Adaptive Prediction Iterative Channel Estimation for
OFDM Signal Reception in a Frequency Selective
Fading Channel
Shinsuke Takaoka and Fumiyuki Adachi
Electrical and Communication Engineering, Graduate School of Engineering, Tohoku University
E-mail: takaoka@mobile.ecei.tohoku.ac.jp, adachi@ecei.tohoku.ac.jp
Abstract- In this paper, pilot-assisted adaptive prediction iterative
channel estimation is proposed for the antenna diversity reception oforthogonal frequency division multiplexing (OFDM) signals.
Adaptive prediction filtering is incorporated into iterative channel
estimation process with decision feedback and reverse modulation.
The filter tap weights are adaptively updated according to changes
in the multipath propagation environment. For the adaptation of tap
weights, the normalized least mean square (NLMS) algorithm isapplied that uses the noisy instantaneous channel gains obtained byreverse modulation and from pilot symbols as the reference signal.
The average bit error rate (BER) performance in a frequency
selective fading channel is evaluated by computer simulation for thefrequency and time multiplexed pilot cases. It is confirmed that
adaptive prediction iterative channel estimation provides better BER
performance than using fixed-tap filter.

1. Introduction
Ultra high-speed (100Mbps~1Gbps) data transmission
capability will be required in the next generation broadband
wireless communication systems. In mobile radio, thetransmitted signal is reflected and diffracted by many
obstacles and is received as a multipath signal with different
time delays at a receiver. The frequency transfer function of
the multipath channel varies randomly both in frequency andin time, that is, the transmitted signal undergoes doubly(frequency and time) selective fading. For achieving
high-speed and high-quality data transmission in such a
doubly selective fading channel, orthogonal frequency
division multiplexing (OFDM) is considered as a promising
wireless transmission technique [1], [2].
For coherent detection of OFDM signals, accurate
channel estimation is necessary. Many channel estimation
methods have been proposed [3]-[6]. Pilot-assisted iterative
channel estimation using decision feedback and reverse
modulation is known to be able to improve the channel
estimation accuracy. In [7], iterative channel estimation with
fixed-tap filter was applied. However, multipath propagation
environment between the transmitter and receiver changes
rapidly according to user’s movement. Therefore, using thetime invariant tap weights cannot always minimize the bit
error rate (BER) in changing propagation environments.
Recently proposed adaptive selection of the tap weights offrequency-domain filter [8] provides superior performance tothe fixed-tap filter. However, its adaptability to rapidlychanging channel conditions is limited.
In this paper, pilot-assisted adaptive prediction iterative
channel estimation is proposed for the antenna diversity

reception of OFDM signals. Adaptive prediction filtering isincorporated into the iterative channel estimation process. The
filter tap weights are adaptively updated according to changes
in the propagation environment. For the adaptation of tap
weights, the normalized least mean square (NLMS) algorithm
is applied that uses the noisy instantaneous channel gains
obtained by reverse modulation and from pilot symbols as thereference signal.
The rest of this paper is organized as follows. In Sect. 2,
the OFDM signal transmission system model is presented.
Section 3 describes the proposed adaptive prediction iterative
channel estimation methods for the frequency and time
multiplexed pilot cases. Then, Sect. 4 presents the computer
simulation results on the achievable BER performance in afrequency selective Rayleigh fading channel. Finally, Sect. 5
offers some conclusions.
2. OFDM Signal Transmission System Model
The OFDM transmission system model considered inthis paper is illustrated in Fig. 1. OFDM signal transmission
using Nc subcarriers is assumed. In the transmitter, the binary
information date sequence is transformed into the quadraturephase shift keying (QPSK) modulated symbol sequence {dn}.
Then, the known pilot symbols are periodically inserted into
the transmitted data symbol sequence. After the transmitted
symbol sequence is serial-parallel (S/P) converted into Nc
parallel data sequences each with a lower rate, OFDM signal
waveform with Nc subcarriers are generated by applying
Nc-point inverse fast Fourier transform (IFFT). Finally, thecyclic prefix of Ng samples is inserted as the guard interval inorder to eliminate the inter-carrier interference (ICI) resulting
from frequency selective fading.
The baseband equivalent representation of the OFDM
signal waveform for the i-th OFDM signaling period is givenbyN −1s (k ) =c
2E sd iN + n( N c + N g )Tc n =0 c

¦

§
·
n
× exp¨¨ − j 2π
(k − N g − i ( N c + N g )) ¸¸
Nc
©
¹

(1)

for k=i(Nc+ Ng) ~ (i+1)·(Nc+ Ng)-1, where Es represents thetransmitted signal energy per data symbol and Tc represents

variance 2(N0/Tc)Nc, respectively, at the nth frequency. Then,after parallel-to-serial (P/S) conversion, antenna diversity
reception using maximum ratio combining (MRC) and data
decision is carried out to obtain the received binary data
sequence {dˆ } .

Transmitter
Data

P/S

IFFT

Pilot
symbol

S/P

MUX

Data
modulatorGI
insersionn

For coherent detection, estimation of {Hm(i,n)} is
necessary. In Sect. 3, we will propose the pilot-assisted
adaptive prediction iterative channel estimation.

Multipathfading channel

Receiver#M-1 antenna

Recovereddata

Remodulator

Fig. 1 OFDM signal transmission system model.
the IFFT sample interval given by T/(Nc+ Ng) when the data
symbol rate per subcarrier is 1/T. The resultant OFDM signal
waveform is transmitted via a frequency selective fading
channel and received by M antennas for diversity reception.
Α Tc-spaced delay-time model of the propagation
channel is assumed. Assuming that the channel has L
independent propagation paths with distinct time delays {lTc},
the discrete-time impulse response hm (t ) of the multipath
channel experienced by the mth antenna, m=0~M-1, may be
expressed ashm (t ) =L −1¦

Adaptive Prediction Iterative Channel Estimation
For channel estimation, the frequency multiplexed pilot
and time multiplexed pilot are considered, as illustrated in Fig.
2. For the former, pilot subcarrier is inserted every Np
subcarriers in frequency, while pilot symbol is inserted every
Np symbols in time for the latter. A 2K-tap adaptive prediction
filter illustrated in Fig. 3 is used in the pilot-assisted adaptive
prediction iterative channel estimation. The operation
principle of the adaptive iterative channel estimation is
described below.
Np
Frequency (n)Iterative
channel estimator

3.

Data
decision

Frequency (n)

P/S

*

FFT

S/P

GI
removalMRC
combiner#0 antenna

#3
#2
#1
#0

Np
#0 #1 #2 #3

･･･

L −1

m ,l

2

] = 1 , where δ(t) is the delta function andE[.] denotes ensemble average operation. Assuming very slow
fading so that the channel impulse response remains constantover one OFDM signaling interval, time dependency of thechannel has been dropped for simplicity. It is assumed that the
maximum time delay of the channel is shorter than GI.
The received signal sample sequence on the mthantenna is
rm (k ) =L −1

¦ h s(k − l ) + µ (k ) , (3)
m ,l

m

l =0

where µm(k) is the additive white Gaussian noise (AWGN)
process with zero mean and variance 2N0/Tc. After removal of
the guard interval, the received signal sample sequence isresolved into Nc-subcarrier components by applying Nc-point
FFT to {rm(k)} to obtain

Rm (i, n) =Time (i)(b) Time multiplexed pilot

Fig. 2 Pilot insertion methods.

hm,l δ (t − lTc ) (2)¦ E[ hl =0#0 #1 #2 #3

Time (i)(a) Frequency multiplexed pilot

l =0

with

#3
#2
#1
#0

2Es
d iN c + n H m (i, n) + Π m (i, n) (4)( N c + N g )Tc

for the n-th frequency component, where Hm(i,n) and Πm(i,n)
are the channel gain and the zero-mean noise sample having

wm(j,−Κ)･･･
wm(j, 1)wm(j,−1)･･･

wm(j,Κ)･･･

Wm ( j + 1)

-+

･･･

NLMS algorithm

Normalization~
H m( p ) (i , n )Fig. 3 Adaptive prediction filter structure.
3.1 Frequency multiplexed pilot caseAt the first iteration stage (p=1), the channel gains(1)ˆ{H m (i, n); n = 0, N p , 2 N p , } at the pilot subcarriers are

estimated using the received pilot subcarriers only. Next, thefirst order interpolation and extrapolation are applied to obtain~the channel gain estimates {H m(1) (i, n); n = 0 ~ N c − 1} at all
subcarriers. Then, antenna diversity using MRC is applied
after P/S conversion. The decision variables {η (1) (i, n)} forthe data symbol sequence {diNc+n; n=0~Nc-1} are obtained as

M −1

¦

out after antenna diversity reception.

~ ∗Rm (i, n)H m( p ) (i, n)η ( p ) (i, n) = m =0M −1¦

2~
H m( p ) (i, n)

(5)

m =0

for p=1, where * denotes complex conjugate operations.
Using decision variables {η (1) (i, n)} , the tentative decisions
data {dˆ (1) (i, n)} are obtained asdˆ ( p ) (i, n) =mind ∈ {exp( jkπ / 2); k = 0 ~ 3}

η ( p ) (i, n) − d (6)for p=1.
At succeeding iteration stages (p≥2), the tentativedecisions {dˆ ( p−1) (i, n)} at the (p-1)th iteration stage are
fedback as pilots to remove data modulation from thereceived signal frequency components {Rm(i,n)} and theinstantaneous
channelgainestimates( p)ˆ{H m (i, n); n = 0 ~ N c − 1} of all subcarriers are obtained :Hˆ m( p ) (i, n) = Rm (i, n)dˆ ( p −1)* (i, n) for p≥2.

(7)Adaptive prediction filtering is applied to theinstantaneous channel gain estimates {Hˆ m( p ) (i, n)} to obtain~
the improved channel gain estimates {H m( p ) (i, n)} . The
adaptively estimated channel gain at the pth iteration stage is
given by­°°Hˆ ( p) (i, n) for n = 0,N − 1c° mk =α (n)°~1°
H m( p) (i, n) = ® k =α (n)wm ( j, k )Hˆ m( p) (i, n + k ) , (8)°wm ( j, k ) kk =≠−0α (n)°° k =−α (n)° k ≠0°¯ otherwise¦¦

where wm(j,k) denotes the filter tap weight and α(n) is definedasfor 1 ≤ n ≤ K − 1­n°α (n) = ® K
for K ≤ n ≤ N c − K − 1 . (9)° N − 1 − n for N − K ≤ n ≤ N − 2cc¯ cAgain, using the adaptively estimated channel gain estimates~{H m( p ) (i, n)} , MRC antenna diversity reception of Eq. (5) iscarried out. After repeating the above-mentioned process for(P-1) times, the final decision variables {η ( P ) (i, n)} at thePth iteration stage are obtained from Eq. (5). In the proposed
iterative channel estimation accurate channel estimation ispossible, since the tentative decision using Eq. (6) is carried

3.2 Time multiplexed pilot case
Adaptive prediction channel estimation is performed
using received pilot OFDM signal waveform at the signaling
period of i mod Np=0. The resultant channel gain estimates are~used as the first stage channel gain estimates {H m(1) (i, n)} forthe iterative channel estimation at time i mod Np=1. For i mod~
Np>1, the Pth stage channel gain estimates {H m( P ) (i − 1, n)} in
the previous signaling period of i-1 are utilized as the first~stage channel gain estimates {H m(1) (i, n)} . As a consequence,~
H m(1) (i, n) is given by­°°°Rm (i − 1, n) for n = 0, N c − 1
k =α (n)°~1°wm ( j, k )Rm (i − 1, n + k ) (10)H m(1) (i, n) = ® k =α (n)°k =−α (n)wm ( j, k ) k ≠0
°
(
)
=
−
α
k
n
°
° k ≠0°¯ otherwise n ≠ 0, N c − 1¦¦when i mod Np=1, and~~H m(1) (i, n) = H m( P) (i − 1, n) , (11)when i mod Np>1; (1+j0) is assumed for the pilot symbol. The
succeeding iteration stages (p≥2) are the same as in thefrequency multiplexed pilot case.
3.3 Tap weight adaptation
Updating of tap weights is incorporated into theiterative channel estimation loop within one OFDM signaling
period. NLMS adaptation algorithm [9] is applied that uses
the noisy instantaneous channel gain estimates {Hˆ m( p ) (i, n)} .
The recursive relation for updating the tap weight vector isrepresented as

Wm ( j + 1)
for 0 ≤ n ≤ K − 1 and­0°
Nc − K ≤ n ≤ Nc −1°°
e m ( n)X *m (i, n) , (12)°µ K
= Wm ( j ) + ®
2
Hˆ m( p ) (i, n + k )°° k =− K° k ≠0°for K ≤ n ≤ N c − K − 1¯¦

where em(n) is the estimation error given bye m (n) = Hˆ m( p ) (i, n) − WmT ( j ) X m (i, n) , (13)µ is the step size, and [.]T denotes transpose. In Eq. (12),­Wm ( j ) = [ wm ( j ,− K ),  , wm ( j ,−1),°wm ( j ,1),  , wm ( j , K )]T°(14)®
ˆ ( p)
ˆ ( p)°X m (i, n) = [ H m (i , n − K ),  , H m (i, n − 1),°
Hˆ m( p ) (i, n + 1),  , Hˆ m( p ) (i, n + K )]T¯

the time multiplexed pilot case, more accurate channel
estimation is available at the first iteration stage. Hence, in thefollowing, only the time multiplexed pilot case is considered.
1.E-01

Without diversity

are the complex tap weight vector and the instantaneous
channel gain vector, respectively.

1.E-02Average BER

4. Computer Simulation
The simulation condition is summarized in Table 1.
The number of OFDM subcarriers is Nc=256. The multipath
fading channel is assumed to have an L=8-exponentiallydecaying power delay profile with time delay separation offour FFT sample 4Tc, as shown in Fig. 4. Each path is subject
to an independent Rayleigh fading. It is assumed that the
maximum time delay is shorter than the guard interval of Ng
samples and the complex channel gain remains almost
constant over the one OFDM signaling period T=(Nc+Ng)Tc.
At the receiver, M=2-branch MRC antenna diversity reception
is assumed. The number of taps of adaptive prediction filter is
2K=24 for both frequency and time multiplexed pilot cases.

Channel model
Antenna diversity

With diversity

τrms /Ts =0.00549 (β=10dB)
τrms /Ts =0.0161 (β=4dB)

1

Relative power (dB)

3

4

5

6

Fig. 5 Impact of number of iterations.
Without diversity

1.E-01

1.E-02

1.E-03#0τrms/Ts=0.0358, Np=100
fDT=0.001, 2K=24, P=5
ProposedSA (3subcarriers)
SA (5subcarriers)
SA (7subcarriers)
Ideal estimation

#1
#2

2

No. of iterations (P)

1.E+00QPSK
Nc=256
Ns/8
L=8-path Rayleigh,Exponential power delay
profileM=2-MRC□ Time mux. pilot (Np =100)△ Freq. mux. pilot (Np =10)

1.E-05

Average BER

OFDM

1.E-03

1.E-04

Table 1 Simulation condition
Data modulation
Number of subcarriers
Guard interval length: Ng

Average Eb /N0 =20dB
fD T=0.001, 2K=24

1.E-04βdB decayWith diversity

1.E-05#L-1
・・・・・・・

Time delay

0

10
20
Average received Eb/N0 per antenna (dB)30

Fig. 4 Power delay profile of multipath channel.

(a) τrms/Ts =0.0358
Fig. 6 Performance comparison.

Figure 5 plots the average BER as a function of
number P of iterations with the normalized delay spreadτrms/Ts of the channel as a parameter at the average received
Eb/N0=20dB, where 1/Ts (=(1+Ng/Nc)/T) is the subcarrier
separation. The pilot insertion of Np=10 (100) for thefrequency (time) multiplexed pilot case and the normalized
Doppler frequency fDT=0.001 are assumed. It is clearly seenthat iterative channel estimation can significantly reduce the
BER. The BER reduces as the number of iterations increases;
however, almost no additional improvement is obtained after
five iterations. Therefore, the use of five iterations (P=5) is
considered to be sufficient. Also, it is observed that the time
multiplexed pilot case provides almost same or smaller BERs
than the frequency multiplexed pilot case. This is because in

The BER performances achievable by the adaptive
prediction iterative channel estimation are compared with theiterative channel estimation using simple averaging (SA) filter
as a fixed-tap filter. Figure 6 plots the average BER
performance as a function of the average received signal
energy per bit-to-background noise power spectrum density
ratio (Eb/N0) per antenna for τrms/Ts=0.00549 (β=10dB),
0.0161 (β=4dB) and 0.0358 (β=0dB). It is found from Fig. 6that the optimum SA filter size is different for different valueof τrms/Ts and that the use of adaptive prediction filter always
provides better BER performance than the SA filter for bothpilot cases. In the case of τrms/Ts=0.0161, the adaptive
prediction iterative channel estimation (time multiplexed pilotcase) reduces the required average Eb/N0 per antenna forachieving BER=10-3 by about 0.3, 0.4 and 1.0 dB than SA
filter with 5, 3 and 7 taps, respectively, and approaches theideal channel estimation case by about 1.9 dB.
Reducing the pilot insertion interval Np improves the
tracking ability of channel estimation against channel
variations in time and frequency, however reduces thetransmission rate. Np=10 (100) produces the reduction in thedata rate of 10 (1) %. Therefore, the use of longer pilot
interval is desirable. Figure 7 plots how the pilot insertion
interval impacts the achievable BER. As the value of Np
increases, the BER increases since the tracking ability ofchannel estimation tends to be lost. It is observed from Fig. 7that for achieving the same BER, the use of the proposed
adaptive prediction filter can enlarge the pilot insertion
interval compared to the SA filter.
1.E+00Without diversity

1.E-01

ProposedSA (3subcarriers)
SA (5subcarriers)
SA (7subcarriers)
Ideal estimation

reception of OFDM signals in a frequency selective fading
channel. The adaptive prediction filter is incorporated into theiterative channel estimation process. The average BER
performance achievable by the use of the proposed
pilot-assisted channel estimation was evaluated by computer
simulation. Frequency multiplexed pilot and time multiplexedpilot were considered. It was found that the use of time
multiplexed pilot provides better BER performance because
the channel estimation accuracy at the first stage is better withtime multiplexed pilot than with frequency multiplexed pilot.
The performance comparison of iterative channel estimation
using adaptive prediction filtering and fix-tap filtering showedthat the former always provides a better BER performance.
Also, confirmed by the computer simulation was that forachieving the same BER, the use of adaptive filtering canenlarge the pilot insertion interval and therefore can increase
the transmission rate.
1.E+00

Average Eb /N0 =20dB
τrms /Ts =0.0161, P=5
2K=24

fD T=0.01

fD T=0.001

Average BER

Average BER

1.E-01
1.E-02

1.E-03
With diversity

1.E-02
Without diversity

1.E-03fD T=0.01

1.E-04
τrms/Ts=0.0161, Np=100
fDT=0.001, 2K=24, P=5

1.E-04

1.E-050

10
20
Average received Eb/N0 per antenna (dB)30

With diversity

1.E-05(b) τrms/Ts=0.0161

0

Average BER

1.E+00

1.E-01

ProposedSA (3subcarriers)
SA (5subcarriers)
SA (7subcarriers)
Ideal estimation

1.E-02

Without diversity

1.E-03

fDT=0.001, 2K=24, P=5
1.E-05
10
20
Average received Eb/N0 per antenna (dB)200
300
Pilot insertion interval (Np)400

Reference

τ rms/Ts=0.00549, Np=1000ProposedSA (3subcarriers)
SA (5subcarriers)
SA (7subcarriers)

Fig. 7 Impact of pilot insertion interval.

With diversity

1.E-04100

fDT=0.001

30(c) τrms/Ts=0.00549
Fig. 6 Performance comparison.
5. Conclusion
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