An Enhanced Channel Estimation Algorithm for OFDM - Combined EM Algorithm and Polynomial Fitting

8/12/2019 An Enhanced Channel Estimation Algorithm for OFDM - Combined EM Algorithm and Polynomial Fitting

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AN ENHAN CED CHANN EL ESTIMATION ALGORITHM FOR OFDM: COMBINED EMALGORITHM AND POLYNOMIAL FITTING

Xiaoqiang M a, Hisashi Kobayashi, and Stuart C. Schwartz

Electrical Engineering Department, Princeton University

Princeton, New Jersey 08544-5263Email: xm a, hisashi, [email protected]

A B S T R A C T

Estimating a channel that is subject to frequency selec-

tive Rayleigh fading is a challeng ing problem in an orthog-

onal frequency division multiplexing (OFDM ) system. In

this paper we propose an enhanced channel estimation al-

gorithm that combines the EM-based algorithms proposed

previously and a Least Squares polynomial fitting (LSPF)

approa ch. Th e combined algorith m can efficiently estimate

the channel response of an OFDM system operating in anenviron ment with m ultipath fadin g and additive white Gaus-

sian noise (AWGN). The algor ithm can improv e the channel

estimate obtained from the EM-based algorithms by poly-

nomial fitting. Simula tion results show that the hit error

rate (BER) as well as the mean square error (MSE) of the

channel can be improved by the algorithm. In particular,

with these additional computation s and demodulation delay,

the MSE can be made smaller than the Cramer-Rao lower

bound (CRLB).

1. INTRODUCTION

An efficient and accurate channel estimation procedure is

necessary for coherent demod ulation in OFD M systems. M any

channel estimation algorithm s have been reported in the lit-

erature [2]-[4]. Recently w e proposed three different EM-

based channel estimation algorithms (refer to [5]-[7]) that

can achieve the CRL B in the high SNR region. In these al-

gorithms the channel is estimated fra me by fram e by assum-

ing the channel is changing from o ne frame to another. If

we further assum e the channel variation is smo oth, then we

might do a better job by considering several OFDM frames

together. Obviously, if the channel does not change dur-

ing several OFD M frames, w e can certainly apply the EM -

based algorithm [5]-[7] rju st average the channel estimates

of those OFDM frames to obtain a better estimate. However,

in a mobile environment with fast moving objects this as-

sumption fails. Fortunately, the channel chan ges smoothly

alone the time axis in a real mobile environment. This

methods or time-domain smoothing), e.g., least squares poly-

nomial fitting, in order to further improve the channel esti-

mation accuracy. Th is basic idea is discussed in [SI and [9].

Similar time-frequency polynomial mode ls are adopted in

these two referenced papers. Both of them concentrate on

the channel frequency response which contains many morecomp onents than the corresponding CIR in a typical OFDM

system. Thus in their methods, the complexity to establish

model coefficients is high. Furtherm ore, they need a large

amount of pilot symbols in the time-frequency grid of the

OFDM systems to minimize the model mismatch. This de-

grades the overall system capacity o r spectral efficiency.

The main objective of this paper is to investigate the

use of the polynom ial fitting method in the time dom ain for

chann el estimation of an OF DM system subject to slow time

varying frequency selective fading. An EM-based channel

estimation algorithm is carried out first to obtain near opti-

mal chann el estimates using th e information of the current

OFDM frame only. Then, polynomial fitting is adopted by

gathering channel estimates of several consecutive OFDMframes. By applying this concatenated channel estimation(EM-based algorithms followed by polynomial fitting), bet-

ter channel estimates can be obtaine d.

2. E M - B AS E D C H A N N E L E S T I M A T I O N

A L G O R I T H M

The most important step in an EM-algori thm is determining

what is known as “complete” data. Different selections of

“com plete” data result in different algorithms. T hree d iffer-

ent EM-based channel estimation and signal detection al-

gorithm s have been proposed in [5]-[7] by ca refully defin-

ing different “complete” data for each algorithm. Their ad-

vantages and disadvanta ges are discussed therein. Basi-

cally, they have almost the sam e performance as measured

by BE R and MSE . The only difference is the rate of conver-

gence and computational burden. The second EM-based al-

gorithm [6] has the fastest convergence speed as seen from

- the simulations. Thus, w e suggest using this algorithm as

the first stage channel estimation method. Furthe rmore, onemooth ness motivates us to use well designed curve fitting

0-7803-7663-3/03/$17.00 02 00 3 IEEE IV - 68 ICASSP 2003

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simplified version that does not need the channel statistics

has negligible performance deterioration. The correspond-

ing simplified algorithm is the one adop ted.

The system model can be ex pressed in vector form for aparticular OF'DM frame as (refer to [6] for details)

7 = XW1.h f N 1)

where L is the channel delay spread, M is the number of

subcarriers and Wr. is the subm atrix of the FFT m atrix with

first L columns. We assume that the channel is static over

the period of one fram e.

The "incomplete" and "complete" data are defined as

11) nd (l',h), respectively. Eac h iterative process p =0 , 1 , 2 ; . ., in the EM algorithm for estimating rom Yconsists of the following two steps:

E - step : &(XIXp) E {bf(Y,hlX)IY,Xp$23

M step : + i = a r g m y x Q ( X I X , ) , 3)

Detail derivations and discussion are included i n [61

3. LEAST SQUARES POLYNOM IAL FITTING

typical mathematical procedure for finding the best fit-

ting cur ve to a given set of points is to minimize the sum of

the squares of the offsets (the residuals) of the points from

the curve. Th e linear least squares fitting technique is the

simplest and most commonly applied form of linear regres-

sion and provides a solution to the problem of finding the

best fitting straight line through a set of points. However,it also introduces large fitting error because of the niodel

mismatch. Polynomial fitting is on e of the nonlinear curve

fitting methods that is often used due to its good proper-ties. First, it is easy to realize and the computation is low

compared to other nonlinear fitting techniques. Mo re im-

portantly, polynomials of different degrees can simulate a

variety of urves.

Suppose we try to fit a set of given da ta with a dh egree

po y nom a

y = a. a x al;xc 4)

A set of observed data {(q,i)ll 5 5 n} are given. The

objective is to find a k t h deg ree polynomial that has the least

offset

n

R = C[y- a0 al z j . +a k & ] ' . 5 )

The coefficients of the least squares polynomial fitting

i=l

can be easily expressed as

- = (XTX)- XTy. 6)

where where - = yl,...,y,) T a = a o , . . . , ~ ) ~nd

x...;-I.11 -

4. CONCATENATED CHANNEL ESTIMATION

A numbe r of OFD M channel estimation metho ds have been

reported in the literature. Sho rtcom ings of the existing al-gorithms are I ) a large error floor that may be incurred by

a mismatch between the estimated and real CIRs, (2) dif-

ficulty in obtaining the correlation function of the channel

impulse response and inevitable e rror due to cha nnel statis-

tics m ismatch, and 3) spectrum inefficiency due to the use

of a high percentage (typically 20 of pilot symbols. On

the other hand, several kinds of blind channel estimation

algorithms have been proposed in order to improve trans-

mission efficiency. These algorithm s are based on the sta-

tistical nature of the received signals, virtual carrier, and

finite-alphabet property of the transmitted signals . How-

ever, most of them need a large amount of data to obtain reli-

able channel estimates. Thus, these blind algorithm s can no t

be used in a high mobility environment (i.e., large Dopplerspread). We, on the other hand, have extended the existing

pilot-based channel estimation alg orithm s and reduced the

number of pilot symbols by using the EM-algorithm [ 5 ]

[7]. These EM-based algorithms are only suitable when

the D blocks of the channel are constant. If the channel is

not constant, the algorithms developed fail to give reliable

channel estimates. T heref ore, it is desirable to use the m for

only one OFDM frame. Consequently, the accuracy of the

channel estimates is limited by these frame-based channel

estimation algorithms. M ore precisely, we observe that the

CRL B for ch annel estimates is given by

7)

where D is the number of frames in which the channel is

constant. The larger the D , the smaller the CRLB will be.

In a typical mobile environm ent D can be as small as 1 .

one coded da ta packet always occupies several OFDM frames.

Conseq uently, channel estimation amo ng consecutive OF DM

frames is required to further improve channel estimation ac-

curacy. In such applications, one stage of channel estima-

tion may not be accurate enough to meet the system per-

formance requirem ent. Although iteration between channel

estimation an d channel decoding is also possible to improve

channel estimation accuracy, the computational complex-

ity is prohibitively high due to the Viterbi trellis search ineach iteration. We, on the other hand, propose a so called

concatenated channel estimation that combines EM-based

channel estimation algorithms and LSPF as shown in Fig.

I . This concaten ated channel estimation can be extended in

an iterative fashion if the output performance is not satis-

factory. Furthermore, this concatenated channel estimation

idea can be extended to other channel estimation methods.

In the packet or block OFD M transmission (e.g. IEEE802. 1 a) ,

IV - 681



Fig. 1. Conca tenated channel estimation algorithm

. . . . . . .

5. SIMULATION RESULTS

In our channel estimation problem, y s the estima te of the

l th tapofCIRatconsecutiveOFDMframes ( ( l ) , k l ( n ) ) ,

xi denotes the i t h ime instant and a is the polynomial co-

efficients from the L SPF 6).

Th e simulation model we use in this paper is the same as

that in [ 6 ] .As previously stated, w e will only show the sys-

tem performance in which the described EM-based channel

estimation algorithm is used in the first stage of the con-

catenated algorithm. Extensive simu lations show that theconcatenated algori thms adopting the other two EM-based

channel estimation algorithms in the first stage result in sim-

ilar performan ce. For the LSP F (the second block), we con-

sider three different scenarios:

1. Th e LSPF is carr ied out every 8 OFDM frames .

2 . The LSPF is carried o ut every 16OFDM frames.

3. The L SPF is carr ied ou t every 24 OFDM frames. Only

the channel estimates of the middle 8 frames are ac-

cepted as the final estimates. Thu s, there is overlap

between two consecutive LSP Fs.

Figs. 2 and 3 illustrate the BER a nd MS E performan ce

by applying the proposed concatenated channel estimation

algorithm in the system described above. The BER im-provement due to the LSPF is negligible since the BER af-

ter the EM-based channel estimation procedure is already

good. However, the M SE improvement due to the LSPF is

significant compared with the MSE of the EM-based chan-

nel estimation algo rithm only. W hen the S N R is larger than

5dB, a MSE that is lower than the CRLB is achieved. Of

course, this MSE improvement is obtained by increasing

the demodulation delay. Theoretically, the longer the ob-

served date, the more improvem ent we can get. In practice,

we should specify the transmission delay according to the

system requirement. For LSPFs with longer observed data

sequence, higher degree polynomials should be considered

to minimize the model mismatch. However, the complex-

ity also becom es higher. Thus, a uadeoff must he carefullyconsidered according to different system parameters and re-

quirements.

Fig. 4 shows a realization of the amplitude of the first

channel tap h,. The curve of initial channel estimates is

a step function, which results in a large deviation fro m the

real channel. The cur ve of EM-based cha nnel estimates os-cillates around the real channel. The channel mismatch is

higher when the deep fades occur. By filtering out spikes

in the estimates of the EM-based algorithm, the curve of

channel estimates after applying the LSPF is very smooth.

It is also fairly close to the real channel respon se. it demon-strates the effectiveness o ft he concatenated channel es tima-

tion algorithm.

The re are two underlying reasons for the MS E improve-

ment . Th e obvious one is that the concatenated algorithm

uses additional information from other OFD M frames which

are located close to the current OFDM frame. T he more im-

portant reason is that the channel estimates obtained by the

EM-based algori thms can he approximated as

EM ( t ) = h(t) E E M ( t ) i (8)

where hEM t ) s the channel estimate obtained from the

EM-based algori thms at t ime t, h(t) s the t rue CIR and

SEM ( t ) s the residual estimation error. After EM-based

iterations, cEh4t ) an be considered as independent Gaus-sian noise for differen t OFD M fram es, as is justified in Fig.

5(a) and 5(h). Th e empirical distribution is close to the

Gaussian distribution with zero mean and variance U &,.

The analytical curve is the Gaussian density function with

zero mean and variance equal to the MSE after the EM-

based channel estimation procedure.

.. ~

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. .. .

2 e 6 8 10 ,I 11 I I mEWNO

<d

Fig. 3. MSE of channel estimates for OFDM systems EM-

based algorithm concatenated with LSPF.

Y 1 0 m Y I . O Y m 7 0 m w n I m

r ,ms

0 s , I , ,

Fig. 4. The realization of the am plitude of hl when SNR

equals to IOdB.

a) PDFs of the estimate of

W h i ) Q h i )

(b) PDFs of the estimate of

Fig. 5 . Simulated and analytical PDFs of the estimate of

R(hl) and D ( h l ) when S NR equals to 1OdB.

dures. Another merit of EM-base algorithms is the good

MSE performance that is close to the CRLB. Otherwise, a

MSE lower than the CRLB could not be achieved. That

is why the c oncatenated channel estimation algorithm with

the variou s EM-based algorithms in the first stage will yield

similar final performance after applying the LSPF.

6. CONCLUSION

In this paper we proposed a concatenated channel estima-

tion algorithm for OFDM systems in order to enhance sys-

tem performance. In particular, EM-based channel estima-

tion algorithms followed by LSPF is proposed and tested.

Simulation results show a significant MSE improvement,

compared to the EM -based algori thm alone. The proposed

concatenated channe l estimation method can he used in other

applications such as single carrier systems with frequency

domain c hannel estimation. In order to make the concate-

nated channel estimation work, G aussian-like estimation er-

rors should be generated from the first stage.

Note: Additional details in a longer paper and large r fig-

ures can be found at http://www.ee.princeton.edurxmdfit.pdf.

7. REFERENCES

[ I ] L. J. Cimini, Jr., * Analysis and simulation of a digital mo-

bile chaneel using orthogonal frequency division multiplex-ing,"IEEE Trans. on Comm., July 1985, pp. 6 65-675.

[2] Ye (Geoffrey) Li, Leonard J. Cimini, Jr., and Nelson R.

Sollenberger, "Robust channel estimation for OFDM sys-tems with rapid dispersive fading channels, IEEE Truns. on

Comm., vol. 46, no. 7, pp. 902-915, Ju l y 1998.

[3] Jae Kyoung Moon and Songn

Choi, "Performance of chan-nel estimation methods for OFDM systems in multipath fad-ing channels," I E E E Trans. on Consumer Elecrronics, vol.

46, no. I , pp. 161-170, Feb. 2000.141 B.M uaue t and M.de Courville, "Blind and semi-blind than-~

ne1 identification methods using second order statistics for

OFDM systems:' Proc. of ICASSP 1999, vol. 5 , pp. 27452748, Mar. 1999.

[5] X. Ma, H. Kobayashi and S . Schwartz, "EM-Based C hannelEstimation for OFDM," in Proc. of PACRIM 2001, vol. 2,pp. 449-452.

[61 X. Ma, H. Kobayashi and S Schwartz, An EM-Based Es-timation of OFDM Signals," in P m c . of W C N C 2002. vol.1,pp.228-232.

[7] X. Ma, H. Kobayashi and S Schwartz, J. Zhang and D . Gu,"Time Domain Channel Estimation for OFDM using EM Al-

gorithm," in Proc. of 3GWireless 02. vol. 2. pp. 696-700.[SI X. Wang and K. J. Ray Liu, "Channel estimation for mul-ticarrier modulation systems using a time-frequency poly-nomial model," I EEE Trans. on Com m . , vol. 50, no. 7. pp.1045-1048. Jul. 2002.

[9] M. X. Chang and Y. T. Su "Model-based channel esti-maion for OFDM signals in Rayleigh fading," I EEE Trans.

on Comm. vol. S O no. 4, pp. 540-544, Apr. 2002.

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An Enhanced Channel Estimation Algorithm for OFDM - Combined EM Algorithm and Polynomial Fitting