1360
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
Estimation of Channel Statistics for Iterative Detection of OFDM Signals Michele Morelli, Member, IEEE, and Luca Sanguinetti
Abstract—Maximum likelihood sequence estimation for orthogonal frequency division multiplexing (OFDM) transmissions over unknown multipath fading channels is analytically infeasible for lack of efficient methods to maximize the likelihood function. A practical solution to this problem has been recently proposed in the context of space–time block-coded OFDM by resorting to the expectation–maximization (EM) algorithm. The resulting detector operates iteratively, exploiting knowledge of the channel statistics and the operating signal-to-noise ratio (SNR). In this work, we address the problem of estimating the above quantities and propose a recursive solution based on ad hoc reasoning. Simulations indicate that the EM detector employing the estimated SNR and channel statistics has better performance than other schemes operating in a mismatched mode. Also, the performance loss with respect to a system with perfect channel knowledge is negligible at SNR values of practical interest. Index Terms—Channel estimation, EM data detection, OFDM.
I. I NTRODUCTION
O
RTHOGONAL frequency division multiplexing (OFDM) is an efficient technique to counteract the effects of multipath fading in high-rate wireless systems. The basic principle of OFDM is to split a high-rate data stream into a number of low-rate streams and transmit the latter in parallel, thereby increasing the symbol duration and reducing the intersymbol interference [1]. This feature has motivated the adoption of OFDM as a standard for the new wireless local area network IEEE 802.11a and the European Telecommunications Standards Institute high-performance local area network (HIPERLAN/2). Coherent OFDM detection requires knowledge of the channel impulse response (CIR). To this purpose, known symbols (pilots) are often multiplexed into the transmitted data stream [2]. In a given application, the number of pilots results from a tradeoff between contrasting requirements. It must be small to limit the system overhead but must be sufficiently large to guarantee accurate CIR estimates. This conclusion holds true as long as channel estimation and data detection are kept as separate tasks. A different approach is taken in [3] in the context of space–time block-coded OFDM. Here, the expectation–maximization (EM) algorithm [4], [5] is exploited to couple the CIR estimation with the decision making process. The resulting scheme operates in an iterative fashion, with channel estimates being derived not just from pilot symbols but from tentative decisions as well. As an averaging over the CIR realizations is involved in the EM algorithm, the
channel statistics and the operating signal-to-noise ratio (SNR) must be measured in some way. Unfortunately, this problem is not addressed in [3] where both the SNR and the channel covariance matrix are assumed known at the receiver. In this work, we reconsider the iterative detector discussed in [3] as applied to regular OFDM systems (i.e., without space–time coding) and propose a method to measure the channel statistics and the operating SNR from the received signal. Simulations indicate that the iterative data detector endowed with the channel statistics estimator has better performance than other existing schemes operating in a mismatched mode and approaches the error rate performance of a system with perfect channel knowledge. The rest of the letter is organized as follows. The next section describes the signal model and introduces basic notations. The EM detector (EMD) for OFDM signals is revisited in Section III while in Section IV we discuss a method for estimating the channel statistics and the SNR. Performance comparisons are made in Section V and conclusions are drawn in Section VI. II. S IGNAL M ODEL A. OFDM System Consider an OFDM system with N subcarriers for the transmission of Nu phase-shift keying (PSK) symbols per block. Let cn = [cn (−P ), cn (−P + 1), . . . , cn (P )]T be the symbols belonging to the nth block with P = (Nu − 1)/2. After insertion of N − Nu zeros (virtual carriers), cn is fed to an N -point inverse discrete Fourier transform (DFT) operator that produces an N -dimensional vector of time-domain samples. The latter is then extended with an NG -point cyclic prefix and fed to a linear modulator with impulse response gT (t) and signaling interval Ts = TB /(N + NG ), where TB is the total length of the OFDM symbol (including the cyclic prefix). At the receiver side, the incoming waveform is first filtered and then sampled with period Ts . Next, the cyclic prefix is removed and the remaining samples are passed to an N -point DFT unit. We assume that the channel variations are negligible over one OFDM block (slow fading), and indicate with h(n) = [h(n, 0), h(n, 1), . . . , h(n, L − 1)]T the Ts -spaced samples of the overall CIR (encompassing the transmit filter, the physical channel, and the receive filter) during the nth block. Denoting H(n, k) =
Manuscript received July 11, 2003; revised December 26, 2003; accepted April 30, 2004. The editor coordinating the review of this paper and approving it for publication is C. Xiao. The authors are with the Department of Information Engineering, University of Pisa, via Caruso, 56126 Pisa, Italy (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TWC.2005.850263
L−1
h(n, )e−
j2πk N
(1)
=0
as the DFT of h(n), the output of the DFT unit at the receiver side is given by [2] X(n, k) = cn (k)H(n, k) + w(n, k)
1536-1276/$20.00 © 2005 IEEE
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
− P ≤ k ≤ P (2)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
where w(n, k) is the thermal noise that is modeled as a white Gaussian process with zero mean and variance σ 2. Letting X(n) = [X(n, − P ), X(n, − P + 1), . . . , X(n, P )]T , from (1) and (2) we have X(n) = A(cn )F h(n) + w(n)
(3)
where A(cn ) = diag [cn (k); −P ≤ k ≤ P ] and F is an Nu × L matrix with entries [F ]k, = e−
j2πk N
− P ≤ k ≤ P,
0 ≤ ≤ L − 1. (4)
Also, w(n) = [w(n, −P ), w(n, −P + 1), . . . , w(n, P )]T is a zero mean Gaussian vector with covariance matrix σ 2 I Nu (we denote I M as the identity matrix of order M ).
1361
The EM algorithm proves to be a way out as it achieves the same final result with a comparatively simpler iterative procedure. In the EM parlance, the observed measurements are replaced with some complete data from which the original measurements could be obtained through a many-to-one mapping. The algorithm iteratively alternates between an E-step, calculating the log-likelihood function of the complete data, and an M-step, maximizing that expectation with respect to the unknown parameters. In our formulation, we view the DFT output X(n) as the incomplete data and define the complete data as the pair [X(n), h(n)]. Then the maximization step takes the form [5] ˜n |ˆ ˆ(i) c(i−1) i = 1, 2, . . . (M-step) c n = arg max Q c n ˜n c
(9) B. Channel Model We assume a multipath channel with Np distinct paths and baseband impulse response c(t, τ ) =
Np
αi (t)δ(τ − τi )
(5)
˜n is a trial value ˆ(i) where c n is the estimate of cn at the ith step, c (i−1) cn |ˆ cn ] is given by of cn , and Q[˜ (i−1) ˜n |ˆ ˆ(i−1) Q c = Eh p X(n)|h(n), c cn n
i=1
where τi is the delay of the ith path, αi (t) is the corresponding complex amplitude, and δ(t) denotes the Dirac delta function. The path gains {αi (t)} are modeled as narrowbandindependent Gaussian processes with zero mean and autocorrelation function Ri (τ ) = σi2 J0 (2πBD τ )
(6)
where σi2 = E[|αi (t)|2 ] is the power associated to the ith path, BD is the Doppler bandwidth, and J0 (x) is the zeroth-order Bessel function of the first kind. Denoting g(t) as the overall response of the transmit and receive filters, from (5) it is seen that h(n) has entries h(n, ) =
Np
αi (nTB )g(Ts − τi )
= 0, 1, . . . , L − 1
i=1
˜n ]} × ln{p [X(n)|h(n), c
1 (πσ 2 )Nu 1 2 × exp − 2 X(n) − A(˜ c)F h(n) σ
˜n ] = p [X(n)|h(n), c
Np
×
σi2 g(1 Ts − τi )g(2 Ts − τi ). (8)
i=1
Notice that contrarily to what it is usually assumed, the covariance matrix of h(n) is not strictly diagonal due to the pulse shaping of the transmit and receive filters.
(11)
where the notation · denotes the Euclidean norm. Modeling h(n) as a zero-mean Gaussian vector (Rayleigh fading) with covariance matrix C h = E[h(n)hH (n)], we have p [h(n)] =
E [h(n1 , 1 )h∗ (n2 , 2 )] = J0 [2πBD TB (n1 − n2 )]
(10)
˜n ] is the conditional In the above equation, p[X(n)|h(n), c probability density function (pdf) of X(n) while Eh {·} is the statistical expectation over the pdf p[h(n)] of h(n). ˜n ] follows from (3) and is The expression of p[X(n)|h(n), c given by
(7)
and autocorrelation function
(E-step).
πL
1 exp hH (n)C −1 h h(n) . det (C h )
(12)
Finally, substituting (11) and (12) into (10) and performing some algebraic manipulations, it is found that (9) is equivalent to the maximization problem (i−1) H ˆ MMSE n, c ˆ(i) ˆ c e X = arg max (n)A(˜ c )F h n n ˜n c
i = 1, 2, . . .
(13)
−1 H H (i−1) ˆn X(n) = F H F + σ 2 C −1 F A c h
(14)
where III. I TERATIVE D ETECTION S CHEMES Maximum likelihood (ML) sequence detection for OFDM systems without channel state information is a challenging task. The reason is that direct maximization of the likelihood function is computationally infeasible for lack of efficient ways to perform the search over the candidate data sequences.
ˆ MMSE n, c ˆ(i−1) h n
is the minimum mean square error (MMSE) estimate of h(n) ˆ(i−1) assuming that c is transmitted [6]. The rationale behind n
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
1362
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
the EM algorithm is easily understood by rewriting (13) as ˆ(i) c n
= arg max ˜n c
P
transmitted data sequence. Correspondingly, the EMD becomes independent of the channel statistics and reads
e {X ∗ (n, k)˜ cn (k)
k=−P
ˆ MMSE n, k, c ˆ(i−1) ×H n
ˆ(i) c n
i = 1, 2, . . .
1) In the absence of coding, maximizing the right hand ˜n is equivalent to maxside of (15) with respect to c imizing each individual term in the sum, i.e., making symbol-by-symbol decisions. In trellis-coded systems, the maximization can be performed by means of the (i) cn (k)] = Viterbi algorithm with branch metrics λk [˜ (i−1) ∗ ˆ ˆn ]}. cn (k)HMMSE [n, k, c e{X (n, k)˜ ˆ(0) 2) Inspection of (15) reveals that an initial estimate c n of the transmitted data sequence is required to initialize ˆ(0) the EMD. An obvious choice for c n comes from the application of conventional coherent detection = arg max ˜n c
P
∗
ˆ (0)
e X (n, k)˜ cn (k)H
(n, k)
k=−P
(16)
ˆ (0) (n, k)} provided that a suitable channel estimate {H is available at the receiver. One method to obtain ˆ (0) (n, k)} consists of multiplexing known symbols {H (pilots) into the transmitted data stream and using any pilot-aided channel estimation scheme available in the technical literature. Alternatively, data decisions provided by the EMD during the current OFDM block can be used to obtain an initial channel estimate for the next block. In this case, the EMD intrinsically performs adaptive chanˆ (0) (n, k)} nel tracking. In the sequel, we assume that {H is computed in the first manner and we call (16) the conventional coherent detector (CCD). 3) From (13) and (14), it is seen that the EMD requires knowledge of C h and σ 2. A simple solution to this problem consists of assuming high SNR values. In this case, σ 2 is vanishingly small and (14) reduces to
˜n c
−1 ˆ LS n, c ˆ(i−1) ˆ(i−1) h = (F H F ) F H AH c X(n). n n (17)
ˆ LS [n, c ˆ(i−1) ] is the least-squares (LS) estiNote that h n ˆ(i−1) mator of h(n) under the assumption that c is the n
e X ∗ (n, k)˜ cn (k)
k=−P
(15)
ˆ MMSE [n, c ˆ MMSE [n, k, c ˆ(i−1) ]} is the DFT of h ˆ(i−1) ]. where {H n As is seen, at the ith iteration, the estimate of cn is computed through conventional frequency-domain detection/equalization techniques [7, p. 337] where channel information is achieved ˆ(i−1) as by means of the MMSE criterion using data decisions c n obtained from the previous iteration. In the sequel, the iterative detector (15) is referred to as the EMD. The following remarks are of interest:
ˆ(0) c n
= arg max
P
×
ˆ LS n, k, c ˆ(i−1) H n
i = 1, 2, . . .
(18)
ˆ LS [n, c ˆ LS [n, k, c ˆ(i−1) ˆ(i−1) ]} is the DFT of h ]. where {H n n In the sequel, (18) is referred to as the least-squares based detector (LSBD). 4) An alternative solution that dispenses with the estimation of C h and σ 2 is adopted in [8], where the channel estimator is designed for fixed nominal values C h and ˆ MMSE (n, c ˆ(i−1) σ 2 . In this case, the estimate h ) in (14) is n replaced by ˆ(i−1) h n, c n
−1 −1 ˆ(i−1) X(n) F H AH c = F H F + σ2 C h n
(19)
and the resulting scheme operates in a mismatched mode. For this reason, we call it the mismatched EM detector (MEMD). 5) Matrix C h is assumed to be diagonal in [3] and no virtual carriers are employed. In these circumstances, the EMD does not involve any matrix inversion as F H F reduces to N × I L and F H F + σ 2 C −1 h becomes diagonal. As shown in (8), however, C h is not strictly diagonal. Also, virtual carriers are usually employed in practical OFDM systems (as in the HIPERLAN/2 and IEEE 802.11a standards) to guarantee sufficiently wide guard bands and avoid aliasing problems.
IV. E STIMATION OF THE C HANNEL S TATISTICS AND N OISE P OWER Applying the Woodbury identity [9, p. 565] to compute −1 in (14) produces (F H F + σ 2 C −1 h ) (i−1) ˆ LS n, c ˆ MMSE n, c ˆ(i−1) ˆ = h h n n
ˆ(i−1) X(n) − σ 2 P −1 F H AH c n
(20)
where P ∈ C / L×L is defined as P = F H F C h F H F + σ2 F H F .
(21)
Inspection of (20) and (21) reveals that computing ˆ MMSE [n, c ˆ(i−1) h ] requires knowledge of σ 2 and P . In pracn tice, these quantities are not known and must be estimated in some manner. In the following, we show how they can be estimated from the received signal.
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
A. Noise Power Estimation We temporarily assume that the symbols cn are known, and we model h(n) as an unknown parameter vector. Then, from (3) we see that the joint ML estimates of σ 2 and h(n) are found by maximizing the log-likelihood function 2 1 ˜ = −Nu ln(π˜ ˜ σ 2 ) − 2 X(n) − A(cn )F h Λ(˜ σ 2 , h) σ ˜ (22) ˜ We begin by keeping with respect to the trial values σ ˜ 2 and h. 2 ˜ σ ˜ fixed and let h vary in the 2L-dimensional space C/L . ˜ achieves a maximum for In these conditions, Λ(˜ σ 2 , h) −1 ˆ h(n) = (F H F ) F H AH (cn )X(n).
(24)
−1
1 X H (n)A(cn )F ⊥ AH (cn )X(n). Nu
1 F ⊥ AH (cn )X(n)2. Nu
(26)
2 reads (the derivation is omitted for The expectation of σ ˆML space limitations)
2 E σ ˆML
Nu − L 2 = σ Nu
(27)
meaning that (26) is a biased estimator. On the other hand, collecting (26) and (27), it is easily seen that an unbiased estimate of σ 2 can be computed as σ ˆ2 =
⊥ H 1 F A (cn )X(n)2. Nu − L
(28)
At this stage, we recall that cn is not known and (28) cannot be directly used. Therefore, we replace cn with the data ˆn provided by EMD. Also, in order to improve decisions c the estimation accuracy, a time-domain filtering operation is applied to σ ˆ 2 leading to the iterative equation 2 ⊥ H µ F A (ˆ σ (n − 1) + cn )X(n) σ ˆ (n) = (1 − µ)ˆ Nu − L (29) 2
σ4 µ . 2 − µ Nu − L
2
in which σ ˆ 2 (n) is the estimate of σ 2 during the nth OFDM block and µ is a positive factor (0 < µ ≤ 1) that controls the
(30)
B. Estimation of P We begin by defining the L-dimensional vector Z(n) = F H AH (cn )X(n).
(31)
Next, substituting (3) into (31) and using the identity AH (cn )A(cn ) = I Nu yields Z(n) = F H F h(n) + F H AH (cn )w(n)
(32)
(33)
Comparing (33) with (21), we see that E[Z(n)Z H (n)] = P . Therefore, an estimate of P can be obtained by low-pass filtering Z(n)Z H (n). Unfortunately, Z(n) is not available as it involves the transmitted data sequence cn . Therefore, we ˆn provided by EMD. again replace cn with the data decisions c This leads to the recursive equation
(25)
It is worth noting that F ⊥ is independent of the transmitted data symbols and, accordingly, can be precomputed and stored in the receiver. Also, bearing in mind that F ⊥ is idempotent (i.e., F ⊥ = F ⊥ F ⊥ ), we may rewrite (25) in the equivalent form 2 = σ ˆML
2 var σ ˆ (n) =
from which it follows that
E Z(n)Z H (n) = F H F C h F H F + σ 2 F H F .
where F ⊥ = I − F (F H F ) F H is the orthogonal complement of F . Finally, the ML estimate of σ 2 is the argument σ 2 ) and reads σ ˜ 2 that maximizes Γ(˜ 2 = σ ˆML
tradeoff between convergence speed and estimation accuracy. ˆn ≈ cn ), it can be Assuming accurate data decisions (i.e., c shown that
(23)
Next, substituting (23) into (22) produces Γ(˜ σ 2 ) = −Nu ln(π˜ σ2 ) 1
− 2 X H (n)A(cn )F ⊥ AH (cn )X(n) σ ˜
1363
ˆ (n) = (1 − λ)P ˆ (n − 1) + λZ(n) ˆ ˆ H (n) P Z
(34)
ˆ (n) denotes the estimate of P at the nth step and in which P ˆ cn )X(n). Z(n) = F H AH (ˆ Inverting both sides of (34) and employing the Woodbury identity yields −1 ˆ ˆ −1 (n − 1) ˆ −1 (n) = 1 ˆ (n − 1) − K(n)Z(n) P P P 1−λ (35) with K(n) =
ˆ (n − 1)Z(n) ˆ λP . H ˆ (n)P ˆ (n − 1)Z(n) ˆ 1−λ+Z
(36)
ˆ −1 in an The above equations allow the computation of P iterative fashion. Actually, as the channel statistics vary slowly ˆ −1 at each (long-term fading), there is no need to update P OFDM block. This suggests an alternative scheme in which ˆ (n) is updated at symbol rate using (34) and is the estimate P inverted only periodically every K blocks, K being a design parameter suitably chosen as a tradeoff between system complexity and tracking capability. V. S IMULATION R ESULTS Simulations have been run to compare the performance of the proposed iterative detectors under the following conditions. 1) The DFT size is N = 256 and the modulated subcarriers are Nu = 213. 2) The CIR is generated as indicated in (7) with four paths (Np = 4). The path delays are kept fixed
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
1364
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
Fig. 1. Performance of EMD with Ni = 1, 2, and 3.
and equal to τ1 = 0, τ2 = 1.4Ts , τ3 = 4.8Ts , and τ4 = 9.7Ts , whereas the path gains vary independently and are modeled as complex-valued Gaussian random processes with zero mean and power σi2 = η exp(−i), i = 0, 1, 2, 3. The constant η is chosen such that the signal energy is normalized to unity, i.e., E{h(n)2 } = 1. Correspondingly, the SNR equals 1/σ 2 , where σ 2 is the variance of the Gaussian noise. 3) The Doppler bandwidth is BD = 10−2 /TB . 4) Pulse g(t) has a raised cosine spectrum with a roll-off of 0.22 and the CIR length is L = 16. ˆ (0) (n, k)] for use in the CCD and to 5) Channel estimates [H initialize EMD, LSBD, and MEMD are computed with the pilot-assisted ML method discussed in [6]. A total of 17 pilot symbols are inserted in each OFDM block at uniformly spaced locations within the signal bandwidth. 6) A rate 1/2 convolutional code is adopted with constraint length four and generator polynomials 15 and 17 (in octals). The coded bits are mapped onto quaternary phase shift keying (QPSK) symbols, which are then interleaved by a 14 × 14 block interleaver to counteract error bursts. A soft-input Viterbi processor is used at the receiver. 7) Unless otherwise stated, parameters µ and λ in (29) and (34) are chosen to be equal to 0.05 and 0.01, respectively. 8) As shown in [8], the channel estimator designed for the uniform power delay profile is robust to the channel statistics mismatch. Thus, we choose C h = diag [(1/L), (1/L), . . . , (1/L)] for the MEMD [10]. The nominal value of σ 2 is fixed to σ 2 = 15 dB. Fig. 1 shows the bit error rate (BER) performance of EMD as a function of the SNR. Marks indicate simulation results while solid lines serve to ease the reading. The parameter of the curves is the number of iterations Ni in the data detection process. The curve labeled ideal channel information (ICI) corresponds to detection with perfect channel knowledge and serves as a benchmark. As is seen, the EMD converges rapidly and in fact only marginal improvements are gained in passing from two to three iterations. Also, the gap between EMD and ICI becomes negligible as SNR grows. Fig. 2 compares the BER of the discussed data detectors as a function of the SNR. Three iterations have been performed with EMD, LSBD, and MEMD. The curve labeled EMD with
Fig. 2.
BER of the data detectors versus SNR.
Fig. 3.
Noise power MSEE versus SNR with µ = 0.1 and 0.05.
perfect channel statistics (EMD-PCS) is obtained from (13) and (20) assuming perfect knowledge of P and σ 2 . As is seen, LSBD and MEMD have virtually the same performance. This means that replacing (C h , σ 2 ) with (C h , σ 2 ) is equivalent to ignoring the channel statistics at all (at least in the considered scenario). Both LSBD and MEMD achieve a gain of nearly 2 dB over CCD, but they are 1.5 dB worse than EMD. Comparing EMD with EMD-PCS indicates that measurement errors in the estimation of P and σ 2 do not impair the performance at the SNR of practical interest. As is intuitively clear, the loss of MEMD with respect to EMD-PCS depends on the degree of mismatch between the true channel statistics and their nominal values and becomes vanishingly small as (C h , σ 2 ) approaches (C h , σ 2 ). Also, simulation results (not shown for space limitations) indicate that increasing the number of pilots from 17 to 27 improves the performance of LSBD and MEMD by approximately 0.7 dB without significantly affecting that of EMD. The reason is that in these circumstances both LSBD and MEMD are initialized with a better channel estimate whereas EMD is very close to the ICI curve and,
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
accordingly, there is little to gain by increasing the number of pilots. Fig. 3 illustrates the mean square estimation error (MSEE) E{[ˆ σ 2 (n) − σ 2 ]2 } versus SNR as obtained with either µ = 0.05 or 0.1. Marks indicate simulation results while thick solid lines represent theoretical values as computed from (30). It is seen that the agreement between simulations and theory is good only at high SNR values. The reason is that the result from (30) has been computed assuming accurate data detection and, accordingly, it is not valid at low SNRs. VI. C ONCLUSION We have discussed a method to estimate the SNR and the channel statistics in OFDM systems operating over frequency selective fading channels. The estimated parameters are employed to perform EM-based iterative ML sequence detection. Simulations indicate that the proposed scheme has good performance and needs only a few iterations to converge. Its gain over similar schemes operating in a mismatched mode is about 1.5 dB for coded QPSK. The performance loss with respect to a system with perfect channel knowledge is negligible at an SNR of practical interest.
1365
R EFERENCES [1] R. Van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Norwood, MA: Artech House, 2000. [2] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb. 1995. [3] B. Lu, X. Wang, and Y. G. Li, “Iterative receivers for space-time blockcoded OFDM systems in dispersive fading channels,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 213–225, Apr. 2002. [4] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Stat. Soc., vol. 39, no. 1, pp. 1–38, Dec. 1977. [5] C. N. Georghiades and J. C. Han, “Sequence estimation in the presence of random parameters via the EM algorithm,” IEEE Trans. Commun., vol. 45, no. 3, pp. 300–308, Mar. 1997. [6] M. Morelli and U. Mengali, “A comparison of pilot-aided channel estimation methods for OFDM systems,” IEEE Trans. Signal Process., vol. 49, no. 12, pp. 3065–3073, Dec. 2001. [7] A. F. Molisch, Wideband Wireless Digital Communications. Englewood Cliffs, NJ: Prentice-Hall, 2001. [8] Y. Li, L. J. Cimini, Jr., and N. R. Sollenberger, “Robust channel estimation for OFDM systems with rapid dispersive fading channels,” IEEE Trans. Commun., vol. 46, no. 7, pp. 902–915, Jul. 1998. [9] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [10] B. Yang, K. Ben Letaief, R. S. Cheng, and Z. Cao, “Channel estimation for OFDM transmission in multipath fading channels based on parametric channel modeling,” IEEE Trans. Commun., vol. 49, no. 3, pp. 467–479, Mar. 2001.
Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on December 4, 2008 at 12:18 from IEEE Xplore. Restrictions apply.
