100624708-ofdm-based-pdf

IEEE Communications Surveys & Tutorials • 2nd Quarter 200718

riven by multimedia based applications, anticipatedfuture wireless systems will require high data ratecapable technologies. Novel techniques such as

OFDM and MIMO stand as promising choices for future highdata rate systems [1, 2].

OFDM divides the available spectrum into a number ofoverlapping but orthogonal narrowband subchannels, andhence converts a frequency selective channel into a non-frequency selective channel [3]. Moreover, ISI is avoided bythe use of CP, which is achieved by extending an OFDMsymbol with some portion of its head or tail [4]. With thesevital advantages, OFDM has been adopted by many wire-less standards such as DAB, DVB, WLAN, and WMAN [5,6].

MIMO, on the other hand, employs multiple antennas atthe transmitter and receiver sides to open up additional sub-channels in spatial domain. Since parallel channels are estab-lished over the same time and frequency, high data rateswithout the need of extra bandwidth are achieved [7, 8]. Dueto this bandwidth efficiency, MIMO is included in the stan-dards of future BWA [9]. Overall, these benefits have madethe combination of MIMO-OFDM an attractive technique forfuture high data rate systems [10–12].

As in many other coherent digital wireless receivers, chan-nel estimation is also an integral part of the receiver designsin coherent MIMO-OFDM systems [13]. In wireless systems,transmitted information reaches to receivers after passingthrough a radio channel. For conventional coherent receivers,the effect of the channel on the transmitted signal must beestimated to recover the transmitted information [14]. As longas the receiver accurately estimates how the channel modifiesthe transmitted signal, it can recover the transmitted informa-tion. Channel estimation can be avoided by using differentialmodulation techniques, however, such systems result in lowdata rate and there is a penalty for 3–4 dB SNR [15 19]. Insome cases, channel estimation at user side can be avoided ifthe base station performs the channel estimation and sends apre-distorted signal [20]. However, for fast varying channels,the pre-distorted signal might not bear the current channeldistortion, causing system degradation. Hence, systems with achannel estimation block are needed for the future high datarate systems.

Channel estimation is a challenging problem in wirelesssystems. Unlike other guided media, the radio channel is high-ly dynamic. The transmitted signal travels to the receiver byundergoing many detrimental effects that corrupt the signal

D

MEHMET KEMAL OZDEMIR, LOGUS BROADBAND WIRELESS SOLUTIONS, INC. AND

HUSEYIN ARSLAN, UNIVERSITY OF SOUTH FLORIDA

ABSTRACT

Orthogonal frequency division multiplexing (OFDM) is a special case ofmulti-carrier transmission and it can accommodate high data rate require-ment of multimedia based wireless systems. Since channel estimation is anintegral part of OFDM systems, it is critical to understand the basis ofchannel estimation techniques for OFDM systems so that the most appro-priate method can be applied. In this article, an extensive overview of chan-nel estimation techniques employed in OFDM systems are presented. Inaddition, the advantages, drawbacks, and relationship of these estimationtechniques with each other are analyzed and discussed. As the combinationof multiple input multiple output (MIMO)-OFDM systems promises higherdata rates, estimation techniques are further investigated for these systems.Although the existing proposed techniques differ in terms of computationalcomplexity and their mean squared error (MSE) performance, it has beenobserved that many channel estimation techniques are indeed a subset ofLMMSE channel estimation technique. Hence, based on a given system’sresources and specifications, a suitable method among the presented tech-niques can be applied.

CHANNEL ESTIMATION FOR

WIRELESS OFDM SYSTEMS

2ND QUARTER 2007, VOLUME 9, NO. 2

www.comsoc.org/pubs/surveys

1553-877X

IEEE Communications Surveys & Tutorials • 2nd Quarter 2007 19

and often place limitations on the performance of the system.Transmitted signals are typically reflected and scattered, arriv-ing at receivers along multiple paths. Also, due to the mobilityof transmitters, receivers, or scattering objects, the channelresponse can change rapidly over time. Most important of all,the radio channel is highly random and the statistical charac-teristics of the channel are environment dependent. Multipathpropagation, mobility, and local scattering cause the signal tobe spread in frequency, time, and angle. These spreads, whichare related to the selectivity of the channel, have significantimplications on the received signal. Channel estimation per-formance is directly related to these statistics. Different tech-niques are proposed to exploit these statistics for betterchannel estimates. There has been some studies that coverthese estimation techniques, however these are limited to thecomparison of few of the channel estimation techniques[21–24]. This paper focuses on an extensive overview of thechannel estimation techniques commonly applied to OFDMbased multi-carrier wireless systems.

OFDM CHANNEL ESTIMATION

Channel estimation has a long and rich history in single carri-er communication systems. In these systems, the CIR is typi-cally modeled as an unknown time-varying FIR filter, whosecoefficients need to be estimated [14]. Many of the channelestimation approaches of single carrier systems can be appliedto multi-carrier systems. However, the unique properties ofmulti-carrier transmission bring about additional perspectivesthat allow the development of new approaches for channelestimation of multi-carrier systems.

In OFDM based systems, the data is modulated onto theorthogonal frequency carriers. For coherent detection of thetransmitted data, these sub-channel frequency responses mustbe estimated and removed from the frequency samples. Likein single carrier systems, the time domain channel can bemodelled as a FIR filter, where the delays and coefficients canbe estimated from time domain received samples, which arethen transformed to frequency domain for obtaining the CFR.Alternatively, radio channel can also be estimated in frequen-cy domain using the known (or detected) data on frequencydomain sub-channels. Instead of estimating FIR coefficients,one tap CFR can be estimated (Fig. 1).

Channel estimation techniques for OFDM based systemscan be grouped into two main categories: blind and non-blind.The blind channel estimation methods exploit the statisticalbehavior of the received signals and require a large amount ofdata [25]. Hence, they suffer severe performance degradationin fast fading channels [26]. On the other hand, in the non-blind channel estimation methods, information of previouschannel estimates or some portion of the transmitted signalare available to the receiver to be used for the channel esti-

mation. In this article, only the non-blind channel estimationtechniques will be investigated.

The non-blind channel estimation can be studied undertwo main groups: data aided and DDCE. In data aided chan-nel estimation, a complete OFDM symbol or a portion of asymbol, which is known by the receiver, is transmitted so thatthe receiver can easily estimate the radio channel by demodu-lating the received samples. Often, frequency domain pilotsare employed similar to those in new generation WLAN stan-dards (802.11a and HYPERLAN2) [27]. The estimation accu-racy can be improved by increasing the pilot density. However,this introduces overhead and reduces the spectral efficiency.In the limiting case, when pilot tones are assigned to all sub-carriers of a particular OFDM symbol, an OFDM trainingsymbol can be obtained (block type pilot arrangement). Thistype of pilot arrangement is usually considered for slow chan-nel variation and for burst type data transmission schemes,where the channel is assumed to be constant over the burst.The training symbols are then inserted at the beginning of thebursts to estimate the CFR (e.g. WLAN and WiMAX sys-tems) [28, 29]. When channel varies between consecutiveOFDM symbols, either the training symbols should be insert-ed regularly within OFDM data symbols with respect to thetime variation of the channel (Doppler spread), or the chan-nel should be tracked in a decision directed mode to enhancethe receiver performance.

In the DDCE methods, to decode the current OFDM sym-bol the channel estimates for a previous OFDM symbol areused. The channel corresponding to the current symbol isthen estimated by using the newly estimated symbol informa-tion. Since an outdated channel is used in the decoding pro-cess, these estimates are less reliable as the channel can varydrastically from symbol to symbol [31, 32]. Hence, additionalinformation is usually incorporated in DDCE such as periodi-cally sent training symbols. Channel coding, interleaving, anditerative type approaches are also commonly applied to boostthe performance of DDCE~techniques.

There are numerous approaches to estimate the channelsfor OFDM subcarriers. The direct estimation of the channelfor subcarriers treats each subcarrier as if the channels areindependent. However, in practice, the CFR is often oversam-pled via the subcarriers, and hence the estimated frequencydomain channel coefficients are correlated. On the otherhand, the noise in these subcarriers can be independent. Byutilizing the correlation of CFR in subcarriers, the noise canbe reduced significantly. Therefore, the channel estimationaccuracy can be improved [28]. Several approaches have beenproposed to exploit this correlation. These approaches andtheir relationship with each other will be discussed in the sub-sequent sections to provide a unified understanding. Similarly,the subcarrier correlation in time and spatial domain can beexploited since the noise can be considered to be independentin time and spatial domain as well.

nFigure 1. Time and frequency domain channels representation for OFDM based systems.

Tap index

CIRCFR

Coe

ffic

ient

s

Coe

ffic

ient

s

DFT/IDFT

Subcarrier index


Although it is a common approach to assume the channelto be constant over an OFDM symbol duration [9, 27], forfast fading channels the same assumption leads to ICI [33],which degrades the channel estimation performance. Hence,the methods employed in data-aided and decision directedchannel estimation need to be modified so that the variationof the channel over the OFDM symbol is taken into accountfor better estimates. External interfering sources also affectthe performance of channel estimation. The effect of interfer-ing sources can be mitigated by exploiting their statisticalproperties. Although most systems treat ICI and externalinterference as part of noise, better channel estimation perfor-mance can be obtained by more accurate modeling [34].

There are basically three basic blocks affecting the perfor-mance of the non-blind channel estimation techniques. Theseare the pilot patterns, the estimation method, and the signaldetection part. Each method covered in this article eithertackles one of the above basic block or several at a time. Thespecific choice depends on the wireless system specificationsand the channel condition. The aspects of each method arepresented such that a suitable method can easily be selectedfor a given wireless system and channel conditions. It can beobserved that each method can be approximated to the othermethods by using the same set of variables. For example, inthis paper it is shown that each estimation method is indeed asubset of LMMSE technique.

In the literature, initial channel estimation methods havebeen mostly developed for SISO-OFDM systems, that is, sin-gle antenna systems. With the emergence of MIMO-OFDM,these methods need some modifications as the received signalin MIMO-OFDM is the superposition of all the transmittedsignals of a given user. In many cases, the methods of SISO-OFDM are easily adopted for MIMO-OFDM but novel meth-ods exploiting space-time codes or other MIMO specificelements are also introduced.

In the rest of the article, starting from a generic systemmodel, the channel estimation techniques will be presentedstarting from the less complicated techniques. More emphasiswill be given on data aided channel estimation as it providessome unique approaches for OFDM systems. Discussions onICI, external interferers, and MIMO systems as well as relatedissues will also be given. Finally, some concluding remarks andpotential research areas will be given at the end of the article.

NOTATION

Matrices and the vectors are denoted with boldface letters,where the upper/lower letters will be used for frequency/time

domain variables; (.)H denotes conjugate-transpose; E{.}denotes expected value; diag(x) stands for diagonal matrixwith the column vector x on its diagonal; 0a×b denotes amatrix of a × b with zero entries; IN denotes N × N identitymatrix; and j=√

——–1.

SYSTEM MODEL

A generic block diagram of a basic baseband-equivalentMIMO-OFDM system is given in Fig. 2. A MIMO-OFDMsystem with Ntx transmit and Nrx receive antennas is assumed.The information bits can be coded and interleaved. The codedbits are then mapped into data symbols depending on themodulation type. Another stage of interleaving and codingcan be performed for the modulated symbols. Although thesymbols are in time domain, the data up to this point is con-sidered to be in the frequency domain. The data is then de-multiplexed for different transmitter antennas. The serial datasymbols are then converted to parallel blocks, and an IFFT isapplied to these parallel blocks to obtain the time domainOFDM symbols. For the transmit antenna, tx, time domainsamples of an OFDM symbol can be obtained from frequencydomain symbols as

(1) (1)

(2)

where Xtx[n, k] is the data at the kth subcarrier of the nthOFDM symbol, K is the number of subcarriers, and m is thetime domain sampling index. After the addition of CP, whichis larger than the expected maximum excess delay of the chan-nel, and D/A conversion, the signals from different transmitantennas are sent through the radio channel.

The channel between each transmitter/receiver link is mod-elled as a multi-tap channel with the same statistics [3]. Thetypical channel at time t is expressed as,

(3)

where L is the number of taps, αl is the lth complex path gain,and τl is the corresponding path delay. The path gains areWSS complex Gaussian processes. The individual paths can be

h t tl l

l

L

( , ) ( ) ( ),τ α δ τ τ= −=

−

∑0

1

x n m IFFT X n k

X n

tx tx

tx

[ , ] { [ , ]}

[ ,

=

= kk e k m Kk

Kj mk K] ,/

=

−

∑ ≤ ≤ −0

12 0 1π

nFigure 2. MIMO-OFDM transceiver model.

Wirelesschannel

S/P

P/S

S/P

X1

XNtx

K

K

Y1

YNrx

K

K

Ant #1 Ant #1IFFTK-

point

Cyclicprefix

Databits

Coding,modulation,interleaving

Deinterleaving,demodulation,decoding Output

bits

P/S

IFFTK-

point

RemoveCyclic prefix

S/P P

/S

IFFTK-

point

CSI

RemoveCyclic prefix

S/P

P/S

Ant #Ntx Ant #NrxIFFTK-

point

Cyclicprefix


correlated, and the channel can be sparse.At time t, the CFR of the CIR is given by,

(4)

With proper CP and timing, the CFR can be written as [3],

(5)

where h[n, l] = h(nTf, kts), FK = e–j2π/K, Tf is the symbollength including CP, ∆f is the subcarrier spacing, and ts = 1/Dfis the sample interval. In matrix notations, for the nth OFDMsymbol, Eq. 5 can be rewritten as

H = Fh (6)

where H is the column vector containing the channel at eachsubcarrier, F is the unitary FFT matrix, and h is the columnvector containing the CIR taps.

At the receiver, the signal from different transmit anten-nas are received along with noise and interference. After per-fect synchronization, down sampling, and the removal of theCP, the simplified received baseband model of the samplesfor a given receive antenna, rx, can be formulated as

(7)

where rx =1, …, Nrx, the time domain effective CIR, hmrxtx[n, l],

over an OFDM symbol is given as time-variant linear filterdepending on the time selectivity of the channel. Please notethat n represents OFDM symbol number, while m denotes thesampling index in time domain so that h m

rxtx[n, l] is the CIR atthe sampling time index m for the symbol n. When the CIR isconstant over an OFDM symbol duration, then h m

rxtx[n, l] willbe the same for all m values, and hence the superscript m canbe dropped. Moreover, irx[n, m] is the term representingexternal interference, wrx[n, m] is the AWGN sample withzero mean and variance of σw

2. After taking FFT of the timedomain samples of Eq. 7, the received samples in frequencydomain can be expressed as,

(8)

(9)

( (10)

where Irx[n, k] and Wrx[n, k] are the corresponding frequencydomain components calculated from irx[n, m]’s and wrx[n, m]’s,respectively. After arranging the terms, and representing the

variables in matrix notation, for rxth receive antenna and nthOFDM symbol, we get

(11)

(12)

Here, Yrx is column vector storing the received signal at eachsubcarrier, F is the unitary FFT matrix with entriese–j2πmk/K√

—K with m and k being the row and column index and

Ψ = FΞrxtxFH, which can be considered as the equivalentchannel between each received and all the transmitted subcar-riers. Moreover Xtx denotes the column vector for transmittedsymbols from txth transmit antenna, Irx is the column vectorfor interferers, Wrx is the column vector for noise, and Ξrxtx isthe matrix containing the channel taps at each m index. Theentries of Ξ are given by

(13)

When the channel is assumed to be constant over oneOFDM symbol and the CP is larger than the CIR length, thenh m

rxtx[n, l] is the same for all m’s, making Ξrxtx a circulantmatrix [35]. The multiplication of FΞrxtxFH then results in adiagonal matrix, and hence no cross-terms between subcarri-ers exist, that is, no ICI occurs. In this case, h is equivalent tothe first column of Ξ. However, when the channel varies overan OFDM symbol, then ICI occurs, and for the equalizationthe channel at each time sample of OFDM symbol is needed,that is, at each m samples. For the frequency domain estima-tion, this requirement translates into the knowledge of thechannel coefficients at each carrier frequency as well as theircross-terms. The number of unknowns in time domain estima-tion are KL, whereas the number of unknowns in frequencydomain (the entries of Ψ) are K2. In either case, the numberof unknowns will be higher than the number of equations, andhence a system of under-determined equations will result in.Simplifications are needed so that the unknowns in the systemof equations are reduced. Different approaches will bedescribed in detail in the subsequent sections.

Once the received signals for each transmit antennas aredetected with the help of channel estimation, the reverseoperation at the receiver is performed, that is, they aredemodulated, de-interleaved, and decoded. As it will be seenlater, the information at different stages of decoding processcan be exploited to enhance the performance of channel esti-mation methods.

Ξrxtx

rxtx

rxtx rxtx

h n

h n h n

=

0

1 1

0

1

0

0

0

0

[ , ]

[ , ] [ , ]

� ��

� � �h n L h n Lrxtx

LrxtxL− −− −

⎡

⎣

⎢

1 11 2 0

0 0 0

[ , ] [ , ]

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

��

h n

h

rxtx0 2[ , ]

rrxtx

rxtx

rxtxn

h n

h n1

0

13

1

2

0 0

[ , ]

[ , ]

[ , ]

� � ��

� hh n L h nrxtxK

rxtxK− −−1 11 0[ , ] [ , ]

.

= + +=∑ ΨX I Wtx rx rxtx

Ntx

1

.

Y F F X I Wrx rxtxH

txtx

N

rx rx

tx

= + +=∑ Ξ

1

,

= ′⎡

⎣⎢

′=

−−∑1

0

12

Kx n k etx

k

Kj m l[ , ] (π )) /′

=

−

=

−

=

∑∑∑⎡

⎣⎢

⎤

⎦⎥k K

l

L

m

K

tx

Ntx

0

1

0

1

1

h n l e I n k W nrxtxm

jkm

Krx rx[ , ] [ , ] [⎤⎦ + +

−2π

,, ]k

+ + ]−

i n m w n m erx rx

jkm

K[ , ] [ , ]2π

Y n kK

y n m erx rx

jkm

K

m

K

[ , ] [ , ]=−

=

−

∑1 2

0

1 π

= −=

−

=

∑∑1

0

1

1Kx n m l h n ltx rxtx

m

l

L

tx

Ntx

[ , ] [ , ]⎡⎡

⎣⎢

=

−

∑m

K

0

1

y n m x n m l h n lrx tx rxtxm

l

L

tx

Nt

[ , ] [ , ] [ , ]= −=

−

=

∑0

1

1

xx

i n m w n mrx rx

∑

+ + [ , ] [ , ],

H n k H nT k f h n l Ff Kkl

l

L

[ , ] ( , ) [ , ] ,≡ ==

−

∑∆0

1

H t f h t e dj f( , ) ( , ) .= −

−∞

+∞

∫ τ τπ τ2


OFDM CHANNEL ESTIMATION TECHNIQUESThere are several basic techniques to estimate the radio chan-nel in OFDM systems. The estimation techniques can be per-formed using time or frequency domain samples. Theseestimators differ in terms of their complexity, performance,practicality in applications to a given standard, and the a pri-ori information they use. The a priori information can be sub-carriers correlation in frequency [36], time [3], and spatialdomains [37]. Moreover, the transmitted signals being con-stant modulus [38], CIR length [39], and using a known alpha-bet for the modulation can also be a priori information [40,41]. The more the a priori information is exploited, in generalthe better the estimates are [42].

For frequency domain channel estimates, MSE is usuallyconsidered as the performance measure of channel estimates,and it is defined by

MSE = E{|H[n, k] – H^[n, k]|2}, (14)

where H^[n, k] is the estimate of equivalent channel at kth sub-carrier of nth OFDM symbol. Although MSE is used exten-sively, sometimes, other measures like BER performance arealso used [43, 44]. BER performance is mainly used when theperformance of OFDM system with the channel estimationerror is to be evaluated [45, 46].

Before introducing the estimation techniques, it is worth-while to look at the data aided channel estimation in generaland the pilot allocation mechanisms.

DATA AIDED CHANNEL ESTIMATION

In this subsection, we will review commonly used methods inthe data aided channel estimation. Initially, we will considerthe methods developed for SISO-OFDM. ICI is assumed notto exist and the CIR is assumed to be constant for at least oneOFDM symbol. Hence, Ψ is a diagonal matrix, where eachdiagonal element represents the channel between the corre-sponding received and the transmitted subcarriers. In thiscase, for the nth OFDM symbol, the channel given in Eq. 5 ateach subcarrier can be related to Ψ as

H[n, k] = Ψ[k, k]. (15)

Furthermore, the external interference is folded into the noisewith noise statistics being unchanged. With the above assump-tion, the expression in (12) can be expressed as

Y = diag(X) H + W, (16)

or

Y[n, k] = H[n, k] X[n, k] + W[n, k]. (17)

Here H and W are the column vectors representing the chan-nel and the noise at each subcarrier for the nth OFDM sym-bol, respectively.

In data aided channel estimation, known information tothe receiver is inserted in OFDM symbols so that the currentchannel can be estimated. Two techniques are commonlyused: sending known information over one or more OFDMsymbols with no data being sent, or sending known informa-tion together with the data. The previous arrangement is usu-ally called channel estimation with training symbols while thelatter is called pilots aided channel estimation (Fig. 3).

Channel estimation employing training symbols periodical-ly sends training symbols so that the channel estimates areupdated [29]. In some cases training symbols can be sentonce, and the channel estimation can then be followed bydecision directed type channel estimation. The details of thedecision directed will be given later in the article.

In the pilots aided channel estimation, the pilots are multi-plexed with the data. For time domain estimation, the CIR isestimated first. The estimate of the CIR are then passedthrough a FFT operation to get the channel at each subcarrierfor the equalization in frequency domain. For frequencydomain estimation, the channel at each pilot is estimated, andthen these estimates are interpolated via different methods.

Pilots Allocation for Data Aided Channel Estimation —For the pilot aided channel estimation, the pilot spacing needsto be determined carefully. The spacing of pilot tones in fre-quency domain depends on the coherence frequency (channelfrequency variation) of the radio channel, which is related tothe delay spread. According to the Nyquist sampling theorem,

nFigure 3. Typical training symbols and pilot subcarriers arrangement.

Time

(a)

Training symbols

Freq

uenc

y

Data symbolsTime

Pilot subcarriers

(b)

Freq

uenc

y

Data subcarriers


the number of subcarrier spacing between the pilots in fre-quency domain, Dp, must be small enough so that the varia-tions of the channel in frequency can be all captured, that is,

(18)

where τmax is the maximum excess delay of channel. When theabove is not satisfied, then the channel available at the pilottones does not sample the actual channel accurately. In thiscase, an irreducible error floor in the estimation techniqueexists since this causes aliasing of the CIR taps in the timedomain [47].

When the channel is varying across OFDM symbols, inorder to be able to track the variation of channel in timedomain, the pilot tones need to be inserted at some ratio thatis a function of coherence time (time variation of channel),which is related to Doppler spread. The maximum spacing ofpilot tones across time is given by

(19)

where fdmax is the maximum Doppler spread and Tf is theOFDM symbol duration. For comb-type pilot arrangements,the pilot tones are often inserted for every OFDM symbols.When the spacing between the pilot tones does not satisfy theNyquist criteria, then the pilots can still be exploited in a com-bined pilot-plus DDCE [48].

The pilots can be sent continuously for each OFDM sym-bol. Since the channel might be varying both in time and fre-quency domains, for the reconstruction of the channel, this2-D function needs to be sampled at least a Nyquist rate.Hence, the rate of insertion of pilots in frequency domain andfrom one OFDM symbol to another cannot be set arbitrarily.The spacing of pilots should be according to Eq. 18 and Eq.19. In general, within an OFDM symbol the number of pilotsin frequency domain should be greater than the CIR length(maximum excess delay), which is related to the channel delayspread. Over the time, the Doppler spread is the main criteriafor the pilot placement.

Many studies are performed in order to get the optimumpilot locations in time-frequency grid given a minimum num-ber of pilots that sample the channel in 2-D at least Nyquistrate. This optimality is in general based on the MSE of the LSestimates [6, 49]. It should be noted that an optimum pilotallocation is a trade-off between wasted energy in unnecessarypilot symbols, the fading process not being sampled sufficient-

ly, the channel estimation accuracy, and the spectral efficien-cy of the system [50]. Hence, an optimum pilot allocation fora given channel might not be optimal for another channel asthe fading process will be different.

In addition to minimizing MSE of the channel estimates,pilots also need to simplify the channel estimation algorithmsso that the system resources are not wasted. For example, itis noted that the use of constant modulus pilots simplify thechannel estimation algorithms as matrix operations becomeless complex [38, 51].

Some other important elements for pilot arrangementsare the allocation of power to the pilots with respect to thedata symbols, the modulation for the pilot tones etc. In manycases, the power for pilot tones and data symbols are equallydistributed. The channel estimation accuracy can be improvedby transmitting more power at the pilot tones compared tothe data symbols [52]. For a given total power, this reducesthe SNR over the data transmission. As for the pilot powerat different subcarriers, studies show that based on the MSEof the LS estimates pilots should be equipowered [6, 53].

Moreover, due to the lack of the pilot subcarriers at theedge of OFDM symbols, the estimation via the extrapolationfor the edge subcarriers results in a higher error [54, 55]. Sim-ulations also reveal that the channel estimation error at theedge subcarriers are higher than those at the mid-bands dueto this extrapolation [56–58]. One quick solution would be toincrease the number of pilot subcarriers at the edge subcarri-ers [58], however this would decrease the spectral efficiency ofthe system [57]. Due to the periodic behavior of the FourierTransform, the subcarriers at the beginning and the end ofthe OFDM symbol are correlated, and this can be used toimprove the channel estimates at the edge subcarriers (Fig. 4).Simulations exploiting this property are reported to enhancethe estimation accuracy of the edge subcarriers [57].

Another issue related to pilot arrangement is the patternof the pilots, that is, how to insert the pilots to efficientlytrack the channel variation both in time and frequencydomains. The selection of a pilot pattern may affect the chan-nel estimation performance, and hence the BER performanceof the system.

Equation 18 states that the pilot spacing in frequencydomain needs to satisfy the Nyquist criteria. More insight intoEq. 18 reveals that the number of required pilots in frequencydomain can be taken as the CIR length. At a first glance, thisdoes not pose any restriction on the pilot spacing that a suffi-cient number of pilots can be inserted in adjacent subcarriers.However, when the MSE of the time domain LS estimation,which is covered in the next subsection, is analyzed, it isobserved that the minimum MSE is obtained when the pilotsare equispaced with maximum distance [6, 31, 39]. This is dueto the reason that when the pilots are inserted in adjacentsubcarriers, then the FFT matrix used in the time domain LSestimation approaches to an ill conditioned matrix, makingthe system performance vulnerable to the noise effect [39].Hence, from the MSE of LS estimation, the pilots in frequen-cy domain need to be equipowered, equispaced, and theirnumber should not be less than the CIR length. Since the useof pilots is a trade-off between extra overhead and the accura-cy of the estimation, adaptive allocation of pilots based on thechannel length estimation can offer a better trade-off [52, 56,59]. As will be seen later in the article, with MIMO and ICIadditional requirements will be observed on the pilot subcarri-ers spacing and properties.

When it comes to the pilot allocation for subsequentOFDM symbols, either the set of subcarriers chosen in a pre-vious OFDM symbol or a different set of pilots can be used(Fig. 3). The use of the same subcarriers as the pilots is a

Dfd Tft ≤

1

2 max

Ddfp ≤

1

τmax∆

nFigure 4. Periodic behavior of subcarriers cross-correlation forK = 64.

Subcarrier index100

0.3

0.4

Cor

rela

tion

coe

ffic

ient

(ab

s)

0.5

0.6

0.7

0.8

0.9

1

20 30 40 50 60 70


widely used pilot arrangement. In such a pilot arrangement,first the channel between subcarriers is estimated via interpo-lation in frequency domain. This is followed by interpolationover OFDM symbols in time domain. In some cases, interpo-lation can be first performed in time domain, followed by thefrequency domain interpolation. The details of different inter-polation techniques will be given later in this section.

The analysis of MSE of time domain LS estimation overseveral OFDM symbol indicates that for a lower MSE, thepilots should be cyclically shifted for the next OFDM symbol[6, 60]. This pilot allocation is similar to those used in DTVapplications, and is similar to the pilot scheme given in Fig.13. In this pilot allocation scheme, the interpolation is firstperformed in frequency domain, followed by the interpolationin time domain. Similar to the pilot scheme used in DTV, ahexagonal type pilot scheme is also proposed [61–63]. In bothschemes, different subcarriers are utilized for each OFDMsymbol, and hence the possibility of sticking into terribly fad-ing subcarriers is eliminated, that is, diversity is exploited.

In addition to the above pilot schemes, different types ofpilot schemes are tested through simulations [56]. The pilotshaving more density than the others, those utilizing differentsubcarriers over time and at the edge subcarriers are expectedto perform better for channels varying both in time and fre-quency domains.

The previous pilot allocation schemes were solely based onthe MSE analysis of the channel estimation. In some cases,other system parameters can also be considered for the pilotsto be used. For example, due to the IFFT block at the trans-mitter side, PAPR of OFDM systems can be very high. It isobserved that different training symbols (not scattered pilots)results in different PAPR [64]. Moreover, different scatteredpilot allocation schemes can result in different PAPR whenmultiplexed with data. Since the data is random, the optimumallocation for minimum PAPR will be different for each trans-mission. However, pre-defined pilot allocation schemes can betested for the best PAPR [65]. With such a scheme however,the information about the pilot scheme needs to be conveyedto the receiver side, and this reduces the spectral efficiency ofthe system.

It is clear from the discussion about the pilot allocationthat a better system performance can be obtained when thesystem is adaptive [52, 59, 60, 66]. In this case, the informa-tion about the channel statistics becomes very critical. Thepilot allocation in the frequency domain requires the delayspread estimation, whereas the one in over OFDM symbols(over time evolution) requires Doppler spread estimation. Ifthese estimates are available, then a pilot scheme using justthe right amount of pilots can yield an acceptable perfor-mance. If this information is not available, then the pilotscheme can be designed based on the worst channel condi-tion, that is, the maximum expected delay and Dopplerspreads. In addition to unknown channel statistics, randomlygenerated pilots can be utilized for the reduction of interfer-ence from adjacent cells. However, it is shown via simulationsthat such pilots cause severe degradation in the channel esti-mation MSE [67].

So far the pilots in the frequency domain are discussed. Insome cases, the estimation can be performed using the data intime domain, that is, data before the FFT block at thereceivers. Training symbols for this case can be set to all 1’s infrequency domain that result in an impulse in the timedomain. When this impulse is passed through the channel,then CIR can be obtained. By careful arrangement of 1’s infrequency domain, the multiple replicas of the CIR can beobtained, and these can be improved through noise averaging.In a similar way, PN sequences superimposed with the data

can be utilized for the channel estimation. In such a case, cor-relators at the receiver can be used for the expected samplesof the OFDM symbols [68–70]. However, it is shown thatsuperimposing training with data is not optimal for channelestimation [71].

Having reviewed the pilot schemes employed in OFDMsystems, it is time to look at the channel estimation tech-niques. Starting from the methods using the least a prioriinformation, in this article we will review channel estimationmethods such as LS estimation, ML, transform domain tech-niques, and LMMSE. Simple interpolation techniques will becovered along with LS estimation technique.

LS ESTIMATION

Before going into the details of the estimation techniques, it isnecessary to give the LS estimation technique as it is neededby many estimation techniques as an initial estimation. Start-ing from system model of SISO-OFDM given in Eq. 17 as[72]

Y[n, k] = X[n, k]H[n, k] + W[n, k], (20)

the LS estimation of H[n, k] is

(21)

In matrix notations,

H^LS = diag(X)–1Y + diag(X)–1W. (22)

Note that this simple LS estimate for H^LS does not exploitthe correlation of channel across frequency carriers and acrossOFDM symbols.

The MSE of LS estimation of Eq. 22 is given by [73]

(23)

where EH = E{H[n, k]}.LS method, in general, is utilized to get initial channel esti-

mates at the pilot subcarriers [72], which are then furtherimproved via different methods.

It is also common to introduce CIR to Eq. 16 to exploitCIR length for a better performance [21, 74]. In this case, Eq.16 can be modified as [74]

Y = diag(X)Fh + W

where H = Fh. The LS estimation of Eq. 24 is then

H^ = QLSFH diag(X)HY (25)

where

QLS = (FH diag(X)Hdiag(X)F)–1. (26)

The above LS estimation will be referred as time domain LS.When no assumptions on the number of the CIR taps orlength are made, then the time domain LS reduces to that offrequency domain, and it does not offer any advantages. How-ever, with the assumption that there are only L number ofchannel taps, which then reduces the dimension of the matri-ces F and hence Q, an improved performance due to thenoise reduction can be obtained [75, 76]. The resultant LSestimation has higher computational complexity than the fre-quency domain LS but the performance increase is the plusside of the approach. The increase in the performance can beconsidered as the exploitation of subcarrier correlation. Acomparison study showed that when the frequency domain LSalso exploits the correlation of the subcarriers, then its perfor-mance can be that of time domain LS (21). Further compari-

MSEK

E SNRLSH

=⋅

HLS[n,k ]=Y [n,k ]

X[n,k ]= H [n,k ]+

W[n,k ]

X[n, k].


son studies showed that based on the SNR information, eithermethod can be used [74]. For example if the SNR is low thenthe time domain LS can be less accurate as additional filteringin time domain is based on less accurate CIR length. In thiscase, the probability of not accounting for all the taps and dis-carding some of them are high. However, for other SNRregions, the time domain LS gives better results as it utilizes amore accurate CIR length. The use of time domain LSbecomes inevitable when OFDM is combined with MIMOsystems [77]. This will be explored more when channel estima-tion techniques for MIMO systems are presented.

Similar to the time domain LS, the ML estimate of theCIR taps for the same system model given in Eq. 24 can bederived. With the assumption of L channel taps and Np num-ber of pilot subcarriers, the ML estimate of the channel coef-ficients is shown to be [58, 78],

H^ML = (FpHFp)–1Fp

H diag(X)HY (27)

where Fp is Np × L truncated unitary Fourier matrix. In theabove formulation, for the sake of simplicity, it is assumedthat pilots symbols are from PSK constellation and hencediag(X)Hdiag(X) = IK, and they do not appear in the paren-thesis for the inverse operation. It can be observed that whenthe number of pilots is greater than the channel length andthe noise is AWGN, the time domain LS estimate in Eq. 25 isequivalent to the ML estimate given in Eq. 27 [58, 79]. Fur-thermore, it should be noted that the ML estimate given in(27) makes the assumption about the CIR length, whichimproves the performance of the estimation accuracy [80].Unlike LMMSE channel estimation, both LS and ML arebased on the assumption that the CIR is a deterministic quan-tity with unknown parameters. This implies that LS and MLtechniques do not utilize the long term channel statistics andhence are expected to perform worse than the LMMSE chan-nel estimation method [58]. However, the computational com-plexity is the main trade-off factor between the two groups ofthe channel estimation techniques.

Before introducing the other channel estimation tech-niques, it is worthwhile to review the methods used for thetraining sequences as well as the pilot subcarriers. The corre-sponding implications on the channel estimation techniqueswill also be covered briefly.

CHANNEL ESTIMATION TECHNIQUES IN TRAINING MODE

As mentioned before, in the training mode, all the subcarriersof an OFDM symbol are dedicated to the known pilots. Insome systems like WLAN or WiMAX, two of the symbols arereserved for the training. If the training symbols are employedover two OFDM symbols, for very slowly varying channels, thechannels at two OFDM symbols for the same subcarriers canbe assumed to be the same. In this case, the estimates can beaveraged for further noise reduction [72]. If the noise vari-ances of the OFDM symbols are different, then Kalman filter-ing can be used such that noise variances are exploited asweighting parameters [81].

Once the channel is estimated over the training OFDMsymbols, it can be exploited for the estimation of the channelsof the OFDM symbols sent in between the training symbols.Depending on the variation of the channel along time, differ-ent techniques can be utilized.

A very common method is to assume the channel beingunchanged between OFDM training symbols [23, 28–30, 69].In this method, the channel that is estimated at training sym-bols is used for the subsequent symbols until a new trainingsequence is received. The channel is then updated by usingthe new training sequence, and the process continues. In fact,

this is one of the algorithms employed for IEEE 802.11a/b/gand fixed WiMAX systems. However, these approaches intro-duce an error floor for non-constant channels, that is, outdoorchannels. The highest performance degradation occurs at thesymbols farthest from the training symbols. For video trans-mission systems, the critical information can be sent over thesymbols closer to the training symbols, while non-criticalinformation can be sent over those farther from the trainingsymbols [29, 30]. It is observed that such an arrangement canimprove the performance without increasing the number oftraining blocks. However, for systems requiring equal prioritypackets like data networks, such an approach cannot be taken.In this case satisfactory results can be obtained by increasingthe rate at which the training symbols are sent at the expenseof system efficiency.

For the fast varying channels, interpolation methods can beutilized in time domain. Interpolating the channel linearlybetween the training symbols is one simple solution [59, 72,82]. The disadvantage with such an approach is the latencyintroduced in the system [83]. Indeed, if the system can toler-ate more latency, then the channel estimation for non-trainingOFDM symbols can be improved by higher order polynomials[66, 84, 85].

CHANNEL ESTIMATION TECHNIQUES IN PILOT MODE

In the pilot mode, only few subcarriers are used for the initialestimation process. Depending on the stage where the estima-tion is performed, estimation techniques will be consideredunder time and frequency domains techniques.

In frequency domain estimation techniques, as a first step,CFR for the known pilot subcarriers is estimated via (22).These LS estimates are then interpolated/extrapolated to getthe channel at the non-pilot subcarriers. The process of theinterpolation/extrapolation can be denoted as

H^ = QH^LS (28)

where Q is the interpolation/extrapolation matrix. The goal ofthe estimation technique is to obtain Q with lower computa-tional complexity but at the same time is to achieve higheraccuracy for a given system. In this subsection, the calculationof matrix Q for simple interpolation techniques will be dis-cussed.

Piecewise Linear Interpolation — Two of the simplest waysof interpolation are the use of piecewise constant [86] and lin-ear interpolation [22, 84, 87, 88]. In the piecewise constantinterpolation, the CFR between pilot subcarriers is assumed tobe constant, while in piecewise linear interpolation the channelfor non-pilot subcarriers is estimated from a straight linebetween two adjacent pilot subcarriers. Mathematically, forpiecewise constant interpolation, Q is a matrix consisting ofcolumns made up from shifted versions of the column vector

where Dp is the spacing of the pilots. For the the piecewiselinear interpolation, Q consists of coefficients that are a func-tion of the slope of the line connecting two pilot subcarriersand the distance of the pilots to the subcarrier for which thechannel is to be estimated.

In the first method, acceptable results can be obtained ifthe CFR is less frequency selective or the CIR maximumexcess delay is very small. Such a constraint makes the CFR atthe subcarriers very correlated that CFR at a group of subcar-riers can be assumed to be the same.

c = … …[ , , , , , , ] ,1 1 1 0 0D

T

p

��


In piecewise linear interpolation some variation is allowedbetween the pilot subcarriers. Such an approach can result ina lower MSE since noise averaging is performed. Moreover,when the channel becomes more frequency selective, thepiecewise linear interpolation results in a better performancecompared to the piecewise constant [86–89]. For a betterinsight into the performance of the piecewise linear interpola-tion, its MSE is derived and is expressed in terms of the chan-nel statistics and the pilot spacing as [87]

(29)

where 1/ζ = Dp, Rf[l] is the frequency domain correlation ofCFR, ℜ denotes the real part of a complex number, and σw

2 isthe noise variance. When the piecewise linear interpolation isto be performed between OFDM symbols over time, then theparameters above need to be replaced with their time domainequivalence. As can be seen from the expression, lower MSEresults in:• When many pilots are used• When the noise is low• When the channel is very correlated

Higher Order Polynomial Fitting — Piecewise linear inter-polation requires more pilot subcarriers for an acceptable per-formance in highly frequency selective channels [52, 86, 89].However, by using a priori information about the frequency ortime selectivity of the channel, the use of higher order polyno-mial can result in better performance. Higher order polynomi-als indeed can approximate the wireless channels accurately,since the channel itself is smooth in both time and frequencydomains [66]. The degree of this smoothness depends on theselectivity of the channel. For highly time and frequency selec-tive channels, the higher the polynomial order, the better theestimation at the expense of higher computational complexity[23]. However, when the channel is changing very slowly bothin frequency and time, then the use of very high order polyno-mials can degrade the performance, as the modelling usesnoise as a means to represent the channel [66]. This behavioralso suggests dynamic polynomial fitting based on the channelstatistics [23]. Simulations show that adaptive polynomial fit-ting performs better than the static polynomial fitting whenthe channels become more selective [23]. In a move towardsreducing the computational complexity of such an adaptation,instead of estimating the true channel statistics, variation ofchannel between two adjacent subcarriers can be monitored,and an idea of how fast the channel is changing can beobtained [90]. Further computational complexity can beachieved if the coefficients of Q are made power of 2 to elimi-nate the multiplication/division via bit shifting. It is observedthat such an approach can yield accurate channel estimates[90].

In the higher order polynomial approaches, the entries ofthe Q are calculated by using more information about thechannel. Higher order polynomial fitting uses more than twopilot subcarriers for the CFR estimation. While some of thepolynomial fitting methods utilize no channel statistics [52,91], others assume to have some information about the statis-tics [66, 85]. The most common higher order interpolationmethods are spline interpolation [22, 60, 89], Gaussian inter-polation [22], and polynomial fitting [66, 85, 91, 93, 94]. In thespline interpolation, basis function of some orders or Beizercurve are defined over a group of subcarriers [60, 84]. These

basis functions are determined such that they are unity at thepilot locations at which they are defined for, and vanishes atthe other pilot locations. The channel at non-pilot subcarrierscan then be found as

(30)

where Np is the number of pilots over a range, Bp[n, k] is thebasis function at subcarrier k, and H[n, p] is the CFR at thepilot location p. The rows of the interpolation matrix, Q, arethen formed using Bp[n, k]’s. For more frequency selectivechannels the order of the basis functions, Bp[n, k]’s, can beincreased for a better performance. This corresponds to hav-ing more columns in Q, and implies the use of more pilot sub-carriers for the estimation of a single subcarrier.

Gaussian interpolation is another interpolation technique,where the coefficients of Q are obtained from a Gaussianfunction [95]. The Gaussian function resembles the sinc func-tion, the ultimate function for ideal low pass filtering. TheGaussian function can be considered as an approximation tothe sinc function. The width of the Gaussian function orequivalently the coefficients used in the interpolation aredependent on the frequency selectivity of the channel. Hence,as with many approaches, the knowledge of the channel statis-tics can improve the performance of the Gaussian interpola-tion.

Similar to the Gaussian interpolation, radial basis functionsutilizing Gaussian function are also used for the interpolationpurpose [96]. The coefficients of the radial basis functions aredetermined through some non-linear training mechanism sim-ilar to those used in neural networks. Overall, the goal is tofind the coefficients of the interpolation using the Gaussianfunction as a basis, and the training process indeed reflectsthe information about the channel statistics to the coefficientsto be used in the interpolation. Hence, the approach of theradial basis function interpolation can be considered as anadaptive low-pass filtering. The improved performance due tothis adaptation comes at the cost of training process usingpilot subcarriers.

2-D regression models for the pilot subcarriers scattered infrequency and time domains are also studied [85, 94]. In thesemodels, a 2-D polynomial whose coefficients are obtainedusing the channel correlation and the initial LS estimates atthe pilot subcarriers is developed. Although higher order poly-nomials can be used, second order approximation is found toyield close to ideal BER performance for certain channels[85].

All of the above interpolators can be seen as a simple low-pass filter. This is due to the fact that CIR has a finite lengththat is in general much smaller than the number of subcarri-ers. The above interpolation methods are not ideal low-passfilters, and hence they introduce an error floor due to eitherthe suppression of some of the channel taps or the inclusionof noise whose effect becomes effective at high SNR regions.A low-pass filtering can eliminate the noise in non-tap loca-tions, which in turn means the elimination of most of thenoise in the estimated subcarriers. For example, it is shownthat the use of raised cosine filter as a low-pass filter providesaccurate channel estimates for WLAN systems [28]. Thesharper the low-pass filtering the better the estimates are.Since the Fourier Transform of a rectangular function (or awindow) is the sinc function, the sinc interpolator with theknown CIR length provides ideal low-pass filtering. However,sinc interpolator is not realizable in practical implementations.Moreover, it is computationally heavy as it requires moreCFR samples.

H n k B n k H n ppp

N p

[ , ] [ , ] [ , ],==∑

1

MSE Rf w= + + + +

− −

1

35 1 0

1

32

4 1

2 2( )( ) [ ] ( )

(

ζ ζ ζ σ

ζ ζ111

31

0

12) { [ ]} ( ) { [ ]}ℜ + − ℜ

=

−

∑ R l R Dfl

D

f p

p

ζ


The low pass interpolation utilizes the extra informationabout the CIR length. Further improvement can be achievedwith the information of other channel statistics [37, 97, 98].However, if the channels are less frequency and time selec-tive, then there is no need for very complicated estimationtechniques, and the use of simple interpolators will do the job.Since computation of the information of channel statistics willneed extra processing, systems unable to get the statistics canassume a worst case scenario for the typical application. Suchsystems can use an interpolator based on the assumed statis-tics throughout the application.

TRANSFORM DOMAIN TECHNIQUES

It was mentioned that in general the CIR length is muchsmaller than the number of pilot subcarriers, that is, L <Np. When an orthogonal transformation is applied to theCFR at the pilot subcarriers, the transform domain containsL number of significant values, that is, values relatively hav-ing more energy or magnitude than the noise. Since thenoise is assumed to be AWGN in frequency domain, it isAWGN in transform domain as well. If the significant val-ues of the transform domain signal are retained, and thenon-significant ones are treated as zero, then the noise termwill be eliminated significantly especially when L << Np[99]. For this operation, some sort of threshold is needed todifferentiate between the significant values of the signal andnoise terms. The CFR can then be obtained by applying theinverse of the orthogonal transformation, since such anoperation will also achieve interpolation for non-pilot sub-carriers [99, 100].

Once the CFR is obtained via a transform domain tech-nique, the channel at subsequent OFDM symbols (over time)can be obtained via different methods. The filtering process oftransform domain is usually followed by linear interpolation intime domain [101]. Wiener filtering is also found to be effec-tive in noise reduction in time domain [102]. With Wiener fil-tering being optimum in the sense of minimizing MSE, it hasbeen applied in both domains as well. This will be covered inthe subsequent sections.

The transform domain techniques exploit the informationabout the number of significant values in the transformdomain and their location. Moreover, more number of pilotsubcarriers are used for the interpolation process. Hence, theyperform better than the simple interpolation techniques ingeneral [103]. Different transform domain techniques are

studied for the channel estimation of OFDM based systems.Fourier [31, 84, 101, 104, 105], Hadamard [106, 107], DiscreteCosine [108, 109], Karhunen-Loeve Transform KLT [110],and 2-D Fourier Transformation [111] are few to name.

In Fourier Transform, two techniques are investigated. Inthe very common approach, first the IFFT of the CFR istaken. The resultant transform domain is the time domain,where typically the channel taps are concentrated into a sub-region. Figure 5 shows a typical CIR with significant taps andnoise. By zeroing the terms out of this subregion that corre-sponds to noise, only the significant taps are retained. Thenoise reduced signal is then transformed back into the fre-quency domain via FFT operation.

Mathematically, the process of the Fourier Transformtechnique for the channel estimation of OFDM based systemscan be represented as [76],

(31)

where

is for the normalization, Fp is the Fourier matrix with the rowscorresponding to the subcarrier index of the pilot tones, andDFT is given by

(32)

The expression for DFT given above is applied in many trans-form domain based approaches using Fourier Transform,which in general neglects the correlation between CIR taps.By incorporating the CIR tap correlation and appropriatelychoosing the coefficients of DL, a full matrix that can result inlower channel estimation MSE can be obtained [76].

When the channel statistics are not available, then DL = ILcan be used. In this case, Eq. 31 can be expressed as [76],

(33)

where FL and FLp contain only the first L columns of F andFp, respectively.

By examining Eq. 31, it can be anticipated that the pilotsneed not be chosen such that they sample CFR uniformly.However, the condition number of Fourier matrices increasesfor closely spaced pilots. Hence, when the pilots are not close-ly spaced, Fourier matrices with higher condition numbers canresult in higher MSE in the presence of noise [31, 39]. Theo-retically, in the presence of AWGN, the MSE of the FourierTransform approach is shown to be minimum when the pilotsare equispaced with maximum distance [39].

In terms of implementation complexity, Fourier Transformapproach is computationally very efficient, thanks to FFTalgorithms. The computational complexity of FFT algorithmsare further reduced via the radix 4 operation with restrictionson the pilot spacing to be a power of 2 [52, 112].

Similar to the above Fourier Transform approach, insteadof taking IFFT as an initial transformation, first FFT is per-formed over CFR samples [59, 29, 102, 113]. The equivalenttransform domain taps are not concentrated into a subregionas they are in the CIR taps. The equivalent taps of this alter-native Fourier Transform approach are shown in Fig. 6 for a 5tap channel.

As seen in Fig. 6, only the center region of the transformdomain needs to be zeroed, meaning the identification of onemore region. Mathematically, the above procedure can be

ˆ / ˆH F F HFT p L LpH

LSK N=

DD 0

0 0FTL L K L

K L L K L K L=⎡

⎣⎢

⎤

⎦⎥

× −

− × − × −

( )

( ) ( ).

K N p/

ˆ / ˆH FD F HFT p FT PH

LSK N=nFigure 5. Typical CIR samples with noise.

Tap index100

-1

0

Tap

coef

ficie

nts

1

2

3

4

5

6

20 30 40 50 60 70


written as

(34)

where the entries of the DFT2 are the cyclically shifted entriesof DFT along its diagonal elements.

Studies are performed to compare the performance ofboth of the Fourier Transform approaches. When the numberof significant taps are estimated correctly, both methods per-form the same. However, since in the second Fourier Trans-form approach three regions need to be identified, in thepresence of noise it is more prone to errors [59, 114].

In both of the Fourier Transform approaches, the noisewithin the significant taps is not eliminated. If the noise in thetransform domain is completely independent, then there isnot much to be done for removing the noise within the taps.However, if the noise in the transform domain is correlated,then by using the information about the noise in the non-taplocations, the noise in tap locations can be reduced. Studiesshow that this additional processing provides further improve-ment in the MSE of the channel estimation [115].

The Fourier Transform approach assumes the knowledgeof the tap location. It is assumed that the taps are equallyspaced with sampling time of the OFDM symbol. When theIFFT of the CFR is taken, the equivalent CIR is given by,

(35)

where BW is the bandwidth of the OFDM symbol. If the CIRtaps are sample spaced, then CFR is band-limited or CIR istime-limited, and hence the performance of the Fourier Trans-form approach is very close to the ideal low-pass filtering.However, if the CIR taps are not sample spaced, that is, τ ≠ τl,then as can be seen from Eq. 35, the energy of the non-samplespaced tap is leaked to the other taps. With this, CFR is notbandlimited anymore, and there is aliasing. When the noise-only taps are eliminated, this leaked energy is also removed,and hence some degradation occur as the total energy of theCIR taps is not preserved. Figure 7 shows the equivalent CIRtaps from CFR when the real taps are non sample spaced.

The aliasing due to non-sample spacing can be consideredas the high frequency terms in the domain from which thetransformation is performed. For this reason, windowing infrequency domain is applied in order to mitigate for the alias-ing effect [116, 117]. In contrast to the conventional Fourier

Transform approaches that use rectangular windowing, otherwindow types like Hamming and Hanning can be used in thefrequency domain. The effect of this windowing is removedwhen the CFR is transformed back from the transformdomain. It is observed that this additional processing improvesthe performance of channel estimation when the taps are notsample spaced. However, since the aliasing effect is not com-pletely removed, still at high SNR regions an error flooroccurs.

Another transform domain technique is studied via DCT[108, 109]. The main reason behind the use of this approach isto get better channel estimates when CIR taps are not samplespaced. In this approach, the effect of high frequency terms ismitigated by exploiting the property of DCT algorithm. DCTequivalently takes the symmetry of the CFR samples andintroduce a 2K sequence with a smoother transition betweenthe elements of 2K sequence. Here, the CFR samples are firstpassed through a DCT operation. Similar to the FourierTransform approach, the energy corresponding to equivalentCIR taps is concentrated in subregions, whose length aremuch smaller than the dimension of the orthogonal transfor-mation. The regions corresponding to non-tap locations arezeroed out for the noise elimination. The resultant transformdomain signal is then passed through an IDCT operation toget CFR. Simulation results show that better results than theconventional Fourier Transform domain approach can beobtained [109]. However, as in the case of windowing opera-tion for CFR samples, since the aliasing is not completelyremoved, an error floor still exists at high SNR region. Themathematical description of the DCT approach is similar tothe second Fourier Transform approach with Fourier matricesbeing replaced by the DCT matrices. Although the sameapproach of taking the symmetry of samples can also beapplied to the Fourier Transform approach, faster DCT algo-rithms than the FFT algorithms can offer a better computa-tional complexity [109]. However, since OFDM based systemsalready has FFT algorithms on IC, additional IC will berequired for the DCT implementation, increasing the cost ofthe modules.

Very similar to the DCT approach, Hadamard and KLTtransforms are also studied [106, 107, 110]. In these methods,the same steps as in Fourier Transform approach are taken sothat the signal and the noise subspaces are separated. Itshould be noted that KLT has not been applied to OFDMbased systems but is tested for single carrier MIMO systems.

h t tB

lW l

ll

L( , ) ( )

sin( ( ))

( ),τ α

π τ τπ τ τ

=−

−=

−

∑0

1

ˆ / ˆ ,H F D F HFT pH

FT p LSK N= 2

nFigure 6. Equivalent taps of fourier transform of CFR.

Tap index100

0

1

Equi

vale

nt t

ap c

oeff

icie

nts

2

3

4

5

6

20 30 40 50 60 70

nFigure 7. Equivalent CIR taps from CFR when the real tapsare not sample spaced.

Tap index50

0

1

Tap

coef

ficie

nts

2

3

4

5

6

10 15 20 25 30 35

Non-sample spacedSample spaced


Analysis showed that SNR improvement can be achieved, sug-gesting its application to MIMO-OFDM systems.

Practical systems such as WLAN and WiMAX introduceguard bands in OFDM symbols via the elimination of the useof the subcarriers at the edge of the OFDM symbols. Thetransform domain techniques suffer from these unused sub-carriers or suppressed subcarriers as this corresponds to rect-angular windowing in frequency domain that results in sincconvolution in transform domain or equivalent time domain.Hence, the taps are leaked to one another due to sinc interpo-lation, and the taps orthogonality is lost. Since the transformdomain techniques assume a certain channel length, L, K – Ltaps are zeroed out during time windowing. When this win-dowing in time domain is applied for reducing the noise plusinterference, it will cause Gibbs phenomenon when the signalin the transform domain is transferred back to the frequencydomain. In other words, the channel frequency response willhave ripples around the edge carriers. The reason for this isthe truncation of the sinc function in time domain. Althoughno studies have been reported to overcome this issue, simpleextrapolation via the use of correlation properties of the sub-carriers (Fig. 4) can be employed before any transformdomain approach.

Although different transform domain techniques are stud-ied, it should be noted that since the noise is assumed to beAWGN in the original domain, it will have its AWGN charac-teristic in the transform domain. The equivalent channel tapswill be concentrated only in a small region. Therefore, as longas the equivalent taps are correctly identified in all the trans-form domain methods, the same performance will be achieved.Since the Fourier Transform approach utilizes fast algorithmsin the OFDM transceiver structure, the use of Fourier Trans-form offers a better trade-off among the transform domainchannel estimation techniques.

When the CFR samples are also available over severalOFDM symbols at the pilot subcarriers, 2-D Fourier Trans-form technique can be used [111, 118]. Here, the CFR sam-ples at the pilot subcarriers are passed through a 2-D FourierTransform. The transform domain signal is expected to have adiamond shape concentrated in a 2-D subregion. By zeroingout the signal values outside this subregion, noise reductioncan be achieved. The resultant signal can then be transformedinto the original domain via an inverse 2-D Fourier Trans-form. The performance of 2-D transform domain approachesdepends on the appropriate transform domain filtering, whichis related to the channel statistics.

The transform domain techniques have been successfullyapplied to the cases where impairments to the system aremodelled as Gaussian noise. For example, an OFDM systemwith significant PAPR can be improved via clipping, wherethe large peaks are replaced with a pre-defined envelope, A.The system model can then be modified as [119]

Y = (αcr × diag(X) + Υ)H + W (36)

where the clipping ratio, αcr = A/√—εx, εx is the average energy

of the input signal, and Υ is the distortion caused by the clip-ping. With αcr known, the effect of distortion can be reducedvia transform domain techniques since the distortionapproaches complex Gaussian distribution with zero mean.Simulations performed for this scenario showed that with theuse of Fourier Transform technique, accurate channel estima-tion can be obtained [119].

The information about the CIR length is important inachieving higher performance in transform domain approach-es. A CIR length taken to be smaller than the actual CIRlength will eliminate the significant taps, while a channeltaken to be longer will result in less noise suppression. How-

ever, the first case is more critical than the second, as andhence in practice the CIR length is usually taken to be thelength of the CP [36, 120, 121].

For more accurate results, algorithms are also developedto estimate the number of significant channel taps. This isneeded especially in high data rate communication where thechannel length can be long but the number of channel tapscan be smaller, that is, sparse channel. The correspondingchannel taps and their location can be searched based on acost function assuming channel tap locations and the corre-sponding coefficients [122–125]. In many cases, since theproblem is similar to finding the tones of a signal, ESPRITand MDL algorithms are employed to get the number and thelocation of the channel taps [11, 124–126]. However, theseapproaches can yield degraded performance when the numberof taps increases. Hence, other approaches based on iterativeML are proposed both to reduce the computational complexi-ty and to get accurate estimation [123]. Moreover, tapssearching based on the energy of the taps is also studied [102,113, 127, 128]. Iterative algorithms like Newton-Lapson’smethod can also be utilized to get the channel taps [129, 130].

The use of PN codes superimposed with the data to revealthe channel taps is also widely applied [70]. The output ofthese correlators is related to CIR, which can be averaged fur-ther when the correlators result in multiple CIR copies. How-ever, the performance of this approach suffer from theinfluence of the transmitted data that suggests an increase inthe power of the PN sequence or time-consuming iterativemethods [31]. Hence, transmitting data and the pilots overdifferent subcarriers, and then using the correlators before theFFT block at the receiver is proposed [131]. In this method, atime domain signal obtained via the IFFT of the pilot signal isutilized to be correlated with the received signal. Ideally, theoutput of the correlator are the delayed impulses whoseamplitude and delay are related to CIR.

Having realized that the performance of the transformdomain techniques are heavily dependent on the CIR taplocations, an inaccurate assumption or calculation of CIR taplocations can degrade any of the transform domain techniquesdrastically. Hence, a transform domain method which inher-ently uses the information of the channel taps is expected toprovide better results. For this purpose, unitary transformbased on the eigenmatrices of the auto-correlation of CFR ofdifferent channel PDP’s is shown to give better results thanthe transform domain techniques presented above [107].When the exact channel PDP is not available, then a channelPDP can be assumed and the transformation can be doneaccordingly. However, if the exact PDP is known then eigen-decomposition of auto-correlation matrix of CFR can providethe optimum transform. In the following section, this opti-mum transform, a special form of LMMSE, is presented indetail.

LMMSE CHANNEL ESTIMATION

LMMSE is widely used in the OFDM channel estimationsince it is optimum in minimizing the MSE of the channelestimates in the presence of AWGN. LMMSE uses additionalinformation like the operating SNR and the other channelstatistics. LMMSE is a smoother/interpolater/extrapolater, andhence is very attractive for the channel estimation of OFDMbased systems with pilot subcarriers. However, the computa-tional complexity of LMMSE is very high due to extra infor-mation incorporated in the estimation technique [22, 36, 58].

For a given linear system model in the form of

y = Ax + w, (37)


LMMSE of the variable x is given by,

x = Ryx Ryy–1y (38)

where Ryx is the cross-covariance between variables y and x.When the expression in Eq. 38 is applied to the OFDM chan-nel estimation given in Eq. 16 with equal pilot spacing, Dp,

H^LMMSE = RHHp ⋅

(RHpHp + σw2 (diag(X) diag(X)H)–1)–1 H^LS (39)

can be obtained. Here, Hp is the CFR at the pilot subcarriers,RHHp represents the cross-correlation between all the subcar-riers and the pilot subcarriers, and RHpHp represents theauto-correlation between the pilot subcarriers. As can beseen in Eq. 39, LMMSE uses additional information in itsestimation process such as the correlation between subcarri-ers and SNR.

The LMMSE estimation of H in Eq. 39 is computationallyvery heavy. For example, the dependency on the transmittedsymbols due to the matrix inversion required at each estimateneeds many operations. Moreover, large sized, full matrixmultiplication required for a single estimate increases thecomputational complexity of LMMSE as well. The non-trivialmatrix inversion required in the LMMSE estimation is anoth-er factor increasing the computational complexity of LMMSE.Therefore, although LMMSE is optimal, without reducing itscomputational complexity, it is hard to realize its applicationin practical systems.

The complexity of LMMSE can be significantly reduced ifthe LMMSE expression is made independent of the transmit-ted symbols. Although the expression inside the inversionoperation also contains the term RHpHp, which is the auto-covariance of the CFR at the pilot tones, RHpHp does notchange for a large number of OFDM symbols since it is afunction of channel PDP. Therefore, for a given large numberof OFDM symbols, the term RHpHp can be assumed to be con-stant, leaving (diag(X)diag(X)H)–1 as the constantly changingparameter from symbol to symbol. By assuming the transmit-ted symbols use the same signal constellation, the expression(diag(X)diag(X)H)–1 in Eq. 39 can be replaced by the expectedvalue of (diag(X)diag(X)H)–1 [36]. That is,

(40)

where β = E{|Xk|2} E{1/|Xk|2}, with Xk’s being the constel-lation points. Then, Eq. 39 becomes,

(41)

It is recommended in some studies that such an approxima-tion should not be assumed for the whole OFDM subcarriers,as the noise level can be different for various portions of thesymbol [132]. In this case, a windowing approach can beapplied to suppress the noise so that over the whole symbolthe noise level is almost constant. The advantages of thisapproach comes at the expense of SNR estimation for eachsubcarrier and the additional filtering. Since it is observed viasimulations that the approximation given in Eq. 40 has negli-gible performance degradation for the OFDM channel esti-mation, the SNR estimation for each subcarrier or subcarrierblock is usually omitted [36, 120].

Although the expression in Eq. 41 is simpler, it still needsto be updated with the changing operating SNR. Moreover,the expression in Eq. 41 needs to be recalculated whenever

the channel PDP changes. Either due to the change in SNRor PDP, the channel estimation via Eq. 41 is computationallycomplex since it requires multiplications in the order of O(K3)for the channel estimate of a single subcarrier.

The complexity of LMMSE is even higher for 2-D channelestimation of OFDM systems since the number of the totalsubcarriers is higher [24, 133, 134]. Hence, the complexity ofLMMSE can be computationally prohibitive for practical 2-Dchannel estimation. For this reason, computationally efficientmethods are proposed so that the benefits of LMMSE arerealized both in 1-D and 2-D channel estimation.

Subspace methods are investigated for the computationalcomplexity and noise subspace reduction for the LMMSEchannel estimation. With subspace methods, the number ofmultiplications required for the channel estimate of a singlesubcarrier is reduced by exploiting SVD [24, 36, 120, 135,136]. Subspace methods applied to the LMMSE channel esti-mation reveal the degree of independency of the subcarriers’auto and cross-correlation matrices. Since the subcarrier cor-relation is a function of the channel delay spread, it ultimatelyreveals long-term significant CIR taps or channel PDP.

Without subspace methods, the complexity of the channelestimation using LMMSE can be reduced significantly byassuming a pre-defined channel length [75]. However, forsparse channels this would mean unnecessary computationwhen the significant number of channel taps is smaller thanthe channel length [122, 126]. With the CIR length beingmuch smaller than the number of the subcarriers, SVD of theauto and cross correlation matrices of CFR result in only asmany significant singular values as the significant number ofCIR taps. As the noise is assumed AWGN in frequencydomain, the SVD decomposition results in equivalent singularvalues for the noise terms. Hence, it can be anticipated thatthe noise in frequency domain is equally distributed in thesubspace domain with equal energy in all dimension of thesubspace. If the subspace due to the noise is eliminated, thennoise reduction is achieved [126]. Moreover, due to the for-mulation of LMMSE, less number of multiplications will berequired after the SVD operation. This will be seen moreclearly with the following derivations.

Starting from an all-pilot case that all the subcarriers arepilot tones, the LMMSE channel estimation can be re-writtenas,

(42)

If SVD is to be performed over Hermitian RHpHp,

RHpHp = UΛUH (43)

can be written. Here U is a unitary matrix and Λ is a diagonalmatrix bearing the singular values λ0, λ1, …, λK–1 in descend-ing orders. Then in Eq. 42 can be re-written as,

H^LMMSE = U∆UHH^LS, (44)

where ∆ is a diagonal matrix with entries

(45)

With the above formulation, the corresponding MSE isexpressed as [36],

δλ

λβi

i

i SNR

i K=+

⎧

⎨⎪

⎩⎪

= … −, , , , . 0 1 1

ˆ ˆ .H R R I HLMMSE H H H H K LSp p p p SNR= +

⎛

⎝⎜

⎞

⎠⎟−β 1

ˆ ˆ .H R R I HLMMSE HH H H N LSp p p pSNR= +

⎛

⎝⎜

⎞

⎠⎟−β 1

E diag diagSNR

HN p

{( ( ) ( ) ) }X X I− =1 β

(46)

In the decomposition of Eq. 44, the number of multiplica-tion is still in the order of O(N3). The number of multiplica-tion can be reduced if only significant singular values of Λ or∆ are considered.

It should be noted that the number of significant singularvalues is related to the number of long-term significant taps.The relationship between these two will be given shortly.Since the number of significant taps is much lower than thenumber of the subcarriers, there will be only a few significantsingular values in Λ or ∆. Therefore, the entries of matrix ∆can be approximated as

(47)

where r represents the number of the significant singular val-ues. The above is nothing but the result obtained from thelow-rank approximation of the RHpHp. With the low-rankapproximation, the number of required multiplicationsreduces from the order O(K3) to O(rK2).

Although the low-rank approximation via a SVD of theauto-covariance matrices reduces the number of multiplica-tion for the channel estimation, obtaining the SVD of theauto-covariance matrices by itself is computationally verycomplex and is in the order of O(K3) [137]. Therefore, it willbe no use to exploit the low-rank approximation if the SVD isto be performed for every estimation process. Although, theauto-covariance matrix, RHpHp, is a function of the channelPDP that can be assumed to be constant for a good numberof OFDM symbols [138], when the channel PDP changes there-computation of SVD of RHpHp can be non-practical espe-cially when the number of the subcarriers is large.

For this reason different approaches are proposed in orderto eliminate the need for SVD operation. The so called robustchannel estimation methods are developed for this purpose[36, 139, 140]. In these methods, a channel PDP is assumedfor the system under the consideration, and the auto-covari-ance matrix and its SVD are then pre-calculated for theassumed channel PDP. The most common assumed channelPDPs are uniform and exponential, with uniform PDP beingused more extensively [13, 126]. Simulation results show thatrobust LMMSE channel estimation results in acceptable per-formance degradation for certain systems when compared tothe LMMSE with perfect channel knowledge [13, 111]. Thedegree of degradation increases as the true channel deviatessignificantly from the assumed channel. By pre-calculating theSVD of the auto-covariance matrices for more possible chan-nel PDPs, this degradation can be mitigated. In this case, bylooking at the delay spread of the channel, the closest channelPDP for which the SVD is pre-calculated can be used. Withadditional computational complexity needed for the delayspread estimation, this approach is found to improve the MSEperformance of the robust LMMSE channel estimator by afactor of 2 dB [139].

The low-rank approximation for LMMSE channel estima-tion has also been investigated for the pilot symbol aidedchannel estimation [24, 36, 136]. It is shown that for the pilotsymbols similar simple expressions to those of all pilot caseare also possible [120]. For the pilot case,

H^rlmmse = PrΓr(Qr)HH^LS, (48)

where the entries of the diagonal matrix Γr are given by,

(49)

Here, the superscript r represents only the first r columns ofmatrices, P = FV and Q = FpV. Here, V is the unitary matrixfrom the SVD of Rhh, the auto-correlation matrix of CIR. Thesingular values of Rhh are denoted by σ0 ≥ σ1 ≥, …, σK–1. TheMSE of the above estimator is shown to be [120],

(50)

The low-rank approximation for the pilot case requiresr(Np + K) multiplications for the channel estimation of all thesubcarriers. Although this is computationally very efficientcompared to the multiplications in the order of O(K3), still theneed for the SVD of the auto and cross-covariance matricesof CFR makes the real-time LMMSE estimation almostimpossible. Therefore, the methods enabling real-timeLMMSE estimation are needed.

A close examination of low-rank LMMSE channel estima-tion shows that for the real-time LMMSE unitary matrices Pr

and Qr are needed as well as the SNR and r significant singu-lar values of RHpHp or Rhh. Since Pr and Qr matrices are relat-ed to each other, if one of them is obtained the other can becalculated easily [120].

It is noted that subspace tracking can enable real time low-rank LMMSE channel estimation of OFDM systems [120].Subspace tracking has been introduced in adaptive filteringwhen the signal under consideration has a subspace with adimension less than the number of the data snapshots. This isa very common case for oversampled systems. In such scenar-ios, the adaptive filtering requires SVD of large-sized matri-ces. Subspace tracking avoids SVD of large-sized matrices bytracking the significant singular values and the correspondingsingular vectors [141, 142]. In OFDM based systems, since thenumber of significant taps determines the dimension of thesignal (CFR samples) subspace, and since in most cases this ismuch smaller than the number of subcarriers and pilot tones,oversampling will be observed. Hence, subspace tracking canbe applied to track the few significant singular values and thecorresponding vectors of the matrices RHH or RHpHp, that is,tracking Pr, Γr, or Qr with a computational complexity in theorder of O(Kr). However, subspace tracking can only be start-ed after some initial channel estimates that need to beobtained via some other methods like transform domain tech-niques.

Subspace tracking has been investigated for OFDM basedsystems in [121] and [138] via different approaches. In the firststudy, for example, the channel estimates at the pilot tonesare transformed into the time domain by using the singularvectors of the auto-covariance of the CFR. This approach isapplied in all the pilot case, and produces no mismatcheswhen the CIR taps are uncorrelated and monotonicallydecreasing. In this case, the P matrix is simply the unitary

MSE rK

KD SNRii

pi( ) = −

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ +

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥11

2

2σγ

γβ

⎥⎥⎥

+

=

−

=

−

∑

∑

i

r

ii r

K

KK

0

1

11 σ .

γ

σ

σβi

ip

pi

p

K

D

KD SNR

i r

i r N

= += … −

= … −

⎧

⎨

⎪0 1 1

0 1

, , ,

, ,

⎪⎪⎪

⎩

⎪⎪⎪

.

δ

λ

λβ

i

i

i SNR

i r

i r K

= += … −

= … −

⎧

⎨⎪⎪

⎩⎪⎪

0 1 1

0 1

, , ,

, ,

,

MSEK SNRLMMSE i i i

i

K= − +

⎛

⎝⎜

⎞

⎠⎟

=

−

∑11 2 2

0

1λ δ

βδ( )

+=

−

∑1 1

K ii r

Kλ


Fourier matrix and is used for the transformation of channelestimates in frequency domain. The transformation using thematrix consisting singular vectors of the auto-covariancematrix of CFR eliminates the noise subspace and henceimproves the performance. The second study, on the otherhand, projects the CIR estimates onto the delay subspace sothat only the significant taps are considered. The methodbears the fact that there will be only small number of timeinstances at which the value of the CIR taps are significant.These time instances are then tracked using the channel esti-mates at the pilot tones. The tracking of the delay subspacecan be considered as a pre-filtering beforeinterpolation/extrapolation that improves the performance ofthe LS estimate at the pilot subcarriers. Different subspacetracking methods are analyzed for the tracking of the delaysubspace, and it is observed that delay subspace tracking inthe order of O(Kr) can give accurate results [143]. Since thesubspace of CFR or CIR are related, both studies essentiallyperform the same task: noise subspace elimination. Since sub-space tracking is more towards oversampled systems, thenumber of pilot subcarriers need to be higher than the num-ber of significant taps, otherwise performance degradation willbe observed especially in high SNR region.

Similar to subspace tracking, LMMSE coefficients aretracked via NLMS and RLS algorithms [144]. In these meth-ods, the CIR taps are updated based on the cost functionsdefined for NLMS and RLS. Although NLMS is less complexand less accurate compared to RLS, care must be taken inRLS algorithm for oversampled systems, as the performancecan be faulty due to implicit matrix inversion needed duringupdate operation [145].

2-D LMMSE — Since the computational complexity of 2-DLMMSE is high, several methods are proposed to eliminatethis heavy computation. Among the methods is the use of twocascading LMMSE filters, thanks to the separability of thechannel correlation in frequency and in time [24, 133, 146,147]. It is demonstrated that the use of two cascaded LMMSEperforms as accurate as the 2-D LMMSE [24, 133, 148]. Thisway the computational complexity is reduced from O((2N)3)to O(N3), that is, an 87 percent decrease in computationalcomplexity. While one of the filters uses the frequency domaincorrelation between subcarriers, the second filter uses the cor-relation of the subcarriers over the time. The filtering caneither be done first in frequency or time domain, followed bythe filtering in the other domain. The correlation of the sub-carriers over the time depends on the mobility of the stations,and hence the Doppler the shift. Therefore, 2-D LMMSErequires channel statistics in time domain as well.

In some cases, the need of channel statistics in timedomain can be eliminated by using simple linear interpolation[149]. Although computationally more efficient, for very timeselective channels more pilots will be needed over the time foran acceptable performance.

Similar to 1-D LMMSE, low-rank 2-D LMMSE channelestimation is also studied [24]. It is observed that the low-rankapproximation yields better results with lesser computationalcomplexity when compared to non-low rank 2-D LMMSE.

Since the computational complexity is highly dependent onthe number of subcarriers, via the use of lesser subcarriers,the computational complexity can be reduced significantly.Such a thinking has brought the use of 2-D LMMSE overmany subregions [24, 147] or piecewise LMMSE concept [87].In fact, piecewise LMMSE is first considered for 1-D channelestimation in frequency domain where the noise level is con-sidered to be different at each subregion [150]. Subregioningimplies that the subcarrier correlation is only limited to the

those within a given neighborhood, ignoring the correlationbetween the subcarriers in different subregions. While formore frequency and time selective channels no significant gainis possible with the use of correlation between all subcarriers,for relatively less frequency and time selective channels, betterperformance can be achieved with the use correlation betweensubcarriers in different subregions. The 2-D LMMSE filterorder is therefore very critical for obtaining more accurateresults. Doppler and delay spread information can be used forthis purpose. Once these two parameters are incorporated, itis observed that more accurate channel estimation than theconventional LMMSE is possible [133, 147, 150]. WithDoppler and delay spread parameters being available, pilotspacing can also be determined so that the pilots sufficientlysample the channel response both in time and frequencydomains. When the Doppler and delay spread estimation can-not be done in real-time, as a rule of thumb, the requiredpilot numbers can be taken twice of that required by the sam-pling theorem [134].

Robust channel estimation is also investigated for the 2-DLMMSE channel estimation [13, 134]. Similar to the channelstatistics assumed in the frequency domain estimation, theauto-correlation of the subcarrier over time evolution isassumed to follow the zero order first kind Bessel function.This function is then based on a maximum assumed Dopplershift, resulting in the well-known Jakes’ Doppler spectrum [13,33]. Simulation results also show that robust 2-D LMMSEchannel estimation results in an acceptable performancedegradation in some mismatch scenarios [13, 111]. The reasonthat the robust channel estimation gives good results even in amismatch scenario is due to the use of channel statistics thatcan be considered as a coarse approximation of the truestatistics. As it will be seen later, the role of the channelstatistics in LMMSE will be visible with accurate estimation ofSNR. When SNR is not available, then the use of channelstatistics alone in robust LMMSE will not bring a desiredimprovement given the computational complexity. This topicwill be more elaborated in the next section.

DECISION DIRECTED CHANNEL ESTIMATION

DDCE is one of the earliest methods studied for OFDM,mainly because of its popularity in legacy systems. In the earli-er studies, DDCE was applied mostly in training based sys-tems, where one or more OFDM symbols were used as thetraining symbols. The main idea behind DDCE is to use thechannel estimation of a previous OFDM symbol for the datadetection of the current estimation, and thereafter using thenewly detected data for the estimation of the current channel[31, 151, 152]. Data detection can be based on hard or softdecision [153]. While for the hard decision a specific constel-lation point is forced, for the soft decision bitwise detection isutilized [154]. Since soft decision utilizes more informationabout the incoming signal and soft-decoding, for the sufficientnumber of iterations, near ideal performance can be observed[155, 156]. Once the data at the subcarriers is detected, anymethods described in the previous subsections can be used toestimate the current channel.

Although DDCE is simple, it inherently introduces twobasic problems: the use of outdated channel estimates, andthe assumption of correct data detection. The use of outdatedchannel estimates does not pose a serious issue when thechannel is varying very slowly. In this case, the channel can beassumed quasi-static over a number of OFDM symbols [32,72]. It was observed that such an approach results in accept-able performance as long as the channel variation is slow overthe time. However, when the channel starts varying faster,



then the outdated channel estimates for the previous OFDMsymbol are no longer valid for the use of the data detection inthe current OFDM symbol [55, 73]. In this case, the datadetection would be incorrect, so are the newly estimatedchannel coefficients. Hence, the error in the channel estima-tion and data detection build up to make the system perfor-mance unacceptable [157]. This error propagation becomesmore critical when the the number of incorrect decisionsincreases at low SNR regions [158, 159].

As a quick solution to overcome the problem related to theoutdated channel estimates, the training symbols can be sentmore often. The time instances at which the training symbolsshould be sent can be based on different criteria. Trainingsymbols can be sent periodically where the period is pre-determined for a given system [159]. Moreover, the change inchannel estimates can be monitored to determine whether achannel estimation is indeed needed [160]. As the channelvaries fast over the time, the need to send the training sym-bols more frequently has a high penalty in terms of the overallsystem efficiency [49]. In this case, training symbols can bereplaced by the pilot subcarriers [161].

When pilot subcarriers are used in DDCE, the convention-al channel estimation methods can be used as long as thenumber of pilot subcarriers is sufficient. The pilots can besent over each OFDM symbol, or with the knowledge of chan-nel statistics, they can be sent with certain OFDM symbols.For example, a reliable subset of data subcarriers can beemployed as the pilot subcarriers [162, 163]. The channel esti-mated at the reliable subcarriers can be used to estimate thechannel at the other subcarriers via interpolation. Althoughsounds to be a good solution, the lack of reliable informationabout the subcarriers and the high probability of the reliablesubcarriers being non-uniformly distributed over the OFDMsymbol reduce the performance of this approach. By increas-ing the power level of the reliable subcarriers, the perfor-mance degradation can be mitigated to some extent [163].

Another approach can be the use of prediction algorithmson the channel estimation. The channel estimated in previousOFDM blocks can be used to predict the channel in the nextblock [73, 164, 165]. Prediction algorithms can be appliedeither on the channel taps or the channels at the subcarriers[81]. The previous has the advantage that the number of vari-ables to predict is much smaller but needs IFFT to get thechannel at the subcarriers [166]. The latter requires no trans-formation but it requires the prediction of more number ofvariables, that is, subcarriers. Besides, since the prediction isperformed individually for each subcarrier, the correlationproperties of the subcarriers are not utilized that results in aworse performance when compared to the prediction for thetime domain channel coefficients [72, 73]. In frequencydomain prediction, the number of subcarriers for which theprediction to be performed can be reduced so that the chan-nel at the other subcarriers can be estimated by simple inter-polation techniques. Depending on the complexity of theprediction algorithm, such an approach can result in a lowercomputational complexity. Linear relationship based predic-tion approaches can give good results in slowly varying chan-nels, but their performance becomes unacceptable once thechannel varies fast over time. Hence, prediction algorithmstaking into account the channel statistics can perform better[6, 72]. Indeed, it was demonstrated that the use of ARMAmodelling and Kalman filtering can result in substantial per-formance improvement in DDCE methods with prediction[159, 165, 166].

Whether it is through prediction or the use of detected sig-nals, the channel estimation at each subcarrier can be passedthrough some filtering to obtain better estimates. Among the

methods, the transform domain methods, LMMSE, or the lowpass filtering are widely applied [132, 167–169]. Moreover,averaging the channel estimates at the subcarriers over anumber of subcarriers is also shown to yield accurate resultsfor slowly varying channels [170].

In the preceding paragraphs, it was stated that the perfor-mance of DDCE can be improved with the information ofchannel statistics. In some cases this information may not beavailable, or it may be desirable to improve the system perfor-mance further. In this case, efficient approaches to the datadetection portion can be introduced for a more reliable chan-nel estimation.

Coding theory is probably one of the most widely fieldsapplied to the data detection portion of the OFDM systems[146]. Figure 8 shows the performance of an OFDM system inRayleigh fading in a multi-path channel without any coding. Itcan be seen that the performance is the same as the singlecarrier systems with flat fading. In the case of multipath, sin-gle carrier systems can use complex equalizers to improve theperformance of the system significantly. The BER perfor-mance of OFDM systems shown in Fig. 8 is therefore unac-ceptable for practical systems employing high ordermodulation [171]. Hence, OFDM systems need to employcoding for an acceptable performance. In most studies,OFDM with coding is called as COFDM [154, 155, 171]. Withcoding available in OFDM systems, DDCE can exploit thisinformation to improve the data detection.

The typical coding mechanisms are RS, convolutional, trel-lis, turbo, and LPDC coding [35, 146, 155, 171–175]. The out-put of the decoder can be further processed for the FEC toincrease the performance of the detection process [32, 168,176]. Simulation results of many studies showed that the BERperformance of the channel estimation with coding is drasti-cally improved compared to those without coding [146, 155,171].

Below is a quick summary of the some of the coding tech-niques applied to OFDM systems. Since the coding tech-niques’ performance in OFDM based systems is out of scopeof the current article, the papers cited in this subject and ref-erences therein can be referred for more detailed information.

Some Coding Techniques Applied to OFDM Systems —RS (outer coding) and CC (inner coding) are usually appliedback to back in OFDM based systems [9, 177]. RS codes arelinear block codes and are suitable for burst type errors. Theirdecoding process is relatively less complex. The CC are binary

nFigure 8. BER performance of an OFDM system in 5 tapRayleigh fading channel with different modulations.

SNR [dB]-5-10

10-3

10-4

BER 10-2

10-1

100

0 5 10 15 20 25 30

BRSKQAM-4QAM-18QAM-84

error correcting codes where input bits are mapped to anotherset of bits, by not only using the present bits to be encodedbut also by the previous information bits. Viterbi decoder isusually employed in the decoding process. The CC are studiedextensively for OFDM based systems, and are shown to yieldimproved BER performance with even improved PAPR per-formance [161, 178, 179].

Trellis codes are introduced by Ungerboeck [180] and area special type of CC. Trellis codes provide a better perfor-mance/complexity tradeoff than lattices in the bandwidth-lim-ited regime, although the difference is not as dramatic. Thekey ideas in the invention of trellis codes were the use of min-imum squared Euclidean distance as the design criterion andthe coding on subsets of signal sets using CC principles [181].Trellis codes are usually combined with STC to improve theperformance of MIMO-OFDM systems [35, 182, 183].

Turbo codes are introduced by Berrou et al., [184], and area new class of iterated short CC. They are built from the par-allel concatenation of two recursive systematic CC using afeedback decoder. Turbo codes can achieve a BER perfor-mance close to the Shannon limit [185] in an AWGN channel[178]. Since there are many subcarriers in an OFDM basedsystem, and that the probability of all subcarriers to fade islow, Turbo codes applied to OFDM systems can lead to ahigh performing wireless system [186].

The LPDC codes are first proposed by Gallager in 1960’s[187], but it has been ignored due to its high computationalcomplexity when long codewords are to be designed [175].Due to its parallelized structure, LPDC codes are easy imple-ment via basic elements. LPDC codes have excellent perfor-mance in AWGN channels, and under certain conditions theycan perform better than the turbo codes [175]. Hence, LPDCcodes are being re-invented for OFDM based systems, as thenew systems require relatively low complex LPDC codes. Forexample, IEEE 802.16e standard employs tile structure inuplink direction, where relatively less number of subcarriersare used for the pilots and data [177]. Moreover, the use ofLPDC coding with STC is shown to yield 5-6 dB coding gainfor MIMO-OFDM systems with reasonable computationalcomplexity [174].

The channel estimation performance has an impact on thedecoding process of the OFDM receivers [188]. Studies in thisarea model the channel estimation error as a Gaussian noise,where the noise power can come from different sources suchas ICI as well as the channel estimation method itself[188–190]. The analyses show that the channel estimationerror becomes effective in system performance when thechannel estimation error is greater that 30 dB [191]. Thisadditional noise reduces the effective operating SINR of thedecoder block in the receiver, thereby causing performancedegradation. In fact, the analysis via simulations show thatunder the channel estimation error, the coding gain of thedecoders reduces significantly [188]. Similarly, the BER analy-ses also indicate performance degradation due to channel esti-mation error (less than 3dB for uncoded case) [45, 192].Hence, as pointed out in the introduction, for a better per-forming OFDM system, channel estimation block in thereceiver should be very accurate.

Iterative channel estimation algorithms can be exploited tominimize the channel estimation errors [156]. In these

approaches, the channel estimation can be found via any ofthe methods described in the preceding sections, and the esti-mates can be improved by remodulating the detected signals[154]. It is clear that when the number of iterations is one,then the approach is the same as the conventional approach-es. However, for more iterations better performance isachieved at the expense of more~computation.

EM ALGORITHM

Among the DDCE channel estimation of COFDM, the EM isone of the most attractive methods. Mostly because EM algo-rithm also utilizes the error probabilities that are alreadybeing utilized by the decoders. For example, the maximum aposteriori (MAP) decoder used in turbo decoding can providethe probabilities of the transmitted symbols, which is exactlywhat the EM algorithm is looking for the channel estimation[193–195]. Hence, computational complexity of EM algorithmis reduced significantly, making EM a good match for theCOFDM channel~estimation.

The EM algorithm consists of two steps: an expectationand a maximization step. The motive of the expectation stepis to estimate the corresponding component of the transmitsignal in the received signal, whereas the motive of the maxi-mization step is to estimate the channel given the transmittedsignals, which can either be the pilots or the detected symbols.With the pilots being available more accurate results can beobtained [196].

In the above description, the detected signal can comefrom a decoder. The decoder itself however, requires channelestimate (Fig. 9). This chicken-and-the-egg problem can besolved iteratively with some initial values either assigned tothe channel or the detected signals. While it is common toemploy EM algorithm in the channel estimation part, for thedata detection part different decoders can be employed. Forexample, in parallel to the mostly applied turbo coding andthe corresponding decoders, it was shown that EM algorithmcan also be integrated with a QRD-M algorithm using CC[197]. Iteratively performing the channel estimation and datadetection with sufficient number of iterations are shown togive very close BER performances to the ideal case [193].

The inherent iterative approach of EM does not necessari-ly need the channel statistics. With a sufficient number of iter-ations, EM algorithm converges to the ML algorithm [40],which was shown in the previous sections to be equivalent toLS. However, due to EM’s iterative nature, the computationalcomplexity is relatively less [40]. For the EM algorithms toconverge rapidly, the initial assumed/estimated values are crit-ical.

The above iterative scheme described for the EM algo-rithms can also be generalized as joint and iterative channelestimation (Fig. 9). Joint and iterative channel estimationtechniques are introduced when the conventional DDCEchannel estimation techniques are unable to estimate thechannel in fast fading, when it is desired to reduce the pilotoverhead, or when the non-linear distortions like poweramplifier non-linearities make the conventional approachesineffective [153, 154]. For example, by using relatively lessnumber of pilots (less than the Nyquist rate), it is shown thatthe joint iterative approaches can detect the symbols, and esti-mate the channel simultaneously with more computationalcomplexity [48]. For non-linear distortions, iterative methodscan extract the distortion, and the channel estimates can beimproved accordingly [153].

Based on Fig. 9, different combination of the coherentdetection, channel estimator, and the decoder algorithms areproposed. For example, Kalman filtering is employed to esti-


nFigure 9. Joint iterative DDCE.

Channelestimator

Channelestimator

Replicagenerator

Coherentdetection Deinterleaver Decoder

Output


mate the time domain channels, while QRD-M is employedfor the data detection part [198]. Due to its inherent iterativeapproach, RLS is also combined with DDCE techniques toestimate the channel, and is shown to provide accurateresults [55, 176, 199, 200]. Since the number of trackingparameters are less in time domain channel and that RLScan be erroneous in oversampled systems [142, 145], the RLSis not recommended over the frequency domain channelparameters.

The transform domain techniques, LMMSE, and MLalgorithms are also studied for the joint iterative channelestimation where the channel estimates and data detectionare improved over each iteration [26, 99, 154, 192]. The useof space-time and space-frequency block decoder is morecommon in MIMO-OFDM systems in these iterativeapproaches [35, 176, 201–203]. For example, Kalman track-ing coupled to the Viterbi decoder in the decoding of aspace-time trellis coded MIMO-OFDM system is observedto give accurate results, where the ambiguity of the start ofthe decoding process is eliminated by using a single pilotsubcarrier [35]. With the space-time codes being orthogo-nal, it is shown that LMS can be successfully applied toMIMO-OFDM systems [204].

Unification of Channel Estimation Approaches The chan-nel estimation techniques proposed so far basically offer atrade-off between the complexity and the performance. Figure10 shows the performance of methods in an exponentiallydecaying 5 tap channel. While LMMSE offers the best perfor-mance especially at low SNR regions, its complexity is thehighest among the given techniques. This can be attributed tothe fact that LMMSE uses more information during the esti-mation process. If LMMSE uses as much information as anyother method, then the performance should be very similar ifnot the same. Hence, LMMSE can be considered as the gen-eralized channel estimation method. In this section, circum-stances for which LMMSE reduces to the other methods willpresented.

UNIFICATION OF LMMSE WITH TIMEDOMAIN LS APPROACH

The formulation of time domain LS was presented previously.By rewriting the OFDM system model in terms of the CIR,

Y = diag(X)Fh + W (51)

Then LMMSE can be written as [75],

H^LMMSE = FRhh

[(FHdiag(X)Hdiag(X)F)–1σw2 + Rhh]–1

(FHdiag(X)Hdiag(X)F)–1

FHdiag(X)HY (52)

With Rhh being invertible,

H^LMMSE = F[(FHdiag(X)Hdiag(X)FRhh)–1σw

2 + IK]–1

(FHdiag(X)Hdiag(X)F)–1FHdiag(X)HY (53)can be obtained. The LS estimation was given by [75],

H^LS = F[FHdiag(X)Hdiag(X)F]–1 FHdiag(X)HY. (54)

Note that when SNR is very high, σw2 is very small, and

LMMSE reduces to time domain LS approach. Hence, if SNRinformation is not available, there is no need to employLMMSE with SNR set to a high value since the same perfor-mance can be achieved with less computational complexityoffered by time domain LS.

As it was demonstrated before that ML is the same as LSwhen the noise is AWGN, LMMSE also reduces to ML whenSNR is very high.

UNIFICATION WITH TRANSFORM DOMAIN TECHNIQUES

In this subsection, the unification of LMMSE with the FourierTransform technique will be presented. The unification ofLMMSE with the other transform domain techniques is alsopossible if the LMMSE estimation is performed in the corre-sponding transform domain technique. Since in the precedingsections frequency domain LMMSE is presented, the unifica-tion with the Fourier Transform technique will be presented.

As most of the transform domain techniques are exploitedfor the pilot subcarrier aided OFDM channel estimation, theLMMSE formulation for the pilot subcarriers with equal spac-ing will be used, and the case of non-equal pilot spacing isvery similar. Starting with the matrix equation,

Hp = Fph, (55)

the auto-covariance matrix of CFR when all the subcarriersare used as the pilots can be expressed as

E {HHH} = E{Fh(Fh)H} (56)RHH = FRhhFH. (57)

The SVD of Hermitian Rhh is in the form of, Rhh = VΣVH.By normalizing F column-wise (making each column unitynorm),

Rhh = FnVKΣ(FnV)H, (58)

where Fn = F/√—K. Let FnV = P, then

RHH = PKΣPH. (59)

The above equation is nothing but the SVD of RHH. Whenthe CIR taps are uncorrelated and are monotonically decreas-ing, then the unitary matrix P is simply the unitary FFTmatrix. Similarly,

(60)

(61)

where Q = FpnV and pilot spacing is chosen to be Dp. With

R Q QH Hp

Hp p

K

D= Σ ,

R P QHHp

Hp

K

D= Σ

nFigure 10. MSE of some of the channel estimation methods inRayleigh fading for a 5 tap channel with QAM-4 modulationand pilot spacing, Dp = 4.

SNR [dB]-15-20

10-2

10-3

MSE

10-1

100

101

102

-10 -5 0 5 10 15 20

LSLinearTransform domainLMMSE


CIR having L taps, Σ is in the form of

(62)

Since it is assumed that Np > L, by truncating Σ to the sizeK × Np to form ΣNp, and Q to the size of Np × Np to formQNp. Then,

(63)

(64)

Since the first Np columns of Q form a unitary matrix, theoverall equations denote the SVD of RHHp and RHpHp. Byreplacing the SVDs of the RHHp and RHpHp into Eq. 64, weget

H^LMMSE = PΓQNH

pH^LS, (65)

where the entries of the diagonal matrix Γ are given by,

(66)

In case of low-rank approximation, only r significant singu-lar values of Rhh will be considered. Then,

H^rLMMSE = PrGr(Qr)HH^LS, (67)

where the entries of the diagonal matrix Γr are given by Eq.49.

For high SNR, Γ approaches to a diagonal matrix withdiagonals being √

——Dp. Moreover, when the CIR taps are uncor-

related and there are only L number of significant taps, thenRhh is a diagonal matrix. In this case, V matrix becomes anidentity matrix, making P and Q matrices simply F and Fp,respectively. Moreover, the SVD of Rhh results in L numberof significant singular values, making r = L. For equal spacedcomb-type pilots

(68)

With the conditions described above, low-rank LMMSEbecomes a transform domain technique using Fourier Trans-form. Here, √

——Dp comes from the normalization due to down-

sampled Fp.As can be seen from different methods, the use of more

information increases the performance of the channel esti-mates at the expense of computational complexity. It is notedin the above sections that when the SNR information is notavailable and is set to a high value, then the performance ofLMMSE reduces to the those of not utilizing SNR, withLMMSE still having high computational complexity. Hence,the use of other methods in case of no SNR informationoffers a better trade-off in terms of the performance and com-putational complexity.

OFDM CHANNEL ESTIMATION WITHINTERFERENCE

So far the effect of ICI, ISI, and external interferers wereignored, and the estimation techniques were performed

accordingly. In this section, the effect of interferers will betreated separately. First the effect of ICI will be considered,followed by the inclusion of external interferers in the channelestimation process. A short discussion of ISI is presentedwhen ICI due to frequency synchronization error is covered.

OFDM CHANNEL ESTIMATION WITH ICI

Again starting from the system of SISO-OFDM, the receivedsignal in the presence of ICI can be expressed as,

Y = FΞFHX + W, (69)

where the external interferers are folded into the AWGNterm. Here, since CIR is not constant over the OFDM sym-bol, Ξ is not a circulant matrix anymore. Hence, the productof FΞFH is not a diagonal matrix [205]. In this case, a receivedsignal at a subcarrier k is affected by the transmitted signals ofall the subcarriers, increasing the number of unknowns byK*(K – 1). This also implies that when the number of subcar-riers increases, the ICI increases as well [148]. The ICI powermainly depends on the product of maximum Doppler frequen-cy and OFDM symbol duration [33]. Hence, while the longsymbol duration of OFDM symbols avoids ISI significantly,under very fast changing channels, this advantageous parame-ter turns into a disadvantageous parameter due to ICIenhancement. The ICI power at the center subcarriers isexpected to be higher than the edge subcarriers since they areaffected more by the ICI of the other subcarriers.

ICI also occurs when there is a frequency offset due trans-mitter/receiver oscillator mismatch, phase noise, and/or thenon-linear power amplifier effect. The oscillator mismatch orthe phase noise cause the received signal to be sampled atincorrect positions, and thereby taking the effect of all thesubcarriers [79, 206], that is, orthogonality loss. When leftwithout compensation, this effect reduces the performance ofchannel estimation methods, especially those based on fixedchannel statistics [25].

Either due to the frequency offset or the fast-varyingnature of the CIR taps, ICI needs to be compensated so thatreliable channel estimation is obtained. When higher ordermodulation techniques are employed, the effect of ICI ismore severe as the detection of the modulated signal needs todifferentiate many closely spaced constellation points. In thisarticle, these two effect will be presented independently, andthe details are given in the subsequent sections.

ICI Due to Frequency Offset — ICI due to frequency offsetmostly occurs due to the loss of synchronization of the subcar-riers or the phase noise of the oscillators. In WLAN andWiMAX standards, in the preamble, two short durationOFDM symbols are provided for the synchronization purpos-es [9, 78, 170]. These short symbols can also be used for thefrequency offset estimation.

Under the synchronization errors (both time and frequen-cy), the correlation properties of the OFDM subcarrierchange in time and frequency domains, the performance ofLMMSE channel estimation algorithms degrade significantlyas these estimation algorithms utilize the correlation proper-ties of the subcarriers. It is shown that synchronization errorcan cause up to 5 dB MSE degradation of LMMSE channelestimation [207]. Hence, the synchronization errors need to becompensated for OFDM based systems.

The compensation of ICI due the frequency offset is rela-tively less challenging compared to the compensation of theICI due to fast channel variation since the value of the fre-quency offset parameter is constant over all the subcarriers.The received signal of a SISO-OFDM in the presence of fre-

H FDF HFT p pH

LSD= .

γ

σ

σβi

ip

pi

p

K

D

KD SNR

i N=+

= … −, , , , . 0 1 1

R Q QH H pp

p NH

p p pN

K

DN= Σ .

R P QHHp

p NH

p p

K

DN= Σ ,

Σ

Σ

= −−− −−−

⎡

⎣

× −

− × − × −

L L K L

K L L K L K L

|

|

|

( )

( ) ( ) ( )

0

0 0

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥,


quency offset can be expressed as [208]

Y = Sεpdiag(X)H + W, (70)

where Sεp is the interference matrix representing ICI due tothe normalized frequency offset εp. Here, the entries of theinterference matrix are given by

(71)

Although frequency offset estimation and its compensationhave been studied in numerous articles, we will only considerthose with the channel estimation. In these studies, the chan-nel estimation, frequency offset estimation, and its compensa-tion are performed jointly.

Bearing the fact that the auto-correlation of CFR decreas-es as the frequency offset increases due to the random behav-ior of transmitted signals, an iterative binary searchingalgorithm based on the diagonal element of the Sep is per-formed by assuming maximum and minimum values for thefrequency offset [208]. At each iteration step the CFR is esti-mated based on the assumed frequency offset and so on. Sim-ulation results show that the frequency offset can correctly beestimated, improving the CFR estimates at the subcarriers.

Moreover, by realizing that the channel estimation error isminimized when the correct length of the CIR is incorporatedinto the frequency offset expression, an iterative method aim-ing to find the first minimum of the MSE of the channel esti-mation based on Fourier Transform is developed [25]. Withthe use of Blackman window for filtering of the CIR taps, it isobserved that frequency offset can be estimated and compen-sated with the proposed iterative method.

Frequency offset compensation can be performed beforethe FFT block in the receiver side [206]. By comparing theCFRs with the compensated and uncompensated received sig-nals, the frequency offset of the current symbol can be detect-ed and then can be linearly interpolated to get the frequencyoffset of the all the subcarriers. The estimated offset value canthen be used to predict the next frequency offset parameter.With a more computational complexity algorithm, studiesexploited Kay filters for the frequency offset estimation byoversampling the pilot subcarriers [79]. Improved perfor-mance can be obtained via a prediction algorithm assuminglinear variation over time. Since the frequency offset isassumed to be the same for all of the subcarrier, averagingcan be introduced to reduce the noise significantly [79].

While frequency synchronization causes ICI, timing syn-chronization destroys OFDM symbol orthogonality and causesISI. Hence, timing synchronization also needs to be consid-ered when performing channel estimation. Timing synchro-nization error causes both carrier and time dependent phaserotations [209]. Therefore, the single pilot tracking used forcommon phase rotation is not sufficient to compensate for thetiming synchronization error. The compensation for this caseneeds at least two OFDM subcarriers to be tracked both infrequency and time domains so that the slope of variation ofthe phase rotation is determined [209]. For efficient systemutilization, time and frequency synchronization and channelestimation can performed jointly [210, 211].

ICI Due to Fast Fading Channel — When the CIR taps varyover the duration of OFDM symbols, for an accurate channelestimation, the CIR taps values corresponding at each sam-pling instance need to be obtained so that the correspondingCFR is estimated. As mentioned earlier, this implies anunderdetermined set of equations as the number of unknowns

is more than the number of equations.In order to reduce the number of equations, the CIR taps

corresponding to each time sample of the OFDM symbol canbe correlated via some basis functions. The knowledge of CIRtaps at couple of sampling points can then be sufficient toestimate the CIR taps at the other time instances. In this case,a set of reduced CIR parameters, Ξr, can be related to thecomplete CIR, Ξ, parameters as [212],

Ξ = QΞΞr (72)

where QΞ is the interpolation matrix. Different approachesare studied for the CIR taps evolution over the time. Themost frequently used method is to assume CIR taps varyinglinearly [82, 212, 213]. Moreover, interpolation via low-passfiltering can be utilized for a better estimate in time selectivechannels [205]. If the CIR taps follow the Jake’s channelmodel [214], taps variation then follows first-kind zero-orderBessel function [212]. In this case, the parameters of theBessel function can be found by locating its first zero crossingvia the examination of the subcarrier correlation evolutionover time. At the expense of more computation, the CIR tapscan be modelled as an AR process [215], whose coefficientcan obtained from the channel statistics.

Some studies tried to model the ICI as AWGN and appliedthe methods which give good performance under AWGN[189]. In one of such studies, 1-D and/or 2-D LMMSE isemployed in the channel estimation of OFDM in the presenceof ICI [149, 216]. It is observed that since ICI increases thenoise level, the number of pilot subcarriers required for thesame MSE performance of no ICI case increases by a signifi-cant amount. Hence, one way of compensation of ICI is toincrease the number of pilots in the frequency domain.

In fact, when the singular values of the auto-covariance ofCFR under the presence of ICI and noise is analyzed, it canbe observed that the singular values can be grouped underthree categories. The first group will have L number of similarsingular values with L being the number of significant CIRtaps. The second group will have I (I < K) number of similarsingular values, and the rest of the singular values correspondto the AWGN. In these grouping, the values of singular valuesare largest for the first group and smallest for the last groupassuming that the CIR taps are not buried in the noise. Figure11 shows this scenario. When a low-rank LMMSE is to beperformed on a system with ICI, then most of the ICI will becancelled except those overlapping with the singular values

S m nm n

KK

m ne

p

p

p

j m nε

ππ επ

ε( , )

sin ( )

sin ( )

(=− +

− +

− +εε p ).

nFigure 11. Classification of singular values representing differ-ent system parameters.

L

CIRtaps

ICI

Noise

K

I

Transform domain

Rela

tive

sin

gula

r va

lues

str

engt

h

that correspond to the CIR taps. Still, the use of low-rankLMMSE would give a low MSE since it eliminates most of thesubspace corresponding to the ICI. Hence, LMMSE is usedwidely in the channel estimation of OFDM with ICI [33, 148,149, 217].

For the OFDM channel estimation using transform domaintechniques, the information about the channel length is there-fore very important in order to reduce the effect of ICI. Witha known CIR length, it is observed that the use of transformdomain techniques reduce the ICI significantly [113, 130].However, efficient methods obtaining the CIR length need tobe developed. In addition to the methods described in trans-form domain techniques about the CIR tap identification,similar methods are proposed when ICI exists. For example,the CIR length under the presence of ICI is found iterativelystarting from a longer CIR length than expected [25]. Similar-ly, the channel length is obtained by correlating the first twoshort OFDM symbols in the preamble of the WLAN systemswith the local short symbols [80, 167]. In this correlation pro-cess, similar to the methods using PN sequences, the channeltaps are revealed, so is their length.

Pilot Spacing in the Presence of ICI — In the previous sub-section, it was mentioned that in the presence of ICI morepilot subcarriers are needed for an acceptable performance. Ifthe number of pilots is to be increased, then it is more appro-priate to place the additional pilots next to the existing onessince the ICI is more severe in the adjacent subcarriers (Fig.12). Bearing this observation in mind, a small subset of sub-carriers are considered to be responsible for the ICI in a sub-carrier within the group [218]. Simulation results show thatsuch pilot arrangement improves the channel estimation per-formance significantly. Similarly, in early studies two out ofphase adjacent subcarriers were employed as the pilots to mit-igate for the effect of ICI [91].

Instead of finding the optimum pilot locations via simula-tions, for a frequency selective channel, theoretical approach-es are carried out for the pilot placement under the presenceof ICI [205, 213]. The approaches showed that in the presenceof ICI the pilots should be all grouped for the optimum elimi-nation of ICI. However, for a frequency selective channel thiswould not sample the CFR appropriately, and hence perfor-mance degradation would occur for the frequency selectivechannels. In order not to have degradation for the frequency

selective channels, the clustered pilot scheme is offered to bethe optimum solution. In this scheme, the group of pilotswould be equispaced over the OFDM symbol. This theoreticalfinding is nothing but the solution found via simulations in[91, 218].

The need for the clustering can be explained as follows.When the CIR taps vary over the OFDM symbol, they needto be sampled frequent enough in time domain so that thecorresponding CFR can be obtained. For example, if uniformtime domain pilots are employed, then their Fourier Trans-form would give concentrated pilots in the transformeddomain. In fact, when all the time domain samples of OFDMare assigned to be pilots, then their Fourier Transform wouldgive an impulse in the frequency domain. Hence, in order tocompensate both time and frequency domains channel varia-tion, the pilots needs to be grouped and then uniformly dis-tributed in the frequency domain [82].

The analysis performed for the channel estimation of ICIdemonstrates that the performance improvement can beachieved with the information of channel statistics. This iseither needed for the optimum pilot allocation and the low-rank LMMSE or the transform domain techniques intendedfor the low-pass filtering.

OFDM CHANNEL ESTIMATION WITHEXTERNAL INTERFERENCES

The channel estimation techniques presented in the previoussections treated the interference from other systems orsources to be part of the AWGN. As long as the interferenceis like AWGN, the methods described in the preceding sec-tions can be utilized safely as they are mostly developed forthe AWGN. However, OFDM systems can suffer from theimpulse noise, which completely destroys the information car-ried over the subcarriers [219, 220]. In such circumstances,instead of trying to estimate the channel at the subcarriers viathe sent data, the estimates at the impulse-free pilot subcarri-ers can be utilized. Based on the channel selectivity, a numberof good estimates at the neighborhood of the destroyed sub-carriers can be used both in time and frequency domains, andusing the past and future estimates. The pilot subcarriersaffected by the impulse noise can be detected by looking attheir energy level, as their energy will be much higher in thepresence of impulse noise.

Similarly, the performance of OFDM channel estimation isinvestigated in the presence of narrowband interference [34]. Bymodeling the narrow-band interference in frequency domain asa complex Gaussian variable, an overall noise term including thenarrow-band interference with a modified variance can beobtained. With the use of a generalized ML estimation, that is,M-estimation method, results better than those not accountingfor the narrowband interference can be obtained when narrow-band interference exists in the system [34].

OFDM channel estimation is also performed for the syn-chronous and asynchronous interference where the noise termin OFDM system model is defined as [221, 222],

(73)

where Ni is the number of interferers, and Iq[n, k] is the qth

interference, which can be synchronous or asynchronous inter-ference. It is assumed that for the synchronous case the inter-ferers’ CPs are aligned with the user’s CP, while forasynchronous case the CPs are not aligned with the user’s CP.A ML estimation algorithm can be applied but the secondorder statistics of the interferers are needed. Efficient non-

′ = +=∑W n k W n k I n kqq

Ni

[ , ] [ , ] [ , ]1


nFigure 12. Typical four orthogonal OFDM subcarriers. Notethat sampling at the incorrect points leads significant interfer-ence.

Subcarriers

-2

-0.2

Cha

nnel

coe

ffic

ient

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8


iterative algorithms are developed for this purpose, and aretested through the simulations successfully [221, 222].

MIMO-OFDM CHANNEL ESTIMATION TECHNIQUES

MIMO-OFDM channel estimation is a challenging task as thereceived signal is the superposition of the signals from multi-ple transmit antennas, (Eq. 12). For the methods to be pre-sented in this section, the ICI and other types of interferenceare folded into the AWGN term for the sake of simplicity.The MIMO-OFDM system model then becomes,

(74)

With the introduction of MIMO, the pilot arrangement hasto be modified so that the existing multiple channels can beestimated. In the subsequent subsections, first pilot allocationfor MIMO-OFDM, and then the corresponding techniqueswill be presented.

PILOT ALLOCATION IN MIMO-OFDM SYSTEMS

When MIMO-OFDM started to draw attention in wirelesscommunication area, pilot allocation schemes that convert thechannel estimation of MIMO-OFDM into the channel estima-tion of SISO-OFDM are proposed widely. In these pilotschemes, at a given pilot scheme, only one of the transmitterantennas sends its pilot signal at a given subcarrier while theothers remain silent [72, 158]. Such a pilot scheme is shown inFig. 13. WiMAX systems also use a similar pilot scheme thatis suitable for two antenna case [9].

With the pilot scheme given in Fig. 13, it can be seen fromEq. 74 that the MIMO-OFDM received signal at the pilotsubcarriers for a given receive antenna is reduced to

Yrx[n, k] = Hrxtx[n, k]Xtx[n, k] + Wrx[n, k] (75)

where k ∈ Ptx with Ptx holding the pilot subcarrier indices forthe transmit antenna tx. With the pilot subcarrier of eachtransmit antenna being disjoint, the received signal for disjointpilot subcarrier indices results in as many SISO-OFDM equa-tions as the number of pilot subcarriers. From that point on,the methods described in the previous sections can be appliedfor the channel estimation. For example, Transform domain

methods are successfully applied in MIMO-OFDM systemsusing pilots as in Fig. 13 [52, 104].

For SISO-OFDM systems, there was an upper bound onthe pilot spacing that the pilot spacing should not be too largeto cause an undersampled CFR function. For MIMO-OFDMsystems using the pilot schemes given in Fig. 13, a lowerbound is dictated so that the interference from the otherantennas is eliminated. The pilot spacing, Dp, is then

Ntx ≤ Dp ≤ K/L. (76)

If WLAN standards are to be employed in a MIMO sys-tem, then the pilot allocation in two of the long OFDM sym-bol in the preamble for the channel estimation can bedesigned for a better performance [12]. Since in a typicalWLAN environment, the channel varies very slowly, it can beassumed that the channel is constant over training OFDMsymbols [103]. In this case, the pilots arranged for the firstOFDM symbol can be cyclically shifted so that the CFR issampled uniformly at more points [54, 103, 223, 224]. Such anarrangement can also mitigate for the edge subcarrier errorssince each antenna can transmit at least one pilot subcarrierclose to the edge subcarriers. Figure 14 shows this scenariofor Ntx = 4. In general, the pilots can be cyclically shifted byNtx/NO,where NO is the number of OFDM symbols over whichthe channel is assumed to be constant. With the assumptionof the channel being constant over the training phase, noisereduction can also be achieved via averaging [12]. During theaveraging, better performance can be achieved if the channelsamples are weighted according to their MSE performance ortheir noise [54. 91].

Although the comb-type pilots given in Fig. 13 for MIMO-OFDM symbols simplify the channel estimation process, theyintroduce some drawbacks. Clearly, they reduce the spectralefficiency since many of the subcarriers are assigned to pilots,with most of them being the silent pilots. Moreover, the useof silent pilots increases the PAPR [60], a critical parameterfor the power amplifier block in the transmitters. Hence, incontrast to the pilot scheme in Fig. 13, the transmission of thepilots for the same set of subcarriers are proposed (Fig. 15).When CFR estimation is to be performed over such a pilotarrangement, Ntx * Np unknowns are at present with only Npequations being available. Hence, instead of direct CFR esti-mation, CIR estimation of each MIMO channel is proposed.A receiver antenna then needs to estimate Ntx CIR, eachassumed to have the same L. It should be noted that this is avalid assumption for MIMO downlink as the transmit andreceive antennas are co-located and hence are expected to

Y X H Wrx txtx

N

rxtx rxdiagtx

= +=∑ ( ) ,

1

nFigure 13. Typical pilots for MIMO-OFDM.

FrequencyPilot subcarriers Data subcarriers

Spac

e

nFigure 14. Cyclically shifted pilots for MIMO-OFDM systems..

Pilots Cyclically shifted pilots

Ant #1

Ant #2

Ant #3

Ant #4

Subcarriers


have the same channel PDP [3, 225]. With this, the number ofunknowns that a receiver antenna has to estimate is Ntx * L.In CIR estimation, the pilots in the frequency domain nowoccupy the same subcarriers, and hence there is interference.Figure 15 shows the overlapped pilots.

By assuming that the CFR for a given transmit antennacan be expressed as in Eq. 5, minimization of MSE in Eq. 74with respect to CIR coefficients for a given receive antenna rxcan be expressed as [77, 226],

(77)

where Xtx and Xv are the pilot subcarriers for the txth and vth

transmit antennas with v = 1,…, Ntx. By rearranging theterms, at the time instant n in matrix notation

Qh = p (78)

or

h = Q–1p. (79)

Here, the entries of the h, Q, and p are expressed as

(80)

where hrxtx is given as in Eq. 3.

(81)

where

(82)

and

(83)

Finally, p can be expressed as,

(84)

where

ptx = (ptx[n, 0], ptx[n, 1], …, ptx[n, L])T (85)

and

(86)

As can be seen, the dimension of the Q matrix is (NtxL ×NtxL), meaning more computational complexity with increas-ing number of transmit antennas and the CIR taps. This alsoimplies that the knowledge of the CIR length is critical inattaining a low complex and more accurate channel estima-tion. It should be also noted that the entries of Q representsome form of cross-correlation between the pilot subcarriers,while those of p represent some form of cross-correlationbetween the received signal and the pilot subcarriers. Differ-ent approaches are proposed in order to eliminate the compu-tational complexity associated with the calculation of the Q–1.One of the approaches is the use of constant modulus signalsfor the pilot subcarriers so that the diagonal submatrices of Qare identity matrices [38, 51]. The matrix inversion is theneliminated by using a single step iterative approach that uti-lizes the previous channel estimates. Although this approachis less complex, performance degradation incur in such anapproach in fast-fading channels. When the cross-correlationbetween the pilot subcarriers is zero, that is, orthogonalsequences, then further simplification arises in the channelestimate as the need for the previous channel estimate is elim-inated [38, 51].

Based on the MSE of LS estimates, the pilots for differenttransmit antennas need to be phase orthogonal in addition tobeing equispaced and equipowered for a given minimum num-ber of pilot tones and power [6]. For this purpose, Hadamard[67, 103, 227, 228], Golay [64], and exponential type [49, 60,63] orthogonal sequences are used in many studies. Hadamardtype codes have good auto and cross correlation characteris-tics and therefore are popular in communication systemsrequiring orthogonality. The Golay codes are found to yieldlower PAPR [64]. The exponential type pilots do not onlyintroduce orthogonal codes, but also simplify the channel esti-mation process [6, 38, 51]. The exponential type pilots arefound to be optimum pilots and are given by

(87)

where ttx represent the pilots for the txth antenna.A close look at the exponential type pilots reveals some

interesting behavior of the pilots when CIR is to be estimated.It is known that a phase shift in frequency domain corre-sponds to a time shift in the time domain. Hence, the phaseof the pilots can be modified in such a way that when theirIFFT is taken, their equivalent CIR representation is delayedin time domain. A careful design of the pilots can put the CIRparameters in distinct positions in time domain so that the

t n k t n k etx

jkL

K[ , ] [ , ]= 1

2π

p n l Y n k X n k Ftx rxk

K

tx Kkl[ , ] [ , ] [ , ] .*=

=

−−∑

0

1

p

p

p

p

=

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

1

�

Ntx

q n l X n k X n k Ftx tx txk

K

tx Kk

1 2 1 20

1[ , ] [ , ] [ , ]*=

=

−−∑ ll .

Qtx tx

tx tx tx tx

tx

q n q n L

q1 2

1 2 1 2

1

0 1

=

− +[ , ] [ , ]�

� � �

ttx tx txn L q n2 1 2

1 0[ , ] [ , ]−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥�

Q

Q Q

Q Q

=

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

11 1

1

�

� � ��

N

N N N

tx

tx tx tx

ˆ

ˆ

ˆ

h

h

h

=

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

rx

rx Ntx

1

�

Y n k h n l F X n krx rxtx Kkl

txl

L

tx

[ , ] ˆ [ , ] [ , ]−=

−

=∑

0

1

1

NN

k

K

Kkl

v

tx

F X n k

∑∑⎛

⎝⎜⎜

⎞

⎠⎟⎟

=

=

−

−

0

1

0*[ , ]

nFigure 15. Overlapping pilots for MIMO-OFDM.

Data subcarriersPilot subcarriers

Time

Spac

eFr

eque

ncy


CIR corresponding to each transmit antenna can be separatedeasily. This property is initially proposed in [6] and [51], andwas later investigated by Auer in different studies [229–231].It is concluded that such pilot schemes indeed provide accu-rate channel estimates when the channel is sample spaced. Itcan be seen in these approaches that for the separation of allthe CIR taps belonging to different transmit antennas, eachCIR tap needs to correspond to a distinct time position, whichsuggests thatNtxL ≤ K. The above idea can be extended toSISO-OFDM systems such that exponential type pilots resultin multiple replicas of the CIR channel in the time domainsignal. These replicas can be averaged in time domain to getbetter estimates [128, 232].

Shifting the phase of the pilots works very well in the sam-ple spaced channels, however, significant performance degra-dation can occur when the channel is not sampled spaced. Inthis case, the paths interfere with each other, and the methodsthat can separate different taps will be needed. Windowingoperation and IPIC-DLL methods studied for single antennasystems can be applied to compensate for the aliasing occur-ring due to non-sample spaced taps [117]. Moreover, Wienerfiltering can be applied in time domain estimates for the sepa-ration of the CIR taps [233].

The CIR channels estimated via Eq. 79 can be furtherimproved if it is passed through an optimum filter. An opti-mum filter coefficient however requires the information aboutchannel PDP. Since in MIMO systems, the existence of multi-ple channels introduces multiple replicas of the same PDP, aquick and more accurate estimation of PDP can be obtainedfor the use in optimum filtering [225].

The space-time and space-frequency codes are also utilizedin the channel estimation of the MIMO-OFDM systems.Before going into the details of these pilots scheme, it isworthwhile to visit the Alamouti type coding that pioneers thespace coding [234]. Starting from two antenna case, the Alam-outi schemes transmits two different signals at the same timeinstances. In the next time instance, a modified version ofthese signals are transmitted from the other antenna. This waytransmitter diversity is achieved both in time and space. Thesetransmitted symbols are called Alamouti codes that are moregenerally termed as STBC. For two transmit antennas, thesecodes are given by,

(88)

where * represent complex conjugate. For the transmittedsymbols to be estimated, the channel need to stay constant byas many OFDM symbols as the number of transmitter anten-nas. Then, the channel at the subcarriers can be obtained viaa sufficient set of linear equations, and can be furtherimproved via enhanced techniques such as Wiener filtering[135, 136]. When the channel is not constant by as many asthe number of transmitter antennas, then this scheme sufferssignificant performance degradation. The Alamouti scheme ismostly investigated for two antenna schemes [10, 11, 135]although it can be generalized for more antennas.

The use of Alamouti codes is mostly applied to the OFDMsubcarriers in frequency domain, resulting in SFBC [128, 235].SFBC’s eliminate the need for the channel to stay constant byas many OFDM symbols as the number of transmit antennasbut requires the channel in frequency domain to be constantby as many subcarriers as the number of transmit antennas. Itcan be observed that when the codes are applied to the sub-carriers over several OFDM symbols, then the diversity due tothe Doppler spread is utilized. In the case of SFBC, the diver-

sity due to the delay spread is exploited [35]. In the applica-tion of space-frequency Alamouti coding, a group of subcarri-ers by as many as the number of transmit antennas areassigned to a group of Alamouti codes. The key assumption isthat the CFR is constant over the group of the subcarriers.Such a scheme results in Ntx equations with Ntx unknowns perCFR for each subcarrier block.

Depending on the system environment either STBC orSFBC coding scheme can be used. When the length of CIR isvery small, then the use of SFBC can result in a good perfor-mance since the assumption of constant CFR over a numberof subcarriers holds. However, for more frequency selectivechannels since the assumption of the constant channel nolonger holds, performance degradation will result in. In thiscase, if the channel is less time selective, then the STBC canbe applied in time domain.

Similar to the SFBC, by assuming that the channel is con-stant by as many subcarrier as the number of transmit anten-nas, the pilot sequence after the IFFT of a transmit antenna isshifted by K/Ntx, and CP is added thereafter as shown in Fig.16 [236, 237]. With such a scheme, only one block of IFFTcan be used instead of Ntx IFFT blocks.

The shift by K/Ntx results in the symbols with differentphase shifts in the frequency domain, which are used to sepa-rate the channel for each transmit antenna. Considering thetwo transmit and one receive antenna system, the received sig-nal can be written as,

Y[n, k] = [H11[n, k] + H12ejkπ] X[n, k] + W[n, k] (89)

where X[n, k] is the only pilot symbol used for both anten-nas. With the assumption that the CFR is constant by asmany subcarriers as the number of transmitter antennas, Eq.89 can be written for two consecutive subcarriers, with twounknowns H11[n, k] and H12[n, k], which can be solved withtwo equations. As can be seen such an approach is nothingbut some special version of the SFBC. This approach is alsosimulated for many transmit antennas, and it is observedthat as long as the channel is not too frequency selective,then the performance of the estimation is acceptable [238,239]. Similar to Alamouti coding, rate-one non-orthogonalspace-time codes based on Hadamard codes are found togive accurate channel estimation with the latter offering lesscomplexity [240].

MIMO-OFDM WITH SPATIAL CORRELATION

The use of multiple antennas in OFDM systems brings anoth-er dimension: spatial dimension. As with the frequency andtime domains correlation, spatial domain correlation can alsobe exploited in the channel estimation of MIMO-OFDM sys-tems. With uncorrelated CIR taps, the spatial correlationbetween the subcarriers having the same indices is just thespatial correlation between the antenna elements [37].LMMSE filtering can be applied to the subcarriers across thespace. It is observed that the use of spatial correlation canprovide additional gain when the correlation is beyond 0.8 as

S =−⎡

⎣⎢⎢

⎤

⎦⎥⎥

s s

s s

1 2

2 1

*

*

nFigure 16. Transmitter diversity with shifted pilots after IFFTby the amount K/2.

Basebandmodulator IFFT CP

CPDK/2


shown in Fig. 17. The use of spatial correlation is also investi-gated via Kalman filtering approach for the channel trackingin time domain [164, 166]. The studies showed that in thepresence of spatial correlation channel tracking can still beperformed with the state equations incorporating the effect ofspatial correlation. In addition to these studies, spatial corre-lation is also found to improve the channel estimate of MIMOsystems via a pre-filtering in time domain [98], where a timedomain LMMSE channel estimation is exploited.

CONCLUSION

In this article, we present the most common methods appliedin the channel estimation of SISO and MIMO-OFDM sys-tems. The SIMO and MISO systems are not covered separate-ly as the methods for SISO and MIMO can be easily modifiedto be applicable to SIMO and MISO systems. Throughout theanalysis it is seen that there are three basic blocks affectingthe performance of the channel estimation. These are thepilot patterns, the estimation method, and the signal detectionpart when combined with the channel estimation. As in manysystems, each block can promise an improved performance atthe cost of additional resources. Hence, the best combinationof these three parameters depends on the typical application[60, 241]. Although the estimation techniques presented inthis article are shown to be a subset of LMMSE channel esti-mation technique, instead of promoting one of the channelestimation techniques, the methods are presented for the sce-narios they perform the best. Thus, a fully adaptive system canbe developed by using each block when necessary.

FUTURE DIRECTIONS

With OFDM now standing as a solid technology for futurewireless systems, OFDM channel estimation techniques canbe improved by incorporating the features of new technolo-gies. It is well-known that one of the promising technologies isMIMO. However, channel estimation methods studied forMIMO-OFDM systems mostly overlook the effect of ICI dueto high speed mobile and external interferers. The modelsthat approximate ICI and external interferers as AWGNmight simplify the estimation process but better results can beobtained by developing more accurate modelings.

Moreover, the standards such as WLAN and WiMAX donot use certain subcarriers known as guard subcarriers. Theuse of transform domain techniques do not provide better

performance with guard bands since transform domain tech-niques introduce CIR path leaks due to the suppression ofunused subcarriers. Methods can be developed to eliminatethe leakage problem by extrapolating the channel for theunused subcarriers, followed by a transform domain tech-nique. Such an approach can reduce the path leaking signifi-cantly. An elegant combination of an extrapolation methodand a transform domain technique can be developed so that apractical estimation method can be realized for WLAN orWiMAX systems.

As adaptation is key to many systems, channel estimationtechniques can be made adaptive by using the informationfrom other physical layer blocks. For example, the informa-tion available at blocks such as timing offset estimation, fre-quency offset estimation, and the output of the decoder canall be used to determine the most appropriate channel estima-tion technique.

Lastly, mobile version of WiMAX uses OFDMA in itsuplink direction. The subcarriers in a given OFDMA symbolare distributed among different users based on a given tilestructure and subchannels [177]. The pilot subcarriers for dif-ferent tiles are no longer adjacent and the subcarrier spacingbetween tiles can vary. Although linear interpolation can easilybe used for the channel estimation, utilization of long termchannel statistics can improve the channel estimation perfor-mance. With the tile assignment changing continuously duringthe uplink transmission of OFDMA, the application of theexisting OFDM channel estimation methods is not straightfor-ward. Research can be performed on how to practically incor-porate long term channel statistics on the uplink channelestimation of OFDMA systems for a better performing system.

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BIOGRAPHIES

KEMAL OZDEMIR ([email protected] )received the B.S. andM.S. degrees in electrical engineering from Middle East TechnicalUniversity, Ankara, Turkey in 1996 and 1998, respectively, and thePh.D. degree in electrical engineering from Syracuse University,Syracuse, NY in 2005. He has worked for Philips between 2000and 2003 and Triverity Inc., between 2004-2005. He was a visitingscholar at University of South Florida between 2003-2004. Current-ly, he is with Logus Broadband Wireless Solutions working onfixed/mobile WiMAX base stations. His research interest are thedevelopment of Signal Processing algorithms and their efficientimplementation on FPGA's, development of MAC algorithms, andthe signal integrity issues for the next generation wireless systems.

HUSEYIN ARSLAN [SM] ([email protected]) has received his PhD.degree in 1998 from Southern Methodist University (SMU), Dallas,Tx. From January 1998 to August 2002, he was with the researchgroup of Ericsson Inc., NC, USA, where he was involved with sev-eral project related to 2G and 3G wireless cellular communicationsystems. Since August 2002, he has been with the Electrical Engi-neering Dept. of University of South Florida. He has alsobeenworking for Anritsu Company, Morgan Hill, CA (as a visiting pro-fessor during the summers of 2005 and 2006) as a part-time con-sulting since August 2005. His research interests are related toadvanced Signal Processing techniques at the physical layer, withcross-layer design for networking adaptivity and Quality of Service(QoS) control. He is interested in many forms of wireless tech-nologies including cellular, wireless PAN/LAN/MANs, fixed wirelessaccess, and specialized wirelessdata networks like wireless sensorsnetworks and wireless telemetry. The current research interestsare on UWB, OFDM based wireless technologies with emphasis onWIMAX, and cognitive and software defined radio. He has servedas technical program committee member, session and symposiumorganizer in several IEEE conferences. He is editorial board mem-ber for Wireless Communication and Mobile Computing journal,and was technical program co-chair of IEEE wireless andmicrowave conference 2004.

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