Blind Channel Identification and Equalisation inOFDM using Subspace-Based Methods

Faisal O. Alayyan, Raed M. Shubair, Abdelhak M. Zoubir, and Yee Hong Leung

Abstract— A subspace-based method is proposed for estimatingthe channel responses of single-input-multiple-output (SIMO)Orthogonal Frequency Division Multiplexing (OFDM) system.Our technique relies on minimum noise subspace (MNS) decom-position to obtain noise subspace in a parallel structure froma set of pairs (combinations) of system outputs that form aproperly connected sequence (PCS). The developed MNS-OFDMestimator is more efficient in computation than subspace (SS)-OFDM estimator, although the former is less robust to noise thanthe later. To maximise the MNS-OFDM estimator performance, asymmetric version of MNS is implemented. We present simulationresults demonstrating the channel identification performance ofthe corresponding OFDM-based SIMO systems employ cyclicprefixing approach.

Index Terms: OFDM, MNS, Blind Channel Identification,Equalisation.

I. INTRODUCTION

OFDM is a multi-carrier digital modulation technique thatfacilitates the transmission of high data rates with a lim-ited bandwidth [33]. It is an effective technique for severalapplications such Digital Audio Broadcasting (DAB) andterrestrial Digital Video Broadcasting (DVB) [18], [32], [35].In addition, OFDM forms the basis for the physical layer inupcoming standards for broadband Wireless Local Area Net-work (WLAN) [24], i.e. ESTI-BRAN HiperLAN/2 [22], IEEE802.11a [24] and Multimedia Mobile Access CommunicationSystems (MMAC) and for Fourth Generation (4G) broadbandwireless systems that will perform multimedia transmissionto mobiles and portable personal communications devices, i.e.European MEMO project and for IEEE 802.16.

Due to increase in the normalised delay spread, multipathfading becomes a major concern as systems with high data rateare more liable to intersymbol interference (ISI). Classically,ISI is eliminated by employing a cyclically extended timedomain guard interval (GI). Thus, each OFDM symbol ispreceded by a periodic extension of the symbol itself. This GIis also known as cyclic prefix (CP) and the system CP-OFDM[10], [11]. Recently, zero-padding OFDM (ZP-OFDM), whichpre-pends each OFDM symbol with zeros rather than repli-cating the last few samples, has been proposed [41], [27],

Manuscript received September 14, 2008; revised December 25, 2008;accepted January 18, 2008.

Faisal O. Alayyan and Abdelhak M. Zoubir are with Signal Pro-cessing Group (SPG), TU Darmstadt, Darmstadt, Germany (email:fayyan,zoubir@spg.tu-darmstadt.de)

Raed M. Shubair is with Khalifa University of Science,Technology andResearch (KUSTAR), Abu Dhabi, UAE (email:rshubair@kustar.ac.ae)

Yee Hong Leung is with Electrical and Computer Engineering De-partment, Curtin University of Technology, Perth WA 6845, Australia(email:Y.leung@exchange.curtin.edu.au)

[28], [26]. ZP-OFDM not only has all the advantages ofthe CP-OFDM, but also guarantees symbol recovery and en-sures finite impulse response (FIR) equalisation. However, theimplementation of a ZP-OFDM system involves transmittermodifications and complicates the equaliser [10], [43].

To maximise the performance advantage of OFDM system,reliable identification of Single Input Multiple Output (SIMO)channels is desired. Currently, the channel identification andequalisation technique used requires a major fraction of thechannel capacity to send a training sequence over the channels[25]. There are practical situations where it is not feasible toutilize a training sequence such as in fast varying channels.To save this fraction of channel capacity, blind identification isan attractive approach. Using the blind channel identificationtechniques, the OFDM-based SIMO receiver can identify thechannel characteristics and equalises the channel all based onthe received signal, and no training sequence is needed, whichhence saves the channel capacity.

Blind identification and equalisation of SIMO channels havebeen a very active area of research during the past few years(see [1], [5], [12], [21], [23], [25] and the references therein).Among the various known algorithms, Second Order Statistics(SOS)-based algorithms are the most attractive due to theirspecial properties [15], [16]. It was, for a while, believedthat the subspace (SS)-based method was the only key to thesurprising success among the existing SOS-based techniques.The SS-based method applies the MUltiple SIgnal Classifi-cation (MUSIC) concept to a relation between the channelimpulse responses and the noise subspace associated with acovariance matrix of the system output. One of the importantadvantages of SS-based method is its deterministic property.That is, the channel parameters can be recovered perfectly inthe absence of noise, using only a finite set of data samples,without any statistical assumptions over the input data. Morerecently, the use of the SS-based method has been suggestedin [10], [11] to accomplish blind SIMO channel identificationin OFDM systems. Despite their high identification efficiency,SS-based method are computationally very intensive, whichmay be unrealistic or too costly to implement in real time,especially for large sensor array systems. The main reason isthat they require non-parallelisable eigen-value-decomposition(EVD) of a large dimensional matrix to extract (estimate) thenoise or signal subspace [2], [4].

In this paper, using a minimum noise subspace (MNS) de-composition concept [4], we introduce several techniques forblind identification and equalisation of OFDM-based SIMOsystems. Our techniques computes the noise subspace via a setof noise vectors (basis of the noise subspace) that can be com-

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puted in parallel from a set of pairs (combinations) of systemoutput, without using reference or pilot symbols. Therefore,an EVD for smaller covariance matrices is required to extractnoise subspace. Ideally, this approach, which relies on theknown structure of the received OFDM symbols, provides aperfect channel estimate in the absence of noise. It is believedto have inspired all the subsequent developments which havetaken place to accomplish unknown parameter identificationsin a wide range array signal processing applications. Further-more, the developed techniques significantly reduce receivercomplexity in wireless broadband multi-antenna systems.

A brief introduction to OFDM-based SIMO system is givenin Section III. We review the block diagram of OFDM-based SIMO system, which employs a CP to mitigate theimpairments of the multipath radio channel. Following the CP-OFDM problem formulation, we then derive a mathematicalmodel corresponding to the ZP-OFDM system. In SectionIV, the basic assumptions are outlined. In Section V, wepresent the original methods [10], [25], [37], [38], [39] thatmarked the beginning of a brand new direction in subspacedecomposition and its application in OFDM-based SIMOsystems. To illustrate the usefulness of the MNS concept,new estimators are derived in Section VI that tradeoff MSEperformance for extra saving in complexity. In Section VII,we highlight the channel estimation based on a SymmetricMNS (SMNS) concept and discuss several important resultsconcerning system identifiability. By employing SMNS-basedmethod, OFDM system guarantee symbol recovery and offersMSE performance close to the system exploiting SS-basedmethod. In Section VIII, equalisation scheme is developedbased on one-IFFT operations. In Section IX, we presentcomputer simulations. In Section X, we discuss the propertiesof the proposed techniques and finally, in Section XI, wesummarise and draw relevant conclusions. Part of the workin this paper has been previously presented in [7], [8], [9].

II. NOTATION

Throughout this paper, the following notations are used.

AT : Transpose of A

AH : Conjugate (or Hermitian) Transpose ofA

A† : Moore-Penrose pseudo-inverse of A

I : Identity matrix of appropriate dimension

0 : Zero matrix of appropriate dimension

E(.) : Statistical expectation operator

‖·‖ : Frobenius norm

III. OFDM-BASED SPATIAL DIVERSITY SYSTEMS

An apparent disadvantage of single-carrier based spatialdiversity systems is the fact that the computational complexityof the receiver will in general be very high. The use of OFDM[14], [29] alleviates this problem by turning the frequency-selective SIMO channel into a set of parallel narrow-bandSIMO channels. This makes equalisation very simple. In fact,

only a constant matrix has to be inverted for each tone [30],[31]. In this section, OFDM-based SIMO is introduced byusing CP and ZP techniques.

A. Standard CP-OFDM System

P/S

Equaliz

er

N M

M

N

CP

N

S/P

S/P

M

Estimator

Re

mo

ve

CP

Re

mo

ve

CP

N

N

FHN

sk uk ucp,k

x(1)cp,k

x(q)cp,k

y(1)cp,k y

(q)cp,k

HcpFHN

FHN

sk

Fig. 1. CP-OFDM system: transmitter and receiver.

Figure 1 depicts the baseband discrete-time block equiv-alent model of a standard CP-OFDM system. The trans-mitted symbols are parsed into blocks of size N : sk =[sk (0) , sk (1) , . . . , sk (N − 1)]T where k = 0, 1, 2 . . . ,K −1. The elements of sk are considered to be independentand identically distributed (i.i.d). We regard these elementsto be in the frequency domain. The symbol block skis then modulated and converted into time domain us-ing the IFFT matrix FHN , where FN has entires fn,d =1N

exp(j2πndN

)and d, n = 0, . . . , N − 1. The data vec-

tor uk = FHN sk = [uk (0) , uk (1) , . . . , uk (N − 1)]T isthen appneded with a CP of length L, resulting in asize M = N + L signal vector: ucp,k = Tcpuk =[uk (N − L) , . . . , uk (N − 1) , uk (0) , . . . , uk (N − 1)]T . Weconsider Tcp is a concatenation of the last L rows of an N×Nidentity matrix IN

(that we denote as Icp

)and the identity

matrix itself IN , i.e., Tcp =[ITcp, I

TN

]T. CP makes the OFDM

appear periodic over the time span of interest. The channelresponse is denoted by h(r) (l) where l = 0, 1, . . . , L(r), andr = 1, 2, . . . , q. To avoid ISI, as indicated previously, the CPlength L is selected to be equal to or greater than the channelorder, i.e., L(r) ≤ L. We consider the upper bound of theSIMO channel order L(r) as a CP length L. The received k-thblock at r-th output for n = 0, 1, . . . ,M − 1, is given by

x(r)cp,k (n) =

L∑l=0

h(r) (l)ucp,k (n− l) + v(r)cp,k (n) (1)

where ucp,k (n− l) and the AWGN, v(r)cp,k (n), is assumed to

be mutually uncorrelated and stationary. Using the following

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notations

xcp,k (n) =[x

(1)cp,k (n) , x

(2)cp,k (n) , . . . , x

(q)cp,k (n)

]Tvcp,k (n) =

[v(1)cp,k (n) , v

(2)cp,k (n) , . . . , v

(q)cp,k (n)

]Th (l) =

[h(1) (l) , h(2) (l) , . . . , h(q) (l)

]T(2)

we can rewrite the input-output relation (1) in vector matrixas

xcp,k (n) =

L∑l=0

h (l)ucp,k (n− l) + vcp,k (n) . (3)

Let q to be number of sensors at the output, H0 be the qM×Mblock-lower triangular Toeplitz matrix and H1 be the qM×Mblock-upper triangular Toeplitz matrix, i.e.,

H0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h(0) 0 . . . . . . 0...

. . .. . .

. . ....

h(L) . . . h(0) . . . 0

0 h(L) . . . h(0)...

.... . .

. . .. . .

...0 . . . h(L) . . . h(0)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (4)

H1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 . . . h(L) . . . h(1)...

. . .. . .

. . ....

.... . .

. . .. . . h(L)

0 . . . . . . . . . 0...

. . . . . .. . .

...0 . . . . . . . . . 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (5)

Based on the aforementioned assumptions and the fact thath(l) = 0,∀l /∈ [0, L], (3) can be written in the block form as

xcp,k = H0ucp,k +

ISI︷ ︸︸ ︷H1ucp,k−1 +vcp,k

= H0TcpFHN sk + H1TcpFHN sk−1︸ ︷︷ ︸ISI

+vcp,k (6)

where

xcp,k =[xTcp,k (0) , xTcp,k (1) , . . . , xTcp,k (M − 1)

]Tvcp,k =

[vTcp,k (0) , vTcp,k (1) , . . . , vTcp,k (M − 1)

]T(7)

Due to the dispersive nature of the channel, ISI arises betweensuccessive blocks and renders xcp,k in (6) dependent onboth ucp,k and ucp,k−1. To obtain an ISI-free data block,we consider a truncated version of xcp,k. This is done by

discarding[xTcp,k (0) , xTcp,k (1) , . . . , xTcp,k (L− 1)

]Twith the

receive-matrix: Rcp = [0qN×qL, IqN ]. The resulting receivedvector can be written as

ycp,k = Rcpxcp,k

= HcpFHN sk + ncp,k (8)

where

ycp,k =[yTcp,k (0) , yTcp,k (1) , . . . , yTcp,k (N − 1)

]Tncp,k =

[nTcp,k (0) ,nTcp,k (1) , . . . ,nTcp,k (N − 1)

]T(9)

The block-Circulant channel matrix (one can verify by directsubstitution from (4) that Hcp = RcpH0Tcp) is defined as

Hcp =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h(0) . . . 0 h(L) . . . h(1)h(1) 0 . . . h(L) . . . h(2)

.... . .

. . .. . .

. . ....

h(L− 1) . . . h(0) 0 . . . h(L)h(L) . . . h(1) 0 . . . 0

.... . .

. . .. . .

. . ....

0 . . . 0 h(L) . . . h(0)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (10)

The linear convolutive channel with ISI in (6) is convertedto be a circular one without ISI in (8). Consider q = 1,N × N circulant matrix Hcp can be diagonalised by pre-and post multiplication with N -point FFT and IFFT matrices;FNHcpFHN = D. Because the FFT (and thus its matrix FN ) isinvertible, we deduce that the circulant matrix Hcp is invertibleif and only if D is invertible and the channel transfer functionhas no zero on the FFT grid.

CP-OFDM system enables one to deal easily with ISIchannels by simply taking into account the scalar channelattenuations. However, it has an obvious drawback that thesymbol sk(l) transmitted on the l-th sub-carrier cannot berecovered when it is hit by a channel zero (h(l) = 0). Thislimitation leads to a loss in frequency domain (or multipath)diversity and can be overcome by the ZP precoder which wereview next [41].

B. ZP-OFDM System

P/S

Equaliz

er

N M

N

ZP

N

S/P

S/P

Estimator

M

M

FHN

sk uk uzp,k

x(1)zp,k

x(q)zp,k

x(1)zp,k x

(q)zp,k

HzpFHN

FHN

sk

Fig. 2. ZP-OFDM system: transmitter and receiver.

Figure 2 depicts the baseband discrete-time block equiv-alent model of a standard ZP-OFDM system. The onlydifference between ZP-OFDM and CP-OFDM is that theCP is replaced by L trailing zeros that are padded at

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each precoded block uk to yield uzp,k = Tzpuk =[uk (0) , uk (1) , . . . , uk (N − 1) , 0, . . . , 0]T where Tzp =[ITN , 0

TL×N

]T. We can write the received block symbol xzp,k

as [41]

xzp,k = H0uzp,k +

ISI︷ ︸︸ ︷H1uzp,k−1 +vzp,k

= H0TzpFHN sk + H1TzpFHN sk−1︸ ︷︷ ︸ISI

+vzp,k (11)

where

xzp,k =[xTzp,k (0) , xTzp,k (1) , . . . , xTzp,k (M − 1)

]Tvzp,k =

[vTzp,k (0) , vTzp,k (1) , . . . , vTzp,k (M − 1)

]T(12)

and the key advantage of ZP-OFDM lies in the all-zero L ×N matrix 0 which eliminates the ISI, since H1TzpFHN = 0.Forming the qM × N matrix Hzp from the first N columnsof matrix H0, (11) can be expressed as

xzp,k = HzpFHN sk + vzp,k (13)

where

xzp,k =[xTzp,k (0) , xTzp,k (1) , . . . , xTzp,k (M − 1)

]Tvzp,k =

[vTzp,k (0) , vTzp,k (1) , . . . , vTzp,k (M − 1)

]T(14)

and Hzp is, a block-Toeplitz matrix, defined as

Hzp =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h(0) 0 . . . 0...

. . .. . .

...

h(L). . .

. . ....

0. . .

. . . h(0)...

. . .. . .

...0 . . . 0 h(L)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (15)

Corresponding to the first N columns of Hzp, the H0 sub-matrix is block-Toeplitz and is always guaranteed to be invert-ible, which assures symbol recovery (perfect detectability inthe absence of noise) regardless of the channel zero locations.This is not the case with CP-OFDM, and precisely the distinctadvantage of ZP-OFDM. In fact, the channel-irrespectivesymbol detectability property of ZP-OFDM is equivalent toclaiming that ZP-OFDM enjoys maximum diversity gain. Inother words, ZP-OFDM is capable of recovering the diversityloss incurred by CP-OFDM.

This section provides a brief review of the OFDM-basedspatial diversity, which sets the scene for the rest of thepaper. For more elaborate introduction to OFDM, the readermay refer to [10], [13], [14], [26], [28], [33], [41], whereinnumerous further references are found. In the next sections, westudy the estimation of Hcp and Hzp from the observationsycp,k and xzp,k.

IV. ASSUMPTIONS

If the distance between each sensor at the receiver is largeenough with respect to the spatial variation of the channel, thepropagation channels between the transmitter and each sensorat the receiver are different one from the other. In this case,it is quite realistic to assume that the components of

h(z) =L∑l=0

h(l)z−l (16)

have no common zeros, i.e.,

h(z) �= 0 for each z( including z = ∞). (17)

The degree of h(z) is assumed to be known and H(z) modelsthe propagation between the emitting sources and the sensor isassumed to be unknown. In this paper, we study the estimationof H(z) from the outputs ycp,k and xzp,k, assuming thatthe transfer function satisfies (17). It can be shown that inthe context of a realistic multipath model, (17) is practicallyreasonable and hold almost surely. The signal parametersof interest are spatial in nature and thus require the cross-covariance information among the various sensors, i.e., thespatial covariance matrix, which is given by

Rn = E{

ycp,kyHcp,k}, Rm = E

{xcp,kxHcp,k

}. (18)

The source samples {sk(n)} is a sequence of i.i.d complex cir-cular random variables with zero mean and bounded momentsup to the fourth-order E

{|sk(n)|4

}:

E{s2k(n)

}= 0, E {sk(n)s∗k(n)} = σ2

s . (19)

The covariance matrix RN of the transmitted symbols sk isexpressed as

RN = E{

sksHk}. (20)

and is assumed to be positive-definite. The additive noisevk(n) is a stationary complex circular white-noise process withzero-mean and second-order moments

E{vk(n)2

}= 0, E {vk(n)v∗k(n)} = σ2. (21)

Moreover, vk(n) is independent from one sensor to another. Inthe general case of spatial independence, the noise covariancematrix has diagonal structure, as follows:

Qn = E{

ncp,knHcp,k}

= σ2In, (22)

Qm = E{

vzp,kvHzp,k}

= σ2Im. (23)

The system is designed such that M > N > L(r) ≥ L. Nochannel state information (CSI) is assumed available at thetransmitter.

V. SS-BASED METHOD

The desire for a more efficient algorithm led to the devel-opment of subspace(SS) methods for the blind estimation ofmulti-channel FIR filters [25]. The basic idea behind thesemethods consists of estimating the unknown parameters byexploiting the orthogonality of subspaces of certain matricesobtained by arranging in a prescribed order the second order

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statistics of the observation. This scheme shares many sim-ilarities with well-known techniques for direction-of-arrival(DOA) estimation in a narrow-band array processing context.The existence of such SS-based methods for blind estimationwas brought to light by Gurelli and Nikias [19] and Moulineset al [25] (see also Hua [21] and Abed-Meraim [1], [5],[6], [20]). Blind channel estimation is particularly importantfor OFDM applications where severe ISI can arise fromthe time-varying multipath fading that commonly exists ina mobile communication environment. The varying channelcharacteristics must be identified and equalised in real time tomaintain the correct flow of information. The use of SS-basedmethods to accomplish blind SIMO channel estimation forOFDM has been proposed for frequency-flat fading channelsin [10], [11]. The extension of it to the general MIMO casehas been successfully introduced by Zeng et al [43]. Morerecently, some SS-based methods have been proposed forsingle-user OFDM systems [27], [34]. The method in [34]can be applied to OFDM systems without CP and, therefore,leads to higher data-rate.

In order to connect our results with [10] and [43], theframework of blind channel estimation in OFDM-based SIMOsystem using CP is now presented. Before going further, wemake the following definitions: h (l) is the true (but unknown)channel response to the SIMO system to be identified, and h (l)is the estimated channel response. For simplicity, we definealso n = qN , n = n − N , and ϑ = q(L + 1). Denoting byRn the covariance matrix of ycp,k and using the signal model(8), Rn can be expressed as

Rn = HcpFHNRNFNHHcp + σ2In. (24)

Since the covariance matrix Rn is unitarily diagonalisable,there exists an n × N matrix Scp and an n × n matrix Gcpsuch that [Scp Gcp] is unitary, and

Rn = ScpΛNSHcp + σ2GcpGHcp (25)

where ΛN represents an N ×N diagonal matrix. By analogywith the narrow-band array processing model, the range spaceof Scp is referred to as the signal subspace and its orthogonalcomplement Gcp is the noise subspace. Determining the basisScp with enough precision constitutes a solution to most detec-tion and parametric estimation problems in array processing.

If the power of the noise over all the sensors is the same,the noise covariance matrix reduces to an identity matrix (upto a scalar). The set of eigenvalues (ordered by magnitude), orsimply eigenspectrum, are thus scaled up by the addition ofthe noise floor σ2. This assumption of uniform noise providesa convenient means to distinguish between the signal andnoise subspaces. If it is not verified, all classical detectionand estimation algorithms fail to perform satisfactorily.

We denote Πn = GcpGHcp as the orthogonal projector

on the noise subspace, and Π⊥n = I − Πn. It is

easy to show that range(Scp)

= range(Hcp), and

range(Gcp)

=(range

(Hcp))⊥

, i.e., the orthogonalcomplement subspace to the column range space of Hcp.The subspace estimation method is ultimately related to thefollowing lemma [5]:

Lemma: Assume that (17) holds and N ≥ L, then, the matrixHcp is of full rank. In addition, the solutions of the matrixequation

GHcpHcpFHN = 0. (26)

under the constraint deg(h(z)) ≤ L are given by h(z) = ηh(z)where η ∈ R is a unique up to a scalar factor.

The knowledge of the column space of GHcpHcpFHN charac-

terises Hcp up to a scale constant (because Qn is full rank).The filter coefficients (which is characterised uniquely by theknowledge of the range of Hcp) can be identified from theknowledge of the range of the signal part of the whitenedcovariance matrix, see [5] for more details. It is not difficultto show that the orthogonality relation (26) between the signalsubspace and the noise subspace is equivalent to

ΠnHcpFHN = 0. (27)

This relation is the keystone of the SS-based method toidentify

h = [h(0)T , . . . ,hT (L)]T . (28)

As an immediate consequence, denote by Qcp the ϑ × ϑsymmetric matrix defined by

h → Tr((

HcpFHN)H

ΠnHcpFHN)

= hHQcph. (29)

The null space of Qcp is reduced to a one-dimensionalsubspace spanned by the vector h associated with h(z). Notethat the above Lemma ensures the consistency of the aboveestimates in the case where the channel order is known orcorrectly estimated. The behavior of the SS-based method inthe case where deg(h(z)) is overestimated has been discussedin [5]. As shown in [25], the SS-based method proceeds byminimising the least squares criterion

h = arg min‖h‖=1

‖χ‖2, χ = BcpHcpFHN (30)

under a suitable constraint, where the obvious choice for Bcpis Bcp = GH

cp, or equivalently, Bcp = Πn. The computationof the whole n noise vectors via eigen-decomposition of Rnis required to estimate the channel parameter. Similar to theweighted subspace fitting (WSF) method [40] in the narrow-band DOA identification context, this identification procedurecan be improved by using the weighted least-squares (WLS)criterion. Using vector notation, (30) can be rewritten as

h = arg min‖h‖=1

‖vec (χ)‖2. (31)

It has been proposed in [1], that the least squares approach canbe generalised by introducing in the above criterion a positiveweighting matrix Wcp. The channel parameter vector is thenidentified by minimising the WLS criterion

hW = arg min‖hW ‖=1

vec (χ)H Wcpvec (χ) . (32)

The choice of Wcp and the statistical performance analysisof this scheme is discussed in [4]. Note indeed if hW is asolution of (32), then eiφhW for any real φ is also solution.

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In practice, a specific value of φ should be chosen using aconstraint such as eHhW ≥ 0, where e is an arbitrary nonzerovector. The ordering of the channel number in hW , of course,will depend on the choice of the vector e. However, mostmeaningful performance criterion are insensitive to this choice.The most convenient choice for e is e = h, which yieldinghHWh ≥ 0.

The above material allows one to analyze the behavior of theSS-based method where CP is used. For ZP-OFDM system,we need some additional notations, such as m = qM andm = m − N . Under assumption (17), the covariance matrixof xzp,k can be written as

Rm = HzpFHNRNFNHHzp + σ2Im. (33)

For M > N , the noise-free covariance matrix is rank-definiteand its EVD is given by

Rm = SzpΛNSHzp + σ2GzpGHzp (34)

where range(Szp)

= range(Hzp)

is the signal subspace while

range(Gzp)

=(range

(Hzp))⊥

is the noise subspace. Notethat the above Lemma can be easily adapted to ZP-OFDMsystem. Therefore, the orthogonality relation (26) can betranslated into a formalism adequate for practical computationusing ZP as follows

GHzpHzpFHN = 0. (35)

Similar to CP-OFDM system, (35) is the keystone of the SS-based method to identify the transfer function Hzp. If we lookat the above weighting matrix approach more carefully, it isnot difficult to see that it works for ZP-OFDM system.

In practice, the weighting matrix can be seen as an approachto compensate for the effect of matrix ill-conditioning due tothe close common zeros of the channel transfer functions. Byusing the optimal weighting matrix, the estimation procedurecan become quite sensitive to the ill conditioning problem (seeFigures 4 and 5 in Section IX). In fact, if the channels haveclose common zeros, the corresponding block-Circulant (orblock-Toeplitz) channel matrices will become nearly singularand consequently result in failure of the SS-based method.

VI. MNS-BASED METHOD

t1

t2

t3

t4

t5

t6

m1

m2

m3

m4

m5

m6

m7

Fig. 3. Tree that connects q = 7 channel outputs as its notes.

Unfortunately, a widely acknowledged problem with theaforementioned techniques is its extensive computational com-plexity due to the EVD of a ’large’ dimensional matrixand rather slow convergence with respect to the number ofblock symbols. In fast changing environments, such as incellular communications, their applications may be limited.This problem is alleviated by the MNS decomposition ap-proach proposed by Abed-Meraim et al [4]. Based on thiscontribution, it is easy to show that, only q − 1 properlychosen noise eigenvectors are just as efficient as using thewhole noise subspace range

(Gcp)

for (26) (or range(Gzp)

for (35)) to yield a consistent estimate of Hcp for the CP-OFDM system (or Hzp for ZP-OFDM system). Furthermore,each of the q−1 noise eigenvectors can be found by using EVDof a ’small’ dimensional covariance matrix corresponding tothe (distinct) pairs of channel outputs given by a properlyconnected sequence (PCS) defined as follows [2]:

Definition 1: Denote the q system outputs by a set ofmembersm1, . . . ,mq . A combination of two (q ≥ 2) members(ti = mi1 ,mi2) is called a pair. A sequence of q−1 pairs is saidto be properly connected if each pair in the sequence consists ofone member shared by its preceding pairs and another membernot shared by its preceding pairs.

Example 1: Consider a system with one input and sevenoutputs. The following sequence of pairs has minimum redun-dancy (six pairs) and spans all system outputs m1, . . . ,m7.

t1 = (m1,m2) , t2 = (m1,m3) , t3 = (m3,m4)

t4 = (m3,m5) , t5 = (m5,m6) , t6 = (m5,m7)

Figure 3 demonstrates an example of PCS with q = 7. Inthe Tree pattern, the notes m2, m4, m6, and m7 are endingnodes while the nodes m1, m3, and m5 are branching nodes.

Remarks:

• MNS-based method can be applied to applications relat-ing to source localisation and array calibration [4].

• In practice a PCS is easy to construct, however, it is nota necessary condition to give the MNS.

• A PCS exploits the diversity of the system outputs withminimum redundancy. This follows, since a sequence hasless than q − 1 pairs or a pair in the sequence has lessthan two members, then the sequence does not give therequired number of independent noise vectors.

• A set of q − 1 pairs span all the system outputs are notnecessarily sufficient to give the required MNS.

A. CP-OFDM Estimator

For the development that follows, it is convenient todefine the i-th pair 2N × N block-Circulant matrix as,Hcp,(i), whose first block-row is given by

[h(i) (0) , 0, . . . ,

0, h(i) (L) , . . . , h(i) (1)]

and first block-column is given by[hT(i) (0) , . . . , hT(i) (L) , . . . , 0T

]Twhere

h (l) =[h(mi1

) (l) , h(mi2) (l)]T, 1 ≤ mi1/mi2 ≤ q. (36)

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Then, for each i-th (1 ≤ i ≤ q − 1) pair of channel outputs,we consider a vector y(i)

cp,k of 2N × 1 successive samples:

y(i)cp,k = Hcp,(i)F

HN sk + n(i)

cp,k (37)

where

y(i)cp,k =

[y(i)Tcp,k (0) , y(i)T

cp,k (1) , . . . , y(i)Tcp,k (N − 1)

]Tn(i)

cp,k =[n(i)T

cp,k (0) , n(i)Tcp,k (1) , . . . , n(i)T

cp,k (N − 1)]T

(38)

and

y(i)cp,k(n) =

[y(mi1

)

cp,k (n) , y(mi2

)

cp,k (n)]T

n(i)cp,k(n) =

[n

(mi1)

cp,k (n) , n(mi2

)

cp,k (n)]T. (39)

The corresponding covariance matrix is expressed as

R(i)n = Hcp,(i)F

HNRsFN HH

cp,(i) + σ2I2N (40)

It is clear that each R(i)n is a sub-matrix of the system output

covariance matrix Rn and has a least eigenvector

vcp,(i) =[vTcp,(i)(0), vTcp,(i)(1), . . . , vTcp,(i)(N − 1)

]T(41)

where each sub-vector is defined according to Definition 1 as2-elements vector (0 ≤ n ≤ N − 1)

vcp,(i)(n) =[v(mi1

)

cp,k (n) , v(mi2

)

cp,k (n)]T. (42)

A set of independent n× 1 noise vectors

Vcp,(i) =[vTcp,(i)(0), vTcp,(i)(1), . . . , vTcp,(i)(N − 1)

]T(43)

is then computed from a set of least eigenvectors{vcp,(i)

}1≤i≤q−1

in the following way [4]: For j = 1, . . . , q,

vi (j) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if the j-th output of the systemdoes not belong to the i-th pair

vi

(j′

)if the j-th output of the system

is the j′

-th member of the i-th pair

(44)

It is well known [17], [42], that Gcp can be uniquely spannedby a basis of q − 1 vectors Vn =

[Vcp,(1), . . . ,V,cp,(q−1)

].

Therefore,

VHn HcpFHN = 0 (45)

where

vHcp,(i)Hcp,(i)FHN = 0. (46)

It has been shown in [25], that h is uniquely identifiable bysolving the linear system of equations (45) or (46). Based on[10], [25], we have

VHn HcpFHN = hHEcpFHN (47)

where Ecp =∑q−1i=1

{Ecp,(i)

}and Ecp,(i) = [E(1)

cp,(i),E(2)cp,(i)]

is a ϑ × N block-matrix, with a ϑ × (N − L)

block-Hankel matrix E(1)cp,(i) with first block-column[

vTcp,(i)(0), . . . , vTcp,(i)(L)]T

and last block-row

[vcp,(i)(L), . . . , vcp,(i)(N − 1)

], and ϑ × L block-Hankel

matrix E(2)cp,(i) with first block-column

[vTcp,(i)(N − L), . . . ,

vTcp,(i)(N − 1), vTcp,(i)(0)]T

and last block-row[vcp,(i)(0), . . . , vcp,(i)(L− 1)

]. Obviously, matrix Hcp,(i)

differs from the filtering matrix Ecp,(i) by interchange ofrows. Hence, the column spaces of Hcp,(i) and Ecp,(i)canonically equivalent. Therefore

∥∥VHn HcpFHN∥∥2 has to be

solved in the least-squares sense leading to the followingquadratic optimisation criterion:

h = arg min‖h‖=1

(hHQcph

)(48)

where

Qcp = Ecp

IN︷ ︸︸ ︷FHNFN EHcp. (49)

This quadratic optimisation criterion allows unique estimationof h up to a scale factor and is thus obtained as the eigenvectorassociated with the minimum eigenvalue of Qcp. In practice,

Vn is used instead of Vn, and Qcp is used instead of Qcp.

B. ZP-OFDM Estimator

Consider the i-th pair 2M × N block-Toeplitz matrix as,Hzp,(i),whose first block-row is given by

[h(i) (0) , 0, . . . , 0

]and first block-column is given by

[hT(i) (0) , . . . , hT(i) (L)

, 0T , . . . , 0T]T

. Then, for each i-th pair of channel outputs,

we consider a vector x(i)zp,k of 2M × 1 successive samples:

x(i)zp,k = Hzp,(i)F

HN sk + n(i)

zp,k (50)

where

x(i)zp,k =

[x(i)Tzp,k (0) , x(i)T

zp,k (1) , . . . , x(i)Tzp,k (M − 1)

]Tn(i)zp,k =

[n(i)Tzp,k (0) , n(i)T

zp,k (1) , . . . , n(i)Tzp,k (M − 1)

]T(51)

and

x(i)zp,k(m) =

[x

(mi1)

zp,k (m) , x(mi2

)

zp,k (m)]T

n(i)zp,k(m) =

[n

(mi1)

zp,k (m) , n(mi2

)

zp,k (m)]T. (52)

The corresponding covariance matrix is expressed as

R(i)m = Hzp,(i)F

HNRsFN HH

zp,(i) + σ2I2M (53)

and its least dominant eigenvector

vzp,(i) =[vTzp,(i)(0), vTzp,(i)(1), . . . , vTzp,(i)(M − 1)

]T(54)

where each sub-vector is defined according to Definition 1 as2-elements vector (0 ≤ m ≤M − 1), i.e.,

vzp,(i)(m) =[v(mi1

)

zp,k (m) , v(mi2

)

zp,k (m)]T. (55)

A set of independent noise vectors

Vzp,(i) =[vTzp,(i)(0), vTzp,(i)(1), . . . , vTzp,(i)(M − 1)

]T(56)

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is then computed from a set of least eigenvectors{vzp,(i)

}1≤i≤q−1

based on (44), we form a m× q− 1 matrix

Vm =[Vzp,(1), . . . ,Vzp,(q−1)

]. Similar to (45) and (46) in

CP-OFDM estimator, we have

VHmHzpFHN = hHEzpFHN . (57)

where Ezp =∑q−1i=1

{Ezp,(i)

}, and Ezp,(i) is a

ϑ × N block-Hankel matrix, with first block-column[vTzp,(i)(0), vTzp,(i)(1), . . . , vTzp,(i)(L)

]Tand last block-row[

vzp,(i)(L) , vzp,(i)(L+ 1), . . . , vzp,(i)(M − 1)]. Therefore∥∥VHn HzpFHN

∥∥2 has to be solved in the least-squares senseleading to the following quadratic optimisation criterion:

h = arg min‖h‖=1

(hHQzph

)(58)

where

Qzp = Ezp

IN︷ ︸︸ ︷FHNFN EHzp. (59)

This quadratic optimisation criterion allows unique estimationof h up to a scale factor and is thus obtained as the eigenvectorassociated with the minimum eigenvalue of Qzp. In practice,

Vm is used instead of Vm, and Qzp is used instead of Qzp.

VII. SMNS-BASED METHOD

It is obvious that MNS-based method does not requireEVD and instead computes the noise subspace via a set ofnoise vectors that can be computed in parallel from a set ofpairs of system outputs, and, the original approach [10], [25],is computationally expensive due to the employment of theEVD. Indeed, depending on the chosen PCS, certain systemoutputs are used more than others. However, this might lead topoor estimation performances particularly if the ’worst systemchannels’ are used most. This leads to the problem of findingthe ’best’ choice of PCS. In the symmetric MNS (SMNS), weavoid that problem by using q noise vectors instead of q − 1as in MNS-based method. The definition is given as follows:

Definition 2: Denote the q system outputs {t1, t2, . . . , tq} bya set of members m1, . . . ,mq . A combination of two (q ≥ 2)members (ti = mi1 ,mi2) is called a pair. A sequence of q pairsis sufficient to give superior channel identification, if each pairof {t1, t2, . . . , tq−1} in the sequence consists of one membershared by its preceding pairs and another member not sharedby its preceding pairs, while the last pair {tq} correspond to theadditional redundancy to guarantee certain symmetry betweenthe system outputs.

Example 2: Consider a system with one input and sevenoutputs. The following sequence of pairs has minimum redun-dancy (seven pairs) and spans all system outputs m1, . . . ,m7.

t1 = (m1,m2) , t2 = (m1,m3) , t3 = (m3,m4)

t4 = (m3,m5) , t5 = (m5,m6) , t6 = (m5,m7)

t7 = (m1,m7) .

VIII. LINEAR EQUALISATION

The problem of extracting communication signals usingsubspace-decomposition approach in OFDM-based SIMO is ofincreasing importance. The proposed SS-, MNS- and SMNS-based methods that are described in the previous sectionsare applicable to solve this problem. In this section, wedevelop an equalisation scheme based on the estimated channelparameters.

Given the estimated multi-channel matrix Hcp, Hcp, ob-tained through h in the previous section by the proposed

estimators, the received signal matrix Ycp ={

ycp,k}K−1

k=0can be written as

Ycp = HcpFHNSK + Ncp (60)

where SK = {sk}K−1k=0 and Ncp =

{ncp,k

}K−1

k=0. From

estimation theory, the maximum likelihood (ML) estimate ofSK is given by :

SK = GZFYcp (61)

where

GZF =(

HcpFHN)†

(62)

and † denotes Pseudo-inverse. The ZF equaliser satisfies thecondition

GZFHcpFHN = IN (63)

and the source symbol can be recovered provided H has fullrank. The ZF equaliser is expected to suffer performancedegradation due to noise enhancement and when there areclose common zeros. To overcome this problem, we nowconsider the MMSE equaliser, which aims to minimise

E{‖sk − sk‖

2}

(64)

where sk − sk is the error in the k-th block of data. Now byallowing G to represent the equaliser matrix:

sk = Gycp,k = GHcpFHN sk + Gncp,k (65)

and thus

sk − sk = Gycp,k − sk

= GHcpFHN sk + Gnk − sk

=(

GHcpFHN − I)

sk + Gncp,k − sk. (66)

The MSE can be written as a function of the equalising matrixG as follows

J (G) = E

{∥∥∥(GHcpFHN − I)

sk + Gncp,k∥∥∥2} . (67)

The MMSE solution is obtained by setting to zero the gradientof J (G) with respect to G and then solving for G (see [36]).We thus obtain

GMMSE = RsyR−1n (68)

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where, using the fact that additive noise is independent of thetransmitted data,

Rsy = E{

skyHcp,k}

= E

{sk[HcpFHN sk + ncp,k

]H}= RN

(HcpFHN

)H. (69)

The linear MMSE estimate of SK is thus:

SK = GMMSEYcp. (70)

Using similar framework of pre-FFT ZF and MMSE equalisersfor CP-OFDM receivers, ZF and MMSE equalisers can alsobe derived for ZP-OFDM receiver. For a given block of dataXzp =

{xzp,k

}K−1

k=0and the estimated multi-channel matrix

Hzp, we have

Xcp = HzpFHNSK + Vzp (71)

where Vzp ={

vzp,k}K−1

k=0. Based on the aforementioned

assumptions and (61)-(70), the linear ZF estimate of SK isthus:

SK =(

HzpFHN)†

Xzp (72)

whereas the linear MMSE estimate is expressed as

SK = RN(

HzpFHN)H

R−1m Xzp (73)

As can be seen, both ZF and MMSE equalisation for CPand ZP are performed through one-IFFT at the correspondingreceiver.

IX. NUMERICAL EXAMPLES

Configuration 1 Configuration 1

0 1 2 0 1 2

0.5-0.5i

0.5-0.5i 0.6-0.6i

0.7-0.7i 0.5-0.5i

0.8-0.8i

0.11-0.11i0.6-0.6i

0.7-0.7i 0.5-0.5i

0.8-0.8i

0.11-0.11i

l

h(1)(l)

h(2)(l)

TABLE I

CHANNEL SET 1: IMPULSE RESPONSE.

20 1 3

-0.049+0.359i

4

0.482-0.569i -0.556+0.587i 1.0000 -0.171+0.061i

0.443-0.0364i 1.0000 0.921-0.194i 0.189-0.208i -0.087-0.054i

-0.211-0.322i -0.199+0.918i 1.0000 -0.284-0.524i 0.136-0.190i

0.417+0.03i 1.0000 0.873 + 0.145i 0.285+ 0.309i -0.049+0.161i

l

h(1)(l)

h(2)(l)

h(3)(l)

h(4)(l)

TABLE II

CHANNEL SET 2: IMPULSE RESPONSE.

In this section, we provide simulation results demonstratingthe performance of the proposed estimators and the corre-sponding multi-channel equalisers. In all simulation examples,

5 10 15 20 25 30−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

SNR (dB)

MS

E (

dB)

Unweighted CP−OFDM Weighted CP−OFDM Unweighted ZP−OFDM Weighted ZP−OFDM

Fig. 4. Performance comparison of CP-OFDM and ZP-OFDMsystems using the SS-based method: SNR vs MSE (Table I, Con-figuration 1).

5 10 15 20 25 30−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

SNR (dB)

MS

E (

dB)

Unweighted CP−OFDM Weighted CP−OFDM Unweighted ZP−OFDM Weighted ZP−OFDM

Fig. 5. Performance comparison of CP-OFDM and ZP-OFDMsystems using the SS-based method: SNR vs MSE (Table I, Con-figuration 2).

the estimator performance was measured in terms of the MSE[3], [21]

MSE =

√√√√ 1

Nr

Nr∑κ=1

∥∥∥hκ − h∥∥∥2 (74)

where hκ denotes the κ-th run estimate of h, and the firstelement of hκ is normalized to be one (the same as h) 1. Nrdenotes the number of runs and was chosen to be 500. Thesignal symbols were drawn from QPSK constellations. TheSNR of the q channels, for CP-OFDM system is defined as

SNRCP = 10log10

⎛⎜⎜⎝E

{∥∥∥ycp,k

∥∥∥2}E{∥∥ncp,k

∥∥2}⎞⎟⎟⎠ (75)

1Note that the expression in the right hand side of (74) is known as MSEin [3], [21]. While in (74), it is known as the normalized root MSE in [10].

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5 10 15 20 25 30 35 40 45 50−70

−60

−50

−40

−30

−20

−10

0

10

SNR (dB)

MS

E (

dB)

MNSSMNSSS

Fig. 6. Performance comparison of CP-OFDM estimators using theMNS-, SMNS-, and SS-based methods: SNR vs MSE.

5 10 15 20 25 30 35 40 45 50−70

−60

−50

−40

−30

−20

−10

0

SNR (dB)

MS

E (

dB)

MNSSMNSSS

Fig. 7. Performance comparison of ZP-OFDM estimators using theMNS-, SMNS-, and SS-based methods: SNR vs MSE.

and for ZP-OFDM system, it is defined as

SNRZP = 10log10

⎛⎝E{∥∥xzp,k

∥∥2}E{∥∥vzp,k

∥∥2}⎞⎠ . (76)

In [21], (75) and (76) has been shown to be given by

SNR = 20log10

(‖h‖σs√qσ

). (77)

Simulation Example 1: We first investigated the influenceof weighting and compare the CP-OFDM and ZP-OFDMreceivers through the implementation of the SS-based methodin terms of their estimation capabilities. We fixed the numberof OFDM symbols to K = 500, and varied the SNR from 5to 30 dB. We simulated the output of a SIMO with q = 2FIR channels of maximum order L = 2. The generatedsymbols are transmitted through 20 sub-carriers and so thesize of the FFT/IFFT was N = 20. Two extreme situations ofchannel coefficients were considered, as shown in Table I. InConfiguration 1, the zeros of the channel are closed together,whereas in Configuration 2, the zeros of the channel are well

100 200 300 400 500 600 700 800 900 1000−45

−40

−35

−30

−25

−20

−15

−10

Number of OFDM symbols

MS

E (

dB)

MNSSMNSSS

Fig. 8. Performance comparison of CP-OFDM estimators using theMNS-, SMNS-, and SS-based methods: Number of OFDM symbolsvs MSE.

100 200 300 400 500 600 700 800 900 1000−50

−45

−40

−35

−30

−25

−20

−15

Number of OFDM symbols

MS

E (

dB)

MNSSMNSSS

Fig. 9. Performance comparison of CP-OFDM estimators using theMNS-, SMNS-, and SS-based methods: Number of OFDM symbolsvs MSE.

separated. In Figures 4 (corresponding to the Configuration 1)and 5 (corresponding to the Configuration 2), we observed thatthe performance of both estimators improves with increasingSNR and in comparison to the CP-OFDM estimator, the ZP-OFDM estimator performs better. It is worthwhile to notethat the gain afforded by optimal weighting is not significantwhen the SNR is large, even when the channel zeros comeclose together: Optimal weighting does not compensate forthe performance loss entailed by the poor condition number ofthe channel matrix. For lower SNR, some improvements canbe observed, especially when the zeros are closely located.The impact of close channel zeros can be quantified in termsof SNR. In fact, badly spaced zeros cause the degradationof the Circulant and Toeplitz matrices condition number, andthus, require higher values of SNR in order to maintain theestimation accuracy.Simulation Example 2: In this simulation study, we simulateda CP-OFDM system with q = 4, N = 64 sub-carriers and aCP of length L = 4 (i.e., M=N+L=68). The simulated channel

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2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

BE

R

ZF CP−OFDMMMSE CP−OFDMZF ZP−OFDMMMSE ZP−OFDM

Fig. 10. Performance comparison of CP-OFDM and ZP-OFDMsystems, using ZF and MMSE-based equalizers: SNR vs BER.

coefficients are given by Table II. Figure 6 illustrates theperformance of the CP-OFDM estimators implementing MNS-, SMNS- and SS-methods. We fixed the number of OFDMsymbols to K = 500, and varied SNR from 5 to 50. We cansee that the performance of CP-OFDM estimators improveswith high SNR. In comparison with the MNS estimator, theSMNS estimator method performs better. Among the threeestimators, the SS estimator is much more robust to noiseand gives superior performance. A similar scenario has beenrepeated for the ZP-OFDM system, with the same setting data.Figure 7 shows the MSE as a function of SNR for MNS,SMNS and SS estimators. It can be seen that the SS estimatorperforms better than the MNS and the SMNS estimators.Simulation Example 3: In this simulation study, we comparedthe performance of the CP-OFDM and the ZP-OFDM systemas a function of the number of OFDM symbols at a SNR of25 dB using similar simulation scenario of Example 2. Theexamples of Figures 8 and 9 compare the performance of theCP-OFDM and ZP-OFDM as a function of the number ofOFDM symbols vs MSE. In both systems, we can see thatthe performance of all the estimators improve with increasingnumber of OFDM symbols. Additionally, the SS estimatoris able to identify channels with much smaller number ofOFDM symbols. SMNS estimator has better performance incomparison with MNS.Simulation Example 4: Figure 10 shows the overall BERperformance of the proposed MNS-based method for theCP-OFDM and ZP-OFDM systems corresponding to SNRrange of 2-20 dB. In order to check the equaliser gain, wesimulated ZF and MMSE schemes which have been discussedin Section VIII. It can be seen that ZF receivers suffer fromperformance penalty in all cases as compared to MMSEreceivers. Moreover, ZP-OFDM receivers perform better thanCP-OFDM receivers.

X. PROPERTIES OF THE PROPOSED TECHNIQUES

In all the aforementioned SOS-based methods for OFDM-based SIMO systems, the focus has been on channel identifica-tion and equalisation. In this section we give some comments

on the above proposed techniques. For uniformity, we subse-quently drop the subscripts m/n and express the covariancematrices as R(i) (corresponding to SS-based method) andR(i)

(corresponding to MNS/SMNS-based method). Moreover,we drop the subscripts cp/zp and express the block-Circulantmatrix Hcp,(i) and block-Toeplitz matrix Hzp,(i) as H(i) . Forthe sake of simplicity, we consider

ρ =

{2N CP-OFDM2M ZP-OFDM

(78)

ν =

{qN CP-OFDMqM ZP-OFDM.

(79)

• The proposed channel estimators can be made to exploitthe signal subspace regardless of the noise subspace andtherefore the minimisation problem can be recast as amaximisation problem [5], [1], [25]. The solution ofmaximisation problem is considered more favorable tothe minimisation problem, as there are fundamental lim-itations on the relative accuracy with which the smallesteigenvalues of the matrix can be computed, and they aremore difficult to compute than the large ones. However, itis shown in [25] that the noise SS-based method exhibitsbetter performance than the signal SS-based method.

• The main advantage of the MNS-based methods is thatthe large matrix EVD is avoided and the noise vectors arecomputed in parallel as the least eigenvectors of a smallersize covariance matrices R(i)

, i = 1, . . . , q − 1, whichrequires only O(ρ2) flops (in contrast with the O(ν3)flops required for the computation of R). Comparatively,the SMNS- and MNS-based methods have almost thesame order of computational cost for SIMO systems 2.

• The proposed ZP-OFDM estimator requires the EVD ofa data correlation matrix of size m × m to extract theorthogonal subspace. In contrast, the proposed CP-OFDMestimator requires the EVD of data correlation matrixof size n × n. Since m > n, the proposed ZP-OFDMestimator is computationally more complex than the CP-OFDM estimator.

• The CP-OFDM estimator is sensitive to channel zerosthat are closed to the sub-carriers, whereas, the ZP-OFDM estimator guarantees symbol recovery and offersa superior BER performance.

• The proposed CP-OFDM based SIMO system rely onthe usual insertion of CP as in standard OFDM systems.Therefore, it does not require transmitter modificationand is applicable to all standardised OFDM systems. Incontrast, the ZP-OFDM based SIMO system presentedrequires transmitter modification to introduce ZP redun-dancy by a filter-bank precoder. Note that ZP is used inthe DVB standard in the form of guard bits.

• By using the optimal weighting matrix W(i), the iden-

tification procedure becomes quite insensitive to the ill-conditioning problem. In fact, if a pair of channels have

2SMNS-based method becomes computationally more expensive if q >>(q − p) (where p is the number of sensors at the transmitter).

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close common zeros, the corresponding block-channelmatrix H(i) becomes nearly singular and consequently

W(i)becomes large. Therefore, the inverse of the weight-

ing matrix,(

W(i))−1

, will qualitatively provide ’moreweighting’ (i.e., larger weighting coefficients) to the noiseeigenvectors associated with a well conditioned H(i)

than to those corresponding to ill conditioned block-channel matrix. However, weighting MNS (WMNS)-based method often incurs high complexity and involveslarge decoding delay, and does not trade well for theaccuracy improvement [5]. For this reason, WMNS-basedmethod has not been considered and investigated forOFDM-based SIMO systems in this paper.

XI. CONCLUSION

This paper presents original reformulation of the SS-basedestimation procedure for the blind identification of OFDM-based SIMO FIR channels. It fully exploits the relations be-tween the noise subspace of a certain covariance matrix formedfrom the observed signals. This reformulation provides someadditional insights into the existing subspace algorithms. Moreimportantly, it allows one to analyse the second order statisticsof the output signals for the case of CP-OFDM and ZP-OFDM receivers. This technique, although reliable and robustin some scenarios, require a computationally expensive andnon-parallelisable EVD to extract the noise subspace. In fastchanging environments, such as in cellular communications,their application may be limited and impossible (too costly)to implement. These problems are alleviated by a MNS-based method which exploits a minimum number of noiseeigenvectors for multi-channel identification. This techniqueof MNS, and especially the concept of PCS, turns out to be apowerful tool that can be applied to other OFDM-based SIMOsystems. The proposed MN-based method is much more com-putationally efficient than the standard SS-based method at theprice of a slight loss of estimation accuracy. However, betterestimates of FIR channels can be obtained by a symmetricversion of MNS with the same order of computational cost.Furthermore, equalisation methods are discussed based on theestimated channels via one-IFFT operation. Simulations haveshown that the proposed method are effective and robust.

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Faisal O. Alayyan attained his B.Sc. degree in Communication Engineeringfrom Etisalat University, College, UAE in 2001. He has also receivedhis M.Sc. degree in Communications from the University of Melbourneand Ph.D. degree from Curtin University, Australia in 2002 and 2008,subsequently. He has worked as a research scientist in Paris Telecom,France. Currently, he is a research fellow in Signal Processing at DarmstadtUniversity of Technology, Germany. His research interests are in OrthogonalFrequency Division Multiplexing, and channel identification and equalization.

Raed M. Shubair received his B.Sc. degree with Distinction and ClassHonors from Kuwait University, Kuwait, in June 1989 and his Ph.D. degreewith Distinction from the University of Waterloo, Canada, in February1993, both in Electrical Engineering. From March 1993 to August 1993 hewas a Postdoctoral Fellow at the Department of Electrical and ComputerEngineering, University of Waterloo, Canada. In August 1993 he joinedKhalifa University of Science, Technology and Research (formerly EtisalatUniversity College), UAE, where he is currently an Associate Professor at theCommunication Engineering Department and Leader of the Communicationand Information Systems Research Group. His current research interestsinclude adaptive array and multi-channel processing, smart antennas andMIMO systems, as well as applied and computational electromagneticmodeling of RF and microwave circuits for wireless communications. Hehas published over 90 papers in refereed technical journals and internationalconferences. He has been a member of the technical program, organizing, and

steering committees of numerous international conferences and workshops.He organized and chaired a number of technical sessions in internationalconferences including IEEE Symposium on Antenna and Propagation (AP-S),IEEE Symposium on Electronics, Circuits and Systems (ICECS), Progress inElectromagnetics Research Symposium (PIERS), and Applied ComputationalElectromagnetics Symposium (ACES). Dr. Shubair was the lead facultymember at Etisalat University College to be elevated to IEEE SeniorMember grade back in 2001. He was honored by the Applied ComputationalElectromagnetics Society (ACES) for his valuable contributions to the ACESsociety in 2005. In recognition of his distinguished research record, Dr.Shubair was elected to become Fellow of MIT Electromagnetics Academyin 2007. Dr. Shubair received the ”Distinguished Research Activities Award”at Khalifa University in March 2008. He has supervised a large numberof undergraduate research projects including Adaptive Beamforming forNext-Generation Wireless Communications, recipient of the ”2004 IEEAward”, as well as Design of Optimum SMI Beamformers for SpatialInterference Rejection, recipient of the ”2005 IEE Award”. Dr. Shubair actedas the main supervisor of the postgraduate research project entitled SmartAntenna Techniques for Broadband Wireless Access recipient of the ”BestM.Sc. Award” in 2005. His papers were selected for three successive years(2004-2006) amongst the ”Selected Best Papers” of the IEEE InternationalConference on Innovations in Information Technology held annually inDubai, UAE. Dr. Shubair is a founding member of the IEEE UAE SignalProcessing and Communications Joint Societies Chapter. He currently acts asSymposium Vice-Chair of the 2008 IEEE Canadian Conference on Electricaland Computer Engineering: Communications and Networking Symposium.Dr. Shubair is Editor for the Journal of Communications, Academy Publisher.He is listed in Who’s Who in Electromagnetics and in several editions ofWho’s Who in Science and Engineering.

Abdelhak M. Zoubir received the Dipl.-Ing degree (BSc/BEng) fromFachhochschule Niederrhein, Germany, in 1983, the Dipl.-Ing. (MSc/MEng)and the Dr.-Ing. (PhD) degree from Ruhr University Bochum, Germany, in1987 and 1992, all in Electrical Engineering. Early placement in industry(Kloeckner-Moeller and Siempelkamp AG) was then followed by AssociateLectureship in the Division for Signal Theory at Ruhr University Bochum,Germany. In June 1992, he joined Queensland University of Technologywhere he was Lecturer, Senior Lecturer and then Associate Professor in theSchool of Electrical and Electronic Systems Engineering. In March 1999, hetook up the position of Professor of Telecommunications at Curtin Universityof Technology, where he was Head of the School of Electrical and ComputerEngineering from November 2001 until February 2003. In February 2003 hetook up the position of Professor in Signal Processing at Darmstadt Universityof Technology. In January 2008, His general interest lies in statistical methodsfor signal processing with applications in communications, sonar, radar,biomedical engineering and vibration analysis. His general interest lies instatistical methods for signal processing with applications in communications,sonar, radar, biomedical engineering and vibration analysis. His currentresearch interest lies in bootstrap techniques for modeling non-stationary andnon-Gaussian signals, for the design of multi-user detectors in the presenceof impulsive interference and for low rank channel estimation for WidebandCDMA. He was the General Co-Chairman of the Third IEEE InternationalSymposium on Signal Processing and Information Technology (ISSPIT) heldin Darmstadt in December 2003, the Technical Chairman of the 11th IEEEWorkshop on Statistical Signal processing held in Singapore in August 2001and Deputy Technical Chairman (Special Sessions/Tutorials) of ICASSP-94held in Adelaide. He served as an Associate Editor of the IEEE Transactionson Signal Processing from 1999 until 2005 and is currently an AssociateEditor of the EURASIP journals Signal Processing and Journal of AppliedSignal Processing. He is a Member of the IEEE SPS Committees on SignalProcessing Theory and Methods and Signal Processing Education. He is aSenior Member of the IEEE and a Member of the Institute of MathematicalStatistics. Prof. Zoubir was elected as an IEEE Fellow for contributions tostatistical signal processing.

Yee H. Leung received his Bachelor (BE) and PhD degrees, in ElectricalEngineering, from the University of Western Australia (UWA). He has workedas a research scientist in DSTO, a lecturer at UWA, and a research fellowin the Western Australia Telecommunications Research Institute (WATRI).Currently he is a Senior Lecturer in the Department of Electrical and ComputerEngineering at Curtin University. His research interests are in adaptive androbust signal processing, and in their applications to telecommunicationsystems.

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