixed product rule, it follows that:

W T W = B1⊗I −B2⊗

IB2⊗

I B1⊗I (57)

where the × matrices B1 and B2 are given by:

B1 = ℜ(W )ℜ(W T ) + ℑ (W )ℑ (W T ) (58)

B2 = ℜ(W )ℑ (W T ) −ℑ (W )ℜ(W T ) (59)

As W is a × unitary matrix, it satis es WW H = I .By expanding th e real and imaginary parts, one gets:

WW H = (ℜ(W ) + ℑ (W )).(ℜ(W T ) − ℑ (W T ))= B1 − B2 = I (60)

By identi cation , it follows that B1 = I and B2 = 0 .Finally (57) can be simpli ed as:

W T W = I2 (61)

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Vincent Choqueuse (S’08-M’09) was born in 1981in Brest, France. He received the Dipl.-Ing. and theM.Sc. degrees in 2004 and 2005, respectively, fromTroyes University of Technology (UTT), France,and the Ph.D. degree in 2008 from University of Brest, France. Since September 2009, he has beenAssociate Professor at the IUT of Brest, France,and a member of the Laboratory LBMS (EA 4325).His research interests focus on signal processing andstatistics for communications and diagnosis.

Ali Mansour (M’97,SM’00) was born at Tripoliin Lebanon in 1969. He received his Electronic-Electrical Engineering Diploma in 1992 from theLebanese University, Tripoli, Lebanon, and hisM.Sc. and the Ph.D. degrees in Signal, Image andSpeech Processing from INPG, Grenoble, France, in1993 and 1997, respectively. From January 1997 toJuly 1997, he held a post doc position at LTIRF-INPG, Grenoble, France. From August 1997 toSeptember 2001, he was a Research Scientist atthe Bio-Mimetic Control Research Center of Riken,

Nagoya, Japan. From 2001 to 2008, he was holding a teacher-researcherposition at the Ecole Nationale Supérieure des Ingénieurs des Etudes etTechniques d’Armement (ENSIETA),Brest, France. Since February 2008, hehas been a senior-lecturer at the Department of Electrical and ComputerEngineering at Curtin University of Technology, Perth, Australia. His researchinterests are in the areas of blind separation of sources, high order statistics,signal processing, COMINT, radar, sonar and robotics.

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Ludovic Collin received the Ph.D. degree in elec-trical engineering from the University of BretagneOccidentale, Brest, France, in 2002. From 1989 to1999 he was with ORCA Instrumentation, Brest,where he developed oceanographical instrumenta-tion and acoustic modems. From 1999 to 2002he was Research and Teaching Assistant at FrenchNaval Academy, Lanveoc, France, and at the In-stitute of Technology of Lannion. From 2003 to2007 he was Assistant Professor at the ENSIETA,

Brest. Since 2007 he has been Assistant Professorat the University of Brest and member of the Laboratory for Science andTechnologies of Information, Communication and Knowledge (Lab-STICC- UMR CNRS 3192). His research interests are in MIMO systems andinterception of communications.

Kof Clément Yao (M’05) received the PhD degreein Optical Signal Processing and computer sciencesfrom University Louis Pasteur of Strasbourg, Francein 1990. After his post-doctorate research on opticalneural networks at Ecole Nationale Supérieure desTélécommunications of Brest, he joined the Frenchnaval academy as assistant professor in statisticalsignal processing in 1992. His research interest wasfocused on Pattern recognition and blind signalseparation in underwater acoustics. Since 2001, hehas been Assistant Professor at University of Brest,

France. His present research interests are in MIMO systems and blindinterception of digital communication signals.

Gilles Burel (M’00-SM’08) was born in 1964. Hereceived the M.Sc. degree from Ecole Supérieured’Electricité, Gif Sur Yvette, France, in 1988, thePh.D. degree from University of Brest, France, 1991,and the Habilitation to Supervise Research degreein 1996. From 1988 to 1997 he was a member of the technical staff of Thomson CSF, then ThomsonMultimedia, Rennes, France, where he worked onimage processing and pattern recognition applica-tions as project manager.

Since 1997, he has been Professor of DigitalCommunications, Image and Signal Processing at the University of Brest.He is Associate Director of the Laboratory for Science and Technologiesof Information, Communication and Knowledge (Lab-STICC - UMR CNRS3192). He is author or co-author of 19 patents, one book and 140 scienti cpapers. His present research interests are in signal processing for digitalcommunications, MIMO systems and interception of communications.

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Blind Channel Estimation for STBC Systems Using Higher-Order Statistics

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