Blind SOS subspace channel estimation and equalization techniques exploiting spatial diversity in...

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Digital Signal Processing 14 (2004) 171–202

www.elsevier.com/locate/ds

Blind SOS subspace channel estimation andequalization techniques exploiting spatial diversi

in OFDM systems

Hassan Ali,a,∗,1 Arnaud Doucet,b and Yingbo Huac

a Communications and Signal Processing Group, Schoolof Electrical and Computer Engineering,Curtin University of Technology, GPO Box U1987, Perth 6845, Australia

b Signal Processing Group, Department of Engineering,University of Cambridge, Trumpington Street,Cambridge, CB2 1PZ, UK

c Department of Electrical Engineering, University of California, Riverside, CA 92521, USA

Abstract

In this paper, we present second-order-statistics (SOS) subspace-based blind channel estechniques, which exploit the receive antenna diversity in each of the following situations:prefix-OFDM (CP-OFDM), zero padded-OFDM (ZP-OFDM), and bandwidth efficient-OF(BWE-OFDM) systems. We also propose a number of combinations of pre-FFT equalizerFFT equalizer, zero-forcing (ZF) equalizer, and minimum-mean-square-error (MMSE) equFor any number of receive antennas, the pre-FFT equalizers require only one FFT at the rAs a result, a considerable reduction in hardware complexity and power saving may be obespecially for systems with higher number of sub-carriers. In contrast, post-FFT equalizers rconsiderable reduction in processing complexity at the cost of one FFT for each antenna. Alinear-complexity implementations of the proposed receivers are considered, along with somefications in the presence of null side carriers. The effectiveness of the new techniques is demothrough simulations. 2003 Elsevier Inc. All rights reserved.

Index terms:Blind channel estimation and equalization; CP-OFDM; ZP-OFDM; BWE-OFDM; Spatial dive

* Corresponding author.E-mail addresses:[email protected] (H. Ali), [email protected] (A. Doucet), [email protected]

(Y. Hua).1 Part of this work was done when he was with the ARC Special Research Centre for Ultra-Broa

Information Networks (CUBIN), Department of Electrical and Electronic Engineering, The UniversiMelbourne, Parkville 3010, Victoria, Australia.

1051-2004/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S1051-2004(03)00008-3

172 H. Ali et al. / Digital Signal Processing 14 (2004) 171–202

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1. Introduction

Orthogonal frequency division multiplexing (OFDM) [1] is now considered an eftive technique for digital broadcasting (terrestrial digital audio, video broadcasting: DDVB [2,3]) but also for high rate modems over twisted pairs. OFDM has a relatively losymbol duration which produces greater protection to multi-path interference and imnoise. It reduces bit rate of each carrier against inter-symbol-interference (ISI) problem andalso provides high bit rate transmission by using a number of those low bit rate caFrequency bandwidth is partitioned into several small parallel independent sub-chand each of them is handled by these low rate carriers. As shown in Fig. 1a, OFDprovide immunity against frequency selective fading environment: bandwidth of the easub-carrier is narrow enough compared with that of conventional single-carrier moduIn an OFDM system, some carriers may be attenuated by frequency selective fading, bother carriers may not be attenuated. Therefore a multi-carrier OFDM system can trdata correctly. Also, due to the ability to transmit different data using several orthognal overlapping sub-carriers an OFDM system increases bandwidth efficiency and scapacity.

In standard OFDM systems (i.e., cyclic prefix-OFDM (CP-OFDM)), the probleminter-block-interference (IBI) arising dueto channel memory is taken care by repeatinglast few samples of each OFDM symbol at its beginning, prior to its transmission. Thadded guard interval is known as the cyclic prefix (CP). The CP converts linear contion into circular convolution. This allows diagonalization of the associated channel mand thus equalization of the channel distortion in the frequency domain by using atap equalizer for each carrier independently. For equalization purposes in OFDM, a knowtraining sequence is sent by the transmitter and a training algorithm is performedreceiver of the observed channel output and the known input to estimate the channel [This solution increases the overhead of the system and consumes the valuablebandwidth. Implementation of blind channel estimation algorithms, on the other hand, apear attractive since they avoid the use of the training sequences, save bandwidthcapable of tracking slow channel variations. A variety of techniques are available forchannel identification (for example, see [5–10]).

Fig. 1. (a) Robustness to frequency selective fading:single-carrier modulation and multi-carrier OFDM modution. (b) Coded-OFDM (COFDM).

H. Ali et al. / Digital Signal Processing 14 (2004) 171–202 173

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Pushed by the recent requirement of tremendous capacity increase for voice, Internand multimedia traffic along with high speed data handling capabilities in future getion mobile wireless systems, OFDM is newly considered for mobile wireless broadsystems (ETSI BRAN, IEEE802.11a [11], MMAC, and HIPERLAN/2 [12]). However,ing OFDM in mobile channels iscritical because of the impactof the time variant naturof the channel and thus resulting impact of frequency nulls (deep fades). That is ifcarrier coincides with the channel null (in other words when the channel has nullsfades) on the FFT grid), then information carried by that sub-carrier is lost.

A number of solutions have been proposed to make OFDM more robust to this nband fading. The first is to employ channel coding together with interleaving and/oquency hopping as in the so called coded-OFDM (COFDM) [13,14] (see Fig. 1b) vaFor example, the conventional trelliscoded modulation (TCM) was used in [13], andcodes were used in [14]. However, COFDM techniques often incur high complexitinvolve large decoding delay [15]. Some of them require channel state informationat the transmitter [16,17], which may be unrealistic or too costly to acquire in wirapplications where the channel changes on constant basis [18]. COFDM techniques futhermore come at the price of bandwidth efficiency reduction which is not compatiblebandwidth expensive mobile and wireless applications. The second way to guaransymbol detect-ability in presence of bad carriers is zero padded-OFDM (ZP-OFDMstead of introducing the CP, after each IFFT processed block a zero padded guardis inserted. Some channel estimation techniques employing ZP precoding to combcaused by multi-path channel have also been reported [19]. However, ZP-OFDMmentation involves transmitter modification and complicates the equalizer. Anotheof robustifying OFDM against random frequency selective fading is to introduce meinto the transmission by linear precoding (LP) [18] across the sub-carriers. LP-OFDMcomes at the price of complicating the transmitter and receiver and furthermore at thof bandwidth reduction.

Our way of robustifying OFDM against faded sub-carriers is to exploit rich spatiaversity by employing multiple antennas at the receiver. The objective that motivateuse of spatial diversity is threefold: first, the maximization of the mean signal powerrespect to any noise thereby resulting in improving link budget, signal quality, and capaity of communication system, second, mitigating the impact of deep fades, and third,additional power or bandwidth consumption. The transceiver system is essentially ainput/multiple-output (SIMO) system with a single antenna at the base station and mantennas at the mobile terminals or vice versa. Multiple antennas are traditionally uthe base station in order to achieve high performance and capacity. The use of mantennas at the mobiles is now being considered. This enables parallel data transmimproved signal-to-interference ratio (SIR) and extended coverage [20].

Figure 2 shows the concept of the proposed spatial diversity based OFDM systemreceiver has multiple antennas with independent fading patterns and whose transma single antenna. The use of multiple antennas at the receiver makes fading to bmore randomized between OFDM tones of each antenna than the conventional systemtwo or more antennas are placed at the receiver, each would have a different set opath signals giving effect to different fading for each carrier at each antenna. Theof each channel would vary from one antenna to the next, therefore, carriers that m


iver an-s withploitnd

grted,cients.

lly ex-28].pt.diver-mon

l basedrst, wechniqueost-

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Fig. 2. Concept of randomized fading due to spatial diversity in OFDM systems.

unusable on one antenna due to fading become usable on another. Though recetenna diversity based beam-forming techniques can be applied in OFDM systemco-channel interference [21]. However, beam-forming is not always able to fully exall the path gains [22]. Furthermore, limitations of beam-forming techniques for broadbawireless Internet access arises from implementation [23]. A couple of diversity combinintechniques [24–26], including Cisco’s vector OFDM (VOFDM) [27] have been repohowever, they use training sequences or pilot tones to estimate the channel coeffiWhen viewed as FIR SIMO system, the receiver antenna diversity can be optimaploited by the blind two-step maximum likelihood (TSML) approach proposed in [But in this paper, we consider a suboptimal approach based on the subspace conce

Motivated by aforementioned challenges and convincing advantages of spatialsity, in this paper, we propose blind spatial diversity combining techniques in a comframework. These techniques can be seen as counterpart to training or pilot symbodiversity combining and above described robust frequency selective techniques. Fipropose a second order statistical (SOS) based blind subspace channel estimation teexploiting spatial diversityin the CP-OFDM. We also propose corresponding pre- and pFFT zero-forcing (ZF) and minimum-mean-square-error (MMSE) equalizers. SecoZP-OFDM offers a number of advantages over CP-OFDM, we also develop a blindspace channel estimator exploiting spatial diversity in ZP-OFDM and corresponding prand post-FFT ZF and MMSE equalizers. A number of equalization techniques [2have recently been proposed for spectrally efficient OFDM systems which do not usor ZP guard interval. We thus extend the idea of spatial diversity to bandwidth efficOFDM (BWE-OFDM) and propose a spatial diversity exploiting blind channel estimatoand corresponding ZF and MMSE equalizers. Linear adaptive implementation of thposed receivers is considered. It is furthermore shown how the proposed estimators t


zerosractical

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ion, wearriers.lsens and

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e CP,

asithpmentheoryoth theitstream

ach

into account known zeros in the input stream for channel identification. These knownin the input stream are referred to as virtual carriers and are sometimes used in pOFDM systems to create frequency guard bands [2].

The remainder of this paper is organized as follows. In Section 2, we describe thechannel baseband signal model for the CP-OFDM system. The new subspace cestimator and corresponding equalizers are presented in this section. In this sectalso propose modifications as to the proposed estimator in presence of null side cSections 3 and 4 similarly deal with ZP- andBWE-OFDM, respectively. Section 5 deawith the linear adaptiveimplementation of the proposed receivers. In Section 6, we describthe properties of proposed techniques. In Section 7, we present computer simulatiofinally, in Section 8, we summarize and draw some conclusions.

2. CP-OFDM spatial diversity exploiting channel estimation and equalization

In standard CP-OFDM systems, a CP equal to or greater than the channel orderally inserted as the guard interval to prevent IBI and allow one tap equalization.

2.1. Multi-channelbaseband data model

The development of the multi-channel baseband data model is based upon the followassumptions:

(1) The transmitted signals(n) goes throughZ finite impulse response (FIR) channe(due to over-sampling in spatial domain by aZ antenna receiver system) with complvalued impulse response vectorsh(r) = [h(r)(0), . . . , h(r)(L)]T , r = 1, . . . ,Z, whereLdenotes the known maximum channel orderL.

(2) Data model satisfiesP > M > L and CP guard intervalP − M � L. HereM andP denote symbol block length and transmitted data vector length including threspectively.

(3) We assume perfect synchronization of carriers and symbol blocks.

Figure 3 shows the concept of the proposedCP-OFDM system, whose transmitter hsingle antenna and receiver hasZ antennas. This wireless communication system wantenna array receiver can be modelled as a SIMO system. To facilitate the develoof the multi-channel model arising due to antenna array, we first visit some basic tfrom [5,32], in case of a mono-sensor system (i.e., we have a single antenna at btransmitter and receiver). Let us consider the CP-OFDM transmitter in Fig. 3a. First, the bstream is mapped to the complex valued data stream. The complex valued input datas(n) is parsed into blocks of sizeM : s(i) := [s(iM), s(iM + 1), . . . , s(iM + M − 1)]T ,where the block indexi = �n/M� and� . � denotes integer floor of the argument. For esymbol indexn, we can writen = iM +m, with m ∈ [0,M −1]. The indexn thus indicatesthe position of a symbol inside the symbol blocks(i) which constitutes theith OFDMsymbol. The symbol blocks(i) is then modulated by the IFFT matrixFH

M , whereFM

stands for the sizeM × M FFT matrix with entries(1/√

M)exp(−j2πmn/M). The data


valuestedply

r

d

ithn-ten

oid

Fig. 3. CP-OFDM: (a) transmitter, (b) pre-FFT antenna array receiver.

vectoru = FHMs(i) is then appended with a CP of lengthP −M, resulting in a sizeP > M

signal vector:u(i) := [u(iP ),u(iP + 1), . . . , u(iP + P − 1)]T . The elements ofs(i) areamplitudes modulating sub-carriers. They are often interpreted as frequency domainbecause they are carried byM normalized frequencies of OFDM modulation (implemenby M point IFFT).s(i) is therefore also called frequency domain OFDM symbol or simOFDM symbol.

DefineFcp := [Fcp,FM ]H as theP ×M matrix, whereFcp stands for theM × (P −M)

matrix corresponding to the lastP − M columns ofFM . The ith transmitted block (afteIFFT modulation and CP insertion) can thus be written as:u(i) := Fcps(i).

The resulting linearly precoded “OFDM chip sequence”u(n) is then pulse shapeto the corresponding continuous time signaluc(t) = ∑∞

n=−∞ u(n)ϕc(t − nT ), whereT

is the chip period andϕc(t) is the chip pulse. The transmitted waveformuc(t) propa-gates through a dispersive channelhc(t) and is filtered by the receive filterϕc(t). Often,hc(t) will have a simple physical characterization as a sum of multi-paths ash(t) =∑NR−1

k=0 αk exp(−jθk)δD(t − τk), whereNR is the maximum number of reflectors,αk isthe attenuation factor,θk is the phase shift, andτk is the propagation delay associated wthekth channel path. Leth(l) := (ϕc �hc � ϕc)(t)|t=lT be the equivalent discrete time chanel impulse response. The received signalxc(t) sampled at the chip rate can then be writas

x(n) := xc(t)|t=nT = h(n) � u(n) + v(n) =L∑

l=0

h(l)u(n − l) + v(n), (1)

wherev(n) := vc(t)|t=nT is the filtered additive white Gaussian noise (AWGN). To avIBI, the CP length is selected to be equal to or greater than the channel order, i.e.,P −M �L. Note that any OFDM system is designed to verify this condition.


oved.

ed

I

ub-

Let H0 be theP ×P lower triangular Toeplitz matrix with first column[h(0), . . . , h(L),

0, . . . ,0]T andH1 be theP × P upper triangular Toeplitz matrix with first row[0, . . . ,

h(L), . . . , h(1)].Based on the assumption thatP −M � L and the fact thath(l) = 0,∀l /∈ [0,L], (1) can

be written in the block form as

x(i) = H0u(i) + H1u(i − 1) + v(i) = H0Fcps(i) + H1Fcps(i − 1) + v(i), (2)

whereP ×1 vectors:x(i) := [x(iP ), x(iP +1), . . . , x(iP +P −1)]T andv(i) := [v(iP ),

v(iP + 1), . . . , v(iP + P − 1)]T .The second term in (2) denotes IBI. To remove IBI (and thus ISI) the CP is rem

This is done by removing the firstP − M entries from the vectorx(i), with the receive-matrix Rcp := [0M×(P−M), IM ]. The resulting received vector can thus be written as

y(i) = Rcpx(i) = RcpH0u(i) + RcpH1u(i − 1) + Rcpv(i)

= RcpH0Fcps(i) + n(i) = HFHMs(i) + n(i), (3)

wheren(i) is the M × 1 truncated noise vector andH is the M × M circulant chan-nel matrix, with first row [h(0),0, . . . ,0, h(L),h(L − 1), . . . , h(1)] and first column[h(0), h(1), . . . , h(L − 1)h(L), . . . ,0]T . Note thatRcpH1 = 0, therefore IBI is completelyremoved among OFDM symbols.

Now consider the proposed multi-channel FIRsystem. The chip rate sampled receivsignal at each antenna can be written as

x(r)(n) =L∑

i=0

h(r)(l)u(n − l) + v(r)(n), r = 1, . . . ,Z, (4)

where for therth channel:x(r)(n) is the output,{h(r)(l)}Ll=0 is the impulse response,v(r)(n)

is the AWGN, andu(n) denotes the common input to theZ channels. In this case, IBremovedith received vector at each antenna can then be written in block form as

y(r)(i) = H(r)FHMs(i) + n(r)(i). (5)

Stacking the outputs of theZ channels gives the composite receive vector:

y(i) = HFHMs(i) + n(i), (6)

wherey(i) = [y(1)T (i), . . . ,y(Z)T (i)]T , H = [H(1)T , . . . , H(Z)T ]T , andn(i) = [n(1)T (i),

. . . ,n(Z)T (i)]T .

2.2. Blind subspace channel estimator

The multi-channel circulant matrixH has full column rankM under the condition:

(a0) the polynomialsH(r)(z) = ∑Ll=0 h(r)(l)z−l do not share common zeros at the s

carrier frequencies.

Proof. The proof of (a0) is deferred to Section 2.3.2.


hannel

nt of

lored.d

irtualroll-offen theknowne beenhether

of

ll

The condition (a0) is assumed to hold throughout the development of proposed cestimator. That isH is assumed to have full rank.

We will furthermore assume the following conditions throughout the developmechannel estimators and equalizers presented in this paper.

(a1) There exists aK � M, such that theM × K matrix SK := [s(0), . . . , s(K − 1)],comprising of symbol blocks of lengthM, has full rankM. Note that asK → ∞,(1/K)

∑K−1i=0 s(i)sH(i) tends to the input correlation matrixCss = E{s(i)sH(i)}. But

this persistence of excitation (p.e.) assumption (a1) will be satisfied even for co(e.g., coded) input provided their spectra are non-zero for at leastM frequencies [19]

(a2) The complex valued source symbolss(n) and AWGNv(n) are mutually uncorrelateand stationary and noise is also assumed uncorrelated among channels.

Now given a block of data{y(i)}K−1i=0 , the objective here is to estimate theZ(L+ 1) × 1

composite channel vectorh = [h(1)T , . . . ,h(Z)T ]T .In practical OFDM systems, some sub-carriers are not modulated [2]. These v

carriers are not used for data transmission but are usually introduced inside theregion (to create a null guard interval) to avoid aliasing effects on data symbols, whsystem operates over multi-path propagation channels [33]. In this capacity they areas null side carriers. By exploiting these null side carriers, a number of schemes havproposed [29,33,34]. We therefore distinguish below two cases that depend on wsome null side carriers are present or not, since this affects the identifiability.

2.2.1. Channel estimation with no null side carriersWe choose to collectK consecutive data vectors{y(i)}K−1

i=0 from (6) in a matrix:

YK = HFHMSK + NK, (7)

where the matrices:YK = {y(i)}K−1i=0 andNK = {n}K−1

i=0 . The channel covariance matrixthe received data is thus:

Cyy = E{y(i)yH(i)

} = HFHMCssFM HH + Cnn, (8)

whereCnn = E{n(i)nH(i)}. By assumption noise is white (Cnn = σ 2n I ) and the input

signal is rich enough thatCss has full rank, i.e., rank(Css) = M.Taking the eigenvalue-decomposition (EVD) ofCyy , we have

Cyy = Ss diag(λ1, . . . , λM)SHs + σ 2

n GnGHn , (9)

whereSs = [S1, . . . ,SM ] andGn = [G1, . . . ,GZM−M ]. The columns ofSs span the signasubspace, while those ofGn, the noise subspace. The columns ofH also span the signasubspace and thus by orthogonality, we have

GHj H = 0, 1� j � ZM − M. (10)

In practice,Cyy is estimated by the sample average in time over, say,K blocks

Cyy ≈ C(K)yy = 1

K

K−1∑y(i)yH(i) (11)

i=0


us steps

-

),

ere

basise

izing

u-

d

and, therefore, (10) can be solved in the least squares sense, by following analogoas (for (13)) in the section below, to uniquely identifyh up to a scale factor.

2.2.2. Channel estimation with null side carriersWe assume the presence ofM −M virtual carriers at the tail end of each OFDM sym

bol. Thus each OFDM symbol consists ofM modulated source symbols andM − Mnon-modulated symbols. The IFFT matrixFH

M thus reduces to a partialM × M ma-trix FH

M = [f0, f1, . . . , fM−1]. The removal ofM − M columns ofFHM correspond to the

M −M virtual carriers ins(i). The received data model is then given by

y(i) = HFHM s(i) + n(i), (12)

wheres(i) denotes the new data vector of reduced lengthM. Under the data model (12the resulting equation (10) no longer has a unique solution because rank(Css) = M. Thisis in contrast to rank(Css) = M in case of no virtual carriers (see assumption (a1)). In thfollowing, the corresponding adjustments which take into account the null side carriers adetailed.

Following the analogous steps (8)–(9) for the data model (12), the correspondingfor the noise subspace is given by:Gn = [G1, G2, . . . , GZM−M]. We observe that thcolumns ofHFH

M also span the signal subspace and thus by orthogonality we have

GHj HFH

M = 0, 1� j � ZM −M. (13)

In practice, since the output data vectors are noisy, this equation is solved by minimthe quadratic form

q(h) =ZM−M∑

j=1

∥∥GHj HFH

M

∥∥2. (14)

Let GHj HFH

M = hH Gj FHM . Here Gj is a ZM × 1 vector defined as:Gj = [g(1)T

j , . . . ,

g(Z)Tj ]T , where g(r)

j = [g(r)j (1), . . . , g

(r)j (M)]T . The Z(L + 1) × M filtering matrix Gj

(formed fromGj ) is defined as:Gj = [G(1)Tj , . . . , G(Z)T

j ]T , where each(L+1)×M matrix:

G(r)j = [

G(r)j,0, G

(r)j,1

], (15)

with M × (M − L) Hankel matrixG(r)j,0 with first column[g(r)

j (1), g(r)j (2), . . . , g

(r)j (L +

1)]T and last row[g(r)j (L+1)g

(r)j (L+2) . . . g

(r)j (M)] and(L+1)×L Hankel matrixG(r)

j,1

with first column[g(r)j (M − L), g

(r)j (M − (L + 1)), . . . , g

(r)j (M), g

(r)j (1)]T and last row

[g(r)j (1), g

(r)j (2), . . . , g

(r)j (L)].

Therefore‖GHj HFH

M‖2 = hH Gj FHM FM GH

j h and (14) can thus be expressed as:q(h) =hH Qh, whereQ = ∑ZM−M

j=1 Gj FHM FM GH

j and the channel estimate can thus be formlated as

h = arg min‖h‖=1hH Qh. (16)

This quadratic optimization criterion allows unique estimation ofh up to a scale factor anh is thus obtained as the eigenvector associated with the minimum eigenvalue ofQ.


eived

hance-e now

f

2.3. ZF and MMSE equalizers

2.3.1. Pre-FFT equalizersNow given a block of data{y(i)}K−1

i=0 , the estimate of the multi-channel matrixH (i.e.,H) (obtained throughh in the previous section by the subspace estimator), the recsignal matrix can be written as

YK = HFHMSK + NK. (17)

The objective of this section is to estimate the block of dataSK according to ZF and MMSEcriteria.

From the estimation theory, the maximum likelihood (ML) estimate ofSK is given by:SK = GZFYK , whereGZF is the ZF equalizer given as

GZF = (HFHM

)† = ((HFHM

)H HFHM

)−1(HFHM

)H, (18)

where † denotes pseudo inverse. The ZF equalizer satisfies the conditionGZFHFH

M = I and

source symbols can be recovered providedH has full rank.The ZF equalizer is expected to suffer performance degradation due to noise en

ment and conditions near to common frequency nulls. To overcome this problem, wconsider MMSE equalizer, which aims to minimizeE{‖s(i) − s(i)‖2}, wheres(i) − s(i)is the error in theith block of data. We assume here that the matrixH, and the correlationmatricesCss andCnn are known.

Now by allowingG matrix representing the equalizing matrix:

s(i) = Gy(i) = GHFHM s(i) + Gn(i) (19)

and thus

s(i) − s(i) = Gy(i) − s(i) = GHFHMs(i) + Gn(i) − s(i)

= (GHFH

M − I)s(i) + Gn(i). (20)

The MSE can be written as the function of equalizing matrixG as

J (G) = E{tr[(

GHFHM − I

)s(i) + Gn(i)

][(GHFH

M − I)s(i) + Gn(i)

]H }. (21)

The MMSE solution is obtained by equalizing to zero the gradient ofJ (G) with respect toG and then solving forG (see [35]). We thus obtain

GMMSE = CsyC−1yy , (22)

where using the fact that additive noise is independent of the transmitted data:

Csy = E{s(i)yH(i)

} = E{s(i)

[HFHMs(i) + n(i)

]H } = Css

(HFHM

)H,

and

Cyy = E{y(i)yH(i)

} = (HFHM

)Css

(HFHM

)H + Cnn = (HFHM

)Css

(HFHM

)H + σ 2n I.

The linear MMSE estimate ofSK is thus:SK = GMMSEYK . The implementation details othe CP-OFDM pre-FFT antenna array receiver are shown in Fig. 3b.


na can

levanth each

ript (

ved:

ero’strictive.l nullsheerZF

r ofcan be

erence

2.3.2. Post-FFT equalizersBecauseH(r) is a circulant matrix, it is well known thatFMH(r)FH

M is aM ×M diagonalmatrix [36]

D(r)H = diag

[H(r)(ej0), . . . ,H (r)

(ej

2π(M−1)M

)] = diag[δ(r)0 , . . . , δ

(r)M−1

], (23)

where the diagonal elements are values of the frequency responseH(r)(z) := ∑Ll=0 h(r)(l)

× z−l of therth channel, evaluated at the sub-carriersz = ej 2πM m for eachm ∈ [0,M − 1].

Therefore, given (5) for each antenna, the FFT of the received symbol at each antenthen be expressed as

y(r)(i) = FMH(r)FHMs(i) + FMn(r)(i) = D(r)

H s(i) + n(r)(i). (24)

Let y(r)m (i), sm(i), andn(r)

m (i) (for m ∈ [0,M − 1]), denote themth element ofy(r)(i), s(i),andn(r)(i), respectively. Equation (24) can then be written in the scalar form as

y(r)m (i) = δ(r)

m sm(i) + n(r)m (i), m ∈ [0,M − 1]. (25)

From above frequency domain equations it is obvious that FIR convolutive channel reto each antenna is converted to parallel flat fading sub-channels independent witother (at each antenna).

From (25) we see that in case of a mono sensor system (we drop the superscr)as we have a single antenna at the receiver), provided the diagonal channel matrixDH

is known, each symbolsm(i) in the OFDM symbol under consideration can be receiindependently on sub-carrier basis assm(i) = ym(i)/δm. Equivalently in the matrix formats(i) = D−1

H y(i) = D−1H DH s(i) + D−1

H n(i). The matrixDH is invertible (or equivalentlycirculant matrixH is invertible) if and only if the channel transfer function has no zon the FFT grid, i.e.,H(ej2πm/M) �= 0, ∀m ∈ [0,M − 1]. The condition that the channelfrequency response does not have a null at any of the data carriers appears resHowever, in such a case, any information on the sub-channel is lost. Even channecloser to FFT grid cause serious amplification of noise and has a major impact on taverage system performance. This loss of information can be avoided by using receivdiversity. This is shown next explicitly in case of spatial diversity combining post-FFTequalizer.

Let y(i) = [y(1)T (i), . . . , y(Z)T (i)]T denote the post-FFT composite receive vectothe antenna array. The compact system description for multiple antenna systemobtained as

y(i) = DH s(i) + n(i), (26)

whereDH = [D(1)TH , . . . ,D(Z)T

H ]T andn(i) = [n(1)T (i), . . . , n(Z)T (i)]T .Now given a block of data{y(i)}K−1

i=0 , the objective of this section is to estimateSK .The block model of the receive data in (26) is

YK = DH SK + NK, (27)

whereYK = {y(i)}K−1i=0 = [Y(1)T

K , . . . , Y(Z)TK ]T andNK = {n(i)}K−1

i=0 . From the estimatehof the multi-channel system, one can easily obtain the estimated multi-channel interf


en

nd

d onlyof the

2e

lativelyse of

st-FFT

x

be

matrixDH . We therefore assume that channel and thus interference matrix has already beidentified through the proposed subspace estimator.

The post-FFT ZF estimateSK is given by:SK = GZFYK , whereGZF is the post-FFTZF equalizer (satisfying the conditionGZFDH = I) and can be obtained as

GZF = D† =(

Z∑r=1

D(r)HH D(r)

H

)−1

DH . (28)

Since channel interference matricesD(r)H are diagonal,GZF can be further decomposed a

we can thus calculate each row ofSK independently. We isolate one row ofSK , saym anddenote it bysH

m andyH(r)m denote themth row of Y(r)

K . The post-FFT ZF estimate ofSK onrow by row basis can thus be calculated as

sHm =

∑Zr=1 δ

(r)∗m yH(r)

m∑Zr=1 |δ(r)

m |2.

The estimated block of transmitted data through the post-FFT ZF equalizer is thus:

SK = sH

0...

sHM−1

=

∑Z

r=1 δ(r)∗0 yH(r)

0∑Zr=1 |δ(r)

0 |2...∑Z

r=1 δ(r)∗M−1yH(r)

M−1∑Zr=1 |δ(r)

M−1|2

. (29)

It is also clear that source symbols can be recovered through the equalizer if anif the frequency responses of all the channels do not have common nulls at anydata carriers. This corresponds to the full rank condition on the multichannel matrixDH

(or equivalently the multi-channel circulant matrixH). The condition (a0) in Section 2.thus holds true. As compared to the occurrence of channel nulls on the FFT grid, thoccurrence of common channel nulls in case of a multiple antenna systems has revery low probability, therefore, the loss of information can be avoided. However, in caoccurrence of (or conditions near to) common frequency nulls, we can resort to a poMMSE equalizer. The post-FFT linear MMSE estimate ofSK is: SK = GMMSEYK , wherelike the pre-FFT equalizer in the previous section, the post-FFT MMSE equalizer:

GMMSE = CsyC−1yy

, (30)

whereCsy = E{s(i)yH(i)} = CssDHH andCyy = E{y(i)yH(i)} = DH CssDH

H + Cnn. Asnoise is uncorrelated among channels therefore theZM × ZM noise covariance matriCnn = E{n(i)nH(i)} is a block diagonal matrix given as

Cnn =

C(1)

nn0 · · · 0

0 C(2)

nn· · · 0

......

. . ....

0 0 · · · C(Z)

nn

, (31)

whereC(r)

nn, is theM × M noise covariance matrix associated withrth channel. Using

matrix inversion lemma, we show in Appendix A that post-FFT MMSE solution canwritten as


duced

rder is

e as-ead

asation

dg

SK = GMMSEYK =(

Z∑r=1

D(r)HH

{C(r)

nn

}−1D(r)H + {Css}−1

)−1

×(

Z∑r=1

D(r)HH

{C(r)

nn

}−1Y(r)K

). (32)

Note thatM × M matrix inversion is required as opposed toZM × ZM matrix inver-sion in case of direct implementation mentioned above. The complexity is thus reconsiderably.

3. ZP-OFDM spatial diversity exploiting channel estimation and equalization

In ZP-OFDM systems, a ZP guard interval equal to or greater than the channel oinserted to prevent IBI and allow one tap equalization.


The development of the multi-channel data model in ZP-OFDM, is based on thsumptions (1)–(3) in Section 2, except in this case we assume a ZP guard interval (instof CP guard interval) of sizeP − M � L.

Figure 4 shows the concept of the proposedZP-OFDM system whose transmitter hsingle antenna and receiver has multiple antennas. First, complex valued informstreams(n) is parsed into blocks:s(i) of lengthM > L. TheM × 1 block is then mappeto a blocku of lengthP > M: first, by takingM-point IFFT of s(i) and then by paddinP − M trailing zeros at the end of the resulting vector.

Fig. 4. ZP-OFDM: (a) transmitter, (b) pre-FFT antenna array receiver.


d

dto ord

n

entor in

e

en-

d

To implement zero padding at the transmitter, the matrixFcp in (2) must be replacewith [19]

Fzp =[

FHM

0(P−M)×M

]. (33)

The corresponding transmitted vectoru(i) = Fzps(i), is then serialized and transmittethroughZ FIR channels. Note that in order to avoid IBI, a ZP guard interval equallarger than the maximum channel order (i.e.,P − M � L) is inserted. Resulting receivevector at each antenna is then:

x(r)(i) = H(r)0 u(i) + H(r)

1 u(i − 1) + v(r)(i)

= H(r)0 Fzps(i) + H(r)

1 Fzps(i − 1) + v(r)(i)

= H(r)0 Fzps(i) + v(r)(i) = H(r)

M FHMs(i) + v(r)(i), (34)

where vectorv(r)(n) has lengthP and H(r)M is the P × M Toeplitz matrix with first

row [h(r)(0),0, . . . ,0] and first column[h(r)(0), . . . , h(r)(L),0, . . . ,0]T . As can be seein (34), with zero padding no IBI appears at the receiver.

Stacking outputs ofZ channels, the composite receive vector can be developed as

x(i) = HMFHMs(i) + v(i), (35)

wherex(i) = [x(1)T (n), . . . ,x(Z)T (n)]T , HM = [H(1)TM , . . . ,H(Z)T

M ]T , andv(i) = [v(1)T (n),

. . . ,v(Z)T (i)]T .


Here the multi-channel matrixHM is full rank and only becomes rank deficient whh = 0. The new estimator follows the analogous steps as of the CP-OFDM estimaSection 2. In case of no virtual carriers,the new estimator is summarized as follows:

(1) Collect K � M blocks of data to form the matrix{x(i)}K−1i=0 = XK and compute

C(K)xx = (1/K)x(i)xH(i).

(2) Determine theZP − M eigenvectorsGj = [g(1)Tj , . . . ,g(Z)T

j ]T of sizeZP × 1 (each

comprising ofZ subvectorsg(r)j = [g(r)

j (1), . . . , g(r)j (P )]T ), which correspond to th

ZP − M smallest eigenvalues of the matrixC(K)xx .

(3) For eachj form the filtering matrixGj = [G(1)Tj , . . . ,G(Z)T

j ]T comprising ofZ Hankel

matricesG(r)j (of size(L + 1) × M) with first column[g(r)

j (1), . . . , g(r)j (L + 1)]T and

last row[g(r)j (L + 1), g

(r)j (L + 2), . . . , g

(r)l (P )].

(4) Estimate the channel vectorh as the eigenvector corresponding to the minimum eigvalue ofQ = ∑ZP−M

j=1 GjGHj .

Corresponding adjustments to accommodate null side carriers are straightforward antherefore not detailed here.


s

ationl matrix

con-

larssed as

n be

mpleguard


3.3.1. Pre-FFT equalizersNow for a given block of dataXK and the estimated multi-channel matrixHM , we have

XK = HMFHMSK + VK, (36)

whereVK = {v(i)}K−1i=0 . After the above procedure,pre-FFT ZF and MMSE equalizer

can be easily constructed. Consequently,GZF = (HMFHM)† andGMMSE = CsxC−1

xx , whereCsx = E{s(i)xH(i)} = Css(HMFH

M)H andCxx = E{x(i)xH(i)} = (HMFHM)Css(HMFH

M)H

+ σ 2n I.

3.3.2. Post-FFT equalizersThe pre-FFT equalizers in ZP-OFDM prevent the use of very simple equaliz

schemes, we thus propose post-FFT equalizers based upon the key result of channediagonalization in [37].

Like [5] (which follows [37] through matrix implementation), by mapping theP × 1vectorx(r)(i) to anM × 1 vector by adding its lastP − M elements to its firstP − M

elements, through the matrixRzp := [IM, Izp], whereIzp denote the firstP − M columnsof IM (matrix implementation of the overlap-and-add (OLA) technique used in blockvolution), we obtain

y(r)(i) = Rzpx(r)(i) = RzpH(r)M FH

Ms(i) + Rzpv(r)(i) = H(r)FHMs(i) + n(r)(i). (37)

This shows that therth linear convolutive channel with IBI is converted to a circuone without IBI. The FFT of the received symbol at each antenna can then be expre

y(r)(i) = FM y(r)(i) = FMH(r)FHMs(i) + FM n(r)(i) = D(r)

H s(i) + n(r)(i). (38)

The composite receive vector of the antenna array can thus be obtained as

y(i) = DH s(i) + n(i), (39)

where y(i) = [y(1)T , . . . ,y(Z)T ]T , DH = [D(1)TH , . . . ,D(Z)T

H ]T , and n(i) = [n(1)T , . . . ,

n(Z)T ]T .Now given a block of data{y(i)}K−1

i=0 and estimate of the channel interference matrix

DH , the matrix model of the received data is thus

YK = DH SK + NK, (40)

whereYK = {y(i)}K−1i=0 = [Y(1)T

K , . . . ,Y(Z)T ]T andNK = {n(i)}K−1i=0 . This model is similar

to the matrix model (27), therefore, similar post-FFT ZF and MMSE equalizers caeasily developed under the same set of assumptions.

4. BWE-OFDM spatial diversity exploiting channel estimation and equalization

In OFDM systems, CP or ZP guard interval is used to combat IBI and allow for a sione tap equalization scheme. However, due to the extra symbols required by the


forample,8],aboutpatialtion,width

ed onon (3),

FDM:

e

interval, the OFDM spectrum is under-utilized. This overhead can be considerably largechannels with long impulse responses and short block transmission formats. For exin wireless ATM network demonstrator (WAND), one of the projects in HIPERLAN [3the efficiency of bandwidth usage is only 66.67%. There are quite a few studieschannel equalization in BWE-OFDM systems, however, they do not consider rich sdiversity for improved performance and mitigating the effects of fading. In this secwe therefore consider OFDM systems without guard interval and efficiency of bandusage achieving 100%.


The development of BWE-OFDM multi-channel data model in this section is basthe assumptions (1)–(3) in Section 2, except we do not take into account assumptiand instead we assume onlyM > L.

Figure 5 shows the concept of the proposed transceiver system. In the BWE-Otransmitter in Fig. 5a, first, complex valued information streams(n) is parsed into blockss(i) of lengthM. TheM × 1 symbol blocks(i) is then mapped to a blocku(i) of lengthM by taking theM point IFFT ofs(i). We express this operation as:u(i) = FH

Ms(i).The vectoru(i) is then serialized and transmitted throughZ FIR channels. Based on th

assumption thatM > L and the fact thatrth FIR channelh(r)(l) = 0, ∀l /∈ [0,L], we writethe received data block form as

x(r)(i) = H(r)0 u(i) + H(r)

1 u(i − 1) + v(r)(i)

= H(r)0 FH

Ms(i) + H(r)1 FH

Ms(i − 1) + v(r)(i), (41)

where the second term denotes IBI.

Fig. 5. BWE-OFDM: (a) transmitter, (b) antenna array receiver.


IBI,

carding

posed

hannel

e

ponses

Unlike CP- and ZP-OFDM, we neither insert CP nor ZP guard interval to removebut to avoid the IBI, we consider a truncated version ofx(r)(n). Ignoring the initialLreceived samples from the observation vector at each antenna is equivalent to disthe firstL rows of H(r)

0 . In matrix form we achievethis with the receive matrixRbwe=[0M×L, IM ], which removes firstL entries fromM × 1 vectorx(r)(i). Note that IBI iscancelled at the receiver becauseRbweH

(r)1 = 0. Thus, we obtain

x(r)(i) = Rbwex(r)(i) = RbweH(r)0 FH

Ms(i) + RbweH(r)1 FH

Ms(i − 1) + Rbwev(r)(i)

= �H(r)0 FH

Ms(i) + v(r)(i). (42)

Note thatv(r)(i) andx(r)(n) are truncated noise and received vectors of lengthQ = M −L.Also, the truncated Toeplitz channel filtering matrix�H(r)

0 is of sizeQ × M with first row[h(r)(L),h(r)(L − 1), . . . , h(r)(1), h(r)(0),0, . . . ,0] and first column[h(r)(L), . . . ,0]T .

Stacking the outputs of theZ channels gives:

x(i) = �H0FHMs(i) + v(i), (43)

wherex(i) = [x(1)T (i), . . . , x(Z)T (i)]T , �H0 = [�H(1)T0 , . . . ,�H(Z)T

0 ]T , andv(i) = [v(1)T (i),

. . . , v(Z)T (i)]T .


The matrix�H0 is known as a generalized Sylvester matrix, which has full column rankM under the conditions [39]

(a0) The polynomialsH(r)(z) = ∑Ll=0 h(r)(l)z−l have no common zero2.

(a1) Q is greater than the maximum degreeL of the polynomialsH(r)(z), i.e.,Q � L (orequivalentlyM > 2L).

(a2) At least one polynomialH(r)(z) has degreeL.

The conditions (a0)–(a2) are assumed to hold throughout the development of prochannel estimator. That is�H0 is assumed to have full rank.

The new estimator also follows the analogous steps as of the earlier proposed cestimators. In case of no virtual carriers,the new estimator is summarized as follows:

(1) Collect K � M blocks of data to form the matrix{x(i)}K−1i=0 = �XK and compute

C(K)xx = (1/K)x(i)xH(i).

(2) Determine theZQ − M eigenvectorsGj = [g(1)Tj , . . . ,g(Z)T

j ]T of sizeZQ × 1 (each

comprising ofZ subvectorsg(r)j = [g(r)

j (1), . . . , g(r)j (P )]T ) , which correspond to th

ZQ − M smallest eigenvalues of the matrixC(K)xx .

2 The condition (a0) for CP-OFDM case (Section 2.2) is comparatively weaker, where channel resH(r)(z) do not share any common nulls at sub-carrier frequencies.


en-

ion of

y

be

to theti-pathquiringrix.callyptiveolved

ned

n-

(3) For eachj form the filtering matrixGj = [G(1)Tj , . . . ,G(Z)T

j ]T comprising ofZ Toep-

litz matricesG(r)j (of size (L + 1) × M) with first row [g(r)

j (Q), g(r)j (Q − 1), . . . ,

g(r)j (1), . . . ,0] and first column[g(r)

j (Q),0, . . . ,0]T .(4) Estimate the channel vectorh as the eigenvector corresponding to the minimum eig

value ofQ = ∑ZP−Mj=1 GjGH

j .

It is to be noted that in case of null side carriers the corresponding modified versthe channel estimator should be applied.


Let �H0 is the estimate of matrix�H0 obtained throughh in the previous section b

the subspace estimator. With�H0 known at the receiver, the received signal matrix canwritten as

�XK = �H0FHMSK + �VK, (44)

where�VK = {v(i)}K−1i=0 . For this block of received data,GZF = (�HFH

M)† andGMMSE =CsxC−1

xx , whereCsx = E{s(i)xH(i)} = Css(�HFHM)H andCxx = E{x(i)xH(i)} = (�HFH

M)×Css(�HFH

M)H + σ 2n I.

5. Blind linear adaptive implementation

The batch channel identification techniques proposed in this paper are linkedequalization stage. These techniques can not easily cope with time varying mulchannels as the subspace decomposition is accomplished via a standard EVD reO(p3) flops of computation, wherep is the dimension of the channel covariance matEstimation of time varying multi-path communication channels for equalization typiinvolves use of fast adaptive algorithms having linear complexity. The problem of adachannel estimation for tracking the multiple FIR channel variations can therefore be sby using fast subspace tracking algorithms.

The standard adaptive implementation of the proposed CP-OFDM receiver is outlinext.

(1) Replace the channel covariance matrixCyy by its recursive version through exponetial windowing:

C(K)yy (i) = αC(K)

yy (i − 1) + (1− α)y(i)yH(i), (45)

or a sliding window of lengthK:

C(K)yy (i) = C(K)

yy (i − 1) +{

(1/J )[y(i)yH(i) − Cyy(0)], for i � K,

(1/J )[y(i)yH(i) − U(i − K)], for i > K,(46)


ata

ing

nt

-

ociated

ute the

dap-bspace

onven-rixed. The

b-cted

at the ith iteration, whereC(K)yy (i) is the sample covariance matrix using the d

available up to timei, U(i − K) = y(i − K)yH(i − K) andα is a forgetting factor(0 < α � 1).

(2) Based onC(K)yy (i), estimate the noise subspaceGn through the noise subspace track

algorithm.(3) Use the estimated noise subspace to computeQ(i).(4) Recursively estimate the minimum eigenvector ofQ(i), using a stochastic gradie

adaptation [40] (a real version is analyzed in [41])

β(i) = hH(i − 1)Q(i)h(i − 1), (47)

h(i) = h(i − 1) − µ[∥∥h(i − 1)

∥∥2Q(i) − β(i)I]h(i − 1). (48)

(5) Use the channel estimateh to obtain the desired signal vectors(i) through the associated equalizer.

The proposed ZP- and BWE-OFDM receivers canbe adaptively implemented in similarfashion by using relevant channel covariance matrices, updating vectors and assequalizers.

In literature, many subspace tracking algorithms have been developed to compsubspace recursively, which require O(p2q), O(pq2), and O(pq) flops of computationat each update (whereq is the dimension of the signal or noise subspace). In our ative implementation, we propose to estimate the noise subspace by an efficient sutracker: normalized orthogonal Oja(NOOja) algorithm of complexity O(pq), proposedin [42]. This method estimates the noise subspace based on Rayleigh quotient. Ctional subspace tracking approaches rely on recursive version of channel covariance mat(see steps 1 and 2 above), however, in case of NOOja, no such procedure is requiralgorithm is outlined next (for CP-OFDM noise subspace tracking).

• Initialization of the algorithm:Gn(0) = any arbitrary orthogonal matrix or noise suspace, which is derived from EVD of the initial channel covariance matrix and injefor initialization.

• Algorithm at iterationi:

f(i) = GHn y(i),

j(i) = Gn(i)f(i),

p(i) = y(i) − j(i),

βopt(i) = +β

‖y(i)‖2 − ‖f(i)‖2 + γ,

φ(i) = 1√1+ β2(i) + ‖p(i)‖2‖f(i)‖2

,

τ (i) = φ(i) − 1

‖f(i)‖2 ,

p(i) = −τ (i)j(i)/βopt(i) + φ(i)p(i),


te of thee,rableu-more

paceach.roxi-

hod.powerxistingf NP

h

al sub-trackerum

alIn

e anydatrix.

at.,

r(i) = p(i)/‖p(i)‖2,

w(i) = GHn (i)r(i),

Gn(i + 1) = Gn(i) − 2r(i)wH(i). (49)

In this algorithm,β is a learning parameter and 0< β < 1.The proposed channel estimators can be made to exploit signal subspace despi

noise subspace and minimization problem canbe turned into maximization problem (sefor example, [39,43,44]). Solution of maximization problem is considered more favoto the minimization problem [44], as there are fundamental limitations on the relative accracy with which the smallest eigenvalues of a matrix can be computed, and they aredifficult to compute than the big ones [45]. However, it is shown in [39] that noise subsbased approach exhibits better performance than the signal subspace based appro

A class of fast signal subspace tracking algorithms include Oja [46], projection appmation subspace tracking (PAST) [47], and novel information criterion (NIC) [48] metThese methods are variations of the power method [49]. The natural version of themethod (NP) proposed in [50], guarantees fast convergence and outperforms all evariations of the power method in this class. In [50], three types of implementation oare presented, which require O(p2q), O(pq2), and O(pq) flops of computation at eacupdate. The O(pq) implementation of the NP method is superior to the O(pq) equivalentof Oja, PAST, and NIC methods. In case of adaptive implementation based on signspace tracking, the batch EVD is therefore proposed to be replaced by NP subspaceof linear complexity (i.e., NP3). The problem of recursive estimation of the maximeigenvector ofQ(i) can be solved by a simple power iteration method [49]

g = Q(i)h(i − 1), (50)

h(i) = g/‖g‖. (51)

The standard treatment of adaptive implementation can be initialized by a random initichannel covariance matrixC(K)

yy (0). This will cause some initial delay in convergence.order to skip this convergence delay, like [51], we can use pilot symbols (which arway present at the beginning of each frameof OFDM symbols for synchronization aninitial channel estimation) to initialize the estimation of the channel covariance mThe initialization process can be described by the following:

(i) Obtain the initial channel estimateh(0) through the pilot symbols.(ii) From h(0), compute the initial estimation of the channel covariance matrixCyy as

Cyy(0) = HQ(0)FHMCssFMHH

Q(0).

6. Properties of proposed techniques

6.1. Robustness to channel zero locations and order over-estimation errors

Proposed spatial diversity exploiting receivers allow unit circle zeros located2πm/M—a case when deep frequency fades deteriorate performance of conventional (i.e


How-rosdiver-ommonbe in-lexity).ro.sed

atial

powersmallyire am-lationber ofonmuch, anduch

r-ethodmberalfg

ovari-osedtrix ofs in

red toa-s inso it ispared

SISO) CP-OFDM and ZP-OFDM techniques relying on simple one tap equalization.ever, except ZP-OFDM spatial diversity based receiver, they fail in case of common zeor common deep frequency fades. The performance of CP- and BWE-OFDM spatialsity based receivers degrades, when they have conditions near to common zeros or cdeep frequency fades. To prevent this deterioration additional spatial diversity cancluded at the expense of increased number of antennas (involving increased compIt is to be noted that it is highly unlikely that all the channels will share a common ze

If an upper bound�L > L is only available on the maximum channel order, the propoCP-OFDM and ZP-OFDM channel estimators and equalizers work very well with�L re-placingL. This distinct advantage is not possible with the proposed BWE-OFDM spdiversity based receiver, which requires exact estimation ofL.

6.2. Hardware complexity and power consumption

Spatial diversity based OFDM receivers should be designed with care as theconsumption, size, and computational complexity are the important factors towardssize mobile and wireless terminals. Post-FFTdiversity combining receivers (involving ancombining technique: from maximum ratio combining to selection diversity) requdifferent FFT processor for each antenna, thusresulting in huge increase in hardware coplexity as well as higher power consumption. It is important to note that FFT demoduis power hungry and requires a lot of processing especially with increasing numsub-carriers. The post-FFT diversity combining receivers thus involve power consumptiwhich increases linearly with the number of antennas. To our knowledge there is notwork available in the literature treating this problem. In [52,53], Okada, HashizumeKomaki showed that in COFDM the computational cost of pre-FFT combining is mless than that of post-FFT and can effectively improve the bit-error-rate (BER) perfomance in multi-path fading channel while requiring only one FFT processor. The min [25] requires only one FFT at the receiver for two branch diversity. For higher nuof antennas, the number of required FFTs when using technique [25] is reduced by has compared to conventional post-FFT receivers. In contrast pre-FFT diversity combininreceivers proposed in this paper are attractive as they only require one FFT processor.

6.3. Computational complexity and performance

The proposed ZP-OFDM estimation technique requires EVD of the channel cance matrix of sizeZP × ZP to determine orthogonal subspaces. In contrast, propCP- and BWE-OFDM subspace estimators require EVD of channel covariance masizeZM × ZM and ZQ × ZQ, respectively. This requirement furthermore reduceZP-OFDM (SISO) subspace estimator in [19] to channel covariance matrix of sizeP × P .This means that proposed ZP-OFDM receiver is computationally complex as compathe rest of the receivers. On the other hand ZP-OFDM (SISO) receiver is most computtionally efficient. Also CP-OFDM receiver is complex than BWE-OFDM receiver athe former case dimension of the received vector is higher than in the later case. Alworthwhile to mention that post-FFT equalizers are computationally simple as com


y are

d to these, the

re-oiset to chan-ent in

rrsband-posed-

l as ind thuseroosed

e any

anceted sys-

were

Chan-1ow thembolsnel

to pre-FFT receivers which require inversion of matrices of large dimensions. Thethus attractive for fast processing.

We expect superior performance by the proposed ZP-OFDM receiver as compareZP-OFDM (SISO) and proposed CP- and BWE-OFDM receivers, as in the former cachannel estimator creates a noise subspace of spanned byZP − M vectors againstP − M

vectors in case of ZP-OFDM (SISO),ZM − M vectors in case of proposed CP-OFDMceiver, andZQ−M vectors in case of proposed BWE-OFDM estimator. The reduced nsubspace dimension in these estimators makes these subspace methods less robusnel noise and thus performance degradation is obvious [54]. Performance improvemproposed diversity combining receivers is obvious as each additional antenna accounts foincreased noise subspace dimension. Performance improvement in ZP-OFDM receiveis also possible by increasing the size of the guard interval, however, this is not awidth efficient approach. The requirement of added guard interval overhead in proreceivers can thus be kept to the minimum level (i.e.,L) in order to avoid additional bandwidth wastage.

6.4. Transmitter modification

The proposed CP-OFDM receiver rely on the usual insertion of CP guard intervastandard OFDM systems. Therefore it does not require transmitter modification anis applicable to all standardized OFDM systems (DAB, DVB, etc.). ZP-OFDM techniqupresented here require transmitter modification to introduce zero padding. Recently, zepadding is used in DAB standard (ETS 400 301) in the form of guard bits [19]. PropBWE-OFDM technique also requires transmitter modification as it does not includguard interval.

7. Simulation results

In this section, we provide some selected simulation results to illustrate the performof proposed spatial diversity based channel estimators and equalizers. The simulatem is as follows.

7.1. System parameters

(i) The input symbols were generated using randomly drawn BPSK symbols andseparated into 20 sub-bands as the size of the FFT/IFFT wasM = 20.

(ii) We simulated the output ofZ = 4 FIR channels of maximum orderL = 4.(iii) CP- and ZP-OFDM included a guard intervalP − M = L to remove IBI.(iv) The channel coefficients were chosen as in [39] for Channel Set 1, whereas for

nel Set 2, channel coefficients were obtained artificially. A realization of Channel Setand Channel Set 2 is given in Tables 1 and 2, respectively. Figures 6a and 6b shzero locations of the Channel Set 1 and Channel Set 2, respectively. Different syrepresent different zeros of different channels of each channel set. Note that Chan


re max-mmonn.

etsses of

elow.

Table 1Channel Set 1 impulse responses

r h(r)(0) h(r)(1) h(r)(2) h(r)(3) h(r)(4)

1 −0.049+ 0.359i 0.482− 0.569i −0.556+ 0.587i 1.000 −0.171+ 0.061i2 0.443−0.0364i 1.000 0.921− 0.194i 0.189− 0.208i −0.087− 0.054i3 −0.211− 0.322i −0.199+ 0.918i 1.000 −0.284− 0.524i 0.136− 0.190i4 0.417+ 0.030i 1.000 0.873+ 0.145i 0.285+ 0.309i −0.049+ 0.161i

Table 2Channel Set 2 impulse responses

r h(r)(0) h(r)(1) h(r)(2) h(r)(3) h(r)(4)

1 0.9186−1.7718i −3.5531+2.0599i 3.9705−2.0167i −5.1877−1.8506i 0.0002 −0.3683−0.7445i 0.4243−1.9184i 1.8028−1.5353i 2.3522−0.4563i 0.0003 1.4356+0.8317i −1.9531−0.7904i −0.1929+0.7736i 1.3794−3.4362i 0.0004 −1.4782+0.1523i −1.7868−0.6220i −0.9031−3.1749i −0.1614−2.1122i 0.000

Fig. 6. Zero locations: (a) Channel Set 1, (b) Channel Set 2.

Set 2 has a common zero and the channel set can be seen as the situation wheimum channel order is over-estimated. Though the Channel Set 1 has no cozeros it is a difficult scenario as channel zeros are close to common zero conditioAlso, channel frequency response magnitudes corresponding to the two channel sare plotted in Fig. 7. Notice the presence of common nulls in frequency responthe Channel Set 2.

(v) We employed Channel Set 1 in all evaluations except in simulation example 2 b


mean-

mBER

t 1 by.re

they over-nel

t 2 by.ts

Fig. 7. Frequency response magnitudes: (a) Channel Set 1, (b) Channel Set 2.

7.2. Performance comparisons

To evaluate the channel estimation error, we employed the normalized-root-square-error (NRMSE) with 100 Monte Carlo runs. This is defined as

NRMSE= 1

‖h‖

√√√√√ 1

Nt

Nt∑iR=1

‖hiR − h‖2, (52)

whereNt is the number of Monte Carlo runs andhiR is the estimate of the channel froith run. The overall performance comparisons were made in terms of achievableaveraged over 100 independent runs.

Simulation example 1. The estimates of the true channel responses of Channel Sethe three subspace estimators with 500 OFDM symbols at SNR=25 dB are shown in Fig. 8True channel coefficients are indicated by “o” and estimates by ZP-OFDM estimator aindicated by “.”; estimates by CP-and BWE-OFDM estimators are indicated by “∇” and“∗”, respectively. The figure demonstrates thatall the estimators are able to estimatechannel coefficients of Channel Set 1. As can be seen, ZP-OFDM estimates exactllap true coefficients, whereas, CP- and BWE-OFDM estimates are close to true chancoefficient locations.

Simulation example 2. The estimates of the true channel responses of Channel Sethe three estimators with the same number of symbols at SNR=25 dB are shown in Fig. 9As can be seen, ZP-OFDM estimates again exactly overlap the true channel coefficien


tesWE-ros of

R ofe of aller of

Fig. 8. Comparison of true and estimated coefficients of Channel Set 1.

Fig. 9. Comparison of true and estimated coefficients of Channel Set 2.

and CP-OFDM estimates are close to the true channel coefficients. BWE-OFDM estimado not match with the true channel coefficient locations. Clearly, this failure of BOFDM estimator is because of the presence of non-invertible configurations of zethe Channel Set 2.

Simulation example 3. Figure 10 shows the performance of the estimators at SN25 dB as a function of the number OFDM symbols. We can see that the performancthe estimators improve with increasing the number of OFDM symbols. Large numb


FDMsym-

bols.

500R for

ith in-orms

gives

seding tolated

deredbased, the

Eever,

erfor-nnel 2nnel 2

Fig. 10. NRMSE v/s number of OFDM symbols.

OFDM symbols are required to obtain good channel estimates by CP- and BWE-Oestimators. In contrast, ZP-OFDM estimator does not need large number of OFDMbols and it is able to identify channels with much smaller number of OFDM symZP-OFDM estimator is thus quite suitable for fast tracking of channel variations.

Simulation example 4. In this simulation study, we fixed the number of symbols atand varied the SNR from 0–25 dB. Figure 11 shows NRMSE as a function of SNthe estimators. We observe that the performance of the three estimators improves wcreasing SNR. In comparison with BWE-OFDM estimator, CP-OFDM estimator perfbetter. Among the three estimators, ZP-OFDM estimator is much robust to noise andsuperior performance at low SNR.

Simulation example 5. Figure 12 shows the over all BER performance of the propospatial diversity based pre-FFT: CP-, ZP-, and BWE-OFDM receivers correspondSNR range of 0–25 dB. In order to check the spatial diversity gain, we also simuZP-OFDM pre- and post-FFT receiver with no spatial diversity. In this case we consichannel 2 of Channel Set 1. We have also shown performance of spatial diversitypost-FFT ZP-OFDM receivers. In contrast to the dotted lines for MMSE receiversBER curves for ZF receivers are given by the solid lines. This is clear from results that:

(i) ZF receivers suffer from performancepenalty in all cases as compared to MMSreceivers. This performance penalty is not visible in CP-OFDM receivers, howthis can be experimented with reduced number of OFDM symbols.

(ii) Antenna diversity results in performance improvement. For example, see the pmance gap between ZP-OFDM receivers employing 1 antenna dealing with chaof Channel Set 1 and ZP-OFDM receivers employing 4 antennas. Note that cha


loser tonter-

Fig. 11. NRMSE v/s SNR.

Fig. 12. BER v/s SNR: CP-OFDM, ZP-OFDM, and BWE-OFDM receivers.

possess frequency response which is too severe and contain nulls which are cFFT grid. The spatial diversity based ZP-OFDM receivers did a good job of couacting the effects of channel nulls.


ncer, their

etterased

ndFTt high

re-ivers,per-

ancengnin highns onance

M

non-mber

M re-rbi-s

dwidthl-with

d blind,de:

chan-ed

(iii) Though CP-OFDM and BWE-OFDM receivers do not result in similar performaas ZP-OFDM receivers with the same number of receive antennas, howeveperformance can be improved by increasing the number of receive antennas.

(iv) At low SNR, spatial diversity based pre-FFT ZP-OFDM receivers perform bthan spatial diversity based post-FFT ZP-OFDM receivers. In spatial diversity bpre-FFT ZP-OFDM receivers, high SNR produces an error floor in resulting BER athey suffer from performance penalty as compared to spatial diversity based post-FZP-OFDM receivers. In no spatial diversity case, this problem does not appear aSNR, but pre-FFT ZP-OFDM receivers perform better than post-FFT ZP-OFDMceivers even at low SNR. Like performance of post- to pre-FFT ZP-OFDM receCP-OFDM post-FFT receivers perform to pre-FFT CP-OFDM receivers. Theirformance is therefore not depictedhere to avoid further complexity.

(v) CP-OFDM receivers perform better than BWE-OFDM receivers. This performdifference is due to processing of higher dimension received data blocks containireceived data samples that were thrownout in the BWE-OFDM channel estimatioand equalization. These data samples possess significant energy resultingperformance. The CP-OFDM channel estimators do not invoke strict restrictiochannel zero locations. This is also another factor contributing in better performof CP-OFDM receivers.

(vi) ZP-OFDM receivers result in superior performance among all the receivers.(vii) As compared to ZP and CP-OFDM receivers, poor performance by BWE-OFD

receivers should not be seen as an impairment due to the following reasons:(a) Their performance degrades only when channels have zeros close to

invertible configurations. This behavior can be avoided by increasing the nuof receiver antennas.

(b) For the same allocated bandwidth, the information rate in ZP and CP-OFDceivers depend on the ratioM/P . Note that, the rate reduction can be made atrarily small by selectingM(and thusP) sufficiently large. Obviously, this comeat the cost of increased decoding delay and greater complexity so that banefficiency is generally traded with computational complexity and maximum alowable delay. We wish to emphasize that these problems do not ariseBWE-OFDM receivers where the information rate depend on the ratioM/M.

8. Conclusion

Building on the receive antennas diversity, we have proposed subspace basemethods for channel identification and equalization in any of the CP-OFDM, ZP-OFDMand BWE-OFDM systems. The important features of the proposed procedures inclu

(a) Recovery of lost information even when deep fades are present.(b) Performance enhancement without excess bandwidth.

The proposed CP-OFDM and BWE-OFDM channel estimators do not guarantee thenel identifiability for certain configurationsof channel zeros. In contrast, the propos


com-FDMofhes toby theTre-

hand,, areTs re-mented

ZP-OFDM estimator do not impose any restrictions on channel zero locations. Aspared with the proposed CP-OFDM and BWE-OFDM receivers, the proposed ZP-Oreceiver has a better performance, i.e., faster convergence rate with a smaller numberOFDM symbols. Proposed CP-OFDM and ZP-OFDM receivers are robust approacmaximum channel order over-estimation errors, where as this feature is not sharedproposed BWE-OFDM receiver. The pre-FFT equalizers are shown to require a single FFprocessor, regardless the number of receive antennas. As a consequence, considerableduction in hardware complexity and power saving may be obtained. On the otherthe post-FFT equalizers, thoughrequiring a separate FFT processor for each antennashown to be computationally simple and support fast processing (apart from the FFquired). The proposed channel estimation and equalization procedures can be impleonline and can cope with OFDM systems where null side carriers are present.

Appendix A

Proof of (32). The linear MMSE estimate ofSK is

SK = GMMSEYK. (A.1)

By substituting (30) in (A.1), we obtain

SK = Csy(Cyy )−1YK = CssDHH

(Cnn + DH CssDH

H

)−1YK. (A.2)

For simplicity, we arbitrarily assumeCss = σ 2s I. Thus (A.2) can be written as

SK = σ 2s DH

H

(Cnn + σ 2

s DH DHH

)−1YK. (A.3)

By using matrix inversion lemma3 on the inverted term in (A.3), we obtain

SK = σ 2s DH

H

[{Cnn}−1 − {Cnn}−1DH

((σ 2

s

)−1I + DHH {Cnn}−1DH

)−1

× DHH {Cnn}−1]YK

= σ 2s

[I − DH

H {Cnn}−1DH

((σ 2

s


)−1]DHH {Cnn}−1YK

= σ 2s

[I − σ 2

s DHH {Cnn}−1DH

(I + σ 2

s DHH {Cnn}−1DH

)−1]DHH {Cnn}−1YK. (A.4)

Letting M = σ 2s DH

H {Cnn}−1DH , we obtain

SK = σ 2s

[I − M(I + M)−1]DH

H {Cnn}−1YK (A.5)

and since(I + M)−1 = I − M(I + M)−1, therefore

SK = σ 2s (I + M)−1DH

H {Cnn}−1YK

= ((σ 2

s


)−1DHH {Cnn}−1YK

= ({Css}−1 + DHH {Cnn}−1DH

)−1DHH {Cnn}−1YK (A.6)

3 For A, B, C, D matrices of compatible dimension:(A + BCD)−1 = A−1 − A−1B(C−1 + DA−1B)−1

× DA−1.


Com-

E

41

er

tis-

hannel

C,

:

and since

DHH {Cnn}−1DH =

D(1)

H

D(2)H...

D(Z)H

H

{C(1)

nn}−1 0 · · · 0

0 {C(2)

nn}−1 · · · 0

......

. . ....

0 0 · · · {C(Z)

nn}−1

D(1)H

D(2)H...

D(Z)H

=

Z∑r=1

D(r)HH

{C(r)

nn

}−1D(r)H (A.7)

and

DHH {Cnn}−1YK =

D(1)

H

D(2)H...

D(Z)H

H

{C(1)

nn}−1 0 · · · 0

0 {C(2)

nn}−1 · · · 0

......

. . ....

0 0 · · · {C(Z)

nn}−1

Y(1)K

Y(2)K...

Y(Z)K

=

Z∑r=1

D(r)HH

{C(r)

nn

}−1Y(r)K , (A.8)

Eq. (A.6) can thus be written as

SK =(

Z∑r=1

D(r)HH

{C(r)

nn

}−1D(r)H + {Css}−1

)−1( Z∑r=1

D(r)HH

{C(r)

nn

}−1Y(r)K

). (A.9)

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