Channel Estimation Lmmse

8/4/2019 Channel Estimation Lmmse

1/31IEEE 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 for

future high data rate systems [1012].

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 34 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 by

undergoing many detrimental effects that corrupt the signal

D

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

HUSEYIN


2/31IEEE 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 to

be 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[2124]. This paper focuses on an extensive overview of thechannel estimat0237 Tc0.0263astatk0f thlivei4er



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. The

specific 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; 0ab denotes a

matrix ofa b

with zero entries; IN denotesNN

identitymatrix; andj=1.

SYSTEM MODEL

A generic block diagram of a basic baseband-equivalentMIMO-OFDM system is given in Fig. 2. A MIMO-OFDMsystem withNtx transmit andNrx 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)

whereXtx[n, k] is the data at the kth subcarrier of the nthOFDM symbol,Kis the number of subcarriers, andm 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)

whereL is the number of taps, l is the lth complex path gain,and l is the corresponding path delay. The path gains are

WSS complex Gaussian processes. The individual paths can be

h t tl ll

L

( , ) ( ) ( ), = =

0

1

x n m IFFT X n k

X n

tx tx

tx

[ , ] { [ , ]}

[ ,

=

= kk e k m K k

K j mk K

] ,/

=

0

1

20 1

n Figure 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

Outputbits

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 = ej2/K, Tf is the symbollength including CP, f is the subcarrier spacing, and ts = 1/Dfis the sample interval. In matrix notations, for thenth 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)

whererx =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 notethatn represents OFDM symbol number, whilem denotes thesampling index in time domain so that hmrxtx[n,l] is the CIR at

the sampling time indexm for the symboln. When the CIR isconstant over an OFDM symbol duration, then hmrxtx[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 ofw2. After taking FFT of the timedomain samples of Eq. 7, the received samples in frequencydomain can be expressed as,

(8)

(9)

( (10)

whereIrx[n,k] and Wrx[n,k] are the corresponding frequencydomain components calculated from irx[n,m]s andwrx[n,m]s,

respectively. After arranging the terms, and representing the

variables in matrix notation, forrxth receive antenna andnthOFDM symbol, we get

(11)

(12)

Here, Yrx is column vector storing the received signal at eachsubcarrier, F is the unitary FFT matrix with entries

ej2mk/KKwithm andk being the row and column index and

= FrxtxFH, 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 rxtx isthe matrix containing the channel taps at eachm index. Theentries of are given by

(13)

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

hmrxtx[n, l] is the same for all ms, making rxtx a circulantmatrix [35]. The multiplication ofFrxtxFH 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. 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 eachm 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 areKL, whereas the number of unknowns in frequencydomain (the entries of) areK2. 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.

rxtx

rxtx

rxtx rxtx

h n

h n h n

=

0

1 1

0

1

0

0

0

0

[ , ]

[ , ] [ , ]

h n L h n LrxtxL

rxtxL

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[ , ] [ , ]

.

= + +

=X I W

tx rx rxtx

Ntx

1

.

Y F F X I Wrx rxtxH

tx

tx

N

rx rx

tx

= + +

=

1

,

=

=

1

0

1

2

Kx n k etx

k

K j m l

[ , ]( )) /

=

=

=

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[ , ] [ , ] [

+ +

2

,, ]k

+ + ]

i n m w n m erx rxj

km

K[ , ] [ , ]

2

Y n kK

y n m erx rxj

km

K

m

K

[ , ] [ , ]=

=

12

0

1

= =

=

10

1

1K

x n m l h n ltx rxtxm

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 d j f( , ) ( , ) .=

+

2



OFDM CHANNEL ESTIMATION TECHNIQUES

There 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)

whereH^[n,k] is the estimate of equivalent channel at kth sub-

carrier ofnth 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, is a diagonal matrix, where eachdiagonal element represents the channel between the corre-sponding received and the transmitted subcarriers. In thiscase, for thenth OFDM symbol, the channel given in Eq. 5 at

each subcarrier can be related to

as

H[n,k] = [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 to

the delay spread. According to the Nyquist sampling theorem,

n Figure 3. Typical training symbols and pilot subcarriers arrangement.Time

(a)

Training symbols

Freque

ncy

Data symbolsTime

Pilot subcarriers

(b)

Freque

ncy

Data subcarriers



the number of subcarrier spacing between the pilots in fre-

quency domain,D

p, must be small enough so that the varia-tions of the channel in frequency can be all captured, that is,

(18)

where max 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 unnecessary

pilot 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 the

channel 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 extrapolation

for 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 [5658]. 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 frequency

domains. 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

D fd Tf

t 1

2 max

Ddf

p 1

max

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

Subcarrier index

1000.3

0.4Correlationcoefficient(abs)

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 over

several 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 [6163]. 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 different

subcarriers 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 1s 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 1s 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 [6870]. 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 priori

information, 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 ofH[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)

whereEH=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^ = QLSFHdiag(X)HY (25)

where

QLS = (FHdiag(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 SNRLS

H

=

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 ofL channel taps andNp num-ber of pilot subcarriers, the ML estimate of the channel coef-ficients is shown to be [58, 78],

H^

ML = (FpHFp)1FpHdiag(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 hence

diag(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, 2830, 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 using

the 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 training

symbols [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^ = QHLS (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

whereDp 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 0

D

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 [8689]. 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/ =Dp,Rf[l] is the frequency domain correlation ofCFR, denotes the real part of a complex number, and w2 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 MSE

results 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 Beizer

curve 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)

whereNp is the number of pilots over a range,Bp[n,k] is thebasis function at subcarrier k, andH[n,p] is the CFR at thepilot locationp. 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 ofQ 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 or

equivalently 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 an

adaptive 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 more

CFR samples.

H n k B n k H n ppp

Np

[ , ] [ , ] [ , ],=

=

1

MSE Rf w= + + + +

1

35 1 0

1

32

4 1

2 2( )( ) [ ] ( )

(

111

31

0

12

) { [ ]} ( ) { [ ]} + =

R l R Dfl

D

f p

p



The low pass interpolation utilizes the extra information

about 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 L, by truncating to the sizeK Np to form Np, and Q to the size ofNp Np to form

QNp. Then,

(63)

(64)

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

H^

LMMSE = PQNH

pH^

LS, (65)

where the entries of the diagonal matrix are given by,

(66)

In case of low-rank approximation, onlyrsignificant singu-lar values ofRhhwill be considered. Then,

HrLMMSE = PrGr(Qr)HH^LS, (67)

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

For high SNR, approaches to a diagonal matrix withdiagonals being

Dp. Moreover, when the CIR taps are uncor-

related and there are onlyL number of significant taps, thenRhh is a diagonal matrix. In this case, Vmatrix becomes anidentity matrix, making P and Q matrices simply F and Fp,respectively. Moreover, the SVD ofRhh results inL numberof significant singular values, makingr=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 WITH

INTERFERENCE

So far the effect of ICI, ISI, and external interferers were

ignored, 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= FFHX + W, (69)

where the external interferers are folded into the AWGNterm. Here, since CIR is not constant over the OFDM sym-bol, is not a circulant matrix anymore. Hence, the productofFFH is not a diagonal matrix [205]. In this case, a receivedsignal at a subcarrierk is affected by the transmitted signals ofall the subcarriers, increasing the number of unknowns by

K*(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 long

symbol 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 H FT p pH

LSD= .

i

ip

pi

p

K

D

K

D SNR

i N=

+

= , , , , .0 1 1

R Q Q H H pp

p NH

p p pN

K

DN= .

R P QHHp

p NH

p p

K

DN= ,

=

L L K L

K L L K L K L

|

|

|

( )

( ) ( ) ( )

0

0 0

,



quency offset can be expressed as [208]

Y= Spdiag(X)H + W, (70)

where Sp is the interference matrix representing ICI due tothe normalized frequency offset p. 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 an

underdetermined 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 tapscorresponding 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, r, can be related to thecomplete CIR, , parameters as [212],

= Qr (72)

where Q 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 selective

channels [205]. If the CIR taps follow the Jakes 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 haveI(I


21/31

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 selective

channels. 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 the

corresponding 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 intended

for the low-pass filtering.OFDM CHANNEL ESTIMATION WITH

EXTERNAL 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)

whereNi 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 users CP, while forasynchronous case the CPs are not aligned with the users CP.A ML estimation algorithm can be applied but the second

order statistics of the interferers are needed. Efficient non-

= +=

W n k W n k I n k qq

Ni

[ , ] [ , ] [ , ]

1

IEEE Communications Surveys & Tutorials 2nd Quarter 200738

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

ence.

Subcarriers

-2

-0.2

Channelcoefficient

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)

wherek Ptxwith 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 applied

for 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 first

OFDM 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 by

Ntx/NO,whereNO 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 and

receive antennas are co-located and hence are expected to

Y X H Wrx txtx

N

rxtx rxdiagtx

= +

=

( ) ,1

n Figure 13. Typical pilots for MIMO-OFDM.FrequencyPilot subcarriers Data subcarriersSpace

n Figure 14. Cyclically shifted pilots for MIMO-OFDM systems..Pilots Cyclically shifted pilots

Ant #1

Ant #2

Ant #3

Ant #4

Subcarriers


23/31



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 [229231].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, which

suggests 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 of

these 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].SFBCs 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 to

the 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 numbe r 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 withNtx 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 number

of 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 byK/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 ofNtx 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] + H12ejk]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 can

provide additional gain when the correlation is beyond 0.8 as

S =

s s

s s

1 2

2 1

*

*

n Figure 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 tracking

in 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 modified

to 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. The

use 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 a

practical 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 spacing

between 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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