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Signal Processing

Signal Processing 93 (2013) 2038–2054

0165-16

http://d

n Corr

E-m1 M2 Fe

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

Review

Preamble-based channel estimation in OFDM/OQAMsystems: A review

Eleftherios Kofidis a,n,1, Dimitrios Katselis b, Athanasios Rontogiannis c,1,Sergios Theodoridis d,2

a Department of Statistics and Insurance Science, University of Piraeus, 80, Karaoli & Dimitriou str., 185 34 Piraeus, Greeceb Signal Processing and Automatic Control Laboratories, School of Electrical Engineering, KTH Royal Institute of Technology, Stockholm,

Swedenc Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, Metaxa & Vas. Pavlou

str., 152 36 Athens, Greeced Department of Informatics and Telecommunications, University of Athens, Panepistimioupolis, Ilissia, 157 84 Athens, Greece

a r t i c l e i n f o

Article history:

Received 5 June 2012

Received in revised form

7 November 2012

Accepted 14 January 2013Available online 23 January 2013

Keywords:

Channel estimation

Filter bank-based multicarrier (FBMC)

Intrinsic interference

Orthogonal frequency division

multiplexing (OFDM)

Offset QAM (OQAM)

Preamble

84/$ - see front matter & 2013 Elsevier B.V

x.doi.org/10.1016/j.sigpro.2013.01.013

esponding author. Tel.: þ30 210 414 2475;

ail addresses: kofidis@unipi.gr (E. Kofidis), d

ember, EURASIP.

llow, EURASIP.

a b s t r a c t

Filter bank-based multicarrier communications (FBMC) have recently attracted increased

interest in both wired (e.g., xDSL, PLC) and wireless (e.g., cognitive radio) applications, due to

their enhanced flexibility, higher spectral efficiency, and better spectral containment com-

pared to conventional OFDM. A particular type of FBMC, the so-called FBMC/OQAM or OFDM/

OQAM system, consisting of pulse shaped OFDM carrying offset QAM (OQAM) symbols, has

received increasing attention due to, among other features, its higher spectral efficiency and

implementation simplicity. It suffers, however, from an imaginary inter-carrier/inter-symbol

interference that complicates signal processing tasks such as channel estimation. This paper

focuses on channel estimation for OFDM/OQAM systems based on a known preamble. A

review of the existing preamble structures and associated channel estimation methods is

given, for both single- (SISO) and multiple-antenna (MIMO) systems. The various preambles

are compared via simulations in both mildly and highly frequency selective channels.

& 2013 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2039

2. System model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2040

3. Preamble-based channel estimation methods: the single-antenna case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2041

3.1. Pairs of pilots (POP) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042

3.2. Interference approximation method (IAM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2042

3.2.1. The IAM-R method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043

3.2.2. The IAM-I method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043

3.2.3. The IAM-C method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043

3.2.4. Optimal preambles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043

. All rights reserved.

fax: þ30 210 414 2340.

imitrik@kth.se (D. Katselis), tronto@noa.gr (A. Rontogiannis), stheodor@di.uoa.gr (S. Theodoridis).

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2039

3.2.5. The extended IAM-C (E-IAM-C) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044

3.3. Interference cancelation/avoidance methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044

4. Preamble-based channel estimation methods: the multiple-antenna case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045

4.1. MIMO POP method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045

4.2. MIMO IAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045

4.3. Optimal sparse preamble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047

5. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047

5.1. SISO systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2049

5.2. MIMO systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2050

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2050

Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2051

Appendix A Time–frequency neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2051

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2052

3 Nevertheless, this advantage was partly given up in [70] and a

CP-based OFDM/OQAM system was proposed for the sake of facilitating

the data reception process. See also [21].

1. Introduction

Orthogonal frequency division multiplexing (OFDM) hasbecome quite popular in both wired and wireless commu-nications [7,41,89], mainly because of its immunity to thefrequency selectivity of the channel, which allows a signifi-cant increase in the transmission rate [86,73]. Using thecyclic prefix (CP) as a guard interval against intersymbolinterference (ISI), OFDM manages to turn a frequency selec-tive channel into a set of parallel flat channels with indepen-dent noises. This greatly simplifies both the estimation of thechannel and the recovery of the transmitted data at thereceiver. However, these advantages come at the cost of anincreased sensitivity to frequency offset and Doppler spread.This is due to the fact that, although the subcarrier functionsare perfectly time limited, they suffer from spectral leakage inthe frequency domain and hence inter-carrier interference(ICI) results under non-ideal conditions. Moreover, the inclu-sion of the CP entails a waste in transmitted power as well asin spectral efficiency, which, in practical systems, can go upto 25% [7].

An alternative to CP-OFDM, that can mitigate thesedrawbacks, is given by filter bank-based multicarrier(FBMC) systems, which have recently attracted increasedinterest [79] for both wired (e.g., power line comms (PLC)[10,15,1]) and wireless (e.g., cognitive radio [30,92,57] andDVB-T [4]) applications. For an excellent presentation ofFBMC systems, including a review of recent applicationareas and related standardization activities, the reader couldrefer to [31]. FBMC enjoys enhanced flexibility, higherspectral efficiency, and better spectral containment com-pared to CP-OFDM [2,54]. Special attention has beenrecently paid to an OFDM filter bank-based variant employ-ing offset quadrature amplitude modulation (OQAM),known as OFDM/OQAM [61,31,48,49]. This was, in itsprinciple, first proposed back in the 1960s [20,81], but theinterest in this scheme was only recently revived due to itsdesirable features including the potential for maximumspectral efficiency as well as implementation simplicityover alternative FBMC schemes [12,13,82]. It consists ofpulse shaping, implemented via an IFFT/FFT-based efficientfilter bank, and the subcarriers are loaded with staggeredOQAM symbols, i.e., real symbols at twice the symbol rate ofOFDM/QAM [40,83]. This allows the pulses to be welllocalized in both the time and the frequency domains [18],thus increasing the system’s robustness to frequency offsets[9,80] and Doppler effects [16,78,27,59] and providing

better spectral containment in bandwidth sensitive applica-tions [3,6,84]. The OQAM-based OFDM scheme can be seento exhibit similar peak-to-average power ratio (PAPR) per-formance with classical OFDM/QAM [88,58]. Moreover,because of the good time localization, a CP does not haveto be included in the OFDM/OQAM transmission,3 whichleads to still higher transmission rates [83,5,78].

However, the subcarrier functions are now only orthogo-nal in the real field, which means that there is always anintrinsic imaginary interference among (neighboring) subcar-riers and symbols [47]. This makes the channel estimationtask for OFDM/OQAM systems more challenging. A numberof training schemes and associated estimation methods haverecently appeared in the literature, including both preamble-based [51,65,68,42,43,25,55,56,69] and scattered pilots-based[47,76,51,66,45,46,91,67,8,38,60,62,36,50] ones. These can beroughly characterized as aiming at cancelling/avoiding theundesired interference or constructively exploiting it toimprove the estimation performance. The focus of this paperis on OFDM/OQAM channel estimation based on a knownpreamble. Existing preamble structures and their associatedestimation methods are reviewed, for both the single-antenna (SISO) and multiple-antenna (MIMO) scenarios.The problem of preamble optimization to achieve minimumestimation error is also shortly discussed, based on recentlyreported results. The estimation performances of the princi-pal methods are compared via simulations for both mildlyand highly frequency selective channels.

The rest of the paper is organized as follows: In Section 2,the discrete-time baseband equivalent model for the OFDM/OQAM system is described. Section 3 reviews the mainapproaches proposed in the literature for structuring thepreamble and exploiting it for SISO channel estimation. TheMIMO case is addressed in Section 4. Comparative simula-tion results are reported in Section 5. Section 6 concludesthe paper.

Notation. The complex conjugate of a complex numberz is denoted by z%. Also, E¼

ffiffiffiffiffiffiffi�1p

. RðzÞ or zR denotes thereal part of z, while its imaginary part is written as IðzÞ or zI.The set of real numbers is denoted by R. Boldface lower andupper case letters are used for vectors and matrices,

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542040

respectively. The nth-order identity matrix is denoted by In.J � J is the Frobenius norm. The expectation is denoted byEð�Þ. � is the (left) Kronecker product.

2. System model

The (QAM or OQAM based) OFDM modulator output istransmitted through a channel of length Lh, which is, asusual in block transmissions, assumed to be invariant inthe duration of an OFDM symbol. At the receiver front-end, noise is added, which is assumed Gaussian with zeromean and variance s2. For CP-OFDM, the classical config-uration is assumed [86]. The discrete-time signal at theoutput of an OFDM/OQAM synthesis filter bank (SFB) isgiven by [83] 4

sðlÞ ¼XM�1

m ¼ 0

Xn

dm,ngm,nðlÞ, ð1Þ

where dm,n are real OQAM symbols, and

gm,nðlÞ ¼ g l�nM

2

� �eEð2p=MÞmðl�ðLg�1Þ=2ÞeEjm,n

with g being the real symmetric prototype filter impulseresponse (assumed here of unit energy) of length Lg, M

being the even number of subcarriers, and jm,n ¼j0þ

ðp=2ÞðmþnÞ mod p, where j0 can be arbitrarily chosen[83]. In this paper, jm,n is defined as ðmþnÞðp=2Þ�mnp asin [83]. The filter g is usually designed to have lengthLg ¼ KM, with K being the overlapping factor. The doublesubscript ð�Þm,n denotes the (m,n)-th frequency–time (FT)point. Thus, m is the subcarrier index and n the OFDM/OQAM symbol time index.

The pulse g is designed so that the associated sub-carrier functions gm,n are orthogonal in the real field [18],that is

RX

l

gm,nðlÞg%

p,qðlÞ

( )¼ dm,pdn,q, ð2Þ

where di,j is the Kronecker delta. This implies that even inthe absence of channel distortion and noise, and withperfect time and frequency synchronization, there will besome intercarrier (and/or intersymbol) interference at theoutput of the analysis filter bank (AFB), which is purelyimaginary (the notation in [65] is adopted)X

l

gm,nðlÞg%

p,qðlÞ ¼ E/gSp,qm,n ð3Þ

and known as intrinsic interference [47]. Making thecommon assumption that the channel is (approximately)frequency flat at each subcarrier and constant over theduration of the prototype filter [65], which is true5 forpractical values of Lh and Lg and for well localized g’s, onecan express the AFB output at the pth subcarrier and qth

4 Alternative configurations exist, generating a real OFDM/OQAM

signal. See [40,87,71].5 This is not an accurate subchannel model in case M, the number of

subcarriers, is not large enough with respect to the frequency selectivity

of the channel [74] (see Section 5 for such examples). This is also the

case in high mobility scenarios, where M should be small, to cope with

ICI from frequency dispersion [32–34].

OFDM/OQAM symbol as

yp,q ¼Hp,qdp,qþ EXM�1

m ¼ 0

Xn|fflfflfflfflffl{zfflfflfflfflffl}

ðm,nÞaðp,qÞ

Hm,ndm,n/gSp,qm,n

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ip,q

þZp,q, ð4Þ

where Hp,q is the channel frequency response (CFR) at thatFT point, and Ip,q and Zp,q are the associated interferenceand noise components, respectively. Zp,q has been shownto be stationary (in fact, also Gaussian with zero meanand variance s2) and correlated among subcarriers (see[17]). In the sequel, this correlation will be mostlyneglected, as it is this case with well localized AFBs.

For pulses g that are well localized in both time andfrequency, the interference from FT points outside aneighborhood Op,q of (p,q) (excluding (p,q) itself) isnegligible. If, moreover, the CFR is almost constant overthis neighborhood, one can write (4) as

yp,q �Hp,qcp,qþZp,q, ð5Þ

where

cp,q ¼ dp,qþEX

ðm,nÞ2Op,q

dm,n/gSp,qm,n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

up,q

¼ dp,qþ Eup,q ð6Þ

is the virtual transmitted symbol at (p,q), with

up,q ¼X

ðm,nÞ2Op,q

dm,n/gSp,qm,n ð7Þ

being the imaginary part of the interference from theneigboring FT points. When known pilots are transmittedat that FT point and its neighborhood Op,q, the quantity in(6) can be approximated. This can then serve as a pseudo-

pilot [65] to compute an estimate of the CFR at thecorresponding FT point, as, for example,

Hp,q ¼yp,q

cp,q�Hp,qþ

Zp,q

cp,q: ð8Þ

This observation underlies the so-called interferenceapproximation method (IAM) to be described in detail inthe next section.

The most common assumption is that, with a welltime-frequency localized pulse, contributions to Ip,q onlycome from the first-order neighborhood of (p,q), namelyOp,q ¼ fðp71,q71Þ,ðp,q71Þ,ðp71,qÞg (see Fig. 1). A spe-cial case is given by Op,q ¼O1

p,q ¼ fðp71,qÞg. This ariseswhen we place three adjacent pilot tones at positionsðp�1,qÞ,ðp,qÞ,ðpþ1,qÞ and zeros at the rest of the first-order neighborhood positions or when we place non-zeropilot tones at all positions in the training vector and zerovectors around it.

If the neighbors of (p,q) carry unknown (data) symbols,one cannot (directly) approximate the imaginary inter-ference in (6). However, by properly choosing one of theneighboring symbols, say at the point (r,s), this interfer-ence can be forced to zero. Then the pseudo-pilot in (6)becomes real, equal to dp,q. The pilot at (r,s) is then knownas a help pilot [47].6

6 This idea was later proposed again in [51].

2,21,2,21,22,2

2,11,1,11,12,1

2,1,,1,2,

2,11,1,11,12,1

2,21,2,21,22,2

qpqpqpqpqp

qpqpqpqpqp

qpqpqpqpqp

qpqpqpqpqp

qpqpqpqpqp

ddddd

ddddd

dddddddddd

ddddd

Time

Freq

uenc

y

Fig. 1. The first-order time-frequency neighborhood of FT point (p,q).

Fig. 2. The MIMO-OFDM/OQAM system.

7 Same principles apply if the pilot blocks are placed in the middle

of the frame (midambles).

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2041

The above formulation can be easily extended to theMIMO case. Consider an Nt � Nr MIMO system, withidentical SFBs (AFBs) at each transmit (receive) antenna(see Fig. 2). Then, adopting the same assumptions asbefore, one can write an equation analogous to (5) foreach receive antenna j¼ 1,2, . . . ,Nr

yjp,q �

XNt

i ¼ 1

Hj,ip,qci

p,qþZjp,q, ð9Þ

where Hj,ip,q is the M-point CFR of the channel from the ith

transmit antenna to the jth receive antenna, cip,q is the

corresponding virtual symbol, namely

cip,q ¼ di

p,qþEX

ðm,nÞ2Op,q

dim,n/gSp,q

m,n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ui

p,q

¼ dip,qþEu

ip,q

and Zjp,q denotes the corresponding noise component.

Clearly, the latter is uncorrelated among different receiveantennas. Collecting the above equations for all receiveantennas, results, for each FT point, in an input–outputequation similar to that for MIMO-OFDM [85]

yp,q �Hp,qcp,qþgp,q, ð10Þ

where

Hp,q ¼

H1,1p,q H1,2

p,q � � � H1,Ntp,q

H2,1p,q H2,2

p,q � � � H2,Ntp,q

^ ^ & ^

HNr ,1p,q HNr ,2

p,q � � � HNr ,Ntp,q

2666664

3777775

is the MIMO CFR at that point and

cp,q ¼ dp,qþ Eup,q

with

dp,q ¼ ½d1p,q d2

p,q � � � dNt

p,q�T

and up,q defined similarly.

3. Preamble-based channel estimation methods: thesingle-antenna case

Known (training) symbols for the purpose of channelestimation can be gathered in the beginning of a frameand/or placed at isolated FT points throughout the frame.These are the two principal pilot configurations, known aspreamble and scattered pilots, respectively. The focus ofthis paper is on the design of the preamble7 for OFDM/OQAM systems and its use in estimating the channel. Thedifficulty in this task, compared to OFDM/QAM, lies inestimating a complex CFR in the presence of the imaginary

intrinsic interference among neighboring FT points. Thepreamble design and the associated estimation methodshould take this scenario into account in some appro-priate way. Three different approaches for preamble-based channel estimation, including their variants, arereviewed in this section, for SISO systems. The firstapproach, pairs of pilots (POP), relies on simple algebraicrelations for the input/output samples in a number of (inpractice consecutive) time instants. The second one,interference approximation method (IAM), aims atapproximating the intrinsic imaginary interference fromneighboring pilots and hence constructing complexpseudo-pilots as above, to accommodate the complexCFR. A number of variants of IAM are considered. In thiscontext, recent results on designing optimal – in the senseof minimizing the channel estimation error – preamblesare also recalled. An alternative approach, that of cancel-ing or avoiding interference, is then discussed. The exten-sion of some of these methods to the MIMO case isaddressed in the following section.

Note that the methods presented here can be appliedto any OFDM/OQAM filter bank, provided, of course, thatthe prototype filter is sufficiently well localized in timeand frequency and the number of subcarriers is largeenough that the models in Eqs. (5) and (10) hold withsufficient accuracy. Thus, methods aiming at enhancingthe channel estimation performance through aninterference-minimizing filter design are omitted; seee.g., [22,72]. Moreover, we do not include methods basedon preambles that are not completely known to thereceiver (as in, e.g., [35], where the training sequence issuperimposed on the data) or are too spectrally inefficient(e.g., made of isolated pilots surrounded by a largeneighborhood of zeros, as in [75–77]) or especially suitedto wideband subchannels, being significantly more com-plicated (e.g., [33]) or significantly longer than thoseconsidered here (e.g., [39,38]).

In the sequel, and in addition to the assumptionsstated in the previous section, the channel is assumed tobe time invariant during the preamble period.

8 This is without loss of generality as analogous patterns result for

other definitions of j0 (see e.g., [26]).

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542042

3.1. Pairs of pilots (POP) method

This method, proposed in [65], aims at computing achannel estimate by using (5) at two different (in practiceconsecutive) time instants, q1,q2, to construct a system ofequations for the real and imaginary parts of Hp,q. Here, itis described in an alternative, equivalent way [23].

With the noise being neglected, denote by Wp,q ¼ 1=Hp,q

the corresponding zero forcing (ZF) equalizer coefficients.Then, one can write

yp,q1Wp,q1

¼ dp,q1þ Eup,q1

,

yp,q2Wp,q2

¼ dp,q2þ Eup,q2

and therefore

yRp,q1

WRp,q1�yI

p,q1W I

p,q1¼ dp,q1

,

yRp,q2

WRp,q2�yI

p,q2W I

p,q2¼ dp,q2

:

Noting that Wp,q1�Wp,q2

due to the (almost) time invar-iance of the CFR,

yRp,q1

�yIp,q1

yRp,q2

�yIp,q2

24

35 WR

p,q1

W Ip,q1

24

35¼ dp,q1

dp,q2

" #,

hence

WRp,q1

W Ip,q1

24

35¼ 1

yIp,q1

yRp,q2�yR

p,q1yI

p,q2

yIp,q1

dp,q2�yI

p,q2dp,q1

yRp,q1

dp,q2�yR

p,q2dp,q1

24

35:

This can be more compactly written as

Wp,q1¼WR

p,q1þEW I

p,q1¼ E

dp,q1y%

p,q2�dp,q2

y%

p,q1

Iðy%

p,q1yp,q2Þ

: ð11Þ

In practice, the preamble would comprise the first twoOFDM/OQAM symbols, that is, q1 ¼ 0 and q2 ¼ 1. In thatcase, and with the preamble suggested in [65], wheredp,0 ¼ ð�1Þp and the second symbol is all zeros, the abovewould simplify to

Wp,0 ¼ Eð�1Þpy%

p,1

Iðy%

p,0yp,1Þ: ð12Þ

The CFR would then be computed as Hp,0 ¼ 1=Wp,0.An advantage of the POP scheme, besides its simplicity,

is that it does not explicitly depend on the employedprototype filter. However, it must be emphasized that theabove derivation only holds when noise is negligible. Onecan see that (see [65]), in the presence of noise, themethod can have unpredictable performance since thedegree of the noise enhancement in general also dependson unknown (hence uncontrollable) data.

3.2. Interference approximation method (IAM)

If the interference Eup,q in (6) is only due to theimmediate neighbors of (p,q) and these FT points carrytraining (hence known) symbols, then one can computean approximation of this interference, as pointed outabove. This can then be used to construct the complexpseudo-pilots cp,q and subsequently use them to get achannel estimate in exactly the same way as in CP-OFDM(see (8)). This method is called (for obvious reasons) the

interference approximation method (IAM) and requiresthat all input symbols in the immediate neighborhood of(p,q) be known. This implies that, for each subcarrier, (atleast) 1.5 complex symbols are required for training,representing an increased (though not significantly) over-head compared to CP-OFDM. The training would need toextend over more than three time instants for less welltime-localized filters.

Writing (8) out,

Hp,q ¼Hp,qþZp,q

dp,qþ Eup,q,

one can see that the pilots should be so chosen as tominimize the effect of the noise term. In [64,65], it isproposed that these be chosen so as to have pseudo-pilotsof maximum magnitude. To this end, the surroundingtraining symbols should be such that all terms in (7) havethe same sign so that they add together. Moreover, thisshould happen at all frequencies p.

To this end, one needs to know the interference weights/gSp,q

m,n for the neighbors ðm,nÞ 2 Op,q of each FT point (p,q)of interest. These can be a priori computed based on theemployed prototype filter g. Moreover, it can be shown(see the Appendix) that, for any choice of g, these weightsfollow a specific pattern, which, for the definition of j0

adopted here, and for all q, can be written as8

ð�1ÞpE 0 �ð�1ÞpEð�1Þpd �b ð�1Þpd�ð�1Þpg dp,q ð�1Þpgð�1Þpd b ð�1Þpdð�1ÞpE 0 �ð�1ÞpE

ð13Þ

with the horizontal direction corresponding to time andthe vertical one to frequency, as in Fig. 1. The abovequantities can be shown (see the Appendix) to be given by

b¼ e�jð2p=MÞðLg�1Þ=2XLg�1

l ¼ 0

g2ðlÞejð2p=MÞl, ð14Þ

g¼XLg�1

l ¼ M=2

gðlÞg l�M

2

� �, ð15Þ

d¼�je�jð2p=MÞðLg�1Þ=2XLg�1

l ¼ M=2

gðlÞg l�M

2

� �ejð2p=MÞl, ð16Þ

E¼ e8 Eð2p=MÞðLg�1ÞXLg�1

l ¼ M=2

gðlÞg l�M

2

� �e7 E2ð2p=MÞl ð17Þ

and are (absolutely) smaller than one, with b,g,d40.Generally, b,g4d49E9. For example, g¼ 0:553, b¼ 0:25,d¼ 0:2172, and E� 0:0004 for the g’s designed in [11]with M¼ 512,K ¼ 3.

Fig. 3. Preamble structures for (a) IAM-R , (b) IAM-I, (c) IAM-C methods, and E-IAM-C method for (d) EZ0 and (e) Eo0. M¼8. OQPSK modulation is

assumed. d0 and d1 are randomly chosen BPSK symbols.

9 With jm,n ¼ ðmþnÞp=2 instead, these symbols must be all equal.

See [53] for more on this, independent of the jm,n definition.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2043

3.2.1. The IAM-R method

In order to simplify the task of generating pseudo-pilots of large magnitude, it was suggested in [64,65,63]to place nulls at the first and third OFDM/OQAM symbols

of the preamble, that is, dp,0 ¼ dp,2 ¼ 0, p¼ 0,1, . . . ,M�1.

Then, the imaginary part of the pseudo-pilot at ðp,1Þ willonly come from the symbols at the positions ðp71,1Þ.Obviously, these pilots must be OQAM symbols of max-imum modulus, d. Moreover, in view of the pattern (13),

they should satisfy the condition dpþ1,1 ¼�dp�1,1 for all p,

in order to yield pseudo-pilots of maximum magnitude,

namely 9dp,1þ E2bdpþ1,19¼ d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ4b2

q. This preamble

structure is shown in Fig. 3(a), for the example of M¼8and with OQPSK modulation. This channel estimationmethod is called IAM2 in [65], to distinguish it fromIAM1, where all symbols, at all three time instants, arerandomly chosen. To prevent possible increase of the peakto average power ratio (PAPR) with IAM2, a randomchoice of the middle symbols in IAM2 is also proposedin [65], giving rise to the IAM3 method. In the sequel, wewill focus on IAM2, due to its ability to maximize themagnitude of the pseudo-pilots with the aid of real pilotsymbols, and use the name IAM-R, coined in [68] tosignify the fact that its pilot symbols are real.

3.2.2. The IAM-I method

It can be verified from (6) that one can do better inmaximizing the magnitude of the pseudo-pilots if imaginary

pilot symbols Edp,1 of maximum modulus d are also allowedin the preamble. The reason is that the correspondingpseudo-pilot would then be imaginary and hence of largermagnitude than in IAM-R, namely 9dp,1þup,19, provided thatthe signs of the pilot symbol and its neighbors are properlychosen so that dp,1up,140. Such an IAM scheme was firstproposed in [68], and was called IAM-I to signify thepresence of imaginary pilots. The middle preamble vectorconsists of triplets, with each one of them following theabove principle, and the pilots in each triplet are otherwiseselected in random and independently of the other triplets.Hence, imaginary pseudo-pilots result in only one third ofthe subcarriers, with magnitude dð1þ2bÞ, whereas the restof them deliver complex pseudo-pilots of smaller magni-tude, d9ð1þbÞþ Eb9. An example, for M¼8 and OQPSKsymbols, is depicted in Fig. 3(b). Note that the above

preamble is, strictly speaking, not OQAM. Nevertheless, itcan still be fed to the synthesis filter bank and perfectlyreconstructed at the analysis bank.

3.2.3. The IAM-C method

With the preamble proposed in [68], imaginarypseudo-pilots will result in only one third of the sub-carriers used, whereas the rest of the subcarriers willdeliver complex pseudo-pilots. With a slight modification,one can improve upon this and obtain pseudo-pilots thatare either purely real or imaginary at all the subcarriers.The idea is to simply set the middle OFDM/OQAM symbolequal to that in IAM-R but with the pilots at the oddsubcarriers multiplied by E, as shown in [24] and inde-pendently in [29]. Again, for the M¼8 OQPSK example,this will result in the preamble shown in Fig. 3(c). Thispreamble was called IAM-C in [29].

3.2.4. Optimal preambles

In [29], the IAM-C preamble was derived as a solutionto the problem of optimal 3-symbol preamble design withzero guard symbols, where optimal is in the sense ofgenerating pseudo-pilots of maximum magnitude. Optim-ality of preambles of this kind, in the sense of attainingthe minimum possible mean squared error (MSE) with agiven transmit energy budget, was recently studied in[52] and the results were later complemented in [53].Two different configurations were considered: sparse,where only Lh isolated subcarriers carry pilots, with therest of them being nulled, and full, with all frequenciesbeing occupied with non-zero pilots. It was proved that:

1.

The pilot-carrying subcarriers in an optimal sparsepreamble are equidistant with the pilots being equi-powered.

2.

With jm,n defined as above and including the phasefactors eEjm,1 in the pilot symbols, they should be all equalbut with alternating signs in an optimal full preamble.9

One can readily see that the IAM-C preamble – proved tobe optimal in [29] – follows the general structure of an

Fig. 4. Preamble structures aiming at interference cancelation: (a) [43],

(b) [51], (c) [91], and avoidance: (d) [42,25]. M¼8. OQPSK modulation is

assumed.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542044

optimal full preamble of this kind, as it was derived in[52,53]. For example, multiplied with ejjm,1 the middlevector in Fig. 3(c) becomes E,�E,E,�E, . . ..

3.2.5. The extended IAM-C (E-IAM-C) method

The optimality of IAM-C is restricted to the 3-symbolpreambles with nulls at their sides. One could wonderwhether an alternative preamble structure whichemploys the side symbols as well could yield pseudo-pilots that are stronger than those of IAM-C. It turns outthat such a preamble can indeed be formed in ananalogous way as previously, by employing the left andright hand sides of the neighborhood (13) as well [55].10

Thus, at an odd-indexed subcarrier p, with the pilot 7dEin the middle, one should place 8d at the right hand side,and its negative, 7d at the left hand side. These sidepilots will then contribute 7dgE each to the interferencefor the central pilot. In an analogous way, at an even-indexed subcarrier p, the middle pilot, say 7d, will get atotal interference of 72dgE from its ðp,0Þ and ðp,2Þneighbors, if these are chosen as 7dE and 8dE, respec-tively. It can then be verified that, because of (13), theinterference components with weights 7d cancel eachother, and hence have no contribution to the pseudo-pilot. However, this loss is not significant since d issignificantly smaller than g. The resulting pseudo-pilotsare again either purely real or imaginary, with magnituded91þ2ðbþgþ2EÞ9, which is clearly larger than thatachieved by IAM-C. For the example filters employed here[11], with M¼512 and K¼3, the corresponding magni-tudes are 2.6076d and 1.5d. An example of the preambleconfiguration for M¼8 and OQPSK modulation is given inFig. 3(d). One can see that the left hand column is built byrepeating each quadruple of the middle pilots in reverseorder, while the right hand column is simply the negativeof the left hand one.

The above hold for prototype filters having EZ0, as itis most often the case. Whenever Eo0, as it is the case forextended Gaussian function (EGF)-based filter banks forexample [63], the E-IAM-C preamble can be modified to

yield pseudo-pilots of magnitudeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ2bÞ2þ4ðg�2EÞ2

q.

Fig. 3(e) shows such a preamble configuration for M¼8,with OQPSK modulation.

It must be emphasized here that, since all three OFDM/OQAM symbols of the E-IAM-C preamble are non-zero, itwill require more power to be transmitted than theprevious ones. This fact is fairly taken into account in thesimulation experiments (see Section 5), along with possibleinterference coming from the data part of the frame.

3.3. Interference cancelation/avoidance methods

An alternative, more straightforward approach to copewith the intrinsic interference is to so design the preamble

10 This idea seems to have first been investigated in [26, Section

5.4]. It must be noted, however, that for a number of pulse types

(including those of [11] employed for the simulations in this paper,

where g4b4d and do0:5), the preamble of [55] results in pseudo-

pilots of a larger magnitude than in the preambles developed in [26].

and the associated estimation method as to somehowcancel or avoid it. This is in contrast to the IAM approachpresented above, where interference is taken advantage ofto improve the channel estimation accuracy. The methodsfollowing this alternative approach can be roughly cate-gorized as follows.

A direct way to cancel the interference is to simply nullthe data surrounding the pilot FT point. Note that this ideawas already used in some of the IAM preambles above,where guard OFDM/OQAM symbols were inserted to(approximately) null the interference from precedingand following data. The problem can be further simplifiedif zeros are also transmitted at the middle symbol, at allodd- (or all even-)indexed subcarriers. In this way, theCFR can be estimated at the even-(odd-)indexed subcar-riers just like in CP-OFDM. The channel at the rest of thefrequencies will then have to be estimated via interpola-tion [37]. Such a preamble design was proposed in [43]and an example (for M¼8) is given in Fig. 4(a). The sameidea was essentially used in [76,77], and it was alsoindependently included in the preamble evaluation studyof [24] (for MIMO). In the latter, only every third sub-carrier was modulated, in accordance with the pilotconfiguration suggested in WiMAX DL-PUSC with seg-mentation [7]. A less direct design, which relies on thesymmetries in (13) to cancel the interference from adja-cent subcarriers, was proposed in [51] and is depicted inFig. 4(b) for the M¼8 example.

A more spectrally efficient preamble design method,which also takes advantage of the symmetries in theneighboring times in (13) to transmit data at these FTpositions instead of nulls, was suggested in [91]. The waythis is achieved is by considering only the first-orderneighbors and canceling the interferences coming fromthem in pairs. Thus, following the pattern in (13), the datamodulating the first-order neighbors would be structuredas follows:

a b c

d � d

�a b �c

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2045

Note that the idea is similar to that of the help pilot [47],although, here, only four, instead of seven informationsymbols can be transmitted per pilot. In fact, this idea isproposed in [91] in the context of scattered pilots. Onecan, however, easily use it to design a preamble with theproperty of interference cancelation, by simply applyingthe above pattern for every third subcarrier and resortingto interpolation for the rest of them. This would result in apreamble containing about 4M=3 data symbols; this is aconsiderable gain in spectral efficiency compared to themethods previously described. To obtain better channelestimates at the intermediate frequencies, one could evenapply this in every second subcarrier. This is madepossible by the fact that, as shown in (13), the interfer-ence weights are the same for every second subcarrier.However, such a preamble construction would imposesome additional constraints to the data that can betransmitted, reducing the data symbols to only 3þðM=2Þ.Fig. 4(c) provides an example for M¼8 and pilots placed atthe even-indexed subcarriers.

Preambles of an even higher spectral efficiency can beconceived, comprising only one reference OFDM/OQAMsymbol surrounded by completely unknown data. Hence,no guard symbols or data of a specific structure aretransmitted along with the known symbols as in theprevious cases. This, however, implies that the interfer-ence from surrounding symbols is completely unknownand its elimination becomes an important issue. Theapproach proposed in [42] is to follow an iterativeprocedure, where in each iteration the available channelestimate is used to approximate the surrounding data andvice versa. The original channel estimate is only based onthe preamble symbol, ignoring the interference effect. Thefollowing channel estimates are computed on the basis ofboth pilot symbols and estimated data symbols (e.g.,using least squares (LS)). The channel and data estimatesare improved from one iteration to the other. In thesimulation results reported in [42], the iterative approachis shown to achieve better performance than the methodrelying on guard symbols, in only two iterations. Moreiterations are shown to lead to further improvements.Such an iterative approach was also proposed in [67],albeit for isolated (scattered) pilots. Observe that one canso choose the pilot symbols so as to, at least, cancel theinterference from neighboring pilots, by respecting thesymmetry of (13). An example, for M¼8, is given inFig. 4(d). A similar preamble is used in the method of[25], where the channel and data symbol estimates areused to approximate the interference and subtract it fromthe AFB output. An alternative and quite effectiveapproach to reducing the interference (and noise) effectson the CFR estimates is through the exploitation of theknowledge of (an upper bound of) the channel delay spread(e.g., via DFT interpolation of the estimated CFR values), asrecently demonstrated in [69] (see also [52]), where theinitial estimates in the method of [42] were shown to admita considerable improvement. Such a filtering of the CFRestimates can be easily applied in other preamble-basedmethods as well, including the IAM schemes (and CP-OFDMchannel estimation) evaluated in Section 5, with significantgains in performance (not shown here).

4. Preamble-based channel estimation methods: themultiple-antenna case

4.1. MIMO POP method

The POP method of Section 3.1 can be easily extendedto the MIMO case, as follows [24]. Consider (10) withnoise being neglected and assume Hp,q is left-invertiblefor all p,q (hence NrZNt). Denote by W the (per-sub-carrier) ZF equalization matrix, that is, Wp,q ¼Hyp,q, whereHyp,q is the left inverse of Hp,q. Write (10) in the formWp,qyp,q ¼ cp,q for 2Nr time instants, sayq¼ 0,1, . . . ,2Nr�1, assuming that the channel (and henceW) can be viewed as invariant in this time period. Then

WRp,0yR

p,q�W Ip,0yI

p,q ¼ dp,q,

for q¼ 0,1, . . . ,2Nr�1. Collecting these equations into one,one can write

½WRp,0 W I

p,0�yR

p,0 yRp,1 � � � yR

p,2Nr�1

�yIp,0 �yI

p,1 � � � �yIp,2Nr�1

24

35

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Yp

¼ ½dp,0 dp,1 � � � dp,2Nr�1�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Dp

or

½WRp,0 W I

p,0�Yp ¼Dp:

The ZF equalizer can then be computed as

½WRp,0 W I

p,0� ¼DpY�1p

and hence the channel matrix can be found as the right

inverse of

Wp,0 ¼WRp,0þ EW

Ip,0 ¼DpY�1

p

1

E

" #� INr

!:

Remark. It must be noted that this scheme is no longerassociated to the concept of ‘‘pairs of pilots’’ and the name‘‘POP’’ can only be used here as a reminder of its SISOcounterpart.

4.2. MIMO IAM

As in MIMO-OFDM, one can consider three differentpatterns for preamble training (see e.g., [7, Section 5.7]):independent (one antenna at a time), scattered (trans-mitting at different frequencies), and orthogonal (transmit-ting orthogonal signals at the same times and frequencies).IAM preambles of the latter category can be easily con-structed from their SISO counterparts if a pattern analogousto that used for MIMO-OFDM in [85] is followed. Such anextension of IAM to the MIMO setup was reported andanalyzed in [56], and is, in fact, the first attempt to addressfull preamble-based channel estimation in the MIMO-OFDM/OQAM context. A sparse preamble design was first consid-ered in [24] and was based on earlier work on orthogonal

MSE-optimal training design for MIMO-OFDM systems. Thiswill be described in the next section.

Fig. 5. IAM-C preamble for a 2� X system, with (a) and (b) correspond-

ing to the two transmit antennas. M¼8. OQPSK modulation is assumed.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542046

It is clear from (10) that one will need at least Nt (non-zero) OFDM/OQAM symbols to estimate the CFR matrix.Assume the channel is time invariant during a period of2Ntþ1 OFDM/OQAM symbols (3Nt for E-IAM-C). Take theexample of Nt ¼ 2, for the sake of simplicity. One antennacan use the same preamble as the one used in the SISOcase, but with a repetition, as shown in the example ofFig. 5(a), for IAM-C (similarly for the other IAM variants).The other antenna uses the same preamble, but withreversed signs at the second non-zero OFDM/OQAMsymbol (cf. Fig. 5(b)). Then, writing (10) at times q¼1,3(q¼1,4 for E-IAM-C) results in

½yp,1 yp,3� ¼Hp,1

c1p,1 c1

p,3

c2p,1 c2

p,3

24

35þ½gp,1 gp,3�:

Taking the structure of this preamble into account andrecalling the assumption about the interference beingmostly contributed from the first-order FT neighbors,one can easily see that c1

p,1 ¼ c1p,3 ¼ c2

p,1 ¼�c2p,3 � cp and

hence

½yp,1 yp,3� ¼Hp,1

cp cp

cp �cp

" #þ½gp,1 gp,3�

¼Hp,1cpA2þ½gp,1 gp,3�, ð18Þ

where A2 is the orthogonal matrix

A2 ¼1 1

1 �1

� �

and the pseudo-pilot cp can be again pre-calculated as inthe SISO case. An estimate of the CFR matrix at thesubcarrier p can then be computed as

Hp,1 ¼ ½yp,1 yp,3�1

cpA�1

2

¼Hp,1þ1

2cp½gp,1 gp,3�A2, ð19Þ

Remarks.

1.

It should be noted that, in all IAM variants except forIAM-I, cp’s have the same magnitude at all frequencies p.

2.

Note that, in view of the orthogonality of the matrixA2, noise enhancement is again controlled by themagnitude of the pseudo pilot cp. Indeed, one canreadily see that the covariance matrix of the noiseterm in (19) equals s2=9cp9

2INr

, which corresponds toan MSE given by

EðJHp,1�Hp,1J2Þ ¼Nrs2=9cp9

2: ð20Þ

3.

The preamble given above is easily generalized tosystems with more than two transmit antennas, withNt being a power of 2. Simply select the correspondingmatrix ANt

to be a Hadamard matrix of order Nt andbuild the preamble accordingly. The MSE will be stillgiven by (20).

4.

For an Nt � Nr system, the IAM preambles with guardsymbols require 2Ntþ1 OFDM/OQAM symbols. This isequivalent to Ntþ1=2 complex OFDM/QAM symbols,that is, half a complex symbol longer than the pre-amble one would use in MIMO-OFDM. Nevertheless,even this extra OQAM symbol can be avoided inpractice, by eliminating the leading null symbol inthe preamble above. This is often possible wheninterference from preceding data can be practicallyignored, as for example in WiMAX [7], where the gapbetween two successive frames can play the role ofthe missing guard symbol. The number of complexOFDM symbols spent for the E-IAM-C preamble issomewhat larger: Nt=2 more than that one would usefor MIMO-OFDM. However, as it is demonstrated inthe simulation examples, this increase in the pream-ble duration can offer considerable improvement inthe estimation accuracy.

5.

Eq. (18) relies on the assumption that the prototypefilter is sufficiently well localized in time so thatinterference from time instants other than the pre-vious and next is negligible. In fact, often this is notthe case in practice and there is non-negligible inter-ference to the (p,q) FT point from time instants q72as well. Hence the ‘‘pseudo pilot’’ matrix in (18) isequal to

c1p c1

p

c2p �c2

p

24

35

instead, with cip corresponding to the ith transmit

antenna and 9c1p9a9c2

p9. The latter implies that the

two columns of the CFR matrix (i.e., the channelsfrom the two transmit antennas) are not estimatedwith the same accuracy. The associated MSE for the

subcarrier p is then equal to ðNrs2=2Þð1=9c1p9

2þ

1=9c2p9

2Þ. Things are analogous, yet somewhat more

complicated, for Nt42. In general, the pseudo pilotmatrix is no longer unitary. It only has (nearly)orthogonal rows. This fact was taken into accountin the simulations; that is, the correct (not theideal) pseudo pilot matrix was used in all cases.Note that this difficulty can be easily overcome,albeit at an extra cost in training bandwidth, if

11 If

smaller12 N

(NtNrLh)

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2047

more than one guard symbol are placed among theNt repetitions of the IAM preamble above.

4.3. Optimal sparse preamble

The MIMO preambles described above are of durationproportional to the number of transmit antennas, Nt.Shorter preambles, consisting of three OFDM/OQAM sym-bols per antenna (independently of Nt) as in the SISO case(with zero guard symbols surrounding a non-zero one),were also reported in [24] and are of the sparse type, i.e.,with nulls among the pilot subcarriers. One of thosepreambles was based on earlier work on orthogonal

training design for MIMO-OFDM systems [90] and shownto be also optimal in the MSE sense. The idea is brieflyrecalled here. Such a channel estimation approach wasalso recently adopted in [19].

Let N¼M=Lh be an integer,11 greater than or equal to2Nt.

12 Consider Nt sets of Lh pilots each, placed at sub-

carriers fpi,piþN,piþ2N, . . . ,piþðLh�1ÞNg, i¼ 1,2, . . . ,Nt,with user-chosen starting positions p1,p2, . . . ,pNt

2

f0,1, . . . ,N�1g and with the rest of the subcarriers beingnulled. Thus, the pilot subcarriers within each set areequispaced. Moreover, the pilot symbols are chosen to beequipowered. pi’s can be so selected as to have at least onezero between two non-zero pilots (sparse preamble), thusavoiding interference among them.

Since the focus will be on the middle non-zero pre-amble vector in each antenna, the time index (normallyq¼1) will be dropped for simplicity. Let

yðpÞ ¼ ½yTp yT

pþN � � � yTpþðLh�1ÞN�

T ,

with yp � yp,1 defined as in (10), contain the receivedsignals at the pth pilot-tone set and similarly define gðpÞ

for the corresponding frequency domain noise compo-nents. Then one can readily verify that

yðpÞ ¼DðpÞ1 F LhW ðpÞh�,1þDðpÞ2 F Lh

W ðpÞh�,2þ � � � þDðpÞNtF Lh

W ðpÞh�,NtþgðpÞ

¼XNt

i ¼ 1

C iph�,iþgðpÞ, ð21Þ

where

C ip ¼DðpÞi F Lh

W ðpÞ

with

DðpÞi ¼ diagðdip,di

pþN , . . . ,dipþðLh�1ÞNÞ � INr

,

F Lh¼ Fð1 : N : M,1 : LhÞ � INr

with Fð1 : N : M,1 : LhÞ being (in Matlab notation) thesubmatrix of the Mth-order DFT matrix F consisting ofits first Lh columns and every Nth of its rows,

W ðpÞ¼ diagð1,wp

M ,w2pM , . . . ,wðLh�1Þp

M Þ

not, one can use for Lh the smallest power of 2 that is not

than Lh, that is, 2blog2 Lhc .

ZNt anyway, since the number of unknown parameters

cannot exceed the number of received signals (MNr).

with wM ¼ e�Eð2p=MÞ denoting the Mth root of unity, and h�,l

contains the impulse responses of the channels fromtransmit antenna i to all receive antennas. Then one cancompute an estimate of the entire system impulseresponse h¼ ½ðh�,1ÞT ðh�,2ÞT � � � ðh�,Nt Þ

T�T by solving the

NtNrLh � NtNrLh system of equations

yðp1Þ

yðp2Þ

^

yðpNtÞ

266664

377775

|fflfflfflfflffl{zfflfflfflfflffl}y

¼

C1p1

C2p1� � � CNt

p1

C1p2

C2p2� � � CNt

p2

^ ^ & ^

C1pNt

C2pNt

� � � CNtpNt

2666664

3777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C

hþ

gðp1Þ

gðp2Þ

^

gðpNtÞ

266664

377775

|fflfflfflfflffl{zfflfflfflfflffl}g

or

y¼ Chþg: ð22Þ

Note that g is white. Thus, the LS solution of (22) will beoptimal in the MSE sense if C is unitary. A sufficientcondition ensuring the latter is that

dip ¼ di

pþN ¼ � � � ¼ dipþðLh�1ÞN for all p,i ð23Þ

and the matrix

D¼

d1p1

d2p1� � � dNt

p1

d1p2

d2p2� � � dNt

p2

^ ^ & ^

d1pNt

d2pNt� � � dNt

pNt

26666664

37777775 ð24Þ

be unitary. For example, with M¼8, Nt ¼ 2 and Lh ¼ 2, wehave N¼4. Choosing D to be the orthogonal matrix

D¼1 1

1 �1

� �

and p1 ¼ 0, p2 ¼ 2, the middle (non-zero) vectors of thepreambles employed at the two transmit antennas will be

1 1

0 0

1 �1

0 0

1 1

0 0

1 �1

0 0

Note that, in the SISO case, the above reduces to theoptimal sparse preamble-based approach of [52] as out-lined previously. In fact, it can be verified that, in thatcase, the pilot symbols need only be equipowered (not asin (23)) as proved in [52].

5. Simulation results

In this section, simulation results are reported thatdemonstrate and compare the performances of the IAMschemes previously discussed. In the SISO scenario, theiterative method of Hu et al. [42] was also tested, with thepreamble built as in Fig. 4(d), preceded by zeros (due, e.g.,

0 1000 20000

200

400IAM−R

Mag

nitu

de s

quar

ed

0 1000 20000

10

20

30IAM−I

0 1000 20000

500

1000IAM−C

Sample no.

Mag

nitu

de s

quar

ed

0 1000 20000

1000

2000

3000E−IAM−C

Sample no.

Fig. 6. Magnitude squared of the modulated IAM preambles. M¼512, K¼3. QPSK modulation.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542048

to the inter-frame gap) and followed by random i.i.d. data.Results of the POP method were not included since, asexpected, it had a quite poor performance compared withthe rest of the methods studied. For the sake of complete-ness, CP-OFDM is also included, with the minimum pos-sible CP length (equal to the channel order). A realisticscenario where all the preambles are followed by pseudo-random data13 was considered. This is of importance forthe results to be presented, since the assumption usedabove that the middle preamble OFDM/OQAM symbolreceives negligible interference from the following data isnot exact, even with the well localized prototype filtersemployed in the experiments. Thus, the late part of thepreamble signal contains a portion of the front tail of thedata burst as well, and this was taken into account when

estimating the required transmit power. Note that thisproblem is not present in CP-OFDM and is due to thelong tails of the OFDM/OQAM prototype filter impulseresponse. It should also be pointed out here that the SFBoutput signal due to the (SISO) 3-symbol OFDM/OQAMIAM preamble is of length ðKþ1ÞM, which is approxi-mately Kþ1 times as long as that of CP-OFDM. One couldconsider shortening the preamble burst by eliminating itstails, as it was recently described in [14]. As for the timeinvariance assumption for the channel and its validitythroughout the preamble signal, one can see that, forscenarios similar to that of WiMAX for example [7], and

13 In fact, what comes immediately after the preamble is usually

control information rather than random data (e.g., [7]). Nevertheless,

whatever structure is found in this control part of the frame is ignored

here for the purposes of the experiments.

for reasonable mobile speeds, the channel coherenceinterval spans many more than Kþ1 M-blocks.

The experiments utilized prototype filters designed asin [11]. Filter banks with M¼512 and K¼3 were used.QPSK modulation was adopted. The comparison results arepresented in terms of the normalized MSE (NMSE) inestimating the channel and/or the corresponding detection(uncoded bit error rate (BER)) performance of zero forcingsingle-tap per subcarrier equalizers working in the fre-quency domain. For the latter, the channels were assumedtime invariant over the whole frame for simplicity. Thesignal to noise ratio (SNR) loss in CP-OFDM due to the CPredundancy in data transmission was taken into accountwhen calculating BER. The NMSE, EðJH�HJ2=JHJ2

Þ, whereH is the CFR and H its estimate, is plotted with respect tothe SNR. The latter is defined as the ratio of the channelinput (i.e., SFB output) power to the power of the noise.Note that in order to be fair with respect to the powertransmission requirements, all the preambles are appro-priately scaled so as to result in the same power at the SFB

output. It must be emphasized that these power valuesturn out to differ much more with each other than they doat the SFB input. (In fact, earlier comparison studies seemnot to have taken this into account and only equalizedpowers at the SFB input.) An example is shown in Fig. 6.The figure also shows that, as pointed out in [63], this kindof IAM preambles result in high PAPR signals at the SFBoutput, due to their deterministic/periodic nature. In fact,as it can be seen in the example, this effect is more evidentin IAM-C and its extended version (PAPR � 33 dB) than inIAM-R (� 32 dB), while it is much less severe in IAM-I(� 18:6 dB) due to its pseudorandom symbols.

0 10 20 30−40

−30

−20

−10

0

10

SNR (dB)

NM

SE

(dB

)

M=512, K=3, VehA

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

0 10 20 30−40

−30

−20

−10

0

10

SNR (dB)

NM

SE

(dB

)

M=512, K=3, VehB

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

Fig. 7. Estimation (NMSE) performance of the preamble-based methods

for SISO systems. CP-OFDM is included for the sake of the comparison.

Filter banks with M¼512 and K¼3 have been used. Time invariant

Rayleigh channels were assumed, with a (a) Veh-A and (b) Veh-B profile.

0 10 20 3010−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BE

R

M=512, K=3, VehA

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

0 10 20 30Eb/N0 (dB)

BE

R

M=512, K=3, VehB

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

10−4

10−3

10−2

10−1

100

Fig. 8. Detection performance (BER) of frequency domain per-subcarrier

single-tap zero forcing equalization with the channel estimates obtained

in Fig. 7.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2049

5.1. SISO systems

Training in CP-OFDM was based on a single-symbolcomplex QPSK pseudo-random preamble. The iterationsin the method of [42] relied on a single data OFDM/OQAMsymbol following the pilot symbol. At moderate to highSNR values, two iterations sufficed for convergence.Fig. 7(a) and (b) shows the MSE performance of themethods under study for Veh-A and Veh-B channels[44], of medium and high frequency selectivity, respec-tively. Observe that in both cases E-IAM-C outperformsthe other schemes in the whole SNR range considered,with the iterative method of Hu et al. exhibiting the worstperformance among the OFDM/OQAM schemes. More-over, all the OFDM/OQAM methods perform better thanCP-OFDM at low to moderate SNR values. At higher SNRs,CP-OFDM takes over while the NMSE curves of the OFDM/OQAM schemes reach an error floor. This is a well knownphenomenon in such systems and is due to the fact thatthe approximation (5) is not exact for channels of

significant time dispersion. Hence there is residual intrin-sic interference, which is hidden by the noise at low SNRsand shows up in the weak noise regime. It should also benoticed that the OFDM/OQAM methods are less wellperforming with channels of higher frequency selectivityand the crossing point with the CP-OFDM curve appearsearlier in that case (see Fig. 7(b)). This behavior is againexplained by the fact that, in the highly frequencyselective case, the assumption of a locally flat CFR, andhence (5), is much less valid. The corresponding BERresults are reported in Fig. 8.

Fig. 9 provides an example of the performancesobtained when the preambles are only normalized toequal powers at the SFB input (instead of at the trans-mitted signal as above), as it is the case in most of theexisting literature. Moreover, the interference from thedata part is considered to be negligible. Comparing withFig. 8(a), one can see that the performance of all OFDM/OQAM methods is then improved while the differencesamong the IAM methods are reduced. In fact, E-IAM-C and

0 10 20 3010−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BE

R

M=512, K=3, VehA

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

13 14 15 16 17

10−2

Eb/N0 (dB)

BE

R

M=512, K=3, VehA

CP−OFDMIAM−RIAM−IIAM−CE−IAM−CHu et al.

Fig. 9. (a) As in Fig. 8(a), but with preamble powers equalized only at

the SFB input. (b) Focus on (a), at the below 10�2 BER level.

0 10 20 3010−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BE

R

M=512, K=3, 2x2, VehA

CP−OFDMIAM−RIAM−IIAM−CE−IAM−C

0 10 20 3010−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BE

R

M=512, K=3, 2x2, VehB

CP−OFDMIAM−RIAM−IIAM−CE−IAM−C

Fig. 10. As in Fig. 8, for 2�2 MIMO systems.

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542050

IAM-C curves almost coincide. Fig. 9(b) zooms on thecurves of Fig. 9(a) at BER of 10�2 and less, to more clearlydemonstrate the performance differences. Note that theIAM schemes using non-real pilots can offer a gain of upto 2 dB compared to CP-OFDM [68].

5.2. MIMO systems

The channel taps were modeled in accordance with theKronecker model, that is hðlÞ ¼ R1=2

r ðlÞhwðlÞR1=2t ðlÞ,

l¼ 0,1, . . . ,Lh�1, where the receive and transmit correla-tion matrices RtðlÞ,RrðlÞ are generated according to theexponential model, and hwðlÞ is a matrix of i.i.d. zero meancircularly symmetric Gaussian entries following a Veh-Aor Veh-B power delay profile. Weak spatial correlationwas assumed at both the transmit and receive sides of thechannel. The CP-OFDM preamble was similarly chosen,namely ANt

� x with x being a pseudorandom M-vector ofcomplex QPSK pilots.

Fig. 10 shows the BER performance of the methodsunder study in 2�2 systems. Veh-A and Veh-B channels

were considered in Fig. 10(a) and (b), respectively. Oncemore, E-IAM-C outperforms the other schemes in thewhole SNR range considered. Moreover, all the IAMmethods perform better than CP-OFDM at low to moder-ate SNR values. At higher SNRs, the error floor effect,which was observed in the SISO scenario, appears heretoo. As previously, it is more severe for more frequencyselective channels (cf. Fig. 10(b)).

6. Conclusions

The problem of preamble-based channel estimation inOFDM/OQAM systems was revisited in this paper, and areview of the preamble structures and associated estima-tion methods was given, for both SISO and MIMO systems.Results on MSE-optimal design of full and sparse pre-ambles with zero guard symbols were also reviewed forSISO systems. Recent results on MSE-optimal sparsepreambles for the MIMO scenario were also reported.The performances of the principal methods (includingIAM variants and the iterative method of [42]) were

E. Kofidis et al. / Signal Processing 93 (2013) 2038–2054 2051

demonstrated via results of simulation for both mildlyand highly frequency-selective channels, also in a realisticscenario including interference from data. A recentlydeveloped method of the IAM type, E-IAM-C, was seento perform at least as well as the rest and at all consideredSNR values. Further investigation is required to assess theloss in performance from tail truncation in the IAMpreambles and mitigate the high PAPR problem associatedwith their periodic structure. Moreover, improved IAMdesigns are needed for MIMO-OFDM/OQAM, as well asoptimality results for full preambles in such systems.

Acknowledgment

Part of this work was done within the FP7 projectPHYDYAS (Contract no: ICT-211887; http://www.ict-phydyas.org). D. Katselis was additionally supported by theEuropean Research Council under the advanced GrantLEARN, Contract 267381, and by the Swedish ResearchCouncil under Contract 621-2010-5819.

E. Kofidis would like to acknowledge fruitful discus-sions with Drs. P. Siohan, C. Lele, and J. Du. Special thanksare due to Dr. T. Hidalgo Stitz for his detailed andconstructive comments.

Appendix A. Time–frequency neighborhood

In this appendix, the interference terms from theneighbors of a FT point are computed in closed form,based on the knowledge of the prototype filter. Moreover,their symmetries are demonstrated.

These are the quantities we have previously denotedas

/gSm,nmþp,nþq ¼If/gmþp,nþq9gm,nSg,

where

/gmþp,nþq9gm,nS¼X

l

gmþp,nþqðlÞg%

m,nðlÞ:

The latter has to be computed for p 2 f�2,�1,0,1,2gand q 2 f�1,0,1g, with ðp,qÞað0,0Þ. In general, it can bewritten as

/gmþp,nþq9gm,nS

¼X

l

g l�ðnþqÞM

2

� �g l�n

M

2

� �eEð2p=MÞp l�ðLg�1Þ=2ð ÞeEDj

� E �/gSm,nmþp,nþq, ð25Þ

where Dj�jmþp,nþq�jm,n ¼ ðpþqÞðp=2Þ�ðmqþpnþpqÞp.Substituting this in (25) and after some algebra, thefollowing simplified form results:

/gmþp,nþq9gm,nS¼ ð�1ÞmqEpþqe�Eð2p=MÞpðLg�1Þ=2

XLg�1�maxðqðM=2Þ,0Þ

l ¼ maxð�qðM=2Þ,0Þ

gðlÞg lþqM

2

� �eEð2p=MÞpl: ð26Þ

Observe that this is independent of n.Let us consider the various cases in detail:(1) q¼ 0, p 2 f71,72g

With p¼ 71, Eq. (26) becomes

/gm71,n9gm,nS¼ 7Ee8 Eð2p=MÞðLg�1Þ=2XLg�1

l ¼ 0

g2ðlÞe7 Eð2p=MÞl

and hence

/gSm,nm71,n ¼ 7e8 Eð2p=MÞðLg�1Þ=2

XLg�1

l ¼ 0

g2ðlÞe7 Eð2p=MÞl: ð27Þ

In view of the fact that the latter quantities are real andhence equal to their conjugates, we can write

b�/gSm,nmþ1,n ¼�/gSm,n

m�1,n: ð28Þ

For p¼ 72, we have

/gm72,n9gm,nS¼�e8 Eð2p=MÞðLg�1ÞXLg�1

l ¼ 0

g2ðlÞe7 Eð2p=MÞ2l

¼�e8 Eð2p=MÞðLg�1ÞGð2Þð72Þ

with Gð2ÞðpÞ denoting the M-point discrete Fourier trans-form of g2. This coincides with the ambiguity function ofg, Agðq,pÞ, calculated at ð0,72Þ. But the latter is known toequal zero for all g’s that can be used as prototype filtersin such filter banks [28]. Hence

/gSm,nm72,n ¼ 0:

(2) q¼�1, p 2 f�2,�1,0,1,2gWith q¼�1, Eq. (26) becomes

/gmþp,n�19gm,nS

¼�Eð�1ÞmEpe�Epð2p=MÞðLg�1Þ=2XLg�1

l ¼ M=2

gðlÞg l�M

2

� �eEpð2p=MÞl:

Thus

/gSm,nm,n�1 ¼�ð�1Þm

XLg�1

l ¼ M=2

gðlÞg l�M

2

� ���ð�1Þmg: ð29Þ

For p¼ 71

/gSm,nm71,n�1 ¼�ð�1ÞmE71e8 Eð2p=MÞðLg�1Þ=2

XLg�1

l ¼ M=2

gðlÞg l�M

2

� �e7 Eð2p=MÞl � ð�1Þmd, ð30Þ

while for p¼72

/gSm,nm72,n�1 ¼ ð�1Þme8 Eð2p=MÞðLg�1Þ

XLg�1

l ¼ M=2

gðlÞg l�M

2

� �e7 E2ð2p=MÞl ¼ ð�1ÞmE: ð31Þ

(3) q¼ 1,p 2 f�2,�1,0,1,2gSubstituting q¼1 in (26) yields

/gmþp,nþ19gm,nS¼

E � Epð�1Þme�Epð2p=MÞðLg�1Þ=2XLg�1�ðM=2Þ

l ¼ 0

gðlÞg lþM

2

� �eEpð2p=MÞl

and after a change of variable in the summation

/gmþp,nþ19gm,nS¼

E. Kofidis et al. / Signal Processing 93 (2013) 2038–20542052

E � Epð�1Þmþpe�Epð2p=MÞðLg�1Þ=2XLg�1

l ¼ M=2

gðlÞg l�M

2

� �eEpð2p=MÞl:

Thus

/gSm,nm,nþ1 ¼ ð�1Þm

XLg�1

l ¼ M=2

gðlÞg l�M

2

� �

/gSm,nm71,nþ1 ¼�E

71ð�1Þme8 Eð2p=MÞðLg�1Þ=2

XLg�1

l ¼ M=2

gðlÞg l�M

2

� �e7 Eð2p=MÞl,

/gSm,nm72,nþ1 ¼�ð�1Þme8 Eð2p=MÞðLg�1Þ

XLg�1

l ¼ M=2

gðlÞg l�M

2

� �e7 E2ð2p=MÞl,

which, in view of the above, result in

/gSm,nm,nþ1 ¼ ð�1Þmg, ð32Þ

/gSm,nm71,nþ1 ¼ ð�1Þmd, ð33Þ

/gSm,nm72,nþ1 ¼�ð�1ÞmE: ð34Þ

It thus turns out that the 5�3 neighborhood of the(m,n) point is as follows:

ð�1ÞmE 0 �ð�1ÞmEð�1Þmd �b ð�1Þmd�ð�1Þmg dm,n ð�1Þmgð�1Þmd b ð�1Þmdð�1ÞmE 0 �ð�1ÞmE

ð35Þ

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