Linear time and frequency domain Turbo equalization

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1 Linear time and frequency domain Turbo equalization Michael Tüchler, Joachim Hagenauer Lehrstuhl für Nachrichtentechnik TU München München, Germany Abstract For coded data transmission over channels introducing inter-symbol interference, one approach for joint equalization and decoding in the receiver is Turbo Equalization. We rederive existing linear equalization algorithms applicable to Turbo Equalization for Ñ -ary signal alphabets and compare their computational complexity. Moreover, by evaluating the algorithm performance properly, we select for each iteration the most suitable of the two algorithms with lowest computational complexity and achieve at low bit error rates a performance close to that of optimal approaches for equalization, i.e., maximum a-posteriori probability symbol detection. 1. Introduction We consider a coded data transmission system, where blocks of data bits are encoded to code bits using forward error correction (FEC), which are subsequently interleaved, mapped to symbols from a Ñ -ary signal alphabet and transmitted over a channel with inter-symbol interference (ISI). The channel is modeled in discrete time with the Å finite-length impulse response filter Ò ¼ ÆÒ,, of length Å. The impulse response has energy Å ¼. The coefficients are assumed to be time-invariant and known to the receiver. The noise process is assumed to be independent and identically distributed (i.i.d.) and independent of the data. This system model is valid for many communication systems with frequency selective or multipath channels. The receiver of such a system can perform joint decoding and equalization using Turbo Equalization (Turbo Equ.), which was pioneered in [4] and enhanced in [1, 2]. However, the used trellis-based detection algorithms (soft-out Viterbi equalization (SOVE), maximum a-posteriori probability (MAP) symbol detection) become prohibitively complex for increasing Å and Ñ. In [5, 6, 9, 12], new equalization techniques based on linear filtering were applied to significantly reduce the computational complexity. Among them, we differentiate between minimum mean squared error (MMSE) linear equalization (LE) and matched filtering (MF). The LE algorithm derived in [9] was also implemented in an approximate version (APPLE). In this paper, we provide a framework to use the linear approaches given a Ñ -ary signal alphabet (LE: Section 4.1, APPLE: Section 4.2, MF: Section 5) and specify how to select the most suitable equalization algorithm for each iteration - an approach, which significantly improves the performance as shown in [9]. In all systems, a convolutional code with MAP-based decoding is used for FEC. We start with a brief system definition, explain next the general approach to derive linear algorithms applicable for Turbo Equ., derive in detail the different algorithms (LE, APPLE, and MF) in the time and, if possible, in the frequency domain, devise an adaptation criterion to switch between the algorithms, compare the computational complexity, and conclude the paper with results and final remarks. data c n Encoder Π c n Transmitter Equalizer x k L e ( c n ) ISI channel Π -1 w k L e ( c n ) z k L ( c n ) L ( c n ) Receiver Π Decoder Figure 1. Coded data transmission system. 2. System definition data estimate Consider the communication system in Figure 1 with a receiver performing Turbo Equ. Binary data is encoded using a binary convolutional code to length Ñ blocks ¼ ¼ ¼ ¼ ¼ Ñ Ì of code symbols ¼ Ò ¼. The interleaver permutes ¼ to ¼ Ñ Ì denoted as ¼ µ. The deinterleaver µ reverses the permutation µ. The modulator maps Ñ code bits Ñ, ¼ Ñ µ, to a complex symbol Ü according to the Ñ -

2 ary symbol alphabet Ë ¼ Ñ, where corresponds to the bit pattern ¼ Ñ µ, ¼. We require that Ñ ¼ ¼ and Ñ Ñ ¼. Transmitted over the channel is the sequence Ü Ü ¼ Ü Ü after the length Å prefix or guard interval Ü Å Ü Å Ü, where we assume that the transmitter knows Å. In many applications, the prescribed prefix is already part of Ü due to fixed header and tail sequences. The receiving process of the transmitted Ü is disturbed by complex-valued additive white Gaussian noise (AWGN), i.e., both the real and imaginary part of the noise samples Û is i.i.d. with pdf Ò ¼ Û Ûµ defined as Ò Ûµ ÜÔ Û µ µ Ô Û Ê Ê Thus, we have Û Û Û and Û Û. The receiver observes the sequence Þ Þ ¼ Þ Þ (the first Å symbols are neglected). Due to the prefix, the channel state at the block ends is equal and we can express Þ as Å Þ Ü µ mod Û ¼ µ ¼ In case all Ë are real, we can design a receiver using Þ only, which yields Û Û and Û ¼. Before proceeding, some frequently used notation is introduced. The matrix ¼ contains all zeros, contains all ones. Á is the identity matrix. The operator is the expectation with respect to the joint probability density function (pdf) of the Ü and Û. The covariance operator Cov Ü Ýµ equals Ü Ý À ÜÝ À, where À is the Hermitian operator. The -value operator µ, ¼, equals µ ÐÒ Ö¼, i.e., µ is the Ö log likelihood ratio (LLR). The operator Diag applied to a length Æ vector returns a Æ Æ matrix with the vector elements along the diagonal. 3. Linear algorithms for Turbo equalization We present here the general framework to rederive LE, APPLE, and MF for a Ñ -ary signal constellation using the results in [9]. At first, the statistics Ü Ü and Ú Cov Ü Ü µ of the symbols Ü are computed using the a- priori information Ò µ provided by the decoder: Ü Ú Ë ÖÜ Ë ÖÜ Ñ Ë ¼ Ö Ñ Ü (1) The equalizer assumes the Ò to be independent (which is locally achieved using interleaving) such that ÖÜ is the product of Ñ terms Ö Ñ, which are determined using Ò µ, Ò Ñ. From the independence assumption follows Cov Ü Ü ¼ µ ¼, ¼, too. Filtering Ü with Ò gives Þ Þ Å ¼ Ü µ mod ¼ µ which is subtracted from the received symbols Þ. This difference is filtered using a length Æ linear FIR filter with possibly time-varying coefficients Æ Æ Æ, (Æ Æ Æ ). The output of this filter are the estimates Ü. The equalizer output LLRs Ò µ, Ò ¼ Ñ µ, are the extrinsic information (a-posteriori minus a-priori information) about Ò given the channel observations: Ò µ ÔÓ Ø Ò µ Ò µ ÐÒ Ö Ò ¼Ü Ö Ò Ü Ò µ ÐÒ Ô Ü Ò ¼µ Ô Ü Ò µ It is shown in [8] that this decomposition of the a-posteriori LLR ÔÓ Ø Ò µ yields the best performance in the more general problem of linear MMSE estimation using a-priori information. We must satisfy that Ò µ and hence also Ü is not a function of Ò µ [8]. This is achieved by extending the approach in [8], which is to remove the influence of all Ñ µ, ¼ Ñ µ, on Ü and to replace it with the influence of Ñ µ ¼,. We assume that Ü exhibits a complex Gaussian distribution Ô Ü Ü Üµ, Ü, conditioned on Ü, ¼ Ñ µ: Ü Ü Cov Ü Ü Ü µ Ô Ü Ü Üµ ÜÔ Ü µ In case all Ë are real, Ô Ü Ü Üµ can be a single Gaussian pdf, i.e., Ô Ü Ü Üµ Ò Üµ, Ü Ê. By averaging over all Ô Ü Ü Üµ with Ò ¼ or Ò, respectively, Ò µ is computed as Ò µ ÐÒ Ô Ü Ü Ü µ ÖÜ Ò ¼ ËÒ¼ Ô Ü Ü Ü µ ÖÜ Ò ËÒ where Ò Ñ and ¼ Ñ µ. This simplifies to ± Ü (2) É Ñ ÜÔ ± µ м Ö Ñ Ð Ð ËÒ¼ Ð Ò µ ÐÒ É Ñ ÜÔ ± µ м Ö Ñ Ð Ð ËÒ Ð

3 using the fact that, as shown later, does not depend on. We assume that no additional a-priori information besides that of the decoder is available. Thus, in the first iteration we have Ò É µ ¼, Ò, and Ò µ can be computed without Ñ the terms Ö Ñ Ð Ð. м Ð 4. Turbo equalization using MMSE linear equalization 4.1. Exact implementation This approach was derived in [8] and applied to Turbo Equ. in [9]. The design rule to obtain the is to minimize the MMSE cost function Ü Ü. In general, the estimate Ü is computed from Þ Þ Æµ mod Þ Æ µ mod Þ Æµ mod Ì a length Æ vector of received symbols, as follows (Appendix A in [8]): Þ Þ Æµ mod Þ Æµ mod Ì Ú DiagÚ Æµ mod Ú Å Æµ mod À ¼ Æ ¼ Æ Å µ Ì Û Á Æ ÀÎ À À µ Ü Ü Ú À Þ Þ µ (in general) where À is the Æ Æ Å µ channel convolution matrix À ¼ Å ¼ ¼ ¼ ¼ Å ¼ ¼... ¼ ¼ ¼ Å Due to the mod operation we circularly equalize on a tail-biting block Þ, which is possible using the prefix in the transmitter. However, Ü depends on Ñ µ, ¼ Ñ µ, over Ü and Ú. As mentioned, we set Ñ µ ¼,, yielding Ü ¼, and Ú and recompute and Ü by replacing Ü and Ú with ¼ and : ¼ Û Á Æ À Î À À Ú µ À µ Ü ¼ À Þ Þ Ü ¼µ µ We can express ¼ as scaled version of using the matrix inversion lemma and Û Á Æ ÀÎ À À : ¼ Ú µ À µ Ú µ À µ In [8], a recursive algorithm was derived to compute from with a number of operations only proportional to Æ and Å. With the final expression for the estimates: Ü Ú µ À µ À Þ Þ Ü µ we can compute the statistics and : À Þ Ü Þ Ü µ À À Cov Þ Þ Ü µ À Ú À µ where Ú µ À. The derivation to obtain Ò µ is completed by finding an expression for ± : ± À Ú À µµ À Þ Þ Ü µ À For the case that Ò µ ¼, Ò, we have Ü Þ ¼ and Ú,, yielding a time-invariant Æ : Æ Û Á Æ ÀÀ À µ ± À Æ À Æ µµ À Æ Þ À Æ where Æ stands for No A-priori information Approximate implementation The costly computation of the vector for each is neglected by simply using the vector Æ despite the presence of non-zero a-priori information Ò µ ¼ [8]. The estimates Ü are now given by Ü À Æ Þ Þ Ü µ However, computing the statistics and becomes more difficult: À Æ Þ Ü Þ Ü µ À Æ À Æ Û Á Æ À Î À À Ú À µ Æ In [8], was approximated by a crude time average. Here, the average is over all ¼ (any can be selected) corresponding to each Ü, ¼ µ: ¼ ¼ Û À Æ Æ ¼ The exponents ± are given by Ú À Æ À ÀÀ À µ Æ ± Ü À Æ (3) for general Ò µ Ê. For Ò µ ¼, Ò, yielding Ü ¼ and Ú,, we have especially À Æ À Æ µ and Ü À Æ Þ. 5. Turbo equalization using matched filtering This approach was first introduced in [4] and modified yielding better results in [9]. The estimator filter coefficients are set to yield a matched filter to the ISI channel

4 response Ò. The algorithm in [9] to compute the estimate Ü is used without adaptation to a Ñ -ary signal alphabet: Å Ð Ð Å µ Å µ м Å Ü Ü ¼ Þ µ mod Þ µ mod µ Û Ú Å Ð Å Ú Ðµ mod Ð The exponents ± for general Ò µê are given by ± Ü (4) Å For Ò µ ¼, Ò, we have especially Ü ¼ Þ and Û Å Ð Å Ð. 6. Algorithm adaptation The APPLE and the MF approach derived in Sections 4.2 and 5 require a number of operations per received symbol increasing only linearly with Æ or Å, which is much better than LE (Æ ) or MAP equalization ( ÑÅ ). However, both APPLE and MF suffer a significant performance loss compared to, e.g., LE. In [9], a scheme was proposed to properly switch between APPLE or MF depending on Ò µ, which is unfortunately based on empirical performance evaluation requiring the transmitted data. We propose a novel approach using the signal-to-noise ratio (SNR) of Ü to decide, prior to equalization, which algorithm to use: SNR Ü Ü Cov Ü Ü Ü µ à where Ã Æ À for APPLE and à for MF. We suggest that the algorithm yielding the highest average SNR Ñ Ñ ¼ ¼ à à ¼ ¼ should be used. The average variance is computed as in Section 4.2 yielding the lower bounds APPLE: MF: Ú Ú Ã ¼ à ¼ ¼ ¼ ¼ À Æ À Æ Û À Æ Æ Ú À Æ ÀÀÀ À µ Æ Ã Û Ú Å Ð Å Ð µ on. Thus, the receiver uses the algorithm with largest à for equalization. The bound is tight whenever ¼ is constant in, e.g., for Ú ¼ (symbols Ü are known to the receiver) and Ú (no a-priori information). The average SNR and its lower bound à are monotonically decreasing in Ú ¼ [8]. The maximum is using MF for Û Ú ¼, which is the SNR of an AWGN channel with noise variance Û. We thus expect the Turbo Equ. system performance to be below than that of coded data transmission over the equivalent AWGN channel, since the equalizer is at most able to provide the same SNR of the estimates Ü. We will not consider LE as alternative algorithm due to the much larger computational effort. For MAP equalization, an analysis using is not possible. We rely here, if applicable at all, on the EXIT charts introduced in [9, 11]. 7. Frequency domain implementation In [12], the APPLE and MF algorithm were implemented in the frequency domain. The adaptation to Ñ -ary signal alphabets is straightforward. Table 1 depicts this implementation (the DFT operator is the Discrete Fourier Transform). ÁÒÔÙØ - Þ ¼ Þ Ì, ¼ µ Ñ µ Ì, Ò, and Û, ÁÒØÐÞØÓÒ ¼ Ì DFT Þ ¼ Þ Ì À ¼ À Ì DFT ¼ Å ¼ ŵ Ì ¼ À Û À ÖÓÖ ØÓ ÕÙÐÞØÓÒ - compute: Ü ¼ Ü Ì and Ú ¼ Ú Ì, - decide: use APPLE or MF by comparing Ã, ÕÙÐÞØÓÒ ¼ Ì DFT Ü ¼ Ü Ì APPLE: À À Û À Û À µ, MF: À À µ, Ü ¼ Ü Ì DFT ¼ Ì Ø ØÓ ÕÙÐÞØÓÒ - compute: ¼ µ Ñ µ Ì. Table 1. Frequency domain equalization. 8. Complexity comparison In this section, the computational complexity of MAP equalization, LE, APPLE, and MF is compared. We assume that the statistics Ü and Ú are available for all and skip the computation to obtain Ò µ including and (both mappings Ü Ú Ò µ and Ò µ Ü strongly depend on Ë). Any overhead due to initialization (one-time computations for all iterations), e.g., to compute Å for APPLE or À,,, is neglected. Table 2 gives the number of real multiplications and additions per iteration required to equalize symbols Þ yielding estimates Ü. The DFT is carried out using a radix-2 FFT requiring roughly ÐÓ µ real multiplications and ÐÓ µ real additions for Ð, Ð, [10]. For the complexity of LE see [8]. For MAP equalization, we considered only the computation of all s and the «, recursion [7].

5 approach domain real multiplications real additions MF time ¼Å µ ¼Å µ MF frequency ÐÓ µ ÐÓ µ APPLE time Æ Å µ Æ Å µ APPLE frequency ÐÓ µ ÐÓ µ LE - Æ Å ¼Å Æ µ Æ Å ¼Æ Å µ MAP equ. - ÑÅ Ñ Ñ Å µ µ ÑÅ Ñ µ Ñ Å µ µ Table 2. Computational complexity of equalization per iteration per block. 9. Results and Conclusions We tested the bit error rate performance (simulation of at least ¼¼¼ data bit errors) of a Turbo Equ.-based receiver. Data is encoded (code generator µ µ) to length ¼ blocks of code symbols Ò including S-random (S=30) interleaving [3]. The Ò are modu- µ ¼¼ ¼ ¼ Ü ß Ô ß Ô ß Ô ß Ô Ü Ô ØÒ µ ØÒ µ ßµ Ú Ü ß Ô µ µ Ô ¼ ¼ Ü µ Ô ¼ ¼ Ü Table 3. QPSK modulation. lated to Ü according to Table 3 (includes also Eqs. (1) and (2)). The time-invariant channel impulse response is Ò ¼ ÆÒ ¼ ÆÒ ¼ ÆÒ ¼ ÆÒ ¼ ÆÒ. The system SNR is Æ ¼. The filter parameters for LE and APPLE are Æ and Æ. Figure Û 2 depicts the BER results after iterations: MAP equalization and LE perform best followed by switched APPLE/MF. Using APPLE and MF alone is not satisfactory. Similar results were obtained for unknown Ò (including training) and/or fading coefficients. In conclusion, the novel switched APPLE/MF approach yields a comfortable gain to one-time MAP equalization and decoding. We think, that this and the LE algorithm are most suitable for low coplexity Turbo Equ. using higher order signal alphabets, e.g., PSK in the the Enhanced Data rates for GSM evolution (EDGE) standard. Part of on-going work is an accurate performance analysis and the implementation of channel parameter estimation into the iterative algorithm. References [1] A. Anastasopoulos and K. Chugg. Iterative equalization/decoding for TCM for frequency-selective fading channels. Record on the 31th Asilomar Conf. on Signals, Systems & Computers, 1: , November [2] G. Bauch and V. Franz. A comparison of soft-in/soft-out algorithms for turbo detection. Proceedings on the Intern. Conf. on Telecomm. (ICT 98), pages , June BER no channel APPLE LE SIC mixed LE/SIC MAP one time MAP equ E b /N 0 in db Figure 2. BER performance comparison. [3] C. Heegard and S. Wicker. Turbo Coding. Kluwer Academic Publishing, Boston, [4] A. Glavieux, C. Laot, and J. Labat. Turbo equalization over a frequency selective channel. Intern. Symposium on Turbo codes & related topics, pages , Sep [5] C. Douillard et al. Iterative correction of intersymbol interference: Turbo equalization. European Trans. on Telecomm., 6(5): , Sep-Oct [6] D. Raphaeli and A. Saguy. Linear equalizers for turbo equalization: A new optimization criterion for determining the equalizer taps. Proc. of the 2nd Intern. Symp. on Turbo Codes & Related Topics, pages , Sep [7] L.R. Bahl et al. Optimal decoding of linear codes for minimizing symbol error rate. IEEE Transactions on Information Theory, 20: , March [8] M. Tüchler, A. Singer, and R. Kötter. Minimum mean squared error (MMSE) equalization using priors. submitted to IEEE Transactions on Signal Processing, [9] M. Tüchler, R. Kötter, and A. Singer. Turbo equalization: principles and new results. submitted to IEEE Trans. on Communications, [10] J. Proakis and D. Manolakis. Digital Signal Processing, 3rd Ed. Prentice Hall, Upper Saddle River, New Jersey, [11] S. ten Brink. Convergence of iterative decoding. Electronic Letters, 35(10): , May [12] M. Tüchler and J. Hagenauer. Turbo equalization using frequency domain equalizers. Proc. of the Allerton Conference, Monticello, IL, U.S.A., October 2000.