3780 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO.

8, OCTOBER 2011

Near-Optimal Channel Estimation for OFDM

in Fast-Fading Channels
Ping Wan, Student Member, IEEE, Michael McGuire, Member, IEEE, and Xiaodai Dong, Senior Member, IEEE

Abstract—In this paper, a low-complexity iterative receiver A standard approach to this channel estimation (CE) problem
combining joint iterative channel estimation (CE) with symbol de- is pilot-symbol-aided modulation (PSAM), where known pilot
tection is proposed for coded orthogonal frequency-division mul- symbols multiplexed with the information symbols are sent
tiplexing (OFDM) systems in fast-fading channels with Doppler
frequencies up to 15% of the OFDM subcarrier spacing. The through the channel at regular intervals so that the radio re-
receiver exchanges information between the channel estimator and ceiver can make direct measurements of the channel variations
detector in an iterative fashion to obtain accurate estimates of [7], [8]. However, in fast-fading channels, an underdetermined
the channel state information (CSI). The channel is modeled as system occurs, where the number of unknown CSI parameters
a weighted sum of fixed basis expansion model (BEM) functions. is larger than the number of the received signal samples [9].
The BEM coefficients are characterized as a multivariate autore-
gressive (AR) processes and estimated with a Kalman filter. The To reduce the number of unknowns, different approaches have
initial channel estimate is performed from sparse pilot signals. been exploited for pilot-symbol-assisted CE schemes. One is
Data are detected and decoded, and the channel is estimated again that the channel is approximated with a piecewise linear model
based on the estimated transmitted data. The cycle of detection, both in the time and frequency domains during the period of
decoding, and CE is repeated until convergence. It is shown that one or two adjacent OFDM symbols [10]. Another approach
with a pilot-to-data ratio of 7/144 and with pilots consuming only
1/145 of transmission power, bit error rate (BER) performance is to model the channel gains with a basis expansion model
within 0.1 dB of that achieved with perfect CSI is obtained for (BEM), where CSI estimation can be converted to estimating
BER ≤ 10−5 . a low number of BEM coefficients, as in [11]–[16]. As the
Index Terms—Extrinsic information transfer (EXIT) chart, fading rate increases, the previously proposed CE techniques
fast-fading channels, joint channel estimation (CE) and symbol require a higher density of pilot symbols to obtain sufficiently
detection, orthogonal frequency-division multiplexing (OFDM). accurate channel estimates for reliable detection. This reduces
the overall data rate during fast fading to unacceptable levels
I. I NTRODUCTION for many applications.
To improve the performance of PSAM, an alternative ap-
O RTHOGONAL frequency-division multiplexing (OFDM)
has been widely applied in wireless communications due
to its high spectral efficiency and its ability to efficiently handle
proach is to use iterative CE and symbol detection. In this tech-
nique, PSAM is used to obtain an initial channel estimate, and
the data are estimated using this initial channel estimate. The
multipath propagation [1]. If the radio channel is time invariant
detected data values are then used to reestimate the channel, and
over an OFDM symbol block, simple single-tap equalizers may
the data are detected again. With good initial channel estimates,
be used to detect the OFDM symbol data. Unfortunately, if the a joint CE and symbol detection method has been shown to
channel state evolves significantly during an OFDM symbol perform well with OFDM in slow-fading channels, where the
period, a condition known as fast fading, intercarrier interfer- channel is nearly time invariant over an OFDM block [17]–[19].
ence (ICI) is created, and more sophisticated data detection Methods for joint CE and symbol detection have been
procedures are required [2]–[6]. proposed for fast-fading channels using a discrete prolate
Common to all OFDM detection schemes is the requirement spheroidal BEM (DPS-BEM) for multicarrier code-division
that accurate channel state information (CSI) is needed at the multiple-access systems in [20] and single-carrier (SC) systems
receiver for low error rates to be achieved. However, in most [21], polynomial BEM for OFDM systems [22], Fourier BEM
wireless communications, the CSI is unknown a priori. In these for SC or OFDM systems [23]–[25], and orthonormal discrete
cases, accurate estimation procedures for the radio channel are cosine transfer BEM for OFDM systems [26]. For SC systems,
required to perform low error rate communications. Kim and Tugnait [24] proposed a nonlinear Kalman filtering
method to estimate the channel and detect data iteratively with
a high pilot-to-data ratio required to get convergence. Within a
single OFDM symbol period, Hijazi and Ros [27] employed the
Manuscript received March 5, 2011; revised June 2, 2011 and July 21,
2011; accepted August 12, 2011. Date of publication September 19, 2011; least squares (LS) estimation algorithm and [23] used the LS or
date of current version October 20, 2011. This work was presented in minimum mean square error (MMSE) estimation algorithm for
part at the IEEE Wireless Communications and Networking Conference, CE. For efficient channel modeling, the previous work in [23]
Sydney, Australia, April 2010. The review of this paper was coordinated by
Dr. T. Taniguchi.
used a long OFDM period as the BEM period. Long OFDM
The authors are with the Department of Electrical and Computer Engi- periods create significant interference between the subcarriers
neering, University of Victoria, Victoria, BC V8W 3P6, Canada (e-mail: since the channel gains change significantly during a single
pingwan@ece.uvic.ca; mmcguire@ece.uvic.ca; xdong@ece.uvic.ca).
Digital Object Identifier 10.1109/TVT.2011.2168248
OFDM period. For efficient data detection in the presence of

0018-9545/$26.00 © 2011 IEEE

WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3781

this effect, more costly detection algorithms such as maximum and error correction code for good performance at a
a posteriori probability (MAP) are employed, which is imple- given signal power level. This EXIT chart analysis shows
mented by using the Bahl–Cocke–Jelinek–Raviv algorithm. It that the performance of this method is nearly optimal to
is possible to shorten the OFDM period so that the channel is the case of the ideal CSI for high signal-to-noise ratios
nearly invariant over the OFDM period. However, if this short (SNRs).
OFDM period is also used as the BEM period, the band-limited
channel fading process will not be modeled accurately because The rest of this paper is organized as follows. Section II
of truncation effects such as the Gibbs phenomenon [28]. To provides a system description. The structure of the iterative
avoid this problem, the previous iterative CE/joint detection receiver is provided in Section III. The convergence behavior
methods extend the OFDM symbol period. The cost of the is analyzed with the EXIT chart technique in Section IV.
required detection algorithms increases exponentially with the Simulation results are shown in Section V. Section VI presents
order of modulation and fading rate, making these algorithms our conclusions.
unsuitable for OFDM using higher order modulation. This Notation: We use I D to denote a D × D identity matrix
limits the practical use of the previous iterative CE/joint detec- and D{z} to denote a matrix with vector z on its diagonal
tion algorithms in fast-fading channels to lower order modula- with all other entries being zero. We use
 and  to de-
tion such as quadrature phase-shift keying (QPSK). Moreover, note the ceiling and floor functions, respectively. E[] is the
these methods require a high pilot-to-data ratio to guarantee expectation operation, and P [X = x] is the probability of event
convergence. {X = x}. Π and Π−1 are denoted as the interleaving and
In this paper, we propose a new low-complexity joint deinterleaving, respectively. We use ⊕ to specify the direct
−1
CE/symbol detection scheme for OFDM subject to fast fading sum of N matrices: ⊕N i=0 Ai = diag{A0 , . . . , AN −1 }, where
with a low pilot-to-data ratio. The key to this method is that the diag is the block diagonal operator [29]. 0L is a row vector
BEM coefficients are found for a transmission block consisting of L zeros. F is the fast Fourier
√ transform (FFT) matrix with
of multiple OFDM symbols. The time-varying radio channel the (m, n)th entry Fm,n = 1/ N e−j2π(m−1)(n−1)/N , and F H
is represented by the linear combination of time-varying BEM is the inverse FFT (IFFT) matrix. We use Matlab notation
functions weighted by BEM coefficients. Within each transmis- E(m : n, :) to extract the submatrix from row m to row n from
sion block, the impulse response of the channel is evolving; the matrix E. Vectors (matrices) are denoted with bold lowercase
impulse response of the channel changes from the beginning (uppercase) letters. Superscripts T and H stand for transpose
to the end of each OFDM symbol. The evolution of the BEM and Hermitian transpose, respectively. The list of abbreviations
coefficients between each transmission block is modeled as a employed throughout the paper is provided in Table I.
multivariate autoregressive (AR) process. A Kalman filter is
used to estimate the BEM coefficients within each transmission
block. By using BEM functions derived from the Karhunen II. S YSTEM D ESCRIPTION
Lòeve transform (KLT), the mean square error of the channel We consider the coded OFDM system depicted in Fig. 1.
modeling is minimized [15]. The period of the BEM functions The proposed iterative scheme is characterized by three types
is set as the duration of multiple OFDM symbols. This allows
of signal blocks in the time domain, as shown in Fig. 2: The
a long BEM duration to be used so that truncation error ef-
OFDM block (OB) is used as the basis for detection, the
fects are mitigated in the CE procedure, and a short duration
transmission block (TB) is the basis for CE, and the interleaving
OFDM symbol period is employed to achieve low-cost data
block is the basis for decoding. The iterative receiver for the
detection. In addition, higher order modulation can be used
interleaving block includes CE and symbol detection loops.
without greatly increasing the cost of efficient detection. For the
Relying on a first-order Kalman filter, the CE is first performed
initial CE, sparse pilot signals are used to estimate the channel
by using the pilot symbols and then the detected data symbols in
gains. The data are detected and decoded based on this initial
subsequent iterations, which is described in detail in Section III-
channel estimate. The detected data are then fed into a decision-
A. The symbol detection (based on OFDM blocks) and decoder
directed CE algorithm to obtain an improved estimate of the
(based on interleaving blocks) are described in Section III-B.
channel. The data are detected and decoded again with the new
At the transmitter, the information bits are first encoded as c̃
channel estimate. The process of detection, decoding, and CE
and interleaved into coded bits c and then mapped to symbols s.
is repeated until convergence.
For an OFDM block with N subcarriers and a cyclic prefix (CP)
The main contributions of this paper are listed as follows:
with the length Ncp being equal to or larger than the maximum
1) a novel joint /symbol detection algorithm for fast-fading channel propagation delay L, the length of the OFDM block in
channels that requires only a small number of pilots and samples is Ns = N + Ncp . Due to the CP, there is no interblock
low pilot power; interference between contiguous OFDM blocks. A transmission
2) demonstration that this new CE symbol detection tech- block consists of Kp pilot blocks uniformly inserted in Kb
nique is robust to deviation of the channel parameters OFDM blocks, where Kb = κKp , κ is the number of OFDM
from the design values for the algorithm; blocks between two adjacent pilot blocks, and κ ≥ 1 is an
3) demonstration that the new approach is efficient when integer. A pilot block (PB) has the form of [0L 1 0L ]. In this
applied to different modulations and coding types; paper, we chose κ = 1 or 2, i.e., one OFDM block followed
4) demonstration of the extrinsic information transfer by one pilot block or two OFDM blocks followed by one pilot
(EXIT) chart use to select the modulation, pilot density, block. Hence, the pilot-to-data ratio is (2L + 1)/(κNs ). In [30]
3782 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

TABLE I iteration’s data detection and decoding phases. The CE, symbol
L IST OF A BBREVIATIONS
detection, and data decoding are repeated until convergence is
reached or until the maximum allowable number of iterations
has been performed.
The computational complexity of the channel estimator for
a transmission block of M samples is related to the number of
the BEM coefficients, i.e., Q + 1, which is bounded by Q ≥
2
fd M Ts , where fd is the maximum Doppler frequency, and
Ts is the sampling interval that equals to the data symbol period
[11]. Therefore, for a fixed Doppler frequency, longer transmis-
sion blocks require a larger number of BEM coefficients and,
hence, the larger the cost of the channel estimator. Because
of truncation effects, the shorter the transmission block is, the
more spectral leakage will be seen as the band-limited fading
process is imperfectly modeled for the short time duration
of the transmission block. To deal with burst error channels
due to temporary deep fading, a long interleaver is used with
a convolutional code to give good performance in multipath
fading channels [31].

A. System Model
It is common in practice to describe a radio channel with a
wide sense stationary uncorrelated scattering (WSSUS) model
[32]. Given the sampling interval Ts , the impulse response of
the radio channel in the discrete time is denoted as h(m, l) for
propagation path l at sample m and h(m, l) = 0 for l < 0 or
l > L. At the receiver, the received signal
in the time domain
at sample m is expressed as yt (m) = L l=0 h(m, l)u(m −
l) + vt (m), where yt (m) is the received signal in the time
and this paper, Kb = 10 is used for simulations. The length of a domain, u(m) is the transmitted time domain signal, and vt (m)
transmission block in samples is then M = Kb Ns + Kp (2L + is AWGN [32]. The received time domain signal for the kth
1). The interleaving block includes Kt transmission blocks. OFDM block of a transmission block is rewritten as
The length of an interleaving block in samples is KL = Kt M .
The relationship between OFDM blocks, transmission blocks 
L
 
y kt = H kt uk + v kt = D hkl S l uk + v kt (1)
and interleaving blocks is shown in Fig. 2, for κ = 1. The
l=0
transmitted signals are then transmitted through a multipath
radio propagation channel and are subject to additive white where hkl is the channel gain vector of the kth OFDM block
Gaussian noise (AWGN) with zero mean and variance σv2 = for the lth path, which is given by hkl = [h(kNs + Ncp +
N0 /2. k/κ(2L + 1), l), . . . , h(kNs + Ncp + k/κ(2L+1)+ N −
At the receiver, the received signal in the time domain ỹ t 1, l)]T for k ∈ [0, Kb − 1], and S l is given by circularly
contains two parts, namely, y P t and y t , where y t is the
D P
shifting the identity matrix I N left by the delay of l samples.
received signals resulting from the pilot symbols, and y D t is uk is the kth OFDM symbol vector in the time domain so that
the received signal created as a result of the data symbols. After uk = F H sk , where sk is the signal vector for the kth OFDM
removing pilots and CP, the received signal in the frequency block in the frequency domain. v kt is an AWGN vector. The
domain is expressed as the vector y D . The channel is modeled received signal vector in the frequency domain for the kth
with a BEM for each transmission block. A dynamic model OFDM block is then expressed as
for the evolution of the BEM coefficients between transmission
blocks is described in Section III-A. The symbol detection is y k = H k sk + v k (2)
described in Section III-B.
The initial channel estimate Ĥ for each interleaving block where y k = F y kt is an N by 1 vector, H k = F H kt F H , and
is obtained based on PSAM in the time domain. Initially, the v k = F v kt . When the channel gains are available, this equation
data are detected for each OFDM block using MMSE with can be used to detect sk .
ICI cancellation based on the initial Ĥ and decoded for each For a single OFDM block in a fast-fading channel, we need
interleaving block using a soft-input soft-output (SISO) MAP N (L + 1) channel gains to characterize the channel matrix H kt .
algorithm. The decoded data from this iteration, after appropri- When dealing with CE over fast-fading channels, the number
ate interleaving and extrinsic information calculations, is used of unknown channel gains exceeds the length of the received
for the estimation of the BEM coefficients to support the next signal vector y k . To solve this problem, a BEM for the channel
WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3783

Fig. 1. Block diagram of a coded OFDM system.

longer blocks providing better estimation performance at higher

computational cost.
The BEM coefficient vector for the lth path in the current
transmission block is given as
 T
xl = x0l , . . . , xQ
l . (5)

The channel gain vector in a transmission block for the lth

path is given by hl = Exl . Therefore, the channel gain vector
of the kth OFDM block for the lth path is then expressed as

hkl = E k xl (6)

where E k is the BEM matrix for the kth OFDM block, which
is given by E k = E[kNs + Ncp + k/κ(2L + 1) : kNs +
Fig. 2. Block structure with κ = 1. Ncp + k/κ(2L + 1) + N − 1, :)] for k ∈ [0, Kb − 1], and
E k = E j for k = j. xl gives all the necessary information to
is used, where the channel gains for a finite transmission block specify the channel gains for path l over the transmission block.
are modeled as a weighted sum of basis functions [11], [33]. Define the BEM coefficient vector as x = [xT0 , . . . , xTL ]T . With
A general BEM expression of h(m, l) for a transmission the use of a BEM, (1) is rewritten as
block is written as

L
y kt = D{E k xl }S l F H sk + v kt

Q
h(m, l) = Eq (m)xql , for m ∈ [0, M − 1] (3) l=0

q=0 
L
= D{S l F H sk }E k xl + v kt
where Eq (m) gives the value for sample m of the qth basis l=0
function, and xql is the BEM coefficient for path l corresponding = C k x + v kt (7)
to the qth basis function.
Define the BEM matrix as where C k = [C 0k · · · C Lk ] with C lk = D{S l F H sk }E k . The
 T received signal in the frequency domain is expressed as
E = eT0 , . . . , eTM −1 (4)
y k = F C k x + v k = Mk x + v k (8)
where the row vector is given by em = [E0 (m) · · · EQ (m)].
Several different BEM matrices are available for the CE pro- where the matrix Mk is defined as Mk = [M 0k · · · M Lk ],
cedure with various robustness properties [12], [34]. For the with M lk = F C lk = F D{S l F H sk }E k . When the transmit-
simulations described in Section VI, the optimal BEM matrix ted symbol vector sk is known, (8) is used to estimate x.
for CE is calculated from the KLT of the covariance matrix
of the channel gains [15]. If the covariance matrix of the
B. Multivariate AR Model
channel gains is not available, a BEM matrix based on the DPS
sequence may be used instead [12]. Selection of the length The evolution of the BEM coefficients for each propagation
of the transmission block must be done with some care with path in a fast-fading channel between transmission blocks can
3784 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

be modeled as a multivariate AR process of order 1 [16].

This AR model allows the estimated BEM coefficients for the
previous transmission block to be used to estimate the BEM
coefficients for the current transmission block, reducing the
number of pilot symbols needed and improving CE accuracy.
The BEM coefficients for each propagation path at transmission
block n for the current interleaving block are given by

xl [n] = Al xl [n − 1] + wl [n], for l ∈ [0, L] (9)

where xl [n] = [x0l [n], . . . , xQ

l [n]] . Al is the (Q + 1) by (Q +
T

1) state matrix of the lth propagation path, and wl [n] is an

AWGN vector with zero mean and the covariance matrix of Ql Fig. 3. Kalman filter equations for one iteration [35].
for path l. These matrices satisfy the multichannel Yule–Walker rent transmission block. After CE, symbol detection followed
equation [35], [36] given as follows: by SISO decoding is then performed. After estimates of the
Al Rl = [Ql 0] (10) transmitted symbols are obtained, the channel is reestimated.
The improved channel estimate is then used for redetection of
where Al = [I Al ], and the data symbols. The redetected symbol information is then
fed into the decoder. CE followed by redetection/decoding is
Rx (0) Rx (1) repeated until the convergence is obtained.
Rl = .
Rx (−1) Rx (0)

Equation (10) can be solved efficiently with the A. Kalman CE

Whittle–Wiggins–Robinson algorithm (WWRA) [37], Our CE system uses a Kalman filter to provide the estimate
[38]. The matrix Rl has a Toeplitz-block structure, i.e., of the channel for each transmission block. The Kalman filter
Rx (−i) = RH x (i), where Rx (i) = E[xl [n + i]xl [n] ].
H
takes an estimate of the BEM coefficients for transmission
To calculate Rx (i), first, we obtain the cross-covariance block n − 1, x̂[n − 1], with its estimated error covariance
matrices of hl [n], the lth path channel gains for transmission matrix P [n − 1] and a measurement vector y[n] and then
block n, as computes an estimate of the BEM coefficients for transmis-
  sion block n, x̂[n], as well as its estimated error covariance
Rh (i) = E hl [n + i]hl [n]H
  matrix P [n] [35]. The Kalman filter algorithm assumes a
= E (Exl [n + i]) (Exl [n])H Gauss–Markov model for the evolution of the BEM coefficients
and a linear measurement model relating available measure-
= ERx (i)E H . (11) ments to the BEM coefficients to be estimated. The dynamic
model for the evolution of the BEM coefficients from transmis-
Based on Jakes’ model, the correlation function of path l sion block n − 1 to n is
is expressed as rl [m] = σl2 J0 (2πfd Ts m), where J0 (·) is the
zeroth-order Bessel function of the first kind, and σl2 is the x[n] = Φx[n − 1] + w[n] (13)
variance of the lth path [39]. In general,

the 2average power of where the state vector at the block n is x[n] =
the channel is normalized to one, i.e., L l=0 σl = 1. Hence, the [xT0 [n], . . . , xTL [n]]T , the state transition matrix is given
cross-covariance matrix Rh (i) is given by
as Φ = − ⊕L l=0 Al , and w[n] is an AWGN vector with the
 · · · r [M (i − 1) + 1]  covariance matrix Q = ⊕L
r[M i] l=0 Ql . The relationship between the
 r[M i + 1] ··· r [M (i − 1) + 2]  measurement y[n] and the state vector to be estimated x[n] is
Rh (i) = 
 .. .. .. .

. . . y[n] = G[n]x[n] + v[n − 1] (14)
r [M (i + 1) − 1] ··· r[M i]
(12) where G[n] is called the measurement matrix, and v[n] is the
Since E H E = I, Rx (i) is calculated from (11) and (12) as measurement noise with covariance R[n]. The Kalman filter
Rx (i) = E H Rh (i)E. In [16], it is shown how to efficiently equations for a single transmission block is given in Fig. 3. The
solve for the Al and Ql matrices using the WWRA. estimate of x[n] given all measurements y[n] up to block m
is denoted by x̂[n|m] with the corresponding error covariance
matrix denoted as P [n|m]. ŷ[n|n − 1] is the predicted mea-
III. I TERATIVE R ECEIVER S TRUCTURE
surement for block n, z[n] is the so-called innovation sequence
To improve the initial channel estimates for each transmis- with a covariance matrix given by M [n|n − 1], and K[n] is
sion block, information from the channel estimates based on the the Kalman gain for block n. Given an initial estimate of
pilot-aided method of the previous transmission block is used. the state x̂[−1] = 0 with error covariance matrix P [−1] =
To reduce error propagation, the channel estimates based on de- ⊕Li=0 σl Rx [0], the Kalman filter estimates the channel state at
2

tected/decoded data of previous blocks are not used in the cur- block n given measurements up to y[n].
WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3785

In the following, we derive the measurement model for Similarly, when (17) is overdetermined, a modified equa-
H
the channel state from the pilot symbols. This measurement tion is given by y D = M̃D x + v D , where y D = M̂D y D ,
equation is used in the initial iteration of the system. We also H H
M̃D = M̂D M̂D , v D = M̂D v D with the variance matrix
derive a measurement model for the channel state based on the H
data symbols. This measurement equation is used in the second RD = σv2 M̂D M̂D . For the following iterations, the Kalman
and following iterations of the system. filter is rerun over all transmission blocks and the BEM coef-
For the initial CE of a transmission block, the pilot measure- ficients are reestimated using the data signals with y[n] = y D ,
ments in the time domain y P G[n] = M̃D and R[n] = RD .
t are used to estimate the BEM
coefficients. The received signals at pilot positions are given The initial conditions of the state vector x̂[−1] and error
by [11] covariance matrix P [−1] are taken from the estimated state
  vector and error covariance matrix for the last transmission
CP 0 block of the preceding interleaving block estimated using the
 .. 
yP
t = .  x + vt = C P x + vt
P P
(15) pilot signals only. Using the pilot signals for initial conditions of
CP the Kalman filters prevents error propagation from the decision
Kp −1
feedback propagating beyond a single interleaving block. From
where the estimated BEM coefficients for a given transmission block
  x̂[n], the estimated channel matrix for the kth OFDM symbol
ePk 0Q+1 ··· 0Q+1 is calculated as
 .. .. .. .. 
CP
k = . . . .  
L
0Q+1 ··· ··· ePk +L Ĥ k = F D(E k x̂l )S l F H , k ∈ [0, Kb − 1]. (18)
l=0
in which Pk = (k + 1)Ns + p(2L + 1) for k ∈ [0, Kb − 1],
p ∈ [0, Kp − 1], and v P
t is an AWGN vector.
The most expensive portions of the CE algorithm are the
Since the length of the unknown state vector x is (Q + computation of the measurement matrices and the matrix inver-
1)(L + 1), we choose Kp ≥ (Q + 1). A tradeoff between sion required in the Kalman filter calculations. For the initial CE
complexity and performance exists in the iterative scheme. from pilot symbols, the Kalman filtering calculation requires
Larger values of Q provide better CE performance at the cost the inversion of the square covariance matrix of order (Q +
of needing more pilots and higher computational complexity. 1)(L + 1) requiring O([(Q + 1)(L + 1)]3 ) operations. This is
When Kp > (Q + 1), (15) is overdetermined, i.e., the number of the same order of operations as the standard MMSE coeffi-
of known measurement signals is larger than the number of cient estimation within one transmission block, but this method
unknown BEM coefficients, the modified equation is given by provides much better CE and data detection performance.
For the iterative-data-based CE, the major computation is
y P = C̃ P x + ṽ P (16) the calculation of the measurement matrix M̂D . Counting
the number of element–element multiplications required for
where y P = C H P y t , C̃ P = C P C P , v P = C P v t with the
P H H P
matrix multiplications and FFT, calculation of E lk requires
variance matrix RP = σv2 C P C P . Therefore, for the initial
H
O(N (Q + 1)) operations and FFT of each column of E lk
iteration of a transmission block, the BEM coefficients are requires O(N log2 (N )) operations. Hence, calculation of M̂ lk
estimated with a Kalman filter using pilot signals with y[n] = requires O(N (Q + 1) log2 (N )) operations. Since calculation
y P , G[n] = C̃ P , and R[n] = RP . H
After the first iteration, the received signal in the frequency of M̂ l1k1 M̂ l2k2 requires O(N (Q + 1)2 ), calculation of M̃D
domain y D is used to estimate the channel state based on the for each OFDM block requires O(N (Q + 1)2 (L + 1)). There-
detected data values from the previous iteration. From (8), the fore, the total computational complexity for data-based CE in
received signal for the kth OFDM block at a given transmission one OFDM block is O(N (Q + 1)2 (L + 1)). In Section III-B,
block is then expressed as y k = M̂k x + v k , where M̂k = it will be shown that this is of the same order as detection,
[M̂ 0k · · · M̂ Lk ], M̂ lk = F E lk , and E lk = D{S l F H ŝ}E k , indicating that this CE method, while more expensive than
where ŝ is the detected data from the previous iteration. The purely pilot-based approaches, will not require an increase in
received signal for a given block is the order of the number of multiplication operations for the CE
    and symbol detection.
y0  
M̂0 v0
    .
y D =  ...  =  ..
.  x +  ..  B. Symbol Detection
y Kb −1 M̂Kb −1 v Kb −1
  The iterative processing technique has been widely applied
M̂ 00 ··· M̂ L0 for symbol detection and decoding [40]. For detection and
 . ..  decoding, information about bit values is specified in terms
x + v
D
= .. ··· .
of the log likelihood ratios (LLRs) of the bits. The SISO
M̂ 0(Kb −1) ··· M̂ L(Kb −1)
detector calculates the a posteriori LLR for each bit, given
= M̂D x + v D (17) the measurements of the received signal and the a priori LLR
calculated in the previous iteration by the decoder. In the first
where v D is an AWGN vector. detection/decoding iteration, all bit values are assumed equally
3786 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

likely [41]. The optimal detector calculation has an exponential The EXIT chart technique computes the mutual information
cost with respect to the length of the bit vector sent over one between the LLR and the transmitted bits c given by [44]
OFDM block [42]. For a large number of subcarriers and/or
high-order modulations, this complexity is overwhelming. To  ∞
1 
solve the problem, we use the low complexity suboptimal de- I= pL (l|c)
tection algorithm, where the data symbols are detected by using 2 c=−1,1
−∞
classical MMSE method based on the measurement equation
2pL (l|c)
in (2). · log2 dl, 0 ≤ I ≤ 1 (20)
We express the estimated frequency matrix Ĥ k of (18) pL (l|c = −1) + pL (l|c = 1)
for the kth OFDM block as Ĥ k = Ĥ k1 + Ĥ k2 , where Ĥ k1
is the diagonal matrix which has its only nonzero values as where pL (l|c) is the conditional probability distribution of the
the main diagonal elements of Ĥ k , and Ĥ k2 is equal to LLR, given the transmitted bits c.
Ĥ k with the exception of its main diagonal elements being IAdet
and IA dec
are the a priori mutual information at the
zero [43]. inputs of the detector and the decoder, respectively. The values
det dec
For the initial detection, symbols are detected by ignoring IE and IE denote the extrinsic mutual information values
ICI, using only the elements on the main diagonal of the at the outputs of the detector and decoder, respectively. In a
1
estimated channel transfer matrix Ĥ k1 and the received signal standard EXIT chart for detector/decoder systems, two curves
y k . To suppress ICI efficiently, we estimate the ICI from the are plotted. One curve is plotted for the detector with the a
estimated symbol ŝi−1 and remove the estimated ICI from the priori mutual information on the x-axis and output extrinsic
k
received signal y k . That is mutual information on the Y -axis. Another curve is plotted
i for the decoder, with the a priori mutual information on the
y ik = y k − Ĥ k2 ŝi−1
k (19) Y -axis and output extrinsic mutual information on the x-axis.
i The performance of the system is a function of the distance
where ŝik is the symbol estimate of the ith iteration, Ĥ k2 is between these two curves and where these two curves intersect
the estimated H k2 of the ith iteration, and ŝ0k = 0 for the [44]. The initial value of the mutual information of the detector
i
first iteration. The values y ik and Ĥ k1 are used to estimate the starts at the intersection of the detector curve with the IAdet
=0
symbol sk for the ith iteration.
i
line. The system mutual information moves horizontally to
The detector and the decoder exchange the extrinsic in- intersect the decoder EXIT curve. In each iteration, the tra-
formation iteratively in each detection scheme [42]. In each jectory moves vertically to intersect the detector line and then
i
detection loop, the detector uses the estimated channel Ĥ k , horizontally to intersect the decoder line.
the received signal y ik , and the a priori LLR Ldet A (c), and
The iterative receiver system proposed in this paper is charac-
then, the a posteriori LLR Ldet D (c|y i
) is obtained. The extrinsic terized by the following three EXIT curves: 1) the EXIT curve
LLR obtained from the decoder, i.e., Ldec E (c̃) = Ldec
D (c̃|y i
)− of the detector using pilot-aided CE; 2) the EXIT curve of the
dec
LA (c̃), is interleaved and then sent back to the detector as detector using data-aided CE; and 3) the EXIT curve of the
the a priori LLR. The relationships of the LLRs between the decoder. Fig. 4 shows these curves for a Rayleigh fast-fading
detector and decoder are Ldet dec
A (c) = Π(LE (c̃)) and LA (c̃) =
dec channel with gray-coded QPSK modulation and the convolu-
−1 det tional code with a rate of 1/2 and the generator polynomial [133
Π (LE (c)). After several iterations, the LLRs pass through
a hard decision to get the estimated information bits. 177] specified in octal. The detector in the initial iteration uses
det
For a single OFDM block, the cost of the SISO detec- the pilot-aided channel estimator so that the point of IE for
det
tion/decoding is O(N log2 N ), with the interference calcula- IA = 0 is for the pilot-aided CE. The trajectory then moves
tion requiring O(N 2 ) operations; therefore, the overall cost is horizontally to the decoder’s EXIT curve. For all following
O(N 2 ). iterations, the standard EXIT curve drawing rules are employed
based on the EXIT curves of the detector with data-aided CE
and the decoder; the trajectory moves vertically to the EXIT
IV. E XTRINSIC I NFORMATION T RANSFER (EXIT)
curve of the detector with data-aided CE and horizontally to the
C HART A NALYSIS
EXIT curve of the decoder.
In EXIT charts, the input versus output mutual information The EXIT curve for the detector using ideal CSI for the
of the data detector and decoder subsystems of a receiver are channel and signal parameters described in Section V is also
plotted. The relative positions of these curves provide a method shown in Fig. 4, where “detector with pilot-aided CE” is the
of determining the final bit error rate (BER) and number of EXIT curve of the detector using channel estimates based on
iterations required for convergence of iterative detection and pilot signals, “detector with data-aided CE” is the EXIT curve
decoding receivers [44]. Here, we will extend EXIT charts to of the detector using channel estimates based on detected data
analyze the performance of our data-aided CE and its impact signals, and “detector with ideal CSI” is the EXIT curve of the
on detection and compare its performance with detection when detector based on the perfect CSI. The theoretical trajectory
ideal CSI is used. These EXIT charts will allow a system de- of the presented scheme is also shown in Fig. 4. It can be
signer to determine the required Eb /N0 values for this iterative seen that the proposed methodology resulting in a final mutual
detection system to work with a preselected modulation type information point is almost as good as the ideal CSI case.
and/or error correction code, where Eb is the bit energy. This allows us to describe the performance of this system as
WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3787

Fig. 4. EXIT chart using Kp = 5 and Kt = 10 for fd T = 0.1 at Eb /N0 = Fig. 5. EXIT chart using Kp = 10 and Kt = 10 for fd T = 0.1 at Eb /N0 =
7 dB. 7 dB.

near-optimal as the BER performance for high SNR is nearly in simulations has a rate of 1/2, a constraint length of 7, and its
as good as the BER performance with ideal CSI. generator polynomials G = [133 171] in octal form.
det
In Fig. 4, when IA > 0.05, the detector using data-aided
det
CE provides better values of IE than the detector using pilot- A. Convergence Comparisons Based on Different
aided CE at this Eb /N0 value with QPSK modulation. A Estimated Channels
receiver using only the detector with pilot-aided CE will have
an EXIT curve trajectory that gets stuck at the intersection We analyze EXIT charts of the decoder and the detector
point of the EXIT curves between the detector and the decoder with different estimated channel schemes. A gray-coded
where IE dec
< 0.5, resulting in a high BER. If the initial mutual QPSK constellation is used in the simulations. The normalized
information provided by the first iteration using only pilot- Doppler frequency is set to fd T = 0.1, where T is the OFDM
det
based channel is good, such that the initial IA is larger than the symbol period. We choose Q = 4 basis functions, and the
intersection point of the EXIT curves between the data-aided corresponding number of pilot blocks for each transmission
detector and the decoder (the region where IE det
of the detector block is Kp ≥ 5.
det
with data-aided CE is higher than the IA of the decoder), there Figs. 5 and 6 depict the EXIT charts at Eb /N0 = 7 dB for
will be enough information provided by the initial detection different numbers of pilot blocks. Fig. 5 shows that the true
using pilot-aided CE to allow the error correction code to trajectory needs only three iterations to reach IE dec
≈ 1 when
provide some correction capability. Kp = 10 and Kt = 10. In Fig. 5, the EXIT charts of the pro-
The relative position of the EXIT curve for the detector using posed CE scheme are compared with [24], where the channel
data-aided CE to the EXIT curve for the detector using pilot- is estimated using nonlinear Kalman filtering for SC systems.
aided CE determines the performance of our algorithm. If the The detector with the estimated channel from [24] and decoder
det dec
EXIT curve of the detector using data-aided CE is below the transfer characteristics do intersect at IA = IE = 0.285, and
EXIT curve for the detector using pilot-aided CE after several the decoding trajectory gets stuck at low mutual information,
iterations, our algorithm will perform poorly; vice versa, if the corresponding to high BER. Therefore, the system cannot find a
dec
EXIT curve of the detector using data-aided CE is above the trajectory leading to a large IE using the code G = [133 177],
EXIT curve for the detector using pilot-aided CE after the initial stops improving the performance after several iterations, and
det dec
iteration, the algorithm will perform well. remains in the fixed point IA = IE = 0.285 in [24]. It
should be noted that the SC performance with ideal CSI
is much better than that of OFDM as its ideal CSI EXIT curve is
V. S IMULATION R ESULTS
higher. With the estimated CSI, the SC detection EXIT curve is
In this section, the performance of the proposed algorithm is much lower, and thus, a very high signal power is needed, or the
tested in a simulated wireless OFDM system with a multipath density of pilots needs to be increased for the detector/decoder
fast-fading radio channel. We simulate a four-path WSSUS system to converge. For the fading rate described here, the
channel with mean propagation powers of 8/15, 4/15, 2/15, OFDM CE/data detection scheme described has a distinctive
and 1/15 for propagation delays of zero to three samples, advantage.
respectively. Each path is subject to independent Rayleigh For a reduced value Kp = 5, Fig. 6 shows that the trajectory
fading generated using the method from [45] with Jakes’ model gets stuck at IEdec
≈ 0.96 after three iterations when Kt = 10.
specifying the autocorrelation function. We select an OFDM In such a case, the receiver’s EXIT trajectory depicted in Fig. 6
system with N = 128 samples for each OFDM block, where does not match the transfer characteristics predicted by the
the length of CP is Ncp = N/8. The convolutional code used EXIT chart due to dependencies of the extrinsic LLR between
3788 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

Fig. 6. EXIT chart using Kp = 5 for fd T = 0.1 at Eb /N0 = 7 dB. Fig. 7. BER resulting with estimated CSI with Kp = 5 for fd T = 0.1 with
QPSK.
different bits at the inputs of the detector and the decoder.
Increasing the interleaver length to Kt = 100 decreases these
dependencies of the extrinsic information and the simulated tra-
jectory matches the EXIT chart transfer characteristics. How-
ever, enlarging Kt will increase the computational complexity
of the proposed scheme. Therefore, finding the optimal number
of pilot blocks is a tradeoff between the number of iterations
and the number of pilots at low SNRs. More pilot blocks will
result in reduced number of iterations but will also require
the transmitters to spend more power transmitting non-data-
bearing pilot signals. In this paper, we choose Kp = 10 for
each transmission block and Kt = 10 since the increase in pilot
power for Kp = 10 over Kp = 5 is only a small percentage
of total power, and the memory requirement in the receiver is
reduced by an order of magnitude.

B. BER Performance
We test the BER performance using the receiver discussed in Fig. 8. BER resulting with estimated CSI with Kp = 10 for fd T = 0.1 with
Section III and analyzed in Section IV. Gray-coded QPSK and QPSK.
64-quadrature amplitude modulation (64-QAM) constellation
are used in the following simulations. The normalized Doppler
frequency is set to fd T = 0.1 or 0.15. For a WiMAX system
operating at the carrier frequency of 5.0 GHz and the subcarrier
space is approximately 10 kHz, these Doppler frequencies map
to radio terminal velocities of 216 and 324 km/h, respectively.
The convolutional code used in simulations is the same as for
the analysis described in Section V-A.
The BER performances over the proposed iterative schemes
are shown in Figs. 7–9 for different normalized Doppler fre-
quencies, respectively. “CE ith” shown in the figures represents
the “CE ith iteration.” Each interleaving block consists of Kt =
10 transmission blocks; each transmission block consists of
Kb = 10 OFDM blocks and Kp = 5 or 10 pilot blocks. The
pilot-to-data ratio is low to 7/288 or 7/144, and the percentage
of total power used on pilots is 1/290 or 1/145, respectively. We
choose Q = 4 for fd T = 0.1 and Q = 6 for fd T = 0.15.
Comparing Fig. 8 with Fig. 7, we can see the BER results
with Kp = 10 are better than those with Kp = 5 at Eb /N0 ≤ Fig. 9. BER resulting with estimated CSI with Kp = 10 for fd T = 0.15 with
7 dB, which matches the EXIT prediction. Hence, we choose QPSK.
WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3789

Fig. 10. BER resulting from estimated CSI with Kp = 10 for fd T = 0.1 Fig. 11. BER resulting for designed fd T = 0.1 with QPSK.
with 64-QAM.

Kp = 10 for the later simulations. In Fig. 8, the BER results

of the proposed CE scheme are compared with [23], where the
channel is estimated using MMSE algorithm in the frequency
domain and where the pilot-to-data ratio is 20/128. The basis
functions used in [23] are Fourier basis functions, which are
subject to Gibbs phenomenon resulting in channel modeling
error. Specifically, the algorithm in [23] uses a BEM over a
single OFDM symbol, whereas the new algorithms extend the
model over several OFDM symbols. It can be seen in Fig. 8
that for the new method when fd T = 0.1, the BER results using
the estimated channel approach the case of the ideal CSI after
the third iteration. For fd T = 0.15, BER results are shown in
Fig. 9.
The proposed algorithm is also tested with the higher order
modulation constellations of 64-QAM, as shown in Fig. 10. In
this case, it can be seen that the performance close to the ideal
CSI case is obtained for higher Eb /N0 values. Fig. 12. BER resulting for fd T = 0.1 and true L = 3 with QPSK.

C. Robustness Analysis
For the approach to be useful in field implementations, it is set the designed number of paths of the Kalman filter to the
necessary for it to be robust to variations of the true channel highest expected level. The channel estimator does not require
parameters from the design values. Robustness to variations knowledge of the relative power levels of each propagation
of Doppler frequency, number of propagation paths, and mean path. Only the relative delays and number of paths are needed
path power will be demonstrated here. The simulation BER to obtain good CE and symbol detection/decoding results.
results shown in Fig. 11 are for Eb /N0 = 8 dB with the
designed normalized Doppler frequency of fd T = 0.1. When
VI. C ONCLUSION
the designed fd T is lower than the true value, the proposed
method will diverge. If this algorithm is used in a radio receiver, This paper has presented a new CE/symbol detection
a conservative design choice would be to set the designed nor- methodology for fast-fading radio channel. This method re-
malized Doppler frequency of the Kalman filter to the highest quires low pilot overhead and computational complexity. The
expected level. EXIT chart technique predicts the convergence behavior of the
The approach is also robust to variations of the number of proposed iterative scheme. EXIT chart analysis can be used
paths in the channel estimator from the true L. The simulation to assist with the selection of pilot signals, modulation type,
BER results shown in Fig. 12 are for Eb /N0 = 8 dB with and error correction code as well as the required Eb /N0 value
the true L = 3. The designed mean power for each path is for the algorithm to converge. The new method provides BER
chosen as 1. When the designed L is lower than the true value, results nearly as good as the detection/decoding with perfect
the proposed method will diverge. If this algorithm is used knowledge of the CSI. This technique is robust to variations of
in a radio receiver, a conservative design choice would be to the channel conditions from the designed values.
3790 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 8, OCTOBER 2011

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WAN et al.: NEAR-OPTIMAL CHANNEL ESTIMATION FOR OFDM IN FAST-FADING CHANNELS 3791

Michael McGuire (M’97) received the B.Eng. and Xiaodai Dong (S’97–M’00–SM’09) received the
M.A.Sc. degrees from the University of Victoria, B.Sc. degree in information and control engineering
Victoria, BC, Canada, in 1995 and 1997, respec- from Xi’an Jiaotong University, Xi’an, China, the
tively, and the Ph.D. degree from University of M.Sc. degree in electrical engineering from National
Toronto, Toronto, ON, Canada, in 2003. University of Singapore, Singapore, and the Ph.D.
After completing the Master’s degree, he spent degree in electrical and computer engineering from
two years at Lucent Technologies, Holmdel, NJ. He Queen’s University, Kingston, ON, Canada, in 1992,
joined the faculty of the Department of Electrical 1995, and 2000, respectively.
and Computer Engineering, University of Victoria, She is currently an Associate Professor and
in 2003. His projects included testing the first gen- Canada Research Chair (Tier II) with the Department
eration of multiprotocol cellular telephones and de- of Electrical and Computer Engineering, University
veloping fault detection and management software for high-capacity optical of Victoria, Victoria, BC, Canada. Between 2002 and 2004, she was an Assis-
transmission systems. His research interests include signal processing for tant Professor with the Department of Electrical and Computer Engineering,
communications, model-based filtering, and nonparametric estimation. University of Alberta, Edmonton, AB, Canada. From 1999 to 2002, she was
with Nortel Networks, Ottawa, ON, where she worked on the base transceiver
design of third-generation mobile communication systems. She is an Editor for
the Journal of Communications and Networks. Her research interests include
communication theory, radio propagation, cooperative communications, ultra-
wideband radio, and signal processing for communication applications.
Dr. Dong is an Editor for the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS, and the IEEE TRANSACTIONS ON VEHICULAR
TECHNOLOGY.