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Hsieh and Wei: Channel Estimation for OFDM Systems Based onComb-Type Pilot Arrangement in Frequency Selective Fading Channels

CHANNEL ESTIMATION FOR OFDMSYSTEMS BASED ON COMB-TYPE PILOT

ARRANGEMENT IN FREQUENCY SELECTIVE FADING CHANNELS

Meng-Han Hsieh' and Che-Ho We?Department of Electronics Engineering

National Chiao Tung UniversityHsinchu, Taiwan 30050, R.O.C.

E-ma il: 'meng Qclab.ee.nctu.edu.tw, *chwe i @cc.ntcu.edu.tw

217

A BsTRACT

In this paper, the channel estimation metho ds for OFDM sys-

tems based on comb-type pilot sub-carrier arrangement are in-

vestigated. The channel estimation algorithm based on comb-

type pilots is divided into pilot signal estimation and chan-

nel interpolation. Th e pilot signal estimation based on LS or

MMSE criteria, together with channel interpolation based on

piecewise-linear interpolation or piecewise second-order poly-

nomial interpolation is studied. Owing to the MM SE est imate

of pilot signa ls, the inter-carrier interference and additive white

Gaussian noise are reduced considerably. The computational

complexity of pilot signal estimation based on MMSE crite-rion can be reduced by using a simplified LMMSE estimator

with low-rank approximation using singular value decomposi-

tion. Phase compe nsators before and after interpolation are also

presented to combat the phase changes of subchannel symbols

arising from the frame synchronization errors. Com paring to

the transform-domain processing based channel estimation al-

gorithm, the MM SE estimate of pilot signa ls together with phase

compe nsated linear interpolation algorithm provides better per-

formance and requires less computations.

I . I N T R O D U C T I O N

Orthogonal Frequency Division Multiplexing (OFDM) tech-

nique has received a lot of interest in mobile communicationresearch lately. For wideband mobile comm unication systems,

the radio channel is usually frequency selective and time vari-

ant. Hereafter, in OFDM system s, the channel transfer function

of radio channel appears unequal in both frequen cy and time do-

mains. Therefore, a dyna mic estimation of the channel is neces-

sary for the demodulation of OFDM signals.

In wideband mobile channels, the pilot-based signal correc-

tion scheme has been proven a feasible method for OFDM sys-

tems [I]. Most channel estimation methods for O FDM trans-

mission systems have been developed under the assumption of

a slow fading channel, where the channel transfer function is

assumed stationary within one OFDM data block. In addition,

the channel transfer function for the previous OFDM data blockis used as the transfer function for the present data block. In

practice, the chan nel transfer function of a wideband radio chan-

nel may have significant chan ges even within one OFD M data

block. Therefore, it is preferable to estimate channel character-

istic based on the pilot signals i n each individual OFDM data

block.

Recently, an elegant channel estimation method for OFDM

mobile communication systems has been proposed by Zhao and

Huang [3]. In this metho d, the additive white Gaussian noise

(AWGN ) and the inter-carrier interference (ICI) in the pilot sub-

carriers are reduced by low-pass filtering in a transform do-

main, and the channel transfer function for all the subcarriers

is obtained by the high-resolution interpolation realized by zero

padding and DFTLDFT. Comparing to the conventional linear

interpolation method, this method p rovides about 1 dB and 3 dB

improvement in Eb/No for the same bit error rate values in

slow- and fast-fading noisy radio channel, respectively.

In this paper, we present a different approach for channel

estimation, which is based on a minimum mean-squared error

(MMSE) estimate of pilot signals. Because of the robustness of

the MMSE estimator, the AWGN and the IC1 components are

reduced significantly in fast- or slow-fading noisy radio chan-

nel environments. The computational complexity of the MMSE

estimator can be reduced by using a simplified linear minimu m

mean-squ ared error (LMM SE) estimator with low-rank approxi-

mation by singular value decomposition (SV D) [6]. In addition,

because of the in-sensitivity to parameter mismatch, a generic

low-rank estimator can be used for the samc kind of channels.

The cha nnel transfer function of data subc arriers is then interpo-

lated based on th c MMSE estimate of pilot signals. Two interpo-

lation methods, the piecewise-linear and the piecewise second-

order polynomial interpolations, are studied. Furthermore, by

taking the frame position error into account, we prop ose a phase

pre-compensator and post-compensator to mitigate the modelmismatch error of the interpolator and the MMSE estimator.

Based on our simulations over fast- and slow-fading mobile

channels, we see that a significant improvement in Eb/No fo r

the same BER is provided.

The paper is organized as follow s. In Sec tion 11, the pilot-

based OFDM system is described, and two type of pilot arrange-

ments, the block-type arrangement and the comb-type arrange-

men t, are discussed. Section I11 discusses the estimation of pilot

signals, and a low-com plexity estimator is studied. Interpolation

methods and a way to mitigate the model mism atch problem are

deliberated in Section IV. Section V presents the simulation

results, which indicate the BER improvements. The computa-

tional complexity of the proposed method is evaluated in Sec-tion VI. Section VI1 concludes the paper.

11. SYSTEMESCRIPTION

Fig. 1 shows a typical block diagram of OFDM system withpilot signal assisted. The binary information data are grouped

and mapped into multi-amplitude-multi-phase signals. In this

paper, we consider the 16 -QA M modulation. After pilot inser-

tion, the modulated data { X ( k ) } re sent to an IDFT and are

Contributed Paper

Manuscript received January 13, 1998 0098 3063/98 $10.00 1998E E E



218 IEEE Transactions on Consumer Electronics, Vol. 44, No. 1 , FEBRUARY 1998

Binary

Source Insertion - Interval

(1 6-QAM) Insertion

Signal

Demapper

(1 6-QAM)

Binary

Data

jqjzJ~q#q&ased on

RemovalPilot

Fig 1 Baseband model of a typical pilot-based OFDM system

LPF

--c

AWGN

,U ( r t )

transformed and multiplexed into { ~ ( n ) }s Supp ose that the guard interval is longer than the length of chan-

ne1 impulse respons e, that is, there is no inter-symbol interfer-

ence between OFDM sym bols, the demultiplexed samples Y ( )-1x ( n ) = IDFT{X(IC)}= X ( k ) e 3 2 n k ? z / N , can be represented by [3]

k= O

71= 0 , I , ,N - I, (1) Y ( k ) = X ( k ) H ( k )+ I ( k ) + W(IC), = 0 , 1 , ,N - I, ( 6 )

where N is the number of subcarriers Th e guard interval is

inserted to prevent possible inter-sym bol interferen ce in OFDM

system s, and the resultant sam ples { z g ( n ) }re

where N g is the number of samples in the guard interval Th e

transmitted signal is then se nt to a frequency selective multi-path

fading channel The received signal can be represented by

where h(n) s the impulse response of channel and w ( n ) is the

additive white Gaussian noise. Th e channel impu lse respons e

h(n) can be expressed as [4]

r-1

where

and W (IC) is the Fourier transform of w ( n )

After that, the received pilot signals {Y,(lc)}re extracted

from { Y ( k ) } nd the channel transfer function { H ( k ) } an be

obtained from the information carried by { H p ( k ) } . With the

knowledge of the channel responses {H ( k ) } , he transmitted

data samples {X (k ) can be recovered by simply dividing the

received signal by the channel response:

where r the total number of propagatio n paths, h, he complex

impulse response of the zth path, f ~ ,he zth-path D oppler fre-

quency shift which causes IC1 o f the received signals, X the de-

lay spread index, and 7%he zth-path delay time normalized by

sampling time.

After removing the guard interval from yg(n),he received

samples ~ ( n )re sent to a DF T block to demultiplex the multi-

carrier signals.

where A ( k ) s an estimate of H ( k ) .After signal dem apping, the

source binary information data are 1-e-constructed at the receiveroutput.

Based on the principle of OFDM transmission scheme, it is

easy to assign the pilot both in time-domain and in frequency-

doma in. Several types of pilot arrang eme nt have been studied

[ I ] . Here we consider two major types of pilot arrangement as

shown in Fig. 2 . Th e first kind of pilot arrang eme nt shown in

Fig. 2(a) is denoted as block-type pilot arrangement. The pilot

signal is assigned to a particular OFDM block, which is sent

periodically in time-dom ain. This type of pilot arrangement

is especially suitable for slow-fading radio channels. Because



Hsieh and Wei: Channel Estimatio n for OFDM Systems Based onCo mb -llp e Pilot Arrangement in Frequency Select ive Fading Channels

_c

--t

Signals -aherFFT -

the training block contains all pilots, channel interpolation in

frequency domain is not required. Therefore, this type of pi-

lot arrangement is relatively insensitive to frequency selectivity.

The estimation of chann el response is usually obtained by LS or

MM SE estimate of training pilots [ 5 ] [6].

The second kind of pilot arrangem ent shown in Fig. 2(b) is de-

noted as comb-type pilot arrangement. The pilot signa ls are uni-

formly distributed within each OFD M b lock. A ssuming that the

payloads of pilot signals of the two arrangements are the same,

the com b-type pilot assig nme nt has a higher re-transmission

rate. Thus, the com b-type pilot a rrangement system is provides

better resistance to fast-fading channels. Since only some sub-

carriers contain the pilot signal, the channel response of non-

pilot subcarriers will be es timate d by interpolating neighboring

pilot sub-channels [ 2 ] [3]. Thus, the comb-type pilot arrange-

ment is sensitive to frequency selectivity when c omp aring to the

block-type pilot arrangement system. That is, the pilot spacing

(Af ) , must be muc h sma ller than the c oherence bandwidth of

the channel (Af)c.

In this paper, we consider the radio channel which changes

rapidly. In such enviro nme nt, the channel transfer function

changes significantly from o ne block to the next block. Th us, the

channel estimation in the present block can not be used as the

channel response of the next bIock. Therefore, the comb-type

pilot subcarrier arrangement is adopted to estimate the channel

transfer function in each block. As shown in Fig. 3, the pilot

signals is first extracted fro m the received s ignal, and the ch an-

nel transfer function is estimated from the received pilot signals

and the known pilot signals. The n, the chan nel responses of

subcarriers that carry data are interpolated by using the neigh-

boring pilot channel responses. In the following sections, the

pilot signal estimation and c hann el interpolation algo rithms are

discussed separately.

-r Channel

Response

---t

Extraction -Received Pilot Signal Pilot Signal t-1 Channel E stimated

111. P I LO T I G N A L S T I M A T I O N

For com b-type pilot sub carrier arrangement, the N p pilot sig-nals X p ( m ) ,m = 0 , 1 , . . . ,Np- 1 are uniformly inserted into

X ( k ) . That is, the total N subcarriers are divided into Np

groups, each with L = N /Np adjacent subcarriers. In each

group, the first subcarrier is used to transmit pilot signal. The

OFDM signal modulated on the kth su bcarrie r can be expressed

as

X ( k ) = X ( m L f 1 )

(8 )1 = o ,

information data, 1 = 1 , 2 , . . ,L - 1.

219

Th e pilot signals { X P ( m ) } an be either equal complex values

c to reduce the computational complexity, or random generated

data that can also be used f or synchronization.

Let

H, = W P ( 0 ) HP(1) " %(NP - IllT

= [ H ( O )H ( L - 1) . . . H ( ( N ,- 1 ) .L - l)]' (9)

y , = CYP(0) YXNP - 111 (10)

be the channel response of pilot subcarriers, and

be the vector of received pilot signa ls. Th e received pilot signal

vector Y, can be expressed as

Yp = X, . H p+Ip +Wp,

X P ( 0 ) 0

(1 1)

where

x, = [0 X,(NP - 1)

I, is the vector of IC1 and W, is the vector o f Gaussian noisein pilot subcarriers.

In conventional comb-type pilot based channel estimation

methods, the estimate of pilot signals, based on least squares

(LS) criterion, is given by:

% , l S = [ H P , l S ( O ) l7,,ls(1) ' . . H P , l S ( N P - 1)IT

= xply,

The LS estimate of Hp is susceptib le to Gaussian noise and

inter-carrier interference (ICI). Because the channel responses

of data subcarriers are obtained by interpolating the channel re-sponses of pilot subcarriers, the p erform ance of OFDM system

based on com b-type pilot arrang eme nt is highly dependent on

the rigorousness of estimate of pilot signals. Thu s a estimate

better than the LS estimate is required.

The minimum mean-square error (MMSE) estimate has been

shown to be better than the LS estimate for channel estimation

in OFDM systems based on block-ty pe pilot arrangement [ 5 ] .

Regarding the mean square error of the estimation shown in [ 5 ] ,

the MMS E estimate has about 10-15 dB gain in SN R over the

LS estimate for the same MSE values. The major drawback

of the MMSE estimate is its high complexity, which grows ex-

ponentially with the observation samples. In [6], a low-rank

approximation is applied to a linear minimum mean-squared er-

ror (LMMSE) estimator that uses the frequency correlation of

the channel. In this paper, the same facilitating process is ap-

plied to estimate the comb-ty pe pilot signals. The key idea to

reduce the complexity is using the singular-value decomposi-

tion (SVD) to de rive an optimal low-rank estimator, where per-

forma nce is essentially preserved. Th e mathematical represen-

tation for M MS E estimator of pilot signals is as follows [6]:



220 lEEE Transactions on Consumer Electronics,Vol. 44, o. 1,FEBRUARY 1998

ji

*;c

g

0 0 0 0 0

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

0000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

.000.000.000.000

*

2-

2

Fig. 2. Tw o different types of pilot sub-carrier arrangement

0000000000000000

0000000000000000................000000000000000

0000000000000000

0000000000000000

0000000000000000

0000000000000000

................000000000000000

0000000000000000

0000000000000000.................................

t000000000000000 (Ai),

where HP,lss the least-squares estimate of HI, as shown in

(12), g i is the variance of W ( k ) , nd the cova riance matrices

are defined by

RH,H, = E{H,HE},

RH,H, ,~, = E { ~ p ~ p " , l ~ >

R H , , ~ ~ H , , ~ ~ E HmHF,lS}

Note that there is a matrix inverse involved in the MMSE es-

timator, which must be calculated every time. This proble m

can be solved by using static pilots such as X , ( ? n ) = c fo r

m = 0, 1, . ,N, - 1. A more generic solution is to average

over the transmitted data, and a simplified linear MM SE estima-

to r of' pilot signals is obtaine d as [6]:

-1

H P , l S

where SNR = E ( X , ( k ) /D : is the average signal-to-noise ra-

tio, and p = E i X p ( k ) i 2 11 / Xp(k )12 s a constant depend-

ing on the signal constellation For 16-Q AM transmission,

p = 17 /9 If the auto-co rrelation matrix R H , H ~nd SNR are

known in advance, RH,H, (RH,H, + &I) needs to be

calculated only once Altho ugh the simplified LMMSE estima

tor avoids the matrix inverse ope rahon, the com putational com

plexity is still very high As shown in (14), the estimation re-

quires N, complex multiplications per pilot tone To reduc e the

number of multiplication opeiations, a low-rank app ioximation

using singular value decomposition IS adopted [6] The channelcorrelation matrix is first decom posed as

-1

RH,H, = UhUH (15)

where U IS a matrix with orthonormal columns U O , u1, ,

U N , - ~ an d A is a diagonal matrix, containing the singular val-

ues X(0) < X(1) < 5 X(Np 1) 5 0 on its diagonal

The best rank-m approximation of the estimator in (14) then be-

comes

H, = U [ 4;.. ; UHHp,ls,

where A, is a diagonal matrix con tainin g the values

. k = O , l , . . . . m (17)(k)

X(k) + SNRS(k)=

After some facilitations, the estimator in (16) thus requires

2mNpmultiplications and the total number of multiplications

per pilot tone becomes 2m. In gen eral, the number of signifi-

cant singular value 772 is much smaller than N,, and the com-

putational complexity is reduced considerably comparing to the

full rank estimator (14).

The low-rank MMSE estimator is insensitive to parameter

mismatch [6]. That is, it is possible to design a generic rank-

m estimator for a wide range of channels.

IV . C H A N N E LNTERPOLATIONAfter the estimation of the ch anne l transfer functions of pilot

tones, the channel responses o f data tones can be interpolated

according to adjacent pilot tones. T he piecewise-linear interpo-

lation metho d has been studied in [2] , nd is shown to be better

.than the piecewise-constant inte rpolatio n. In this paper, we con -

sider a piecewise-linear and a piecewise second-order polyno-

mial interpolation me thod which d ue to their inherent simplicity

are easy to implem ent.

A. Linear interpolarion method

In the linear interpolation algorith m, two successive pilot sub-

carriers are used to determine the c hannel response for data sub-

carriers that are located in be twee n the pilots [ 2 ] .For data sub-carrier k , mL < k < (m+ l)L, he estimated channel response

using linear interpolation m ethod is given by

I 7 ,

H ( k ) = H(mL+ ) = 1- - Ei,(m)+ L Q mL e) + 1)

The linear channel interpolation can be implemented by using

digital filtering su ch as the Farrow-structure [81. Furthermore,



Hsieh and Wei: Channel Estimation forOFDM Systems Based onComb-Type Pilot Arrangement in Frequency Selective Fading Channels 22 1

by carefully inspecting (1 8), we find that if L is chosen as the

power of 2, the multiplication operations involved in (18) can

be replaced by shift opera tions, and therefore no multiplication

operation is needed in the linear ch anne l interpolation.

B. Second-order polynomial interpolation method

Theoretically, using hig her-ord er polynomial interpolation

will fit the channel respon se better than t he linear interpolation.

However, the computational complexity g rows as the orde r is in-

creased. Here we consider the seco nd-ord er polynomial interpo-

lation for its acceptable co mpu tationa l complexity. A piecewise

second-order polynomial interpolation can be im plem ented as a

linear time-invariant FIR filter [9]. T he interpolator is given by

where

a(oi+1)

2

cl) = - ( C Y - l)(CY+l)

c-l = 4 a - 1 ){ ” ’ = 2

and CY = l /N .

C. Phase compensated interpolation

For a realistic OFDM system, the frame position error will

affect the channel estimation. Bec ause of the low-pass filtering

in the receiver, the sampled channe l impulse is dispersed and

introduces energy leaks, especially, when th e path delay time is

non-T-spaced [ 5 ] .Due to the multipath propagation and the en-

ergy leaks due to sam pling, the guard interval will be affected by

the preceding symbol. To avoid the inter-symbol interference,

the frame synchronization must pro vide a portion {g(n)} f the

samples {yg(n)} f length N that is influenced by one transmit-

ted symbol only. Let the correct position of the FFT window is

p = q(N, + N ) where q is a positive integer and p E he offset

with respect to this position. In [lo], a frame m isalignment is

identified as two situations as sho wn in Fig. 4.

Suppose that the start position of the FFT window is within

region A, then no IS1 occurs. Th e only effect suffered by the

subchannel symbols is a chan ge in phase that increases with the

subcarrier index. If the start position is within regions B, th e

subcarrier symbols will suffer from IS1 in addition to the pha se

rotation, and the orthogonality of the system is disturbed. In

this paper, we assume that the coa rse frame synchronization has

pulled the fram e start position p , within region A. Thus the re-ceived signal after FF T beco mes [101

Y ( k ) = X ( k ) H ‘ ( k )+ W ’ ( k )= x k ) j y ( +--324Pe - l V g ) / N

+W(j+-”2”(P‘-Ns)IN. (20 )

From Fig. 4, we see that p E 5 Ng when p , is in region A.

Therefore, a group delay exists at the OFD M receiver before de-

multiplexing. The group delay in time-domain introduces pha se

rotations after demultiplexing, which introdu ce mod el mismatch

AT IVhannel

impulse -+

response \

I B ~ A /B14 ’ I

I I *; ,

{ Y ( / c l }

FFTWindow

Fig. 4. Principle of frame synchronization [lo]

Fig. 5. The influence of frame position error to interpolations.

error for channel interpolation algorithms. As shown in Fig. 5 ,

the linear interpolation algorithm introduces significant errors

when the group delay is present. In additio n, the model mis-

matc h error grows as the group time delay increases. Th e model

mismatch problem is more serious for the comb-ty pe pilot sys-

tem than the block-type pilot system, since the comb-type pi-

lot system uses interpolation to determ ine the channel response

of the data carriers, and the pilot carriers are separated more

widely. In addition, the MM SE estimator discussed in Sec-

tion I11 is sensitive to the timing error. Th e reason is that the

MM SE estimator uses a pre-estimated c hann el statistics to esti-

mate the pilot signals. If the frame position error is present, the

MMSE estimator will fail due to mism atch of channel statistics.

Here we propose a method to solve the model mismatch prob-

lem as shown in Fig. 6. Th e estimated channel transfer function

of pilot signals when a group delay in time domain is present

can be represented by

(21)

where 6, is the change in phase caused by frame error and

E H ~m ) s the estimation error. We can see that {f i , (m)} s a

sequence that with a single frequency. There fore, the estimators

presented in [111 can b e applied to estimate O P . The estimator in

[111 that provides the best overall performan ce is

Gp(nz) H , ( m ) . e-i*P” + E f f p (m ) ,

N p - 2

e, = w,Lfi;(m)H,(m + 1) (22)nz=O



222 IEEE Transactions on Consumer Electronics,Vol. 44 , No . 1,FEBRUARY 1998

0 1

qYQ

0.01

Z l a r s e E I DFT Pilot 1stimates- rame Signal--)- of pilot- ync. Extraction- signals

. . . . . . . . . ........................;-.

/-..- ~

-.-,_-I

, ,

,,’,I’

:

~

,._-__.j...........................................................

H p , k ( k ) H p , l s ( q f 4 L l m m s e ( k ) fi(W

low-rankInter-

MMSE compen-polation

estimator sation

Fig 6 Channel estimation w ith phase compensation

where

After 4 is obtained , the estimated channel transfer function of

pilot subcarriers is compe nsate d by it as:

Hp(m) = I Z , ( m ) . e j Q p m

M H,(m) +EL, m ) (24)

where ELp m )= E H ~m) e j opm .

After the phase pre-com pensation, the channel transfer func-

tions of data carriers a re interpolate d by using linear or higher-

order polynomial interpolations. Because the group delay is

compensated by (24), the model mismatch error is reduced con-

siderably. Then, after interpola tion, the pha se cha nge is restored

as

Fig. 7 shows the influence of fra me position error on the cha n-

nel estimation. The FFT size N = 1024, total number of pilot

subcarriers N, s 1 28, and Eb/No equals 10 dB. The AWGN

channel is adopted, with the delay time h o m 0 to 100 samples.

From Fig. 7, we can see that interpolations without phase c om-

pensation will suffer from frame position error, even there is no

ISI. The phase com pensator is not affected by fram e position er-

ror for 8, < rr. When QP is larger than r (corresponding to the

64th sample shown in Fig. 7), the phase estimator suffers from

phase ambiguity and therefore the compensator failed. How-

ever, since the phase change 0, is mo re probable to be positive,

the applicable range of theA phase om pensator can be enlarged

by changing the range of 8, from [ - T , T ) o [- T -k 4 , ~4)

From Fig. 7, we can see that the performances of linear andpolynomial interpolations a re not sensitive to fram e position er-

rors when the frame offset is small. This enlight us to use a

simpler estimator of phase cha nge instead of (22). We choose a

simpler estimator from [ l11 and the phase estimator is

where 0 5 & 5 71.

- N p - 21

e, = 1 fi;(m)fiP(7rL-)

m=O- 1

According to our own simula tions, the performa nces of channel

estimator applying the phase compensators shown in (22) an d

(26) are almost the same .

Linear intbrpo~ation-nd-order polynomial interpolation .......

Linear nterpolation with phase compensation . .

0001 I I0 20 40 60 80 . 100

Delag ( s a m p l e s )

Fig 7 The influence of gioup delay to the interpolation Eb/No = 10dB,N =

1024, N p = 128

V. SIMULATIONS

The proposed channel estimation method is evaluated underslow- and fast-fading radio chan nels. As in [ 3 ] , he sam e simula-

tion environments a re used in this paper. An OF DM system with

symbols modulated by 16-QA M is used. The carrier frequency

is 1 GHz, and the bandwidth is 2 MHz. Th e total number of sub-

channels is N = 1024, and the n umb er of uniformly-distributed

pilot carriers is N p = 128. The channel models adopted are

Rayleigh and Rician as recommended by GSM Recommenda-

tions 05.05 [12], with parameter shown in TABLE I. Here we

assume that the guard intervals are longer than the maximum

delay spread of the channel. To avoid IS1 arising from energy

leakages caused by sampling, a frame position offset of 30 sam-

ples is applied. The sampled channel impulse response in time-

domain is shown in Fig. 8 . We can see that there are energy

leakages ahead the main path sam ple.

Several channel estimation metho ds, LS estimate of pilot with

linear interpolation, MMSE estimate of pilot with linear inter-

polation, MMSE estimate of pilot with 2nd-order polynomial

interpolation, MMSE estimate of pilot with phase compensated

interpolations, and channel estimation using transform domain

proce,ssing technique, are simulated and compared. The sim-

ulations are carried out for different noise and IC1 levels, and

the. results are shown in Fig. 9-12. Th e horizontal variable is

chosen as th e signal energy per bit-to-noise power density ratio

(Eb,”,-,). T he IC1 levels are controlled by the vehicle speed.



Hsieh and Wei: Channel Estimation forOFDM Systems Based onComb-Type Pilot Arrangement in Frequency Selective Fading Channels

Path

Number

1

223

-

Rayleigh Channel Rician Channel

Average, Average.

Power Delay (ps) Power Delay (ps)

dB

-3.0 0.0 0.0 0.0

(dB)

2

3

4

0.0 0.2 -4.0 0.1

-2.0 0.5 -8.0 0.2

-6.0 1.6 -12.0 0.35

6

For low vehicle speed V = 6 k d h r , the corresponding max-

imum Doppler spread is 0.3 % of the subcarrier spacing. Thus,

the IC1 can be neglected. For high vehicle speed V = 150k d h r, the corresponding maximum Doppler spread is then 7.5

% of the subcarrier spacing. Th us, the system performancesare strongly affected by ICI. After decomposition of the auto-

correlation matrices of Rayleigh and Rician channels, we find

the sufficient rank of M MS E estim ators are are 6 and 4, respec-

tively. However, when we use the rank-6 estimator designed

for Rayleigh fading channel as the pilot signal estimator of Ri-

cian fading channel, the perform ance degradation is neglectable.

Thus, a generic low-rank estimator can be used for slow- and

fast-fading Rayleigh and Rician fading channels.

From Figs. 9-12, we see that the performan ce is improved

significantly when the low-rank MMSE estimator is applied.

The phase compensated interpolation successfully solves the

model mismatch problem, and gain s a significant improvem ent

in Eb/No for the same BER values in all cases. We als o noticethat the higher-order polynomial interpolation only improves the

performance little. It is suspectable to use higher-order poly-

nomial interpolation instead of linear interpolation for channel

with small delay sprea d. How ever, we expect that the improve-

ment in performance will be more if the delay spread of channel

is larger. Th e perfor man ce of Zhao and Huang 's algorithm is

limited by the low-pass filtering in transform d oma in. In higher

SNR , the low-pass filtering operation discards som e channel in-

formation and thus a BER floor is formed.

From the simulations shown ab ove, we see for comb-ty pe pi-

lot assisted OFDM systems, the channel estimation based low-

rank MMSE pilot estimator with phase compensated linear in-

terpolation is suggested. Th e proposed me thod elimina te IC1and AWGN by using the low-rank M MS E pilot estimation, and

the model mismatch error is reduced by using linear interpola-

tion with phase compensation. It should be note that although

the proposed algorithm provided better performance, the SNR

and channel statistics have to be estimated in advan ce. Wherea s

the channel estimation based on transform domain processing

needs not to know these informations in advance.

J

-8.0 2.3 -16.8 0.4

-10.0 5.0 -20.0 I 0.5

V I. C O M P U T A T I O N A LO M P L E X I T Y

In this section, we com pare the co mputational complexity of

the proposed algorithm and the channel estimation algorithm

' " t i

I0 16 32 48 64 80 96 112 128

le-06

snnrples

(b )

Fig. 8. Typical channel impulse of (a) Rician and (b ) Rayleigh fading channels.

presented in [ 3 ] . We evaluate the computational complexity

roughly by calculate the number of complex multiplications

used per tone to estimate the cha nnel transfer function . The pro-

posed method uses low-rank MMSE estimator, which requires

2mNp complex multiplications where m is the rank of the es-

timator. Th e phase estimator requires Np complex multiplica-

tions, and the phase compensator requires N complex multi-

plications. As shown in (18), the linear interpolation requires

no multiplications. Th us, the prop osed metho d totally requires

2mNp+Np+N com plex multiplications, w hich is equivalent

to 1+ complex multiplications per tone. The chan-

nel estimator based on transform domain processing uses twoFFTs to estim ate the channel response, one is of size N p and

the other is of size N . By using the decimation-in-frequency

radix-4 implementation of FFT, the total number complex mul-

tiplications required is roughly Np og , Np + Nlog, N . Note

that about one or two multiplications can be saved for FFT be -

cause of som e trivial multiplications. By using the parameters in

Section V , the proposed channel estimator only requires about

2 complex multiplications per tone, while the channel estima-

tor proposed by 131 requires at least 4-5 complex multiplica-

tions per tone by conservative esti mate. Therefore, the proposed

chan nel estimator is com putati onal efficient and is attractive for

(2nz+ 1 Np



224 IEEETransactions on ConsumerElectronics, Vol. 44, No. 1, FEBRUARY 1998

0 1

0.01

0.001

Fig. 9. Comparisons of BERs in slow Rayleigh fading channel (vehicle speed

= 6 k d h r ) .

0 1

w

2

MMSE +l inearMMSE + poly

MMSE + Phase compensated linearMMSE + Phase compensated poly

O o l5Y0 15 20 25 30

&/No

Fig. 10. Comparisons of BERs in fast Rayleigh fading channel (vehicle speed

= 15 0 kmkr) .

a;YQ

0 1

0 1

0 001

Fig 11 C ompansons of BER s in slow Rician fad ing channel (vehicle speed =

6 k d h r )

0 1

wNQ

0 1

5 10 i s 20 25 30

&/N o

Fig. 12. Com parisons of BER s in fast Rician fading channel (vehicle speed =

150 kmihr).

OFDM systems in mobile commu nications

V I I . C O N C L U S I O N

In this paper, a low-complexity channel estimator based on

comb-type pilot arrangement is presented. T he channel transfer

function of pilot tones are estimated by using low-rank MMSE

estimator, and the chan nel transfer fun ction of data tones are in-

terpolated by piecewise linear interpolation method with phase

compen sation. Simulations show that the performance of the

proposed channel estimator o utperfo rms the channel estimator

based on transform domain processing presented by Zhao and

Huang [ 3 ] .Th e computational complexity of the proposed chan-

nel estimator is also studied, and is lower than the previous chan-

nel estimator in [ 3 ] . n co nclusion, the proposed channel estima-

tor provides a practical channel estimation for OFDM systems

based on com b-type pilot a rrangement.

A C K N O W L E D G M E N T

This work was supported by the N ational Science Council of

Republic of China under contract NSC 86-2221-E-009-053.

REFERENCES

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tronics, vol. 42 , no. 4, Nov. 1996.Y. hao and A . Huang, “A novel channel estimation method for OFDM

mobile communication systems based on pilot signals and transform-domain processing,” in P~ o c .EEE 47th Vehicular Technology Con ference,Phoenix, USA , May 1 997, pp. 2089-2093.

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[9 ]

BIOGRAPHY

Meng-Han Hsieh (S’96) was born in Taiwan,R.O.C., in 1968. He received the B.S. and M.S.degrees in electronics engineering form NationalChiao Tung University,R.O.C., in 1990 and 1992,respectively.

He is currently working toward the Ph.D. de-

gree at the same school. His research interestsinclude multi-camer signal processing, synchro-nization and channel estimation algorithms.

Che-Ho Wei (S’73-M’76-SM’S7-F196)was bornin Taiwan, R.O.C., in 1946. He received his B.S.and M.S. degrees in electronics engineering fromthe National Chiao Tung University, Hsin Chu,Taiwan, in 1968 and 1970, respectively, and hisPh.D degree in electrical engineering from the

University of Washington, Seattle, in 1976.From 1976 to 1979, he was an associate pro-

fessor at the National Chiao Tung University,where he is now a Professor with the Departmentof Electronics Engineering and Dean of Collegeof Electrical Engineering and Computer Science.From 1979 to 198 2, he was the engineering man-ager of Wang Industrial Company in Taipei. He

was the Chairman of the Department of Electronics Engineering of NCTU from1982 o 1986, and Director of Institute of Electronics from 1984 to 1989. He wason leave at the Ministry of Education and served as the Director of the Advisory

Office from September 1990 to July 1992 . His research interests include digitalcommunications and signal processing, and related VLSI circuits design.

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