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Soft-Iterative Channel Estimation: Methods and Performance Analysis Monica Nicoli, Member, IEEE, Simone Ferrara, Student Member, IEEE, and Umberto Spagnolini, Senior Member, IEEE
Abstract—In this paper, we propose a soft-iterative approach for the estimation of block-fading channels based on multiblock (MB) processing. Coded transmission over fading channels is usually performed by interleaving the code word over a large time interval and arranging it into data packets (or blocks) that include also pilot symbols to allow the estimation of the time-varying channel. In this paper, we propose to exploit this block-based structure, and also the inherent latency of soft-iterative receivers due to data interleaving, to improve the channel estimate accuracy. Within the observed frame of blocks, the MB approach uses the invariance of some multipath features, such as delays and mean powers, to estimate the covariance matrix of the channel vector and to identify the corresponding channel subspace. Here, the MB method is extended to incorporate soft information, which is available in iterative receiver structures. The performance of this soft-iterative method is derived in terms of mean square error of the estimate by using a general framework that applies also to other soft-based methods in the literature. A performance comparison is carried out among these estimation methods, both analytically and by numerical simulations, showing the benefits of the proposed approach on the turbo equalizer convergence. Index Terms—Channel estimation, communication system performance, equalization, maximum likelihood estimation, mobile communication, multipath channels, radio communication, soft-iterative receiver, subspace methods, time-varying channels, turbo processing.
I. INTRODUCTION
S
OFT iterative (turbo) equalization is a powerful technique that can be adopted at the receiver when data, protected by an error correction code, are transmitted over a frequency selective channel causing intersymbol interference. The equalization and decoding tasks are performed separately and iteratively on the same block of received signals, with exchange of soft information [1]–[3]. The aim is to refine the data estimate by iterations so as to achieve near-optimal performance with reasonable complexity.
Manuscript received January 10, 2006; revised October 16, 2006. This work was partially developed within the IST-FP6 Network of Excellence in Wireless Communications (NEWCOM). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Petr Tichavsky. M. Nicoli and U. Spagnolini are with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, I-20133 Milano, Italy (e-mail: nicoli@elet. polimi.it;
[email protected]). S. Ferrara was with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, I-20133 Milano, Italy. He is now with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2007.893979
The convergence of turbo receivers depends on the quality of channel state information (CSI) [4]–[6]. To improve the CSI accuracy, soft-iterative techniques have been proposed to refine the channel estimate using soft information provided by the channel decoder. Soft feedback is the preferred approach for decision-directed estimation, as it allows to properly weight reliable and unreliable symbols, thus avoiding error propagation effects that usually arise in hard decision feedback [7]. Recently, several soft-iterative channel estimation methods have been developed for single-user [4]–[15] and multiuser [16]–[24] (such as CDMA, MC-CDMA) systems, with either single-antenna [4]–[13], [16]–[23] or multiple-antenna [14], [15], [24] transceivers, over frequency-flat [4]–[6], [15]–[19] or frequency-selective [7]–[14], [20]–[24] channels. Channel estimates are obtained using different criteria: the least squares (LS) criterion [8], [14], [16], [20], [21]; the minimum-mean-square-error (MMSE) method [9], [17], [18], [21], [23], [24]; the maximum a posteriori (MAP) estimation [15]; algorithms based on the expectation-maximization (EM) approach [7], [10], [19]; and adaptive methods for the estimation of time-varying channels [8], [11]–[13], [21], [22], such as least mean squares (LMS), recursive least squares (RLS), or Kalman filtering (KF). In this paper, we focus on soft-iterative estimation methods that combine hard-valued pilot symbols with soft-valued data symbols (such as in [4], [8], and [21]). The soft-iterative approach is here applied to block-based transmission systems where coding and interleaving are performed on a set of data blocks, each containing both pilot and data symbols, transmitted over a block-fading channel. In these systems the CSI is usually evaluated block-by-block, either from pilot symbols (training-based estimate) or from both pilot and data symbols (soft-based estimate). An example of block-by-block soft-based estimation from block-interleaved data and pilots can be found in [21], where the MMSE method is used for the estimation of block-fading channels in MC-CDMA systems. This conventional approach will be referred to as single-block (SB) channel estimation. On the other hand, if the channel can be modeled as a zero-mean stationary Rayleigh process having unknown covariance matrix and varying from block to block, a multiblock (MB) approach can be adopted to improve the accuracy of the SB estimate, as shown in [25]. The underlying idea of this approach is drawing the stationary features of the channel structure from a sample covariance matrix obtained from MB measurements, and then constraining the channel estimate in each block to fulfill the corresponding algebraic structure.
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Fig. 1. Signal model for a convolutionally coded system.
Since turbo processing is performed on a -block set, in this paper, we propose a soft-based MB approach that makes use of both pilot and data signals contained in all the blocks. This channel estimation method takes advantage of the inherent latency of the turbo receiver to improve the estimate accuracy for those channel parameters that are slowly varying within the observed time interval. The estimate relies on the assumption that the covariance matrix of the multipath channel has a deterministic (but unknown) structure that depends on delays and powers. These terms remain constant within the blocks, while the fading amplitudes vary from block to block (block-fading channel). Instead of estimating the delay pattern, the MB approach converts the stationarity of the delays into the stationarity of the subspace spanned by the corresponding channel impulse responses, here referred to as the channel subspace. This stationary subspace is estimated by averaging the signals received over the blocks, while the fast varying parameters are calculated block-by-block. As proved in [25], for increasing , the estimation error for the stationary parameters (i.e., the channel subspace) becomes negligible; thus, the overall channel estima) on the fading tion error depends asymptotically (for amplitudes only. In this paper, the MB approach is modified to combine hardvalued training with soft-valued information-bearing data. At first iteration, the channel estimate is obtained as in [25] from the training signals only, while in the subsequent iterations it is refined using a priori statistics on the information-bearing symbols. An approximate maximum likelihood (AML) estimation is carried out by extending the training set with soft-valued symbols and by modeling the additional noise due to soft-estimation errors as a white Gaussian process. The AML approach is here compared to other similar soft-based estimates using pilot and data symbols, specifically: a LS-based estimate and two other linear approaches named, as in [4], the mixing and the combining methods. The LS and the mixing methods have been widely adopted in the literature (see, e.g., [8] and [21] for the former method, and [7] and [12] for the latter one), while the combining estimate was originally developed in [4] for flat fading channels and it is modified in this paper to be used for the estimation of frequency-selective channels. An analytical performance comparison is carried out between the proposed AML approach and the other above mentioned methods, by deriving the mean square error (MSE) of the estimate for all the considered methods. The derivation of the estimation methods and the MSE analysis are here performed for frequency-selective fading channels in single-carrier block-based transmission
systems, with single antenna at the transmitter and the receiver. Nevertheless, the proposed methods can be generalized and extended to any arbitrary multiple-antenna and/or multiple-carrier system (see also [26]). The extension of the soft-based MB method to the estimation of time-varying channels by means of subspace tracking can be found in [27]. The paper is organized as follows. Section II defines the signal model and the receiver structure. Soft-iterative channel estimation is discussed in Section III, where the AML SB and MB methods are derived. Some other soft-based methods, combining pilot and data signals, are recalled from the literature and adapted to the framework of this paper. The analytical MSE evaluation and comparison is presented in Section IV, and it is validated in Section V by simulation results. Section VI draws some concluding remarks. II. SYSTEM DESCRIPTION We consider a block-based transmission of convolutionally coded data over a frequency-selective channel, of binary inas depicted in Fig. 1. A sequence , is convoformation symbols, lutionally encoded with code rate . The output code are permuted by a random interleaver , bits , and mapped into quadrature phase-shift-keying (the (QPSK) symbols analysis can also be extended to larger constellations). The transmission of the so obtained QPSK sequence is organized, as shown in Fig. 2, symbols each: , in blocks of with being the th data . Before transmission, an uncoded block for is added training sequence as preamble to each block yielding an overall block length of symbols. The blocks are then transmitted over a frequency-selective channel, modeled here as a causal , that includes the transmitter/receiver filter filter, is responses and the multipath effects. The channel vector assumed to be constant within each block interval but varying from block to block (block-fading channel). The equivalent complex baseband model of this system is shown in Fig. 1. denote the th sample measured (after matched filLet tering and sampling at symbol-rate ) at the receiver within the th block; the vectors and gather, respectively, the and the
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Fig. 2. Block-based transmission.
samples received within the training and the data-transsammission phases. For analytical convenience, the first ples at the beginning of each phase have been discarded to avoid overlapping between training and data (see Fig. 2). The received signals are modeled as Training Data
(1)
and where the Toeplitz matrices denote the convolution with, respectively, the training and data symbols: and . The vectors and collect uncorrelated complex Gaussian noise samples with zero mean and variance . The signal-to-noise ratio (SNR) is . defined as A. Subspace Channel Model Within the -block time window, a multipath propagation scenario is considered having constant delays and block-fading amplitudes (2) According to the Rayleigh fading and the wide sense stationary uncorrelated scattering (WSSUS) assumptions (see [28, ch. 14]), the amplitudes are complex normal, , with covariance matrix defining the power delay profile. The amplitudes are also assumed to be uncorrelated . from block to block, i.e., The vector in (convolution of the (2) contains the pulse waveform and transmitter and receiver filter responses) delayed by . Furthermore, the matrix sampled at symbol frequency collects the pulse waveforms for all the paths. We observe that fading uncorrelation is experienced when blocks are distributed in different time frames (e.g., in time-slotted communications such as 3G TD-SCDMA systems [26]) or in different frequency bins (e.g., in frequency-hopping systems), provided that the interblock time/frequency interval is larger than the channel coherence time/bandwidth. Such an assumption is here needed to carry out the performance analysis of the proposed estimation methods,
but the algorithms can be applied also in case of correlated fading (see [25]). Under the above assumptions, the channel is a stationary (up to the second-order statistics) random vector , with covariance matrix and . block-correlation is related to the The structure of the covariance matrix invariant power-delay features of the multipath pattern [25]. , is a In particular, its rank, measure of the number of equivalent paths (or macro-paths) that the system can resolve given the limited bandwidth of the transmitted signals. When a system has to be tailored to operate in many propagation situations, it is common to oversize the , so that , and, thus, results channel support , here rank deficient. Its -dimensional column-space referred to as the channel subspace, represents the subspace spanned by the channel impulse responses that are experienced over the propagation paths. Based on the reduced-rank and the block-fading features of can now be the multipath structure, the channel vector re-parameterized in terms of a constant full-column rank ma(containing the so called channel modes) and a trix (containing the corresponding block-fading vector modal amplitudes) [25] (3) A possible (but not unique) choice for is the matrix containing the eigenvectors corresponding to the non-null eigenvalues . The corresponding modal amplitudes are given by the of components of with respect to these eigenvectors: , where is the diagonal matrix . Another example containing the positive eigenvalues of of basis expansion for channel estimation can be found in [22]. B. Iterative Receiver Structure The iterative receiver structure adopted in this paper is shown in Fig. 1. It consists of a soft-in channel estimator, a sliding-window soft-input-soft-output (SISO) LMMSE equalizer [3] and a log-MAP SISO decoder [29], separated by an ). interleaver and a de-interleaver (of size At each iteration, the soft channel estimator derives (as described later on in Section III) a new estimate for the chanby exploiting both the training symbols and nels the a priori log-likelihood ratio (LLR) for the data-bearing bits . The channel estimates are then used to perform equalization and decoding. Specifically, the SISO equalizer (or subsequently the decoder) uses the a priori information to evaluate the extrinsic (or ), as difference between the LLR of each code bits (or ) and the a priori LLR a posteriori LLR (or ). This refined soft information is de-interleaved (or interleaved) and it is passed to the decoder (or the (or ) for further equalizer) as new a priori LLR processing. In the following, we focus on channel estimation. For further details on the equalization and the decoding tasks, in addition to the review above, we refer to the cited papers.
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III. SOFT-BASED CHANNEL ESTIMATION In the following, we address the problem of estimation of the from the ensemble of blocks . channel At first iteration the estimation is carried out from the training only, using the knowledge of the pilot symsignals (the a priori information on is missing). After bols the first channel estimation, equalization and decoding of the blocks, the LLRs about the code bits are available at the input of the channel estimator and they can be exploited to refine the initial estimate. Namely, the a priori LLRs are used and the variance to compute the mean value , with , for each , . We recall that for QPSK code symbol modulation these statistics can be obtained as [3] (4) (5) . Within the th block, the quantiwith and , respecties (4) and (5) will be indicated as tively. The convolution matrix built from the soft-valued data is , while sequence is the convolution matrix obtained from the data estimation errors . Some technical assumptions will be used in the remainder of the paper to perform channel estimation and to evaluate the performance. A1) The training sequence is assumed to be the same in all , so that is indeblocks, pendent of the block index. is asA2) The information-bearing sequence sumed to be uncorrelated and the block size is assumed long enough so that the following approximations hold: (6) (7) (both matrices are constant over the blocks). The paramin (7) is obtained from the approximation eter
(8) yielding
where (9)
is the sample variance of the information-bearing symbols . Since is constrained to be constant over the blocks, the sample average estimation of the symbol variance can be obtained with higher accuracy . using all the blocks as
represents the effecNotice that the parameter tive number of known data symbols that can be used for channel estimation in each block. It is indeed , with for missing prior informaand for perfect a priori tion . information is assumed to be a staA3) The error sequence tionary white process, independent from and with variance . This implies that (10) Before proceeding with channel estimation, it is convenient to rewrite the model (1) as Training (11) Data where, based on the assumptions above, the soft-valued data is known on every iteration and can be treated as an extension of the training sequence, while is an additive zero-mean noise, independent from . For the derivation of the estimation methods, we will conas a white Gaussian vector with covariance sider . The variance can be obtained as from
where we used the assumption A3. We, thus, get , being the SNR defined as in Section II. As expected, the power of the additional noise term depends on the symbol variance and, thus, on the data estimate reliability. In particular, the noise is null for , while it affects the data perfect a priori information , making the field for uncertain a priori information signal model (11) not homogeneous. The overall noise variance in the training-phase, while it is increased by a is indeed factor in the data-transmission phase to account for the . unreliability of the soft values Below we consider soft-based channel estimation from (11), at first without imposing any specific constraint on the channel structure and then imposing the channel-subspace invariance constraint (3) (for known ). The unconstrained estimate of is obtained block by block from the single-block (SB) mea. We will briefly recall the following surement SB methods proposed in the literature to combine hard-valued pilot and soft-valued data symbols: the least squares (SB-LS) approach (see, e.g., [8] and [21]); the mixing (SB-M) method (see [4], [7], and [12]); the combining (SB-C) method, proposed in [4] and modified here for the estimation of frequency-selective channels. Next, we propose the approximate maximum likelihood (SB-AML) method based on the white Gaussian assump(whiteness is an approximation tion as it holds only for diagonal, e.g., for symbol-spaced
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delays and Nyquist impulse waveform). A structured soft-based paapproach is finally obtained by the AML estimation of . rametrized as (3), from the MB ensemble A. SB Channel Estimation is 1) SB-LS Method: The LS estimation of obtained from the signal model (11) by minimizing with , yielding (12) This approach has the advantage of a very simple implementation but it is suboptimal as it does not take into account the noise ). (neither the additional term 2) SB-M Method: The estimate of the channel vector is obtained as the minimizer of the mixed training-data objecwith tive , where stands for the expectation with recalculated using the a priori statistics spect to the data about the data bits at the input of the channel estimator. After straightforward computations, it can be shown that the minimizer reads (13) where the matrix
is defined as (14)
here approximated according to the assumption A2. As pointed out in [4], this estimate has the disadvantage that it is biased. 3) SB-C Method: The combining method, originally proposed in [4] for the estimation of a scalar channel, is here re-proposed for the estimation of the frequency-selective . The SB-C method is based on a linear combinachannel tion between the unbiased training-based LS estimate
(see Appendix I for the proof). Nowhere ; tice that the solution depends on the unknown symbols needs to be somehow approximated. Acthus, the matrix cording to the suggestion in [4], we set where is the bit error probability at the output of the SISO de. The disadvantage of coder and this approach is the need of estimating . We also observe that , the SB-C estimate (17) reduces to the SB-LS esfor , and timate (12), as it is: . 4) SB-AML Method: The unconstrained maximum likelifrom the nonhomogeneous model (11) hood estimate of and under the Gaussian approximation for is the minimizer of (20) which yields (see Appendix II) (21) where
and (22)
This method will be referred to as single-block approximate maximum likelihood (SB-AML) estimate. As expected, for perand the AML estifect a priori information, in (20), it is mate reduces to the usual LS approach. B. MB Channel Estimation By generalizing the results in [25], here we consider the maxfrom the MB measurements imum likelihood estimation of modeled as in (11), under the Gaussian apand under the channel structure conproximation for straint (3). The MB approximate maximum likelihood estimate , (MB-AML) is obtained by the minimization of defined according to (20), constrained to (3), yielding with
(15) (23)
and the biased data-based estimate (16) the latter being the minimizer of of the estimates (15) and (16)
. The linear combination
represents the estimate of the projector onto the stationary subspace spanned by the channel covariance matrix weighted , and it is obtained from the leading eigenvectors of by the sample covariance matrix
(17) is obtained according to two weighting matrices and , that are chosen so as to minimize the objective under the unbias constraint . In case of frequency-flat channel , the scalar weighting factors have a close form that the optimization is given in [4]. On the other hand, for with respect to the weighting matrices yields (18) (19)
(24) As in the training-based case [25], the MB-AML estimate (23) can be interpreted as a postprocessing of the whitened , so SB-AML estimate called because it has an identity matrix as covariance matrix. In fact, the covariance of the SB-AML estimate is (as it will be proved later on in next section) and, thus, that of the whitened vector is . This whitening operation is performed before the eigenvalue analysis to get uniform noise eigenvalues and, thus,
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TABLE I MSE OF SOFT ITERATIVE SB AND MB ESTIMATES
allowing the separation of the channel subspace from the noise one. The MB-AML estimation (23) consists, indeed, of the projection of the whitened SB-AML estimate onto the subspace spanned by the leading eigenvectors of the matrix , computed from the ensemble of whitened SB-AML estimates. Notice that, differently from the training-based estimate [25], here the preliminary SB-AML estimate, the projection matrix and the whitening factor depend not only on the training signals but also on the a priori statistics of the data symbols. The soft MB-AML estimate (23) reduces to the training-based one only and ) or perfect ( and in case of missing ( ) a priori information.
In this section, we evaluate a close form for the correlation matrix of the estimation error (27) where denotes the correlation matrix of the argument vector. The expectation is taken with respect to the noise and the data symbols . The MSE is finally obtained as . The resulting MSEs for all the estimation methods are summarized in Table I, for the general case (second column) and for uncorrelated training sequence (third column). 1) SB-LS Method: The estimation error (26) for the SB-LS method can be obtained from (11), (12), and (25), as
IV. PERFORMANCE ANALYSIS AND COMPARISON
(28)
In this section, we evaluate and compare the MSE for the SB methods (SB-LS, SB-M, SB-C, SB-AML) and the MB-AML (as for perfect knowledge of the channel method for ). We recall that , and are covariance considered as independent. To simplify the notation, we also introduce the following Gaussian random vectors:
Recalling (25), (22), and the independence assumption for the noise vectors in (28), it can be shown by straightforward calculations that the error correlation matrix is given by
(29) The corresponding MSE is shown in Table I (row 7). 2) SB-M Method: Using (11), (13), and (25), we can write the estimation error for the SB-M method as (25) We further define the parameter
.
A. Performance Analysis
(30) where , and the term appears due to the derived from (7) and (14). approximation The correlation matrix of (30) is, from (25)
Let be the channel estimate for any method presented in denote the estimation error Section III, and (31) (26)
The resulting MSE is given in Table I (row 8).
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3) SB-C Method: The estimation error for the SB-C method . (17) is derived assuming perfect knowledge of the matrix , the estimation Let us introduce the definition error can be written, as proved in Appendix I, as (32) , and are indepenwhere dent random vectors. It follows that the error correlation matrix is
(33) depend on the data symbols , while the where and correlations for and are obtained from (25) as
(see Appendix III). The This result equals the CRB for corresponding MSE is given in Table I (row 11). B. Performance Comparison To simplify the performance comparison, in the following we will focus on the case of uncorrelated training sequence, i.e., for and . Under this assumption, the MSE performances simplify as shown in the third column of Table I. The comparison between the SB performances (rows 7–10 in Table I) shows that the AML approach has some definite advantages with respect to the mixing, combining and LS techniques. First of all, let us consider the estimation errors (28), (30), (32), and (36). The noise terms have zero mean; thus, all the estimates are unbiased, except the SB-M estimate, as it is
(34)
(41)
(35) By plugging these last results into (33) and taking the trace of the resulting matrix, we get the MSE shown in Table I (row 9), and . where 4) SB-AML Method: The estimation error is derived from (11), (21), and (25), as
for . As highlighted in [4], for (unreliable soft data) and , the bias (41) can prevent the turbo receiver from bootstrapping. The results in rows 3-4-5 of Table I show that, for (missing a priori information), the MSEs for the SB-LS, SB-C, and SB-AML methods reduce to
(36)
(42)
Recalling (25) and the definition for , the correlation matrix simplifies according to
i.e., the performance of the conventional training-based estimate (defined as in Section III). On the other hand, the ratio between the MSE of the SB-M method and the MSE of the training-based estimation is given by
(37) The corresponding MSE is given in Table I (row 10). 5) MB-AML Method: The MSE for the MB-AML method . Recalling from [25] that the is here calculated for SB-AML and MB-AML estimates are unbiased, we notice that asymptotically the correlation matrix (24) tends to (38) . This implies that whose subspace is simply tends for to the projector onto the projector , as for a perfect knowledge of the channel , subspace. As a consequence, it is and the estimation error can be written as a function of the SB-AML error as
(43)
that, for large
and
, simplifies to (44)
This implies that in the SB-M receiver the use of data symbols can even worsen the estimate accuracy with respect to the conventional training estimate, thus making the turbo-equalizer deteriorate its performance through the iterations (as it will be shown later by simulation results in Section V and, particularly, dB). On the other hand, for in Fig. 6 for (i.e., for and ), all SB methods reduce to the LS estimate calculated from the overall sequence of known symbols, reaching the lowest MSE value
(39)
(45)
(40)
. By Let us now consider any intermediate values comparing row 3 (corresponding to the MSE of the estimator ) and row 10 in Table I, it can be seen that (i.e., the SB-AML estimate is always more accurate than the conventional training-based one). This improved accuracy,
Recalling (37), it follows that:
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provided for any level of reliability on the data estimate, is another advantage of the AML approach. On the other hand, the under the SB-LS and SB-M methods perform worse than settings
In addition, from rows 10–11 in Table I, we can notice that the for number of unknowns, affecting the MSE performance, is the SB estimator while for the MB estimator it is reduced to the (as the projector number of block-dependent amplitudes is perfectly estimated for [25]), leading to
(46) (55)
and (47) i.e., for unreliable data and large SNR. Besides, the AML method outperforms the LS one for any , as it is (48) being (recall that ). Also, notice that the MSE ratio (48) reduces to 1 only when data symbols . are perfectly known So far, we have compared only the SB methods’ performances. Let us now focus on the AML approach, and consider the MSE of the SB-AML and the MB-AML methods in rows 10–11 of Table I. We observe that in case of missing prior and the information (i.e., at the first iteration), it is MSE reduce to the training-based performance (49) (50) On the other hand, for perfect prior information (i.e., close to the convergence of the iterative approach) it is and the MSE equals the MSE for a training block symbols of (51)
(52) Finally, let us compare the MSE of the SB-AML and the (superMB-AML methods for the training-only case (superscript ) script t), the training-plus-data case and the training-plus-soft-data case . We observe that the following relationships hold: (53) (54) The first inequality can be easily proved by comparing the MSE of the SB-AML estimate in Table I (row 10 and column 2) with (49) and (51), and observing that it is , and, thus, (inequality between matrices is in the sense that the difference matrix is positive semidefinite). Recalling that is a projection matrix, this last result also implies that , proving (54).
To conclude, the AML method is unbiased and outperforms the training-based estimate for any . Numerical results in next section validate this conclusion. They also show that the AML-SB approach outperforms both the LS and the mixing methods, and the combining method in the case of unreliable data estimates. Furthermore, when integrated with MB processing, the proposed method allows a further MSE reducwith respect to the AML-SB approach, tion by a factor yielding the most accurate estimate among all other methods considered in this paper, even for block-wise varying channels. As a final remark on the computational cost, we point out that all the SB methods have similar costs while the MB-AML requires an additional computational burden due to the postprocessing of the SB estimates. This cost is essentially due to requiring a complexity the EVD of the correlation matrix . A more efficient implementation is obtained order of by means of subspace tracking algorithms [27], which can re, where is duce the computational complexity up to the rank order. The subspace tracking approach has also the advantage of adaptively estimating the channel basis when the multipath pattern gradually changes over the blocks.
V. NUMERICAL RESULTS In this section, we compare by numerical simulations the estimation methods considered in Section III in terms of MSE and bit-error-rate (BER) performance. In all the simulations, a frame of 10 000 randomly-chosen equiprobable information bits is coded by a four-state convolutional code with generaand it is permuted by a random interleaver. The tors code bits are then mapped into 10 000 QPSK symbols and arblocks with symbols each. A ranged into QPSK symbols is obtraining sequence of symbols to the setained by adding a cyclic prefix of quence of length defined in [8]. The symbol duration , and the waveform used for the transmission is is a raised cosine pulse with roll-off factor 0.22. The block-fading Rayleigh channel (2) is simulated as a vector, uncorrelated from block to block, using the following sets of multipath parameters. resolvable paths with delays • Model C1: , mean powers and tem. poral support resolvable paths with delays • Model C2: , mean powers and temporal support
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• COST-207 Hill Terrain (HT) propagation environresolvable paths with delays ment [32]: , mean powers , and . temporal support The SB and MB methods defined in Section III will be used for the estimation of these channels. The MB method will be simulated under two different conditions: using as the projection blocks or the true projector matrix estimated from (as for an asymptotic estimate from blocks). Fig. 3 compares the analytical (lines) and simulated (markers) , , , MSE for all SB methods ( ) and the AML-MB method both for and for the asymptotic case . The number of data symbols used for channel estimation is in Fig. 3(a) and in Fig. 3(b). dB, and the channel is modeled The SNR is here set to according to the multipath model C1. The MSE is shown between versus the mutual information the input code bits and the output a priori information . According to [30], the LLR is simulated and variance as a Gaussian variable, with mean value . The relation between and is given by [30] (56) The MSE results for the SB-C method are obtained either by approximating (as described in Section III-A-3 for a practical implementation) or by assuming perfect knowledge of (as in Section IV-A3). In the former case, the MSE values are only simulated (markers without lines), while in the latter case the MSE values are both simulated (markers) and evaluated analytically (lines) from Table I, using the true values of the transfor the computation of the weighting factors mitted data . The MSE values for this ideal case of known have to be considered as a theoretic lower bound of the SB-C performance. As regards the MSE of the MB-AML method, the values are only simulated (as the analytical results in Secfor ), while those for are tion IV-A-3 are valid for both simulated and analytical. For the analytical derivation of the MSE for the MB estimate with finite number of blocks, see [25]. or , The results in Fig. 3(a) confirm that, for all SB methods have the same performance (except SB-M that is affected by a remarkable bias for low ). The MSE degradation of the SB-M method with respect to the other SB follows from the analytical result (43): methods for dB. The MSE reached for and equals the performance of the LS estimate evalufor ) and ated from the training sequence only ( the whole training-data sequence when all symbols are known for ). As pointed out in Sec( for all values, while for tion IV, it is for both the SB-C low the MSE of the estimate is above ). For instance, for method and the SB-LS methods (for ( and ), the simulation confirms the
Fig. 3. (a) MSE for soft-based SB/MB estimates versus the mutual information I for N = 200. (b) MSE for soft-based ML estimates versus the mutual information I for both N = 100 and N = 200. Both figures have = 3 dB and N = 31. Simulated results have been obtained under the Gaussian approximation for the a priori LLRss at the channel estimator input.
MSE gains/losses evaluated from Table I: dB, dB, dB. It is also shown that dB for . Notice that for the SB-M case, the analytical and the simulated curves are slightly different, namely for intermediate values of mutual information due to the Gaussian approximation used in Section III. Fig. 3(b) compares the MSE of the AML-based methods and versus the mutual information for [the SNR and the channel model are the same used for Fig. 3(a)]. Both the analytical (lines) and simulated (markers) values are , shown. As expected, the MSE decreases for increasing being the MSE inversely proportional to the number of symbols (see the analytical results in Table I). Furthermore, the figure shows that the MB method can be more accurate than the SB one even using less soft symbols for the estimation. Fig. 4 shows the MSE versus the SNR, for all the estimation methods and for three different values of mutual infor(top figure), (middle figure) and mation: (bottom figure). The channel is simulated according to the model C1. Again, the SB-M method performs worse than all and the other methods. Its MSE is particularly high for it does not improve for increasing SNR. This behavior is confirmed by the analytical performance in Table I (row 8, column , leading to 3) where the dominant term in the MSE is
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Fig. 5. MSE for soft-based SB/MB estimates versus mutual information I for N = 200 using the channel COST 207 HT. = 3 dB and N = 31. For the MB estimator, the rank r is ranging from 4 to 6. Simulated results have been obtained under the Gaussian approximation for the a priori LLRs at the channel estimator input.
Fig. 4. MSE for soft-based SB/MB estimates versus SNR, for different values of mutual information: (top) I = 0:2, (middle) I = 0:5, and (bottom) I = 0:8. N = 31 and N = 200. Simulated results have been obtained under the Gaussian approximation for the a priori LLRs at the channel estimator input.
, being the data symbol decrease for and the . This effect can be channel power normalized to (bottom figure) for large SNR. We can, noticed also for thus, conclude that the use of a longer data sequence can lead to MSE degradations for the SB-M method, while it is always advantageous for the AML approach. A similar error-floor effect can be noticed also on the MSE performance for the SB-LS and . On the other hand, the SB-C methods, especially for
MSE of the SB/MB AML estimates decreases linearly with the SNR providing a remarkable MSE gain with respect to the other methods, especially for low values of when using the MB approach. The benefits of this improved accuracy on the convergence of the iterative equalizer are shown later on through BER simulations. Fig. 5 shows the MSE of the SB methods and the AML-MB and 4, 5, 6. The simulation parameters method for are the same used for Fig. 3(a), apart from the channel that is here modeled according to the standard model COST 207 HT. With respect to the results in Fig. 3(a), the MSE gain achieved by the MB approach with respect to the SB one is higher here, as the is larger then for the model C1. This gain increases factor for decreasing from 6 to 4, as the first four paths have similar delays (the delay difference is below the sampling interval), and, thus, the effective number of resolvable paths is smaller than the . Notice, however, that for the simnumber of paths ulated performances start to differ from the analytical ones, as the channel distortion induced by the reduced parameterization becomes relevant especially for moderate values of mutual information. The optimal rank value should be estimated from the as a trade-off between the essample correlation matrix timate bias due to the reduced parameterization and the noise variance (see, e.g., the rank estimator adopted in [25]). for Figs. 6 and 7 show the BER performance versus the complete iterative receiver for iterations going from to . The channel model is simulated according to model C2 (Fig. 6) and C1 (Fig. 7). The number of data symbols used . Fig. 6 compares for soft-based channel estimation is the receiver for known channel with the receiver based on SB-M (top figure) and SB-C/LS/AML (bottom figure) channel estiall SB methods have mation. It can be seen that for large comparable performance, with SB-AML performing slightly dB better than the others. On the other hand, for the bias of the SB-M estimate is shown to worsen the receiver performance for increasing . Fig. 7 compares the AML-SB
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Fig. 8. MSE for soft-based SB/MB estimates versus number of iterations.
Fig. 6. BER performance for soft-based SB methods versus E =N for varying number of iteration (n): (top) SB-M; (bottom) SB-C, SB-LS, and SB-ML.
Fig. 7. BER performance for soft-based SB-ML and MB-ML methods versus E =N for varying number of iterations (n).
and AML-MB estimators highlighting the advantages of the MB approach: the performance of the MB receiver is far below the SB one and it is very close to the known-channel lower , the MB approach has an bound. For instance, at advantage of 0.4 dB over the SB approach. Finally, for the channel model C1, Figs. 8 and 9 show the , , , MSE of all the estimates ( , ) for different values of iteration number and . It is important noticing that, for low mutual information between the data bits and the LLRs, i.e., either at the first iterations of the turbo processing or for mod, the matching between the analytical and numererate ical performances for the SB-LS and SB-AML methods is not
Fig. 9. MSE for soft-based SB/MB estimates versus E =N at the second (top) and fourth (bottom) iteration.
as good as in the previous analysis, due to the Gaussian assumption used in the MSE derivation. We recall that the analytical MSE for SB-C method is not shown here as it cannot be calculated in case of unknown transmitted data. The simulated blocks, while the MSE values for MB-AML refer to analytical ones were obtained assuming asymptotic conditions . Looking at the SB-M performance in Fig. 8, it is evident the reason why the BER performances are not satisfactory . In fact, it turns out that the MSE perfor low values of formances deteriorates for increasing number of iterations. This affects the behavior of the overall structure. On the other hand, all the SB estimators behave similarly. for high value of The advantage of using the AML approach (either SB or MB) is
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confirmed in all figures, where this method is shown to achieve always better (or at least equal) performances than the other approaches.
and are the covariance matrices defined, where respectively, in (34) and (35). The minimization of the MSE under the unbiased-estimate constraint can be expressed as
VI. CONCLUDING REMARKS This paper proposes the use of a MB-based estimate [25] for block-fading channels in the context of iterative (turbo) equalization. The soft MB-AML method exploits the invariance of the channel subspace across blocks and it estimates the channel using the soft statistics on the information-bearing data. This technique was demonstrated to be advantageous with respect to other methods proposed in the literature (SB-M, SB-C, and SB-LS) owing to the modal analysis performed on the channel covariance matrix. The analytical performances expressions were derived for all the methods and then compared with the results of numerical simulations. It was shown that, for each value of mutual information and SNR, the performances of the proposed technique in terms of MSE are always overcoming those of the other methods. This improvement reflects on the behavior of the overall iterative structure; in fact, the BER results demonstrate how the proposed estimator can achieve performances closer to those obtainable with a complete knowledge of the channel and in a limited number of iteration (i.e., a faster convergence).
(63) where denotes the generalized derivative with respect [31]. It follows that: to the complex matrix (64)
(65)
APPENDIX II SB-AML ESTIMATION The solution (21) can be derived by rewriting the system (11) according to
APPENDIX I SB-C WEIGHTING MATRICES DERIVATION
(66)
In [4], the SB-C method is derived for the estimation of flatfading channels, thereby the weighting factors are scalar coefficients. In case of a frequency-selective channel, we need to extend the method using two weighting full (nondiagonal) matrices, one for the training-based estimate and one for the soft-based estimate. The training-based (15) and data-based (16) estimates can be expressed, highlighting their dependence on , as (57) (58) where the noise terms are defined as , while the matrix
, is given by (59)
with (60) From (57) and (58), we can write the estimation error, as (61) Imposing now the unbiasedness constraint , we get , and, thus, . are obtained as the miniThe weighting matrices and mizers of the estimate MSE
(62)
yielding the homogeneous set of measurements (67) where mate is now the minimizer of
. The AML estiwhich
equals (20). APPENDIX III CRAMÉR–RAO LOWER BOUND Here we prove that the asymptotic MB-AML performance from all the (40) coincides with the CRB for the estimate of signals when . As observed in Section IV-A, the MB sample correlation matrix tends for to the weighted channel covariance (38). Thereby, the eigencorresponding to the positive eigenvalues of the vectors matrix can be considered as perfectly known by the MB estimator. The projector onto the -dimensional subspace spanned by those eigenvectors (i.e., the sta. tionary channel subspace) is given by The channel decomposition (3) is here performed by selecting the known matrix . as stationary component We get (68) Since is known, the unknown parameters for the MB estimator are only the entries of the vector . These have to be estimated from the signal (67) which is rewritten here as (69)
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Based on the parametrization (68), we can express the CRB as [31] (70) where
denotes the matrix of sensitivities (71)
while is the Fisher information matrix (FIM) for the measurement (69) (72) From (71), the FIM matrix simplifies to (73) and the CRB reads
(74) This equals the asymptotic MB covariance (40). REFERENCES [1] C. Douillard, M. Jézéquel, and C. Berrou, “Iterative correction of intersymbol interference, turbo-equalization,” Eur. Trans. Telecommun., vol. 6, pp. 507–511, Sept.–Oct. 1995. [2] H. V. Poor, “Iterative multiuser detection,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 81–88, Jan. 2004. [3] R. Koetter, A. C. Singer, and M. Tüchler, “Turbo equalization,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 67–80, Jan. 2004. [4] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference cancellation with SISO decoding and EM channel estimation,” IEEE J. Sel. Areas Commun., vol. 19, no. 8, pp. 1450–1460, Aug. 2001. [5] B. Mielczarek and A. Svensson, “Improved iterative channel estimation and turbo decoding over flat-fading channels,” in Proc. IEEE VTC, Sep. 2002, vol. 2, pp. 975–980. [6] M. C. Valenti and B. D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1697–1705, Sep. 2001. [7] M. Sandell, C. Luschi, P. Strauch, and R. Yan, “Iterative channel estimation using soft decision feedback,” in Proc. IEEE GLOBECOM, Nov. 1998, vol. 6, pp. 3728–3733. [8] M. Tüchler, R. Otnes, and A. Schmidbauer, “Performance of soft iterative channel estimation in turbo equalization,” in Proc. IEEE ICC, Apr.–May 2002, vol. 3, pp. 1858–1862. [9] H. Mai, A. G. Burr, and S. Hirst, “Iterative channel estimation for turbo equalization,” in Proc. IEEE PIMRC, Sep. 2004, vol. 2, pp. 1327–1331. [10] R. Lopes and J. Barry, “The extended-window channel estimator for iterative channel-and-symbol estimation,” EURASIP J. Wireless Commun. Netw., vol. 2, pp. 92–99, Apr. 2005. [11] N. Nefedov, M. Pukkila, R. Visoz, and A. O. Berthet, “Iterative data detection and channel estimation for advanced TDMA systems,” IEEE Trans. Commun., vol. 51, no. 2, pp. 141–144, Feb. 2003. [12] S. Song, A. C. Singer, and K. Sung, “Soft input channel estimation for turbo equalization,” IEEE Trans. Signal Process., vol. 52, no. 10, pp. 2885–2894, Oct. 2004. [13] R. Otnes and M. Tüchler, “Iterative channel estimation for turbo equalization of time-varying frequency-selective channels,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1918–1923, Nov. 2004. [14] R. Visoz and A. O. Berthet, “Iterative decoding and channel estimation for space-time BICM over MIMO block fading multipath AWGN channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1358–1367, Aug. 2003.
[15] X. Deng, A. M. Haimovich, and J. Garcia-Frias, “Decision directed iterative channel estimation for MIMO systems,” in Proc. IEEE ICC, May 2003, vol. 4, pp. 2326–2329. [16] L. M. A. Jalloul and R. M. Misra, “Data-aided channel estimation for wideband CDMA,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1622–1634, Jul. 2005. [17] H. El Gamal and E. Geraniotis, “Iterative multiuser detection for coded CDMA signals in AWGN and fading channels,” IEEE J. Sel. Areas Commun., vol. 18, no. 1, pp. 30–41, Jan. 2000. [18] A. Lampe, “Iterative multiuser detection with integrated channel estimation for coded DS-CDMA,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1217–1223, Aug. 2002. [19] A. Kocian and B. H. Fleury, “EM-based joint data detection and channel estimation of DS-CDMA signals,” IEEE Trans. Commun., vol. 51, no. 10, pp. 1709–1720, Oct. 2003. [20] J. Wehinger, R. R. Muller, M. Loncar, and C. F. Mecklenbrauker, “Performance of iterative CDMA receivers with channel estimation in multipath environments,” presented at the IEEE 26th Asilomar CSSC, Pacific Grove, CA, Nov. 2–6, 2002. [21] T. Zemen, “Multi-user communication over time-variant channels” Ph.D. dissertation, Vienna Univ. Technol., Vienna, Austria, 2004 [Online]. Available: http://www.userver.ftw.at/~zemen/papers/Zemen04-Dissertation.pdf [22] T. Zemen and C. F. Mecklenbrauker, “Time variant channel estimation via generalized prolate spheroidal sequences,” IEEE Trans. Signal Process., vol. 53, no. 9, pp. 3597–3607, Sep. 2005. [23] T. Zemen, M. Loncar, J. Wehinger, C. Mecklenbrauker, and R. Muller, “Improved channel estimation for iterative receivers,” in Proc. IEEE GLOBECOM, Dec. 2003, vol. 1, pp. 257–261. [24] M. Loncar, R. Muller, J. Wehinger, C. Mecklenbrauker, and T. Abe, “Iterative channel estimation and data detection in frequency-selective fading MIMO channels,” Eur. Trans. Telecommun., vol. 15, no. 4, pp. 459–470, Sep. 2004. [25] M. Nicoli, O. Simeone, and U. Spagnolini, “Multi-slot estimation of frequency-selective fast-varying channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1337–1347, Aug. 2003. [26] M. Nicoli and U. Spagnolini, “Subspace-methods for space-time processing,” in Smart Antennas—State of the Art, ser. EURASIP Book series on Signal Processing and Communications. New York: Hindawi, 2005. [27] S. Ferrara, T. Matsumoto, M. Nicoli, and U. Spagnolini, “Soft-iterative channel estimation with subspace and rank tracking,” IEEE Signal Process. Lett., vol. 14, no. 1, pp. 5–8, Jan. 2007. [28] J. Proakis, Digital Commun., 3rd ed. New York: McGraw-Hill, 1995. [29] 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. 20, no. 3, pp. 284–287, Mar. 1974. [30] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [31] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [32] Commission of the European Communities, “Digital Land Mobile Radio Communications—COST 207,” Final Rep., ECSC-EEC-EAEC 1989.
Monica Nicoli (M’99) received the M.Sc. degree (with honors) and the Ph.D. degree in telecommunication engineering from the Politecnico di Milano, Milano, Italy, in 1998 and 2002, respectively. During 2001, she was a Visiting Researcher at Signals and Systems, Uppsala University, Sweden. Since 2002, she has been an Assistant Professor in the Dipartimento di Elettronica e Informazione, Politecnico di Milano. Her research interests are in the areas of signal processing for wireless communication systems: antenna arrays processing, MIMO systems, channel estimation and equalization, multiuser detection, turbo processing, multicarrier systems, Bayesian tracking for wireless localization, and remote sensing applications. Dr. Nicoli received the Marisa Bellisario Award in 1999.
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Simone Ferrara (S’06) received the M.Sc. degree in telecommunication engineering from the Politecnico di Milano, Milano, Italy, in 2005, with a thesis entitled “Soft-iterative estimation of structured channels for wireless communication systems.” He is currently pursuing the Ph.D. degree in the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO. From January to July 2005, he was a Research Scientist at the Centre for Wireless Communications, Oulu, Finland. His research interests are in channel estimation for wireless communication systems, as well as statistical inference and sensor signal processing.
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Umberto Spagnolini (M’99–SM’06) graduated as Dott. Ing. Elettronica (cum laude) from the Politenico di Milano, Milano, Italy, in 1988. Since 1988, he has been with the Dipartimento di Elettronica e Informazione, Politenico di Milano, as an Assistant Professor (1990–1998), Associate Professor (1998–2006), and Full Professor (since 2006) in the area of statistical signal processing and communication systems. His general interests are in the areas of signal processing, estimation theory, and system identification. The specific areas of interest include channel estimation, array processing and cross-layer optimization (PHY/MAC) for communication systems, parameter estimation and tracking, wavefield interpolation with applications to UWB radar, and remote sensing. Dr. Spagnolini served as an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING from 1999 to 2006. He was awarded the AEI Award (1991), the Van Weelden Award of EAGE (1991), and the Best Paper Award from EAGE (1998).
