Performance of Soft Iterative Channel Estimation in Turbo Equalization

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1 Performance of Soft Iterative Channel Estimation in Turbo Equalization M. Tüchler Ý, R. Otnes Þ, and A. Schmidbauer Ý Ý Institute for Communications Engineering, Munich University of Technology, Arcisstr. 21, D München, Germany Þ UniK - Center for Technology at Kjeller, PO box 70, N-2027 Kjeller, Norway Abstract To combat the effect of intersymbol interference (ISI) while transmitting data over an ISI channel in a coded data transmission system, the impulse response of the channel is required. As part of the turbo equalization approach, which facilitates iterative equalization and decoding, we introduce a method to iteratively improve the quality of the estimate of the channel characteristics. This is done by incorporating soft information fed back by the decoder to improve the initial estimate, obtained for example using a training sequence. Decision criteria based on the analytical calculation of the variance of the channel estimation error are derived to decide whether the soft information improves the quality of the estimate. The considered estimation algorithm is the well-known recursive-least-squares algorithm. It turns out that incorporating soft information for iterative channel estimation does not always improve the quality of the estimate. If it does, the bit-error-rate performance improves significantly over a system not using soft iterative channel estimation. I. INTRODUCTION Many practical communication systems encounter the problem of data transmission over a channel with intersymbol interference (ISI). To protect the integrity of the data to be transmitted, a controlled amount of redundancy is added (encoding) using an error-correction code (ECC). In this paper, we assume a coherent symbol-spaced receiver front-end, and precise knowledge of the signal phase and the symbol timing, such that the channel can be approximated by an equivalent, discrete-time, baseband model, where the transmit filter, the channel, and the receive filter are represented by a discrete-time È linear filter, Å ½ with the finite-length impulse response Ò ÐÆÒ Ð of length Å. To combat the effects of ISI, linear (linear equalizer, LE), or non-linear processing (decision feedback equalizer, DFE, or detector) of the received symbols can be applied [1, 2]. Optimal methods for minimizing the sequence- (SER) or bit-error rate () are non-linear and are based on maximum likelihood (ML) estimation, e.g., the SER-optimizing Viterbi algorithm [1, 3, 4]. ML estimation turns into maximum a-posteriori probability (MAP) estimation, e.g., the optimizing BCJR algorithm [5, 6], in presence of a-priori information about the transmitted data. Estimation of the transmitted data is here refered to as equalization. An optimal receiver approach for a coded data transmission system would be joint decoding and equalization treating the concatenation of ECC-encoder and ISI channel as one code. However, the computational complexity of such an approach is most often prohibitive, especially if encoder and ISI channel are separated by a permuter called inter. A successful approach to approximately perform joint decoding and equalization was pioneered in the turbo equalization scheme [7], where equalization and decoding tasks are repeated on the same set of received data while feedback information from the decoder is incorporated into the equalization process. A number of such systems are proposed in the literature [7 11]. Most previous work on iterative equalization and decoding assumes that the channel impulse response Ò is known to the receiver. In [10], a least-mean-square type of channel estimation algorithm is used besides a LE to estimate and track Ò, where both the estimator and the equalizer incorporate the feedback information from the decoder. Other approaches use this information for estimation and equalization simultaneously, e.g., using a non-linear Kalman filter based on soft-statistics in [12 14]. We propose in this paper a recursive-least-square (RLS) type of channel estimation algorithm based on soft information, which is distinct from the equalization algorithm. Main part of the paper is the analysis of the quality of the provided channel estimate. We show that incorporating the soft feedback information does not always give a better estimate of the channel impulse response. Based on these results, an optimal algorithm for iterative channel estimation for turbo equalization is devised. The paper is organized as follows. We define in Section II a coded data transmission system applying turbo equalization. The RLS algorithm [15] is rederived (to incorporate soft information) and its performance is analyzed in Section III. Simulation results to verify the analysis are shown in Section IV. The paper ends with Section V. II. SYSTEM DEFINITION We consider a coded data transmission system depicted in Figure 1. The (binary) data is encoded with a (binary) convolutional encoder to length Ä sequences ½ ¾ Ä T of codesymbols Ò already mapped to the symbol alphabet Ë. We assume for simplicity binary phase shift keying (BPSK), i.e., Ë ½ ½, and that the channel impulse response coefficients Ð and the noise samples Û Ò are real-valued. A framework to develop algorithms for higher-order constellations and complex-valued Ð and Û Ò is presented in [16]. An inter permutes the bits in and outputs the sequence Ü Ü ½ Ü ¾ Ü Ä T, Ü Ò ¾ Ë. The deinter reverses the inter permutation. Transmitted over the ISI channel is Ý Ý ½ Ý ¾ Ý Ä Ì T Ø ½ Ø ¾ Ø Ì Ü ½ Ü ¾ Ü Ä T i.e., Ü preceded by the training sequence Ø Ø ½ Ø ¾ Ø Ì T, Ø Ò ¾ Ë, of length Ì, which is known to the receiver. The noise is modeled as additive white Gaussian noise (AWGN), i.e., the noise samples Û Ò are independent and identically distributed (i.i.d.) with normal probability density function (PDF) Æ ¼ Û ¾ µ, where Æ ¾ µ denotes a Gaussian distribution with mean and variance ¾. Input to the receiver is the length Ä Ì sequence Þ Þ ½ Þ ¾ Þ Ì Þ Ì ½ Þ Ä Ì T, Å ½ Þ Ð Ý Ò Ð Û (1)

2 w n data Encoder c n Inter x n yn ISI channel zn Equalizer L e ( xn) Deinter L ( cn) Decoder data estimate training symbols t n Channel estimator L ( xn) Inter L d ( cn) Fig. 1. Coded data transmission system applying turbo equalization in the receiver. where the subsequence Þ X Þ Ì ½ Þ Ì ¾ Þ Ä Ì T of Þ corresponds to the transmitted code symbols Ü Ò. The receiver consists of two soft-in soft-out (SISO) components, the equalizer and the decoder. We consider only the MAP approach [7 9] for both equalization and decoding and note that other less complex SISO equalizers were introduced in [10, 11, 17, 18]. The MAP equalizer computes the a-posteriori probabilities (APPs), È Ü ÜÞ X µ, Ü ¾ Ë, and outputs the loglikelihood ratio (LLR) Ä Ü Ò µ Ä Ü Þ X µ Ä Ü Ò µ ÐÒ È ÜÒ ½ÞXµ È ÐÒ È ÜÒ ½µ ÜÒ ½Þ Xµ È (2) ÜÒ ½µ which is the a-posteriori LLR Ä Ü Þ X µ minus the a-priori LLR Ä Ü Ò µ [11, 19]. The MAP decoder outputs Ä Ò µ Ä Öµ Ä Ò µ given the sequence Ö Ä ½ µ Ä ¾ µä Ä µ T of a-priori LLRs about the code bits Ò and estimates of the transmitted data bits. Applying the turbo principle, Ä Ü Ò µ deinterleaved to Ä Ò µ is considered a-priori LLR Ä Ò µ by the decoder and Ä Ò µ interleaved to Ä Ü Ò µ is considered a-priori LLR Ä Ü Ò µ by the equalizer. The receiver performs equalization and decoding tasks on the same Þ X until a suitably chosen termination criterion stops the iterative process. For initial equalization of Þ X, the symbols Ü Ò are assumed to be equally likely ½ or ½, i.e., Ä Ü Ò µ ¼, Ò. III. PERFORMANCE OF SOFT CHANNEL ESTIMATION The receiver, in particular the equalizer, requires estimates of the unknown parameters of the channel: the noise variance Û ¾ and the channel impulse response Å ½ Å ¾ ¼ T. Estimation algorithms using only the received symbols Þ Ò are referred to as blind algorithms, but the majority uses training symbols Ø Ò known to the receiver. We devise in this section an estimation algorithm based on the RLS algorithm [15] using training symbols and possibly LLRs Ä Ü Ò µ fed back the decoder. The RLS algorithm computes estimates Ò ÒÅ ½ Ò¼ T of for any time step Ò as follows: Ý Ý H ½ (3) Ý Þ where Ý Ý Å ½ Ý Å ¾ Ý T, ¾ ¼ ½ is a forgetting factor to enable the estimator to adapt to a time-varying Ò, and µ H is the Hermitian operator. A number of mostly timerecursive algorithms exists to efficiently compute (3). We assume that Ý ¼ for Ò½. When Ò is time-invariant, is set to ½. The impulse response Ò is assumed to vary so slowly such that it can be considered time-invariant for the analysis carried out in this section, whereas may still be less than ½ to enable the RLS algorithm to track the channel characteristics. Even though this is a severe restriction on the applicability of the shown analysis, we note that the extension to a more realistic channel (variation) model is possible in principal and currently investigated. Since only the first Ì symbols Ý Ò are known to the receiver, the estimation process stops at Ì or continues while using decided estimates Ü Ò from the transmitted symbols Ü Ò provided, e.g., by a DFE processing Þ X. Latter approach is inevitable if the channel characteristics vary during transmission of a sequence Ý. However, no matter what type of equalizer is used to obtain the estimates Ü Ò, errors in doing so are inferior to the estimation algorithm, which assumes that the Ü Ò are exactly Ü Ò. This effect is referred to as error propagation. We propose to use an estimation algorithm, which is able to utilize soft information about the Ü Ò, in this context the expectation Ü. Such an algorithm will not suffer from error propagation. Another way to obtain soft information about Ü Ò is to use the APPs È Ü ÜÞ X µ produced by a MAP equalizer. However, the equalizer produces these APPs after processing the entire received sequence Þ X. This problem was overcome in [12, 20] by altering the original MAP algorithm to produce intermediate APP estimates to run channel estimation in parallel to MAP equalization. A drawback of the existing solutions is that they rely on MAP equalization, which becomes prohibitively complex for channels with long impulse response and/or a largesized signal alphabet. SISO equalization algorithms with less complexity based on linear filters are devised in [11]. However, the techniques for joint equalization and estimation devised in [12, 20] cannot be applied here. To be independent from the actually implemented equalization algorithm, our approach runs estimation in serial concatenation to the equalizer without using its soft-output to improve the estimation. Instead, the estimate is improved by possibly using the LLRs Ä Ü Ò µ provided by the decoder, which is determined by comparing the quality of the estimate in case the Ä Ü Ò µ are used or not used. The quality measure is the covariance matrix Ò H Ò H Ò ÒÅ ½ Ò¼ T of the estimation error Ò. For the first equalization task, the equalizer relies on the estimate obtained using the training symbols, only. Error propagation is thus avoided. To estimate beyond Ì, the LLRs Ä Ü Ò µ are incorporated into the RLS algorithm via the mean Ü Ò Ü Ü¾Ë Ü È Ü Üµ ØÒ Ä Ü Ò µ¾µ of the transmitted symbols Ü Ò. Using the vector Ý Ý ½ Ý ¾ Ý Ä Ì T Ø ½ Ø ¾ Ø Ì Ü ½ Ü ¾ Ü Ä T

3 the estimates Ò are now obtained for ½ ¾ Ä Ì, by Ý Ý H ½ Ý Þ (4) where Ý Ý Å ½ Ý T. We define Ò ÈÒ Ý Ý H and È Ò Ò ½. The same time-recursive updates known for (3) can be used to efficiently compute (4). From Þ H Ý Ò Û Ò equivalent to (1) and the identity È Ò Ò follows that the estimation error Ò is given by È Ò Ý Ý H Û È Ò Ý Ý H Ý H È Ò Ý Ý H ÈÒ Ý Û From Ý Ý, Û ¼, and the assumption that È Ò is independent from all other quantities follows that ¼. Thus, the channel estimate Ò is unbiased and H Ò : Ò È µ µ Ý Û Û ÝH (6) µ µ Ý H Ý Ý µ Ý Ý µ H Ý H È H From the noise statistics follows that Û Û ¾ ÛÆ. The LLRs Ä Ü Ò µ fed back from the decoder can be modeled as i.i.d. samples from a random variable with the PDF Æ ¾ ¾ ¾ µ for some ¾ [21]. It follows that the independence assumption can be imposed on the Ü Ò and the Ü Ò as well yielding Ý Ý µ Ý Ý µ H ¼ for all and Ú Ý Ý µ Ý Ý µ H ½ Ü Ü ¼ (5) Ì ½ Ä ½ ¾ Ì When the Ü Ò are used as training symbols, we thus increase the measurement noise, since the transmitted Ü Ò can be thought of being generated by Ü Ò plus discrete noise with variance Ú Ò. The scalar expression H Ý Ý µ Ý Ý µ H in (6), where ½ ¼ ½, can be written as H Ý Ý µ Ý Ý µ H which yields Ò È Ò Å ½ Å ½ Å Å ½ µ ¾ Û µ Ý Ý H µ Ð Ð Ú Ð Ý Ý H Ð Ð Ú Ð È H (7) Approximating Ò with (7) is computationally expensive. Using again the independence assumption on the Ü Ò, È Ò becomes a diagonal matrix for increasing Ò and a forgetting factor close to ½, since the non-diagonal entries of Ò average out. We observe that the bottom entry Ò on the main diagonal of Ò is given by È Ò ½ ¾ Ì where Ý Ü Ì. For a sufficiently large Ò, the time average E x ½ Ä Ä Ì Ì ½ Ý Ý ½ Ä Ä Ü Ü of the term Ý Ý can be used to approximate Ò: È Ò ½ ¾ Ì È Ì ÈÒ Ì ½ E x ÒÌ ½ The other entries on the main diagonal of Ò are shifts of Ò, e.g., the top entry is given by Ò Å ½, Å. The matrix È Ò is finally approximated as follows: È Diag Ò Å ½ Ò ½ Diag ½ ½ Å ½ Ò where Diag applied to a length Å vector gives back a diagonal matrix with the vector elements on the diagonal. We are interested only in the main diagonal entries of Ò, the error variances of the taps ÒÐ, Ð ¼ Å ½. Since È Ò is approximately a diagonal matrix, only the summations over Ý Ý H ( ¼) in (7) affect the main diagonal È of Ò. The bottom entry ½µ Ò Ò on the main diagonal of the term ¾ µ Û ¾ Ý Ý H in (7) is approximated by È Ò ¾ Û ¾ µ ½ Ì Ò È Ì ¾ Û ¾ µ ÈÒ ½µ Ì ½ ¾ µ E x as it was done for Ò. The bottom entry ¾µ Ò on the main diag- Ð Ð Ú Ð Ý Ý H in (7) is onal of the term È Ò ¾ µ È Å ½ approximated by ¾µ Ò ¼ È Ò Ì ¾ µ ½ E h ½ E x µ E x where Ú Ð was replaced by the time average ½ Ä Ì Ä Ì ½ Ú ½ Ä Ä ÒÌ ½ ½ Ì ÒÌ ½ ½ Ü Ü µ ½ E x È Å ½ and E h is the channel energy Ð Ð. The variance ¾ ÒÐ ÒÐ ÒÐ of the error ÒÐ in the Ðth estimator filter tap at time step Ò (the Šе-th entry on the main diagonal of Ò ) is given by ¾ ÒÐ ½µ Ò Ð ¾µ Ò Ð µ¾ Ò Ð Å (8) When ½, the calculation of ÒÐ ¾ simplifies to ¾ Ò¼ ¾ Û Ò ¾ Û Ì E x Ò Ì µ ½ Ì Eh ½ Exµ Ex Ò Ì µ Ì E x Ò Ì µµ ¾ ÒÌ ½ where ÒÐ ¾ Ò ¾. The quantities E h and Û ¾ required to calculate ÒÐ ¾ are not known to the estimator. For example, E h can be estimated using Ì after processing the entire training sequence at time step Ì : (9) È Ì ÈÒ Ì ½ Ý Ý ÒÌ ½ Å ½ E h ÌÐ ÌÐ

4 The noise variance Û ¾ can be estimated by observing the error È Ò Å ½ Þ Ò ÌÐ Ý Ò Ð. Assuming that Ì, we have ¾ Û ¾ Û ¾ and thus ¾ Û Ò ¾ ½ Ì T ¾ (10) since Ý Ø Ò is known for ½ Ì. For brevity, we only note here that any equalization algorithm, linear or non-linear, should take the error in the channel response estimate Ò into account, i.e., the algorithm using Ò is different from that using a perfectly known Ò. Basically, using Ò instead of Ò yields a larger effective noise variance ¾ Û used inside the equalization algorithm. IV. RESULTS A. Performance of channel estimation using soft information In this section we want to verify the performance analysis of RLS-based channel estimation using soft information in Section III. We simulated channel estimation using the length ½ training sequence Ø ½½½½¼¼¼½½¼½½½¼½¼½¼¼¼¼½¼¼½¼½½¼¼½ chosen according to [22], followed by a length Ä ½¼¼ data sequence Ü of symbols Ü Ò ¾ Ë. The estimator knows only the LLRs Ä Ü Ò µ about Ü Ò and thus applies (4). The Ä Ü Ò µ were generated as i.i.d. samples of a random variable distributed with the PDF Æ ¾ ¾ ¾ µ for some ¾, which is a good model for the LLRs fed back from the decoder in a turbo equalizationbased receiver [11, 21, 23]. The Ø Ò and Ü Ò are transmitted over a channel with impulse response ¼¼ ¼¼ ¼¾½ ¼ ¼ ¼¾ -¼ -¼¾½ ¼¼ -¼¼ ¼¼ (Å ½½) taken from [1], which causes mild ISI but is hard to estimate due to its length. The variance Û ¾ of the noise was set to ¼½ for this experiment. The estimator produces estimates Ò for ½ Ä Ì. In Figure 2, we compared the squared estimation error Ò¼ ¾ for the (representative) tap Ð ¼ averaged over ¾¼¼¼ estimation tasks with the analytical result Ò¼ ¾ in (9) for ½. The results show that not for all soft information constellations it is worth using the LLRs Ä Ü Ò µ for channel estimation. For ¾ ½, the estimate at Ä Ì is noisier than that at Ì, where only the Ø Ò are used for estimation. For ¾ ½¼, there is an improvement using the LLRs Ä Ü Ò µ. Thus, for turbo equalization, channel estimation including the LLRs fed back from the decoder should be performed only for a sufficient quality of the soft information to be determined using (9) ( ½) or (8) in general. We also observed a good match of the simulated and analyzed variance of the estimation error for. B. Performance of iterative channel estimation We present performance results by simulating data transmission using the system depicted in Figure 1. The (timeinvariant) channel and the training sequence are taken from Section IV-A. The chosen ECC is a convolutional rate-½¾ code specified by the polynomial generator matrix ½ ¾ ½ ¾ µ. Transmitted are length Ä ½¾ blocks of code symbols Ü Ò, which were interleaved from Ò using an S-random inter error variance error variance Soft information: LLRs with σ 2 =1 measured value analytical value time index n Soft information: LLRs with σ 2 =10 measured value analytical value time index n Fig. 2. Performance of RLS estimation using training symbols and soft information about the transmitted data with varying quality. [24] with optimal Ë. The noise variance Û ¾ is determined according to Æ ¼ ½Û ¾ in db. We consider different systems, which differ only in the channel estimation procedure: Mode I: The receiver knows Ò and Û ¾ precisely. No channel estimation is used. Mode II: Both Ò and Û ¾ are unknown. Channel estimation ( ½) using the training sequence Ø is applied once to specify Ò (3) and Û ¾ (10). Mode III: Same as Mode II, except that possibly the LLRs Ä Ü Ò µ are used for channel estimation at each iteration when È Å ½ ¾ Ä ÌÐ È Å ½ ¾ ÌÐ. Figure 2 shows the performance after one-time equalization and decoding and 1,2, or 3 iterations. Due to the short block length Ä, more iterations do not improve the performance significantly. As expected, the performance of the receiver knowing Ò and Û ¾ precisely is superior to all other systems. When channel estimation using only Ø is applied, roughly a ¾ db loss in Æ ¼ occurs for all iterations and s. Incorporating soft information into the estimation process improves the performance of the system yielding only a ¼ db loss in Æ ¼ at low s. V. CONCLUSION We showed that iterative channel estimation using the soft feedback information from the decoder does not always give a better estimate of the channel when the channel is timeinvariant. A simple criterion to decide whether this soft information should be used for estimation was devised. In case of a time-variant channel, it is inevitable to track the characteristics of the channel using information about the transmitted

5 10 0 one time equalization and decoding 10 0 after one iteration after two iterations 10 0 after three iterations Fig Mode I: 8known channel characteristics Mode II: one time channel estimation Mode III: iterative channel estimation performance of turbo equalization using various approaches for channel estimation. symbols. For this scenario we are able to compute the quality of the estimate when soft information is used but more research is necessary to compare this scheme to other approaches for joint equalization and estimation. A similar analysis for other estimation algorithms, e.g., the least-mean-square algorithm, using soft information is desired. Final goal is the comparison of various schemes via their error variance performance and to find optimal patterns to where to place training symbols in the transmitted data. REFERENCES [1] J. Proakis, Digital Communications, 3rd Ed. Singapore: McGraw-Hill, [2] S. Haykin, Communication Systems, 3rd Ed. Canada: John Wiley & Sons, Inc., [3] S. Lin and J. J. Costello, Error Control Coding. Englewood Cliffs, New Jersey: Prentice Hall, [4] G. Forney, Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference, IEEE Trans. on Information Theory, vol. 18, pp , May [5] L.R. Bahl et al., Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. on Information Theory, vol. 20, pp , March [6] Y. Li, B. Vucetic, Y. Sato, Optimum soft-ouput detection for channels with intersymbol interference, IEEE Trans. on Information Theory, vol. 41, pp , May [7] C. Douillard et al., Iterative correction of intersymbol interference: Turbo equalization, European Trans. on Telecomm., vol. 6, pp , Sep/Oct [8] G. Bauch and V. Franz, A comparison of soft-in/soft-out algorithms for Turbo detection, in Proc. Intern. Conf. on Telecomm., pp , June [9] A. Anastasopoulos and K. Chugg, Iterative equalization/decoding for TCM for frequency-selective fading channels, in Conf. Record 31th Asilomar Conf. on Signals, Systems & Comp., vol. 1, pp , Nov [10] A. Glavieux, C. Laot, and J. Labat, Turbo equalization over a frequency selective channel, in Proc. Intern. Symp. on Turbo codes, Brest, France, pp , Sep [11] M. Tüchler, R. Koetter, and A. Singer, Turbo equalization: principles and new results, accepted for IEEE Trans. on Comm., Sep [12] E. Baccarelli and R. Cusani, Combined channel estimation and data detection using soft statistics for frequency-selective fast-fading digital links, IEEE Trans. on Comm., vol. 46, pp , April [13] A. Anastasopoulos and K. Chugg, Adaptive soft-in soft-out algorithms for iterative detection with parameteric uncertainty, IEEE Trans. on Comm., vol. 48, pp , Oct [14] S. Song, A. Singer, and K. Sung, Turbo equalization for unknown channel, manuscript, ssong1@uiuc.edu, June [15] S. Haykin, Adaptive Filter Theory, 3rd Ed. Upper Saddle River, New Jersey: Prentice Hall, [16] M. Tüchler and J. Hagenauer, Linear time and frequency domain turbo equalization, in Proc. IEEE Vehic. Techn. Conf., pp , May [17] D. Raphaeli and A. Saguy, Linear equalizers for Turbo equalization: A new optimization criterion for determining the equalizer taps, in Proc. 2nd Intern. Symp. on Turbo codes, Brest, France, pp , Sep [18] Z. Wu and J. Cioffi, Turbo decision aided equalization for magnetic recording channels, in Proc. Global Telecomm. Conf., pp , Dec [19] J. Hagenauer, The turbo principle: Tutorial introduction and state of the art, in Proc. Intern. Symp. on Turbo Codes, Brest, France, pp. 1 11, Sep [20] L. Davis, I. Collings, and P.Hoeher, Joint MAP equalization and channel estimation for frequency-selective and frequency-flat fast-fading channels, IEEE Trans. on Comm., vol. 49, pp , Dec [21] S. ten Brink, Convergence behaviour of iteratively decoded parallel concatenated codes, IEEE Trans. on Comm., vol. 49, Oct [22] S. Crozier and D. Falconer and S. Mahmoud, Least sum of squared errors (LSSE) channel estimation, IEE Proceedings, Pt. F, vol. 138, pp , Aug [23] S. ten Brink, Convergence of iterative decoding, IEEE Electronic Letters, vol. 35, pp , May [24] C. Heegard and S. Wicker, Turbo Coding. Boston: Kluwer Academic Publishing, 1999.