An Iterative Channel Estimator for Fast-Varying Channels Using Successive OFDM Symbols

8/11/2019 An Iterative Channel Estimator for Fast-Varying Channels Using Successive OFDM Symbols

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An Iterative Channel Estimator for Fast-Varying

Channels Using Successive OFDM Symbols

Youssef El Hajj Shehadeh(1)(2)and Serdar Sezginer(1)Sequans Communications, Paris, France

(1)

TELECOM PARISTECH/University of Paris VI, Paris, France(2)

{yelhaaj,serdar}@sequans.com

AbstractThis paper deals with pilot-based channel estimation

for fast varying channels in Orthogonal Frequency Division

Multiplexing (OFDM) systems. Due to the variation of the

channel during one OFDM symbol, one-tap channel estimation

and the corresponding equalization is no longer the optimum

solution and such a variation results in inter-carrier interference

(ICI). One of the possible approaches to predict this interfering

effect is to use basis expansion model (BEM) with which thevariation of the channel can be approximated successfully.

However, as the pure BEM does not solve completely the

problem, we investigate the estimation problem using decision

feedback to enhance the performance. In particular, we propose

a simple algorithm based on using two successive OFDM symbols

to filter channel coefficients and improve not only the

convergence of pure decision feedback based estimation but also

the system performance. Simulation results based on Jakes

channel model with a high Doppler spread and a practical high

data rate system sustain our claims.

Index Termsbasis expansion model (BEM), channel

estimation, decision feedback, fast-variant channel, OFDM.

I. INTRODUCTION

Orthogonal frequency-division multiplexing (OFDM) isone of the major transmission techniques used to achieve highdata rates over wireless mobile channels. Indeed, OFDM basedmultiple access techniques have been included in manywireless communications system specifications such as IEEE802.11 standard for local area networks, IEEE 802.16 for

broadband wireless access systems, and 3GPP E-UTRA formobile wireless access systems.

In OFDM, the available channel bandwidth is divided intoNoverlapping narrow banded subchannels. The serial high-ratedata stream is thus converted into N parallel low-rate

substreams, which are modulated onto the N orthogonalsubcarriers. A cyclic prefix (CP) is inserted before eachtransmitted data block. If the length of the cyclic prefix is equalto, or longer than, the delay spread of the channel, inter symbolinterference (ISI) is completely eliminated. For time-invariantfading channels, the channel matrix in the frequency domainwould be diagonal such that each subcarrier is simplyattenuated by the corresponding frequency-domain channelresponse. In this case, a simple pilot-based channel estimatorusing interpolation may be adopted to estimate the channel inthe frequency domain [2]. However, for rapidly varying

channels, the variation of the channel within one OFDMsymbol destroys the orthogonality between the subcarriers andthis introduces inter-carrier interference (ICI). In this case, anestimation of the channel in the frequency domain becomescomplex due to the large number of unknowns in the channelmatrix. For this reason, it is more appropriate to estimate thevariation of the channel in the time domain [3]. Basis

expansion Modeling (BEM) is a way to approximate the time-variation of the channel within a certain time window.Basically, BEM reduces the complexity as the problem isreduced to estimating the basis coefficients [3]. It has beenrecently adopted to estimate the variation of the channel inOFDM systems and many BEMs have been proposed in thiscontext. The optimal one in the Mean Squared Error (MSE)sense is the Discrete Karhuen-Love BEM (DKL-BEM) [4, 5]as it takes into account the channel statistics to find the bestfitting basis functions. But it is suboptimal in case where thereal channel statistics deviate from the assumed ones. Othersuboptimal approaches, not depending on channel statistics, arethe Complex Exponential BEM (CE-BEM) [7] which leads to a

strictly banded frequency-domain matrix, the Generalized CE-BEM (GCE-BEM) which is a set of oversampled complexexponentials [8], and the Polynomial BEM (P-BEM) [9] whichapproximates the channel variation by means of a polynomialfunction. In all these methods, the number of basis functionsdepends on the normalized Doppler frequency [3].

In this paper, we will focus on the BEM approach toestimate the channel in high-rate OFDM systems based on anequispaced pilot distribution. In particular, the duration of anOFDM symbol in such systems is relatively short leading to avery small normalized Doppler frequency. In this case, a linearapproximation (P-BEM with 2 basis functions) of the channelvariation shows an interesting fitting [12]. Based on this

property, we will propose an algorithm using two consecutiveOFDM symbols. In fact, the relation of channel variation

between two OFDM symbols has been recently introduced in[10]. It is shown that using piecewise linearity betweenconsecutive OFDM symbols helps ICI mitigation. Authors firstget the least squares channel estimates at the pilots forconsecutive OFDM symbols and use them to estimate channelvariation. In this paper, we introduce an explicit usage of BEMin time domain, then an update of channel coefficients is

performed in frequency domain in a simple way usingconsecutive OFDM symbols. In addition, we investigate theeffect of DF in a practical pilot distribution.

978-1-4244-5213-4/09/ $26.00 2009 IEEE 2404


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The rest of this paper is organized as follows. Section IIpresents the general system model and how BEM can be usedto approximate the channel variations. Section III discussesBEM approach for an equispaced pilot distribution, the effectof decision feedback, and proposes a new algorithm based onmultiple OFDM symbols. Simulation results are presented inSection IV and we conclude the paper in Section V.

Notation:We use upper (lower) bold face letters to denotematrices (column vectors), ()T and ()H denote transpose andcomplex conjugate transpose operators, respectively. E[]

stands for the expected value and represents the Kroneckerproduct. We denote an NN identity matrix by IN.Furthermore, we useXi,kto indicate the (i+1, k+1)

thentry of the

matrix Xand diag(x) to indicate a diagonal matrix with xon itsdiagonal.

II. SYSTEM MODEL

A. General System Model

In an OFDM system, the symbols collected in symbolvector s are first transformed from the time domain to thefrequency domain using an IFFT. Then, a CP consisting of thelast symbols is added before the parallel to serial converter. Atthe receiver side, a serial to parallel conversion is applied, CPis removed and an FFT operation is performed to obtain thereceived symbols in the frequency domain.

Figure 1: Simple OFDM system diagram.

The expression of the received vector for the (p+1)th

OFDM symbol can be expressed as:

)()()(

)()()()()(

ppp

ppHptp

zsG

zsFFHr

+=

+= (1)

where)()( pt

H and )(pG represent the channel matrices

respectively in time and frequency domains for the (p+1)th

OFDM symbol; Fdenotes FFT operation, and z is the complexadditive white Gaussian noise vector.

For an FFT size of N, it can be easily shown that)()( pt

H has the entries

),mod(,)(

)(

,

)(

NkiiLNp

p

ki cp

t

hH ++= (2)

where Lcp is the CP length. In the sequel, the index p will bedropped for the sake of clarity.

Due to the time variation of the channel during the OFDMsymbol, the frequency domain channel matrix Gwould not bediagonal. In fact, it will have the entries

[ ]1,0,,1 1

0

)(2

,, =

=

NkieHN

GN

n

N

iknj

nkki (3)

where { }1,,0, = Nknk

H

denote the Fourier transform of the

channel impulse response { }1,,0, = Llnl

h

at the time instant nfor

a channel length ofL and given by

NkehN

HL

l

N

klj

nlnk 1 is the oversampling

ratio, and finally for P-BEM we use qqp pB =, .

If we collect all the channel taps in a single column vector

as [ ]T

NLNL hhhh 1,11,00,10,0 ,...,,...,,..., =h and similarly all basis

coefficients in a single column vector as

[ ]TbQL

b

Q

b

L

b hhhh,1,00,10,0

,...,,...,,...,

=b

h , then neglecting the

modeling error we obtain

bhIBh )( L= . (6)

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As a result, after some algebra, the received symbol vectorcan be expressed in terms of the BEM as

=

+=

Q

q

qq

0

zsDr (7)

where

H

qq diag FbFD )(= and q= diag(FL [ ]

Tb

qL

b

q hh ,1,0 ,..., )=diag(FL

b

qh ). Here, FLcollects the firstLcolumns of the matrix

.FN

Equation (7) can also be written as

zhFsDr += =

b

qL

Q

q

qdiag )(0

. (8)

If we define QDDDD .....10= and )).((1 LQ diag FsIS = + ,

then we obtain

zDShr +=b

. (9)

Using (9), channel estimators can be derived either basedon the whole knowledge of input symbol vector s (i.e., the full

preamble case) or based on only a part of symbol vector s(i.e.,in the presence of pilot signals). We note clearly that BEMsimplifies the estimation as the problem is reduced toestimating h

bof size (Q+1)Lrather than estimating all the NL

channel coefficients.

III. ALGORITHM

We first present the pilot distribution that is used in the

simulations. Next, we discuss the effect of applying BEMdirectly on such a pilot distribution. Then, we propose analgorithm based on decision feedback considering the use oftwo consecutive OFDM symbols.

A.Pilot Distribution

It has been observed in a number of attempts that theoptimal placement of pilots appears to be equispaced clustersand more precisely zero-padded ones [3, 6, 11]. But for nextgeneration mobile wireless systems, a sparse distribution in

both frequency and time is being considered. In the sequel, wewill mainly focus on a pilot distribution which resembles theones recently adapted in mobile wireless systems [14, 15]

where pilots are located equispaced on the FFT grid.

B.Direct BEM approach

As an initial step, we apply directly a basis expansion onthe channel and try to estimate the basis coefficients just usingthe assumed pilot distribution. We will show that it is notefficient to calculate the variation of the channel which isnormally deduced from the interference terms.

Separating the input symbol vector s into a vector spcontaining pilots and a vector sdcontaining data symbols, wecan rewrite (9) as

zhDShDSr ++=b

d

b

p . (10)

Then, considering the channel realizations and the datasymbols as independent stochastic processes, the expression ofthe LMMSE estimator can be obtained as [3]

1

0 )(

++= NxH

hb

H

hbLMMSE N IRRRC (11)

where [ ]Hbbhb E hhR = is the autocorrelation matrix betweenthe basis expansion coefficients which can be concluded fromthe autocorrelation matrix of the real channel coefficients, and

N0 denotes the noise spectral density. In (11), pDS=

depends on the pilots and [ ] HHdbbdxH

E DShhSDR = is

calculated using the assumed statistical properties of thechannel and data.

C.Decision Feedback

As shown in the sequel, using pure BEM with equispaced

pilots may not be sufficient to have a satisfactory estimation;one may use an iterative approach based on detected symbolsto feedback the estimator. In order to improve the performance,we make use of decision based feedback such that a firstestimation just using the pilots can be used for equalization andthen the detected symbols serve as a preamble to re-estimatethe basis coefficients. For the sake of simplicity and withoutloss of generality, our study will be based on hard decisionfeedback while it is clear that with soft-decision basedestimators one may provide better performance. Successiveiterations can be performed in order to enhance the

performance. However, it will be shown that with the proposedapproach two iterations are enough to converge rapidly to a

performance limit.

D. Linear Filtering Between Successive OFDM Symbols

In this subsection, we propose a linear filter to furtherimprove the estimation of the interference coefficients (non

diagonal terms) in the frequency domain channel matrix G. It

is mainly based on the initial information from successive

OFDM symbols. As explained below, this improvement is due

to the fact that ICI terms come mainly from the channel

variation during the OFDM symbol and thus can also beestimated by the variation of the frequency domain channel

matrix through consecutive OFDM symbols.

Let us first investigate the behavior of the coefficients in

the G matrix assuming a satisfactory approximation.

Particularly, we use P-BEM with Q = 1 (linear variation) forthe channel variation during two consecutive OFDM symbols.

In this case, each channel coefficient for a certain tap lcan beexpressed as:

Nnnhh llnl


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line representing the variation of the channel in a certain time

window (i.e., over several OFDM symbols).

It can be easily proved that for the (p+1)th

OFDM symbol

one obtains

icpi

p

ni ELNpnHNH ))((0,)(

, +++= (13)

whereEis are simply the Fourier transform of the ls defined

in (12), i.e.,

NieN

EL

l

N

ilj

li


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SNR value of 40 dB. This motivates the idea of employing

these diagonal coefficients to estimate the non diagonal ones.

Figure 3: Effect of decision feedback, MSE of diagonal terms in Gmatrix.

Figure 4: SER performance of the proposed algorithm

based on successive OFDM symbols.

As explained in Section III, at each iteration of the

proposed algorithm, we use the estimated diagonal

coefficients in the G matrices corresponding to successiveOFDM symbols in order to recalculate the ICI coefficients.

Comparing the iterations in Figure 4 with those of Figure 2,

one can easily see the remarkable performance improvement

just by using this simple linear filtering after each iteration.

Our algorithm makes use of the BEM approach first toestimate the channel in the time domain then to filter the

frequency-domain channel matrices estimated corresponding

to multiple OFDM symbols. This leads to a further gain of 5

dB at high SNR compared to the case where only decision

feedback is employed. It is also worth noting that addition of

such a simple filtering allows a faster convergence to a better

performance limit due to better estimates obtained from thefirst iteration.

V.CONCLUSION

In this paper, we investigated a linear modeling to estimate

rapidly varying channels using an equispaced pilot

distribution. We further studied the effect of decision feedback

and we proposed a new method to improve the estimationperformance based on two successive OFDM symbol

observations. It has been shown by simulations that the

proposed method provides a considerable performance

enhancement by means of a simple linear filtering.

ACKNOWLEDGMENT

The authors would like to acknowledge the support of theEuropean Commission through the FP7 project WiMAGIC

(see www.wimagic.eu).

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An Iterative Channel Estimator for Fast-Varying Channels Using Successive OFDM Symbols