ITERATIVE processing in communication systems according

Transcription

1 1918 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER 2004 Iterative Channel Estimation for Turbo Equalization of Time-Varying Frequency-Selective Channels Roald Otnes, Member, IEEE, and Michael Tüchler, Member, IEEE Abstract We investigate turbo equalization, or iterative equalization and decoding, as a receiver technology for systems where data is protected by an error-correcting code, shuffled by an interleaver, and mapped onto a signal constellation for transmission over a frequency-selective channel with unknown time-varying channel impulse response. The focus is the concept of soft iterative channel estimation, which is to improve the channel estimate over the iterations by using soft information fed back from the decoder from the previous iteration to generate extended training sequences between the actual transmitted training sequences. Index Terms Channel estimation, high-frequency communications, iterative channel estimation, turbo equalization. I. INTRODUCTION ITERATIVE processing in communication systems according to the turbo principle [1], [2] has gained a lot of attention in recent research. Turbo equalization was first proposed by [3] as a receiver technology for systems where the information bits are protected by an error-correcting code (ECC) and shuffled by an interleaver before being transmitted over a frequency-selective channel imposing intersymbol interference (ISI). In a receiver employing turbo equalization, soft information [most often log-likelihood ratios (LLRs)] on the code bits is exchanged between the equalizer and the decoder in an iterative fashion, where both devices are so-called soft-in soft-out (SISO) modules. Most literature on turbo equalization has considered the channel to be known and time invariant. This paper considers systems where the channel is unknown and time varying, with a fading rate so high that tracking of the channel is required between the training sequences. Channel estimation can be performed jointly with equalization or using a separate channel estimation algorithm. All algorithms proposed for joint SISO equalization and channel estimation we found [4] [7] require the equalizer to be trellis based, which imposes a complexity problem. We, therefore, propose to use separate channel estimation alongside the equalization task because it usually saves complexity and a wide range of different equalization and estimation algorithms can be applied. The quality of the channel Manuscript received November 12, 2002; revised July 2, 2003; accepted July 30, The editor coordinating the review of this paper and approving it for publication is I. B. Collings. This paper was presented in part at the Military Communications Conference (MILCOM), Anaheim, CA, Oct. 2002, and at the International Conference on Communication Systems (ICCS), Singapore, Nov R. Otnes was with UniK University Graduate Center, NO-2027 Kjeller, Norway, and with Kongsberg Defence Communications, Norway. He is now with the Norwegian Defence Research Establishment (FFI), NO-3191 Horten, Norway. M. Tüchler is with the University of Applied Sciences, 5210 Windisch, Switzerland ( micha@lnt.ei.tum.de). Digital Object Identifier /TWC estimate can be improved over the iterations by incorporating fed back information from the decoder as input reference signal to the channel estimator. This can be either hard decisions or soft information on the code bits leading to hard or soft iterative channel estimation, respectively. Hard iterative channel estimation has been proposed in [8] [11]. Soft iterative channel estimation has been proposed in [10] and [12] [14], where the channel estimate is constant over a block of data, and in [15] and [16], where the channel estimate is varying from symbol to symbol. Soft-input channel estimation has also been proposed in [17] and [18] using soft symbols fed back from the equalizer rather than from the decoder as input. We use the following nomenclature. All variables may be complex valued. Bold small letters such as denote column vectors, and bold capital letters such as denote matrices. Dependencies on the time index are denoted or. Special matrices are the identity matrix, the all-zero matrix, and the diagonal matrix with the elements from on the main diagonal. The transpose, the complex conjugate, and the complex-conjugate transpose of are denoted,, and, respectively. is the expectation of, and is the covariance of and. The expectation, covariance, and probability conditioned on the a priori code bit LLRs fed back from the decoder are denoted,, and, respectively. II. SYSTEM MODEL Consider the system model shown in Fig. 1. A block of data bits is encoded with a rate- ECC. The code bits are interleaved to according to a permutation function. The bits are mapped to in general complex data symbols,, by mapping consecutive interleaved code bits,,to from the -ary signal constellation. Common constellations are phaseshift keying (PSK), where for all, or quadrature-amplitude modulation (QAM). Besides the data symbols, training and synchronization symbols, which are known to the receiver, are multiplexed with the to form a block of transmitted symbols,. The are modulated onto a carrier at a symbol rate of symbols per second. Given the frame pattern efficiency, the overall data rate of the system is (bits per second). A symbol-spaced discrete-time model for transmitting the symbols over the ISI channel yields the received symbols (1) /04$ IEEE

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER Fig. 1. Reference system model using adaptive turbo equalization in the receiver. where is the length possibly time-varying discrete-time channel impulse response (CIR) and. The are samples of zero-mean circularly symmetric complex white Gaussian noise with variance, which may be varying with. The receiver iterates equalization and decoding tasks on a received frame of symbols. The SISO equalizer outputs the extrinsic LLRs, which, deinterleaved to, are input to the SISO decoder. The decoder produces estimates of the information bits and also outputs the extrinsic LLRs, which, interleaved to, are fed back to the equalizer for the next iteration. Soft iterative channel estimation is included in the receiver by using the LLRs fed back from the decoder to compute extra training information besides the already known training symbols. IV. SOFT ITERATIVE CHANNEL ESTIMATION It is natural to incorporate channel estimation into the iterative process when the CIR is unknown and possibly time varying, e.g., by computing additional training information for estimation purposes from the fed back LLRs. Such training information could be the soft symbol as in (2) (soft iterative channel estimation) or the hard decision calculated from the code bit decisions (hard iterative channel estimation). The idea of using soft symbols for estimation purposes is quite new [10], [15], [16] and classical theory on channel estimation needs to be reexamined for this case. The channel estimator provides a time-varying channel estimate at each time step. Three different error signals, measured at time using the channel estimate at time, are defined III. SISO EQUALIZER The SISO equalizer in the first proposals for turbo equalization utilized trellis-based equalization algorithms [3], [19]. The number of trellis states becomes prohibitively large when the CIR is long and/or the size of the signal constellation is large. Less complex alternatives are, e.g., suboptimal SISO equalizers based on soft ISI cancellation and linear filtering [20] [23], which have in common that the soft information fed back from the decoder is converted into a priori means and variances of the transmitted symbols (2) (3) where and, which correspond to the true estimation error (using precisely known training information) and that using soft symbols or hard decisions, respectively. We use the shorthand notations,, and in the sequel. The simple least mean square (LMS) estimation algorithm, a crude approximation of the stochastic gradient algorithm [26], computes as follows when all are known ( is available): where is the step size. The use of rather than is motivated later in this section. The recursive least square (RLS) algorithm is a generalized least squares algorithm, which minimizes the cost, anex- ponentially windowed sum of the squared error (4) (5) Next, the a priori mean of the ISI is computed from the and subtracted from, and the residual signal is fed through a linear filter of length. Reference [21] shows how to take the a priori variances into account when computing the minimum mean-square error (MMSE) optimal filter coefficients, with computational complexity per symbol interval proportional to. The same algorithm is extended in [24] and [25] to incorporate a time-varying CIR, known or estimated. When all are known, the solution for is given by [26], where (6) (7)

3 1920 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER 2004 The recursions and yield an efficient time-recursive algorithm to compute, which exploits that is updated with the rank-1 matrix. Both the LMS and RLS algorithm can be extended to use soft feedback by replacing with and with or to use hard feedback by replacing with and with. The performance of RLS channel estimation using soft feedback has been analyzed in [16]. We focus our attention now on soft feedback. The code bit LLRs and the training symbols govern the statistics and of each symbol via (2) and (3). We may regard the transmitted symbols as noisy observations of, or in vector form where is discrete-valued noise with zero mean and variance. We assume that for all, which yields that the covariance is given by. The ordinary RLS algorithm using soft feedback minimizes the exponentially windowed sum of the squared error. This is obviously suboptimal for, since the observed noise variance when the real CIR is replaced with is and not, and we are thus minimizing the wrong error variance. We address this problem by deriving a novel RLS-type cost function. To do that, we note that the squared error in the ordinary RLS cost function (6), assuming that all are known, can be written as the expectation of conditioned on all known quantities in the expression for (8) The channel estimate minimizing defined by (10) and (12) is again given by, where (13) (14) Unfortunately, an efficient update rule to calculate from as for the ordinary RLS update is not known to us, since the matrix has full rank in contrast the the rank-1 matrix, and a direct computation has a complexity proportional to. The modified RLS algorithm can be simplified by assuming to be diagonal, which is plausible because the transmitted symbols are assumed to be uncorrelated such that the off-diagonal terms in average out to zero as long as is close to one. Computing is now trivial because is diagonal. Given a PSK signal constellation, we furthermore have regardless of the LLRs such that, which yields Using the assumption together with, (14), and (15), we find the following: (15) Our new idea for soft feedback is to condition the expectation on the actually known quantities and instead of, which yields the alternative cost function Using the correlation matrix, it follows from (4) that (9) (10) (11) where. Since is not known, we approximate it by to evaluate (12) (16) Thus, the modified RLS algorithm is reduced to the ordinary LMS algorithm (5) using the soft symbols. We note that this equivalence does not hold for non-psk signal constellations but a diagonal still simplifies the estimation algorithm tremendously, since the computational complexity per symbol interval is proportional to. So far, we presented three different RLS-like channel estimation algorithms using soft input symbols : ordinary RLS, modified RLS, and approximated modified RLS (which is reduced to LMS for PSK signal constellations). All these algorithms compute the channel estimate as, where and are time-varying estimates of the correlations and. The differences, tabulated in Table I, arise in how and are updated. When the transmitted symbols are uncertain such that, ordinary RLS computes wrong estimates of and. The other algorithms compensate for this by weighting the terms and lower when the uncertainty, represented by the diagonal terms of, is large. Terms depending on are added, such that and approach the true values of and. We also note that a Kalman-based algorithm for soft iterative channel estimation

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER TABLE I UPDATE EQUATIONS FOR 8 AND IN DIFFERENT SOFT-INPUT CHANNEL ESTIMATION ALGORITHMS AND THE CORRESPONDING COMPUTATIONAL COMPLEXITY PER SYMBOL INTERVAL. OPERATOR Diag[X] APPLIED TO THE MATRIX X SETS ALL OFF-DIAGONAL ELEMENTS TO ZERO Fig. 2. Simulated error variance at each symbol interval n in a frame of training and data symbols. Simulation setup: E =N =10dB;, f =f, and as shown above the plots. Legend: A is LMS, B is ordinary RLS, and C is modified RLS (unapproximated version). Solid lines denote soft feedback, and broken lines denote hard feedback. is proposed in [15]. In [25], it has been compared to the algorithms presented in this paper. Before applying the channel estimate to an equalization algorithm, it is desirable to investigate the correlation between the error signal and the transmitted symbols. Most equalizer algorithms assume that the error (noise) signal is uncorrelated with the transmitted symbols. First, we note that for all channel estimation algorithms considered here, is a function of only for. Assuming that the channel estimate is unbiased, and that the channel is varying slowly such that for small,wefind that for but not necessarily for. Writing the error as and using the assumptions above, we find that or must hold to ensure that. For example, if the equalizer requires the error signal to be uncorrelated with, but not necessarily with, a proper choice is the channel model (17) where as previously defined. We verified by simulations (not included in this paper) that this model performs better than in an iterative receiver, even though the variance of is significantly smaller than that of. The derivation of SISO equalizers for known channel conditions is based on the channel model (1). When the CIR is estimated, we replace (1) by (17). Thus, the noise signal is replaced by the error signal and should therefore be replaced by, which is time varying even for a constant. For example, varies depending on

5 1922 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER 2004 Fig. 3. Simulated BER performance over two different channels using the MIL-STD b/s waveform with an interleaver length of code bits. Solid lines: proposed turbo equalization-based receiver using iterative channel estimation. Dashed lines: turbo equalization with one-time channel estimation. Dotted lines: turbo equalization with known channel. Dashed-dotted line: conventional noniterative receiver using decision feedback equalization. whether training or data symbols are transmitted. Since is not known, we have to find an estimate of it, e.g., using a time-average of the squared error. However, the error signal is not available in soft iterative channel estimation, but only the error signal given by (4). To the best of our knowledge, there is no algorithm to estimate from which has a good mathematical foundation. In [25] is devised an ad hoc error variance estimation algorithm with a weak mathematical foundation, which is used in the simulation setup for Fig. 3. V. SIMULATION RESULTS The introduced channel estimation algorithms have been compared through simulations using the following setup. Data symbols and training symbols are generated randomly from a Gray-coded 8-PSK constellation as discussed in Section II. A frame consists of 160 initial training symbols followed by three sequences of 150 data symbols and 30 training symbols. The energy per transmitted symbol is one. The symbols are transmitted over a time-varying ISI channel of length. are mutually independent Rayleigh fading taps with a Gaussian Doppler spectrum [27], where the fading rate is defined as the value of the Doppler spectrum. The tap energies are equal and normalized to yield. The noise has variance. The LLR s fed back from the decoder are modeled as independent outcomes of a Gaussian random variable with mean and variance. This is an accurate model for the LLRs produced by a decoder during iterative decoding [28], where an increasing yields more reliable soft information. Each channel estimation algorithm has been used to generate a time-varying channel estimate of length (to account for the fact that is not known in reality) and the error signal has been recorded. In total, frames have been simulated (,,, and are randomly different for each frame) to estimate the (ensemble) error variance at each symbol interval. Fig. 2 shows the variation of throughout the frame. decreases with time when training symbols are used for estimation and increases when (soft or hard decided) data symbols are used. Although LMS shows slowest convergence during the initial training sequence, it performs well in the remainder of the frame. Using soft inputs, LMS performs only slightly worse than modified RLS and much better than ordinary RLS using soft or hard-decided inputs, especially at low. Estimation with soft inputs generally performs better than with hard-decided inputs. An exception is ordinary RLS at low and fast channel variations (upper right plot), an effect explained in [16]. Other parameter combinations than those presented in Fig. 2 were also simulated, and the general trend is that soft feedback performs better than hard feedback and that LMS using soft feedback is always close to the best performance. Since LMS is the simplest algorithm, our advice for iterative channel estimation is therefore to use LMS with soft feedback. We also tried 16- and 64-QAM constellations, where approximated modified RLS is not identical to LMS. Still, the simulated performance of those two algorithms was indistinguishable. As a case study, adaptive turbo equalization has been applied to a standardized 8-PSK waveform for communications in the high-frequency (HF) band (3 30 MHz). The adaptive turbo equalization-based receiver uses the linear SISO equalizer of [24] for equalization, the LMS algorithm with soft feedback from the decoder for channel estimation, and the bitwise a posteriori probability (APP) algorithm for decoding. Simulation results are shown in Fig. 3 (details on the setup are given in [25] and [29]). We see that a substantial portion of the gain over the iterations is due to iterative channel estimation, in fact for the highest fading rate (rightmost plot) an error floor of 7 10 without iterative channel estimation is reduced to below 10. REFERENCES [1] C. Berrou, Near optimum error correcting coding and decoding: Turbocodes, IEEE Trans. Commun., vol. 44, pp , Oct [2] J. Hagenauer, The turbo principle: Tutorial introduction and state of the art, in Proc. Int. Symp. Turbo Codes and Related Topics, Brest, France, Sept. 1997, pp [3] C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, Iterative correction of intersymbol interference: Turbo-equalization, Eur. Trans. Telecommun., vol. 6, pp , Sept. Oct [4] L. M. Davis, I. B. Collings, and P. Hoeher, Joint MAP equalization and channel estimation for frequency-selective and frequency-flat fastfading channels, IEEE Trans. Commun., vol. 49, pp , Dec

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 6, NOVEMBER [5] X. Wang and R. Chen, Blind turbo equalization in Gaussian and impulsive noise, IEEE Trans. Veh. Technol., vol. 50, pp , July [6] A. Anastasopoulos and K. M. Chugg, Adaptive soft-input soft-output algorithms for iterative detection with parametric uncertainty, IEEE Trans. Commun., vol. 48, pp , Oct [7] J. Garcia-Frias and J. D. Villasenor, Combined turbo detection and decoding for unknown ISI channels, IEEE Trans. Commun., vol. 51, pp , Jan [8] 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, pp , Feb [9] A. O. Berthet, B. S. Ünal, and R. Visoz, Iterative decoding of convolutionally encoded signals over multipath Rayleigh fading channels, IEEE J. Select. Areas Commun., vol. 19, pp , Sept [10] M. Sandell, C. Luschi, P. Strauch, and R. Yan, Iterative channel estimation using soft decision feedback, in Proc. Global Telecom. Conf. GLOBECOM, Sydney, Australia, Nov. 1998, pp [11] S. Tantikovit, A. U. H. Sheikh, and M. Z. Wang, Code-aided adaptive equalizer for mobile communication systems, IEE Electron. Lett., vol. 34, pp , Aug [12] K.-D. Kammeyer, V. Kühn, and T. Petermann, Blind and nonblind turbo estimation for fast fading GSM channels, IEEE J. Select. Areas Commun., vol. 19, pp , Sept [13] C. H. Wong, B. L. Yeap, and L. Hanzo, Wideband burst-by-burst adaptive modulation with turbo equalization and iterative channel estimation, in Proc. 51st IEEE VTS Vehicular Tech. Conf. VTC, vol. 3, Tokyo, Japan, May 2000, pp [14] B. L. Yeap, C. H. Wong, and L. Hanzo, Reduced complexity in-phase/quadrature-phase M-QAM turbo equalization using iterative channel estimation, IEEE Trans. Wireless Commun, vol. 2, pp. 2 10, Jan [15] S. Song, A. C. Singer, and K.-M. Sung, Turbo equalization with an unknown channel, in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing ICASSP, vol. 3, Orlando, FL, May 2002, pp [16] M. Tüchler, R. Otnes, and A. Schmidbauer, Performance of soft iterative channel estimation in turbo equalization, in Proc. IEEE Int. Conf. Communications ICC, vol. 3, New York, NY, May 2002, pp [17] S. J. Nowlan and G. E. Hinton, A soft decision-directed LMS algorithm for blind equalization, IEEE Trans. Commun., vol. 41, pp , Feb [18] E. Baccarelli and R. Cusani, Combined channel estimation and data detection using soft statistics for frequency-selective fast-fading digital links, IEEE Trans. Commun., vol. 46, pp , Apr [19] G. Bauch, H. Khorram, and J. Hagenauer, Iterative equalization and decoding in mobile communications systems, in ITG-Fachbericht 145 (1997): 2nd Eur. Personal Mobile Communications Conf. Together With 3rd ITG-Fachtagung Mobile Kommunikation, Bonn, Germany, Oct. 1997, pp [20] C. Laot, A. Glavieux, and J. Labat, Turbo equalization: Adaptive equalization and channel decoding jointly optimized, IEEE J. Select. Areas Commun., vol. 19, pp , Sept [21] M. Tüchler, A. C. Singer, and R. Koetter, Minimum mean squared error equalization using a priori information, IEEE Trans. Signal Processing, vol. 50, pp , Mar [22] M. Tüchler, R. Koetter, and A. C. Singer, Turbo equalization: Principles and new results, IEEE Trans. Commun., vol. 50, pp , May [23] X. Wang and H. V. Poor, Iterative (turbo) soft interference cancellation and decoding for coded CDMA, IEEE Trans. Commun., vol. 47, pp , July [24] R. Otnes and M. Tüchler, Low-complexity turbo equalization for timevarying channels, in Proc. 55th IEEE VTS Vehicular Technology Conf. VTC, vol. 1, Birmingham, AL, May 2002, pp [25] R. Otnes, Improved receivers for digital high frequency communications: Iterative channel estimation, equalization, and decoding (adaptive turbo equalization), Ph.D. dissertation, Norwegian Univ. Science and Technology, Trondheim, Norway, Dec [26] S. Haykin, Adaptive Filter Theory, 3rd ed. Upper Saddle River, NJ: Prentice Hall, [27] W. N. Furman and J. W. Nieto, Understanding HF channel simulator requirements in order to reduce HF modem performance measurement variability, in Proc. 6th Nordic Shortwave Conf. HF, Fårö, Sweden, Aug. 2001, pp [28] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. Commun., vol. 49, pp , Oct [29] R. Otnes and M. Tüchler, Improved receivers for digital high frequency waveforms using turbo equalization, in Proc. MILCOM 2002, vol. 1, Anaheim, CA, Oct. 2002, pp