Iterative Joint Channel Estimation And Decoding

Iterative Joint Channel Estimation and Decoding Using Superimposed Pilots in OFDM-WLAN Ting-Jung Liang, Wolfgang Rave and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Dresden University of Technology, D-01062 Dresden, Germany {liang,rave,fettweis}@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/MNS Abstract— A preamble technique is used in present OFDMbased WLAN systems to accomplish synchronization and channel estimation. The preamble introduces transmission delay and wastes capacity, especially when a data packet is short. An alternative is a superimposed pilot technique which can provide similar BER vs. Eb /N0 performance by specially designed receiver and optimized system parameters. In this paper we study the conditions under which superimposed pilots can be successfully used for jointly estimating the channel and detecting data. It is shown that the transmission delay or throughput of wideband WLAN systems can be significantly improved when such a technique is employed at the price of higher complexity due to the iterative receive algorithm.

I. I NTRODUCTION In present packet-based OFDM WLAN, such as IEEE802.11a/g, the synchronization and channel estimation functions are accomplished by a preamble technique, which is a reliable receiver algorithm with acceptable performance degradation and low complexity. But the preamble introduces transmission delay and wastes capacity unavoidably due to the time loss before data transmission, especially when a data packet is short (such as an ACK or web page request). An alternative is a superimposed pilot technique. In the literature, iterative channel estimation algorithms have been proposed [1], but there are still open questions, concerning the best receiver architecture and under which conditions wireless systems using superimposed pilots can outperform a preamble technique. A wireless system using a superimposed pilot technique can be considered as a case of the wider class of ”turbo” equalization problems, first discussed in [2]. Analytical tools developed for turbo decoding [3] or turbo equalization [4], [5] can be applied to investigate the iterative behavior of a wireless system employing superimposed pilots. A superimposed pilot technique has been proposed in a continuous transmission system with long data packets [6], such as DVB-T. In this paper, we focus on high throughput short range wireless systems, such as the one studied in the German research project WIGWAM1 , where a non negligible amount of bursty data traffic is expected in office or public access scenarios. The performance of a preamble technique will serve as a benchmark. 1 This work was partly supported by the German ministry of research and education within the project Wireless Gigabit with Advanced Multimedia Support (WIGWAM) under grant 01 BU 370

Sketches in Fig. 1 illustrate the preamble and superimposed pilot techniques. The preamble technique sequentially accomplishes channel estimation functions and the transmission of a data payload. Throughout the paper we assume that the preamble has a duration of 1 OFDM symbol and is transmitted 2 with power σx,pre that is maintained for the M OFDM data symbols. In contrast, superimposed pilots are transmitted parallel in time with M unknown data symbols using a fraction 2 ρ of the total power σx,sup . The parameter ρ is to be optimized. (a) 2 σ x,pre

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M

Preamble

Data

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Fig. 1. Alternative burst structures for an OFDM-WLAN system: (a) a preamble technique, (b) a superimposed pilot technique.

For a fair comparison of a preamble and superimposed pilots, the energy per information bit Eb , taking into account the energy of the pilots and the preamble length, should be kept constant. This leads to the condition 2 2 σx,sup σx,pre Eb Eb 1 1 M+1 = · = · · = . (1) 2 2 N0 qR σ qR σ M N0 sup

w

w

pre

A QAM constellation with alphabet size Q = 2q , the code 2 rate R and Gaussian noise with variance σw are assumed. After this motivation of our interest in OFDM-WLAN with superimposed pilots, the rest of the paper is organized as follows: In section II, the OFDM system model and the iterative receiver for joint channel estimation and decoding is presented. Section III analyzes the performance of the channel estimator using a-priori information assuming a decoder producing likelihood values for the data bits modelled as a Gaussian random variable. In section IV the joint operation of channel estimator and soft decoder is investigated and suitable system parameters are determined before summarizing the paper with the conclusions. II. S YSTEM M ODEL AND R ECEIVER A RCHITECTURE

Throughout the paper a complex baseband model of a timesynchronous OFDM system with N subcarriers is considered. At the transmitter, the data bits are first encoded with a nonrecursive convolutional encoder using the generator G[7, 5]oct , subsequently permuted by a random interleaver and modulated by a Gray-mapping QAM modulator to OFDM data symbols

in the frequency domain. These data symbols are transformed to a time domain signal by an FFT operation to which a guard interval is prepended and transmitted. At the receiver side, we are mainly interested in the case where superimposed pilots are used for which we investigate the receiver structure shown in Fig. 2, similar to the one described in [7]. The channel state is initially estimated only from the known pilot portion within the received signal. Based on the initial estimate, likelihood values λDecIN for the data bits are calculated according to the Generalized LogLikelihood Ratio (GLLR) algorithm proposed in [8]. Then a code word is decoded with the BCJR algorithm [9], [10]. y,σ w2

λ D ecIN

Y, σ W2 FFT

GLLR Hˆ

σ E2

(c ) '

S N R L L R , D ecIN H

Channel Estimation

λ D ecO U T ( c ' ) SNR LLR , DecOUT

Π −1 BCJR Decoder

BER

Π

Fig. 2. Iterative channel estimation and decoding at the receiver for the superimposed pilot technique.

Starting from the 2nd iteration, the current channel state (assumed to be constant during a burst) is re-estimated using both parts of the received signal (pilots and data) exploiting a-priori knowledge λDecOUT obtained from the decoded 2 data symbols. This changes the variance σE of the channel H estimation error, otherwise the metric computation (GLLR) and decoding operations remain the same. Under appropriate conditions the density evolution of the LLR values λDecIN and λDecOUT leads to an increase of their associated SNR values SN RLLR,DecIN and SN RLLR,DecOUT and to correct channel estimation and decoding. The transmission model for a single subcarrier for the superimposed pilot technique can be written as follows2 : p √ (2) Y = ( ρP + 1 − ρS)H + W ˆ = H + EH H (3) the received signal, pilot, power ratio, sent data symbol, instantaneous channel state and noise in the frequency domain are denoted as Y , P , ρ, S, H and W , respectively. The ˆ estimated channel state and channel estimation error are H and EH . The channel transfer function equal to the Fourier transform of the channel impulse response (CIR) h is given by H = F h with the DFT matrix F . The zero-mean complex Gaussian random variables W and EH are assumed to be 2 2 independent. Based on Y and the variances σW and σE , H generalized LLRs for imperfect channel knowledge taking into account the joint distribution of channel coefficients and data 2 symbols are computed (how the estimate of σE is obtained H will be described in section III). For QAM symbols the GLLR 2 Notation: Upper (lower) case letters will be generally used for frequency(time-) domain signals; boldface letters represent vectors; letters with both boldface and underline represent matrices.

algorithm [8] leads to the following expression for the bit metric values required at the decoder input:   √  ′ √ |Y −( ρP + 1−ρS − )Hˆ |2 λDecIN ck = min′ 2 2 |S − |2 σE +σW H S − ∈{S:ck =1}   √ √ 2 |Y −( ρP + 1−ρS + )Hˆ | − min′ (4) 2 2 |S + |2 σE +σW H S + ∈{S:ck =0}

(S + and S − represent the symbol sets, where bit k assumes the values 0 and 1 in the symbol alphabet, respectively). The ′ coded interleaved bits are denoted as ck . For the IEEE802.11a-like reference system under investigation, it is assumed that all 64 subcarriers are available for data transmission. The modified IEEE802.11a long training symbols, with equally probable pilots symbols ±1 through all 64 subcarriers, are used as a preamble and also without change for the superimposed pilot technique with appropriate power scaling. The channel impulse response h is described by an exponential power delay profile which is assumed to be known at the receiver (we used a maximal number of L = 8 taps in numerical examples with tap power ∝ exp(−i) at tap i and the rms value of the sum of all channel taps normalized to one). The LLR values λDecIN,OUT are assumed to be i.i.d. and Gaussian-distributed according to the so-called p Gaussian ′ Density Model [3]. Therefore λ = µλ (1 − 2ck )+ σλ2 N (0, 1) ′ with ck ∈ {0, 1} and µλ being the mean value of λ which is related to its variance by σλ2 = 2µλ . Using this model, the SNR of the LLR distribution at the decoder input calculated with the GLLR algorithm or the one corresponding to the LLR distribution at the decoder output used as a-priori information by the channel estimator (denoted as SN RLLR,DecIN or SN RLLR,DecOUT in Fig. 2) can be defined as: µλ µ2 SN RLLR , λ2 = . (5) σλ 2 Independence between W and EH and the i.i.d. assumption for the LLRs λDecIN,OUT are ideal assumptions. Although they are not perfectly valid for the superimposed pilot technique, we use them to characterize the performance of the iterative channel estimation and decoding algorithm in the next section. The deviations of this assumption are investigated later on by numerical simulations. III. C HANNEL E STIMATION FOR S UPERIMPOSED P ILOTS A. Estimate without a priori Information during first Iteration The channel estimation using the pilot part of the signal only is obtained from the following transmission model [11]: p √ Y = ρP F h + 1 − ρSH + W (6) | {z } | {z } X ef f

W ef f

The diagonal matrices P and S (σP2 = σS2 = 1) contain pilot and data symbols on their diagonal entries. The effective pilot matrix becomes X ef f while the effective noise is W ef f 2 2 2 with variance σef f = (1 − ρ)σX,sup + σW which is due to the AWGN part augmented by a data dependent term. The

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time domain LS channel estimator and its estimation error are given as

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1 2 / (ρσ 2 N ) . (10) 1 + σef i f

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Sup. Pilot(LS, M=1) Sup. Pilot(LS, M=4) Sup. Pilot(MMSE, M=1) Sup. Pilot(MMSE, M=4) Preamble(LS, M=1) Preamble(MMSE, M=1)

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Fig. 3. Error variance of the channel estimate using only superimposed pilots with M = 1 or M = 4 OFDM symbols or a preamble for Eb /N0 = 10 dB and 16-QAM.

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Sˆ and S by a Gray-mapping QAM modulator. We note that, if SN RLLR,DecOUT is known (we will evaluate it numerically at the decoder output), the preamble-based channel estimator and estimation error theory can be directly applied to determine the channel estimation error by replacing a preamble as the effective pilot and Gaussian noise as the effective noise [11], [12]. We also investigated the deviation due to idealized assumptions of the theoretically expected channel estimation error and numerically evaluated values. For the MMSE channel estimator with soft decision in Fig. 5 (a), we observe that the theoretically expected variance according to eq. (10) with soft decision agrees with the simulated results, when SN RLLR,DecOUT > 6. For the MMSE channel estimator with hard decision, Fig. 5 (b) shows that the error variance 2 can only slightly be reduced by averaging over sevσE H eral symbols, when SN RLLR,DecOUT is small (the smallest SN RLLR,DecOUT in simulation is 10−3 ). But again we note the theoretically expected error variance with hard decision agrees with simulation, when the likelihood values become more reliable, in this case for SN RLLR,DecOUT > 8. 0

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(11) ˆ is evaluated either by hard (HD) or soft The estimate S decision (SD) [5] with the symbol estimation error ES = ˆ The effective pilot matrix X S − S. ef f becomes a random variable which is not constant over the subcariers. The vari2 ance of the effective noise W ef f is described by σef f = 2 2 2 (1 − ρ)σES + σW . Simulated variances σES of the symbol estimation error for different constellation sizes and decision methods are shown in Fig. 4. LLR values at the decoder output for given SN RLLR,DecOUT were simulated with the Gaussian Density Model by random source bits, followed by calculating

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The channel estimation model using both pilots and decoded ˆ data [11] exploits the data estimate S: p p √ ˆ h + 1 − ρ(S − S)H ˆ Y = ( ρP + 1 − ρS)F +W {z } | | {z } X ef f

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Fig. 4. Symbol estimation error for different constellation sizes and decision methods obtained with the Gaussian Density Model for given SN RLLR,DecOU T .

(9)

Here C h denotes the covariance matrix of the CIR h, which is available, if the channel statistics are known. Again a factor ρ is the only difference compared to the formula in [12]). The value σi2 corresponds to the variance of tap i of the CIR h. Fig. 3 shows the variance of the estimation error as a function of the power fraction ρ used for the superimposed pilots and the preamble (independent of ρ) for the channel with exponential power delay profile. The variance of the channel estimates found by simulation agree exactly with the analytical 2 expressions derived from estimation theory. Apparently, σE H should be as low as possible.

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which is the same formula as given in [11] apart from the factor ρ. Similarly, the time domain MMSE channel estimator using only pilot symbols and the associated channel estimation error read −1 H 2 −1 ˆ = F [(σef H X ef f H Y ] , f C h + X ef f X ef f )

HD, QPSK SD, QPSK HD, 16QAM SD, 16QAM

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Fig. 5. Error 2 variance σE vs. H SN RLLR,DecOU T 10 for the MMSE channel estimator with soft (top) and hard decision (bottom) (Eb /N0 = 10 dB, 16-QAM, ρ = 0.1). 10

SNR

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The LS theory according to eq. (8) leads to similar results as the MMSE theory (soft decision: SN RLLR,DecOUT > 6 and hard decision: SN RLLR,DecOUT > 8). Generally speaking, channel estimation theory under the present assumptions matches the real case well at high SN RLLR,DecOUT . The difference between theory and simulation can be modelled as an additional Gaussian noise, but it improves iterative BER performance very little.

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Fig. 6. (a) Convergence behavior of iterative detection and decoding for different Eb /N0 and channel estimation techniques in the BER / SN RLLR,DecOU T plane (non-recursive convolutional codes using generator G[7, 5]oct , 16-QAM, ρ = 0.15 and LS / MMSE channel estimator with soft decision for a superimposed pilot technique), (b) Effect of time domain LS or MMSE channel estimator with soft decision (M =4 and ρ = 0.15), (c) Effect of time domain MMSE channel estimator using hard decision or soft decision (M =4 and ρ = 0.15), (d) Effect of power ratio using superimposed pilot technique (M =4), (e) Effect of packet length using superimposed pilot technique (ρ = 0.15), (f) Preamble vs. superimposed pilot techniques (M =4 and ρ = 0.15) - note: Eb /N0 = 18 dB and 16-QAM assumed for all cases in Fig. (b)-(f), max. 20 iterations for a superimposed pilot technique

IV. J OINT C HANNEL E STIMATION AND D ECODING U SING S UPERIMPOSED P ILOT A. Iterative Channel Estimation and Decoding Algorithm 2 The estimation error variance σE is required to evaluate H the GLLR when using a superimposed pilot technique, see Fig. 2 2. As discussed in section III, the variance σE in the first H iteration can be exactly determined by theory. In subsequent 2 iterations, σE is evaluated using the likelihood ratios at the H decoder output. A practical receiver using a superimposed pilot technique can be implemented as follows: • estimate SN RLLR,DecOUT at the input of channel estimator, 2 • calculate the symbol estimation error variance σE for S 2 given SN RLLR,DecOUT (σES vs. SN RLLR,DecOUT can be well approximated by linear functions), 2 • calculate σE according to eqs. (8) or (10). H ′ The coded and interleaved bits ck are assumed to take on the values ”0” and ”1” with equal probability and the probability density function (pdf) of λ(c′ ) is symmetric with respect to λ = 0. Based on the symmetry and consistency conditions on likelihood values [13], SN RLLR,DecOUT is obtained by evaluating the second moments of the pdf of the absolute value of λ and by exploiting the relation σλ2 = 2µλ . The value SN RLLR,DecOUT is uniquely related to a BER value [4] regardless of the specific pilot design. For a specific

power ratio ρ = 0.15 of superimposed pilots, we demonstrate in Fig. 6 (a) the convergence of the density evolution of the LLR-values at the decoder output during the iterative detection and decoding process for different Eb /N0 -values comparing in addition LS and MMSE channel estimation. A reference case was determined for a receiver with perfect channel knowledge. With the preamble technique this unique mapping is maintained due to the GLLR algorithm and the 2 relatively good channel estimate, i.e. low σE values. H With the superimposed pilot technique, numerical results show that the distribution of LLR values can be approximated by a Gaussian density function. But the relation of BER vs. SN RLLR,DecOUT only holds roughly, due to an imprecise 2 estimate (σE is estimated only from the current burst in H our proposed system) and possible correlation between noise W and channel estimation error EH . This leads to a bias of the LLR values (λDecIN ) computed with the GLLR algorithm and subsequently also to a bias of the output LLRs from the decoder (λDecOUT ). This distortion was removed from the observed SN RLLR,DecOUT values by mapping them according to the observed BER to a correspondent SN RLLR,DecOUT value with the same BER on the reference curve with perfect channel knowledge. The deviation of SN RLLR,DecIN was removed with the same method (for different QAM modulation another ’master’ or reference curve was used). These corrected SN RLLR,DecOUT and SN RLLR,DecIN were used to track

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the density evolution of LLR values from half-iteration to halfiteration (Half-Iteration Dynamic Model named in [3]) and to optimize the system parameters discussed in the next section. B. Choices of System Parameters The choices of possible combinations of ρ and M are deduced from the smallest Eb /N0 value, for which a BER ≤ 10−4 is achieved. For QPSK we need Eb /N0 ≃ 16 dB (see Fig. 7 (c)), for 16-QAM Eb /N0 ≃ 18 dB is required (see Fig. 7 (d)). An additional comparison is made to a genie system with perfect channel knowledge needing no pilot power. 1) LS vs. MMSE Channel Estimator and HD vs. SD: At first, we should decide which channel estimator is adequate. Fig. 6 (b) shows that, as expected, a time domain MMSE channel estimator can increase both the initial and final SN RLLR values with respect to an LS estimate. Also soft decisions improve the final SN RLLR but not dramatically as can be seen in Fig. 6 (c). Because complexity is not an issue the time domain MMSE channel estimator using soft decisions was selected for the investigations in the following sections. 2) Effect of Power Ratio ̺ and Packet Length M : Power ratio and packet length are the two most important system parameters in a superimposed pilot technique. In Fig. 6 (d), the power ratio decreases from 15% to 10%. As a consequence, the decoder performance with perfect channel knowledge increases, but the aggregate performance decreases due to worse channel estimate and error propagation in this case.

Generally, if the power ratio becomes too small or too large, the aggregate performance is bad because either the channel estimator or the decoder performs very bad. This means that an optimal power ratio exists for a fixed burst length M . Fig. 6 (e) shows for M = 1, 3 and 4 that, if ρ is fixed, the aggregate performance using a superimposed pilot technique increases monotonously by averaging more symbols to improve the quality of the channel estimate. 3) Preamble vs. Superimposed Pilot Techniques: In addition to the performance of a superimposed pilot technique for different M and ρ, the performance of a preamble technique decreases monotonously as M decreases, because relatively more energy is invested in estimating the channel. In the following, our interest is in the relative performance of both Eb Eb techniques under N0 = N . 0 sup

pre

The optimal range of system parameters (M and ρ), for which a superimposed pilot technique can outperform the preamble technique, was investigated by simulation, stepping through the range M =[1:1:5] and ρ=[0.1:0.05:0.5] provided 1 that ρ < M+1 (It is reasonable to apply smaller energy for superimposed pilots than for a preamble with the same total energy). To give a fair comparison, an iterative preamble technique was studied as well. It estimates first the instantaneous channel state using only the preamble, then using both preamble and decoded data at subsequent iterations. Fig. 6 (f) shows that such a preamble technique converges after three iterations whereas a superimposed pilot technique requires at least ten iterations.

The dark shaded areas in Fig. 7 (a) and Fig. 7 (b) indicate the optimal range of system parameters determined by simulation. The value BERpre denotes the BER of a preamble technique after three iterations. The first number in Fig. 7 (a) and Fig. 7 (b) indicates the normalized bit error rate difference (BERpre −BERsup )/BERpre with BERsup after ten iterations. Positive values correspond to advantages for the superimposed pilot technique. The second number gives the same quantity with BERsup after twenty iterations. There is no significant difference between the performances with ten and twenty iterations. The optimal range of 16QAM is smaller than for QPSK due to the smaller distance between constellation points. The BER of a superimposed pilot technique is close to that of a preamble technique only for optimized parameter values. C. Iterative Behavior of Systems Using Superimposed Pilots The iterative behavior of QPSK (M =2, ρ = 0.25) and that of 16-QAM (M =4, ρ = 0.15) is compared by BER curves in Figs. 7 (c), (d) and by plotting the variance of the channel estimation error in Figs. 7 (e), (f), respectively. With optimized system parameters, the BER of the superimposed pilot technique for both 16-QAM and QPSK is practically the same as the one achieved with the preamble technique through all Eb /N0 . With superimposed pilots the BER converges after around ten iterations for both constel2 lations to the one found for the preamble technique (σE H converges after about three iterations for QPSK and after five iterations for 16QAM). After the first iteration, the values of 2 are located at around 0.1 regardless of Eb /N0 for both σE H examples in Fig. 7 (e) and (f). D. Two Application Examples Two applications in Fig. 8 illustrate cases of uplink users employing optimal system parameters (M =2, ρ = 0.25) for QPSK. As reference, we take the case that three symbols (one preamble, two data) with power P = 1 are sent by a uplink user at time index zero. In Fig. 8 (a), the data of another uplink user using the superimposed pilot technique with the same average power as in preamble technique (the instantaneous power with superimposed pilot will then be P = M+1 M = 1.5) can be transmitted during two symbol durations. According to Fig. 7 (c), both techniques achieve the same BER. The transmission delay is shortened with superimposed pilots to two thirds at the price of at least ten iterations at the receiver. In Fig. 8 (b), packets with one more data symbol are sent with the same power (P = 1) as before. All three data symbols Eb are superimposed with a pilot. A value of N = 15 dB 0 pre

corresponds to BER|pre = 4 × 10−4 in Fig. 7 (c). Under the Eb 2 2 assumptions of σx,sup = σx,pre and M =2, we find N0 = sup Eb M + 10 log( M+1 ) = 13.24 dB according to eq. (1). It N0 pre

refers to BER|sup = 1.5 × 10−3 in Fig. 7 (c). In this case, the throughput|sup = (3 × data symbols) × (1 − BER|sup ) is improved by almost 1.5 time from the throughput|pre = (2 × data symbols) × (1 − BER|pre ).

P =1.5 P =1

P =1

0 1 2

3

t

(a)

0 1 2

3

t

(b)

0 1 2

3

t

Fig. 8. Uplink users applying a superimposed pilot technique (QPSK, M=2 and ρ = 0.25) (a) with same average power, or (b) with same signal power as a preamble technique.

V. C ONCLUSIONS A practical joint channel estimation and decoding algorithm using a superimposed pilots is proposed. Its iterative characteristics and performance are investigated and optimized. The results show that a superimposed pilot technique and a preamble technique have similar BER performance through all Eb /N0 under optimized power ratio and data packet length (under the assumption that an MMSE channel estimator and soft decisions are used, although this is not essential) with limited complexity (iteration ≦ 10). The superimposed pilot technique is suitable and flexible for systems including high percentage of short bursty data traffic to reduce transmission delay or increase throughput. The system performance with more powerful codes or other transmission channels are interesting topics in the future. Also channel tracking using the superimposed pilot technique is a potential application. R EFERENCES [1] C. Ho, B. Farhang-Boroujeny, and F. Chin, “Added Pilot Semi-Blind Channel Estimation Scheme for OFDM in Fading Channels,” in IEEE GLOBECOM, vol. 5, pp. 3075–3079, Nov. 2001. [2] C. Douillard, A. .Picart, P. .Didier, M. Jezequel, C. Berrou, and A. Glavieux, “Iterative Correction of Intersymbol Interference: Turbo Equalization,” Europ. Trans. Telecomm., vol. 6, no. 5, pp. 507–511, 1995. [3] D. Divsalar, S. Dolinar, and F. Pollara, “Iterative Turbo Decoder Analysis Based on Density Evolution,” IEEE J. Sel. Areas Comm., vol. 19, no. 5, pp. 891–907, 2001. [4] M. T¨uchler, R. Koetter, and A. C. Singer, “Turbo Equalization: Principles and new Results,” IEEE Trans.Comm., vol. 50, no. 5, pp. 754–767, 2002. [5] M. T¨uchler, A. C. Singer, and R. Koetter, “Minimum Mean Squared Error Equalization Using A Priori Information,” IEEE Trans. Signal Proc., vol. 50, no. 3, pp. 673–683, 2002. [6] V. Jungnickel, T. Haustein, E. Jorswieck, V. Pohl, and C. von Helmolt, “Performance of a MIMO System with Overlay Pilots,” in IEEE GLOBECOM, vol. 1, pp. 594–598, Nov. 2001. [7] R. Otnes and M. T¨uchler, “Iterative Channel Estimation for Turbo Equalization of Time-Varying Frequency-Selective Channels,” IEEE Trans. Wireless Comm., pp. 1918–1923, 2004. [8] M. M. Wang, W. Xiao, and T. Brown, “Soft Decision Metric Generation for QAM with Channel Estimation Error,” IEEE Trans. Comm., vol. 50, no. 7, pp. 1058–1061, 2002. [9] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, 1974. [10] W. E. Ryan, Concatenated Convolutional Codes and Iterative Decoding. Wiley Encyclopedia of Telecommunications, J. G. Proakis Editor, 2001. [11] T.-J. Liang and G. Fettweis, “MIMO Preamble Design With a Subset of Subcarriers in OFDM-based WLAN,” in Proc. VTC Spring, 2005. [12] M. Morelli and U. Mengali, “A Comparison of Pilot-Aided Channel Estimation Methods for OFDM Systems,” IEEE Trans. Signal Proc., vol. 49, no. 12, pp. 3065–3073, 2001. [13] I. Land, P. A. H¨oher, and S. Gligorevi´c, “Computation of Symbol-Wise Mutual Information in Transmission Systems with LogAPP Decoders and Application to EXIT Charts,” in 5th ITG Conference on Source and Channel Coding (SCC), (Erlangen Germany), pp. 195–202, Jan. 2004.