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L1 1I RBF Turbo Equalization This chapter presents a novel turbo equalization scheme, which employs a RBF equaliser in- stead of the conventional trellis-based equaliser of Douillard et al. [ 1.531. The basic principles of turbo equalization will be highlighted. Structural, computational cost and performance comparisons of the RBF-based and trellis-based turbo equalisers are provided. A novel ele- ment of our design is that in order to reduce the computational complexity of the RBF turbo equaliser (TEQ), we propose invoking further iterations only, if the decoded symbol has a high error probability. Otherwise we curtail the iterations, since a reliable decision can be taken. Let us now introduce the concept of turbo equalization. 11.1 Introduction to Turbo equalization In the conventional RBF DFE based systems discussed in Chapter 10 equalization and chan- nel decoding ensued independently. However, it is possible to improve the receiver’s per- formance, if the equaliser is fed by the channel outputs plus the soft decisions provided by the channel decoder, invoking a number of iterative processing steps. This novel receiver scheme was first proposed by Douillard et al. [l531 for a convolutional coded binary phase shift keying (BPSK) system, using a similar principle to that of turbo codes and hence it was termed turbo equalization. This scheme is illustrated in Figure 1 1.1, which will be de- tailed during our forthcoming discourse. Gertsman and Lodge [308] extended this work and showed that the iterative process of turbo equalization can compensate for the performance degradation due to imperfect channel estimation. Turbo equalization was implemented in conjunction with turbo coding, rather than conventional convolutional coding by Raphaeli and Zarai [309], demonstrating an increased performance gain due to turbo coding as well as with advent of enhanced IS1 mitigation achieved by turbo equalization. The principles of iterative turbo decoding [ 1521 were modified appropriately for the coded M - QAM system of Figure 1 l .2. The channel encoder is fed with independent binary data d, and every log, (M) number of bits of the interleaved, channel encoded data Ck is mapped to an M-ary symbol before transmission. In this scheme the channel is viewed as an ’inner encoder’ of a serially concatenated arrangement, since it can be modelled with the aid of 4.53 Adaptive Wireless Tranceivers L. Hanzo, C.H. Wong, M.S. Yee Copyright © 2002 John Wiley & Sons Ltd ISBNs: 0-470-84689-5 (Hardback); 0-470-84776-X (Electronic) 454 CHAPTER 11. RBF TURBO EOUALIZATION Channel Decoder output Figure 11.1: Iterative turbo equalization schematic Channel Interleaver Mapping Discrete Channel Equaliser Figure 11.2: Serially concatenated coded M-ary system using the turbo equaliser, which performs the equalization, demodulation and channel decoding iteratively. a tapped delay line similar to that of a convolutional encoder [ 153,3101. At the receiver the equaliser and decoder employ a Soft-Idsoft-Out (SISO) algorithm, such as the optimal Maximum A Posteriori(MAP) algorithm [l621 or the Log-MAP algorithm [288]. The SISO equaliser processes the a priori information associated with the coded bits ck transmitted over the channel and - in conjunction with the channel output values Vk -computes the aposteriori information concerning the coded bits. The soft values of the channel coded bits ck are typically quantified in the form of the log-likelihood ratio defined in Equation 10.6. Note that in the context of turbo decoding - which was discussed in Chapter 10 - the SISO decoders compute the a posteriori information of the source bits only, while in turbo equalization the a posteriori information concerning all the coded bits is required. In our description of the turbo equaliser depicted in Figure 1 1.1, we have used the notation LE and CD to indicate the LLR values output by the SISO equaliser and SISO decoder, respectively. The subscripts e, i, a and p were used to represent the extrinsic LLR, the combined channel and extrinsic LLR, the a priori LLR and the a posteriori LLR, respectively. Referring to Figure 1 1.1, the SISO equaliser processes the channel outputs and the a priori information C," (ck) of the coded bits, and generates the a posteriori LLR values Cf(ck) of the interleaved coded bits ck seen in Figure 1 1.2. Before passing the above a posteriori LLRs generated by the SISO equaliser to the SISO decoder of Figure 11.1, the contribution of the decoder - in the form of the a priori information C," ( ck) - from the previous iteration must be removed, in order to yield the combined channel and extrinsic information C? (Q) seen in Figure 1 1.1. They are referred to as 'combined', since they are intrinsically bound and cannot be separated. However, note that at the initial iteration stage, no a priori information is available yet, hence we have Cf(ck) = 0. To elaborate further, the a priori information C,"(ck) was removed at this stage, in order to prevent the decoder from processing its own output information, which would result in overwhelming the decoder's current reliability- estimation characterizing the coded bits, i.e. the extrinsic information. The combined channel 11.2. RBF ASSISTED TURBO EQUALIZATION 455 and extrinsic LLR values are channel-deinterleaved - as seen in Figure 1 1.1 - in order to yield L?(C,), which is then passed to the SISO channel decoder. Subsequently, the channel decoder computes the a posteriori LLR values of the coded bits C:(c,). The a posteriori LLRs at the output of the channel decoder are constituted by the extrinsic LLR C: (c,) and the channel-deinterleaved combined channel and extrinsic LLR C?(C,) extracted from the equaliser’s a posteriori LLR -C: (Q). The extrinsic part can be interpreted as the incremental information concerning the current bit obtained through the decoding process from all the information available due to all other bits imposed by the code constraints, but excluding the information directly conveyed by the bit. This information can be calculated by subtracting bitwise the LLR values C?( c,) at the input of the decoder from the a posteriori LLR values C: (c,) at the channel decoder’s output, as seen also in Figure 1 1.1, yielding: C,D(c,) = C,D(c,) - Cf(c,). (11.1) The extrinsic information C: (c,) of the coded bits is then interleaved in Figure 1 1.1, in or- der to yield C:(ck), which is fed back in the required bit-order to the equaliser, where it is used as the a priori information Cf(ck) in the next equalization iteration. This constitutes the first iteration. Again, it is important that only the channel-interleaved extrinsic part - i.e. C:(ck) of C:(.,) - is fed back to the equaliser, since the interdependence between the a priori information Cf(ck) = C: (ck) used by the equaliser and the previous decisions of the equaliser should be minimized. This independence assists in obtaining the equaliser’s reliability-estimation of the coded bits for the current iteration, without being ’influenced’ by its previous estimations. Ideally, the a priori information should be based on an independent estimation. As argued above, this is the reason that the a priori information Cf(ck) is sub- tracted from the a posteriori LLR value C: (ck) at the output of the equaliser in Figure 1 1.1, before passing the LLR values to the channel decoder. In the final iteration, the a posteri- ori LLRs C:(&) of the source bits are computed by the channel decoder. Subsequently, the transmitted bits are estimated by comparing C:(&) to the threshold value of 0. For C:(&) < 0 the transmitted bit d, is deemed to be a logical 0, while d, = +l or a logical 1 is output, when CF(d,) 2 0. Previous turbo equalization research has implemented the SISO equaliser using the Soft- Output Viterbi Algorithm (SOVA) [ 1531, the optimal MAP algorithm [3 1 l] and linear filters [172]. We will now introduce the proposed RBF based equaliser as the SISO equaliser in the context of turbo equalization. The following sections will discuss the implementational details and the performance of this scheme, benchmarked against the optimal MAP turbo equaliser scheme of [31 l]. 11.2 RBF Assisted Turbo equalization The RBF network based equaliser is capable of utilizing the a priori information Cf(ck) provided by the channel decoder of Figure 1 1.1, in order to improve its performance. This a priori information can be assigned namely to the weights of the RBF network [312]. We will describe this in more detail in this section. For convenience, we will rewrite Equa- tion 8.80, describing the conditional probability density function (PDF) of the ith symbol, 456 CHAPTER 11. RBF TURBO EQUALIZATION i = 1, . . . , M, associated with the ith subnet of the M-ary RBF equaliser: nf (11.2) i =l, ,M, j =l, ,n: where cf , W;, p(.) and p are the RBF’s centres, weights, activation function and width, respectively. In order to arrive at the Bayesian equalization solution [85] - which was high- lighted in Section 8.9 - the RBF centres are assigned the values of the channel states r; defined in Equation 8.83, the RBF weights defined in Section 8.7.1 correspond to the apriori probability of the channel states p; = P(.;) and the RBF width introduced in Section 8.7. l is given the value of 20; where 0; is the channel noise variance. The actual number of channel states n: is determined by the specific design of the algorithm invoked, reducing the number of channel states from the optimum number of Mm+L-l, where m is the equaliser feedfor- ward order and L + l is the CIR duration [246,286,287]. The probability pf of the channel states rf , and therefore the weights of the RBF equaliser can be derived from the LLR values of the transmitted bits, as estimated by the channel decoder. Expounding further from Equation 8.2 and 8.10, the channel output can be defined as rj = Fsj, (11.3) where F is the CIR matrix defined in Equation 8.11 and sj is the jth possible combination of the (L+m) transmitted symbol sequence, sj = [ sjl . . . sjp . . . s~(L+~) ] . Hence - for a time-invariant CIR and assuming that the symbols in the sequence sj are statistically independent of each other - the probability of the received channel output vector rj is given by: T P(rj) = P(sj) = p(sjl n . . . slP n . . . s~(~+~,) j = 1,. (11.4) p=l The transmitted symbol vector component sjP - i.e. the pth symbol in the vector - is given by m = log2 M number of bits cjplr cjP2,. . . , clpm. Therefore, P(sjp) = P(cjPl n . . . cjPq n . . . cjprn) m = nP(cj,,) j = 1 , ,nt? p= l, . ,L+m. (11.5) q= 1 We have to map the bits cjps representing the M-ary symbol sjP to the corresponding bit {ck}. Note that the probability P(rj) of the channel output states and therefore also the RBF weights defined in Equation 11.2 are time-variant, since the values of Cp(ck) are time-variant. 11.3. COMPARISON OF THE RBF AND MAP EOUALISER 457 Based on the definition of the bit LLR of Equation 10.6, the probability of bit ck having the value of +l or -I can be obtained after a few steps from the a priori information lF(ck) provided by the channel decoder of Figure 1 1.1, according to: (11.6) Hence, referring to Equation 1 1.4, 1 1.5 and 11.6, the probability P(rj) of the received chan- nel output vector can be represented in terms of the bit LLRs CF(cjpq) as follows: P(rj) = P(q) L+m p= I p=l q=1 L+m m ,. where the constant = rI::y n;=1 l+exp(-L,E(cjpq)) exp(-L,E(c3pq)’2) is independent of the bit cjpq. Therefore, we have demonstrated how the soft output Cf(Ck) of the channel decoder of Figure 1 1. l can be utilized by the RBF equaliser. Another way of viewing this process is that the RBF equaliser is trained by the information generated by the channel decoder. The RBF equaliser provides the a posteriori LLR values of the bits ck according to (11.8) where fkBF(vk) was defined by Equation 11.2 and the received sequence vk is shown in Figure 11.2. In the next section we will provide a comparative study of the RBF equaliser and the conventional MAP equaliser of [3 131. 11.3 Comparison of the RBF and MAP Equaliser The a posteriori LLR value ‘c: of the coded bit ck, given the received sequence Vk of Fig- ure 11.2, can be calculated according to [3 1 l]: (11.9) 458 CHAPTER 11. RBF TURBO EQUALIZATION - -1 -> +l - - - - - - - k-3 k-2 k-l k k+l Figure 11.3: Example of a binary (M = 2) system’s trellis structure where S’ and S denote the states of the trellis seen in Figure 11.3 at trellis stages IC - 1 and IC, respectively. The joint probability p(s’, S, vk) is the product of three factors [31 l]: ds’, S, Vk) = ds’, Uj<k). P(SlS’) . P(UkIS’, S) .P(q>kls), (11.10) -p- ak-l(S‘) Yk(S‘,S) Plc (3) where the term cq-1 (S’) and /3k (S) are the so-called forward- and backward oriented transi- tion probabilities, respectively, which can be obtained recursively, as follows [31 l]: (11.11) (11.12) Furthermore, ~k (S’, S), IC = 1, . . . , 3 represents the trellis transitions between the trellis stages (IC - 1) and IC. The trellis has to be of finite length and for the case of MAP equal- ization, this corresponds to the length 3 of the received sequence or the transmission burst. The branch transition probability 71; (S’, S) can be expressed as the product of the a priori probability P( S 1 S’) = P( Q) and the transition probability p( uk 1 S’, S) : %(S’, S) = P(Ck) .P(4S’, ST). (11.13) 11.3. COMPARISON OF THE RBF AND MAP EQUALISER 459 The transition probability is given by: (11.14) where 6k is the noiseless channel output, and the a priori probability of bit ck being a logical 1 or a logical 0 can be expressed in terms of its LLR values according to Equation 11.6. Since the term 1 in the transition probability expression of Equation 1 1.14 and the term l+exp(-LE(ck)) exp(-L:(ck)’2) in the a priori probability formula of Equation 11.6 are constant over the summation in the numerator and denominator of Equation 11.9, they cancel out. Hence, the transition probability is calculated according to [3 1 l]: &Gq yk(S’, S) = wk ’ ?;(S’, S), (11.15) (11.16) (11.17) Note the similarity of the transition probability of Equation 11.15 with the PDF of the RBF equaliser’s ith symbol described by Equation 10.3, where the terms wk and y* (S’, S) are the RBF’s weight and activation function, respectively, while the number of RBF nodes n: is one. We also note that the computational complexity of both the MAP and the RBF equalisers can be reduced by representing the output of the equalisers in the logarithmic domain, utilizing the Jacobian logarithmic relationship [288] described in Equation 10.1. The RBF equaliser based on the Jacobian logarithm - highlighted in Section 10.2 - was hence termed as the Jacobian RBF equaliser. The memory of the MAP equaliser is limited by the length of the trellis, provided that decisions about the kth transmitted symbol Ik are made in possession of the information related to all the received symbols of a transmission burst. In the MAP algorithm the recur- sive relationships of the forward and backward transition probabilities of Equation 1 1.1 l and 1 1.12, respectively, allow us to avoid processing the entire received sequence vk everytime the a posteriori LLR ,Cf(ck) is evaluated from the joint probability p(s’, S, vk) according to Equation 11.9. This approach is different from that of the RBF based equaliser having a feedforward order of m, where the received sequence Vk of m-symbols is required each time the a posteriori LLR Cf(ck) is evaluated using Equation 11.8. However, the MAP algorithm has to process the received sequence both in a forward and backward oriented fashion and store both the forward and backward recursively calculated transition probabilities ctk ( S) and Pk ( S), before the LLR values Cf (Q) can be calculated from Equation 1 1.9. The equaliser’s delay facilitates invoking information from the ’future’ samples uk, . . . , u~lc-~+l in the de- tection of the transmitted symbol Ik-7. In other words, the delayed decision of the MAP equaliser provides the necessary information concerning the ’future’ samples uj>k - rela- tive to the delayed kth decision - to be utilized and the information of the future samples is generated by the backward recursion of Equation 1 1.12. The MAP equaliser exhibits optimum performance. However, if decision feedback is used in the RBF subset centre selection as in [246] or in the RBF space-translation as in 460 CHAPTER 11. RBF TURBO EQUALIZATION Section 8.1 1.2, the performance of the RBF DFE TEQ in conjunction with the idealistic assumption of correct decision feedback is better, than that of the MAP TEQ due to the in- creased Euclidean distance between channel states, as it will be demonstrated in Section 1 1.5. However, this is not so for the more practical RBF DFE feeding back the detected symbols, which may be erroneous. 11.4 Comparison of the Jacobian RBF and Log-MAP Equaliser Building on Section 1 l .3, in this section the Jacobian logarithmic algorithm is invoked, in order to reduce the computational complexity of the MAP algorithm. We denote the forward, backward and transition probability in the logarithmic form as follows: which we also used in Section 1 1.3. Thus, we could rewrite Equation 1 1.1 l as: and Equation 1 1.12 as: (11.18) (11.19) (1 1.20) (11.21) (11.22) LFrom Equation l 1.21 and 1 1.22, the logarithmic-domain forward and backward recursion can be evaluated, once I‘k(s’, S) was obtained. In order to evaluate the logarithmic-domain branch metric rk ( S’, S), Equations 1 1.15-1 l. 17 and 1 1.20 are utilized to yield: (1 1.23) 11.4. COMPARISON OF THE JACOBIAN RBF AND LOG-MAP EQUALISER 461 By transforming ak(s), yk(s’, S) and Pk(S) into the logarithmic domain in the Log-MAP algorithm, the expression for the LLR, C: (Ck) in Equation 1 1.9 is also modified to yield: In the trellis of Figure 1 1.3 there are M possible transitions from state S’ to all possible states S or to state S from all possible states S’. Hence, there are M - 1 summations of the ex- ponentials in the forward and backward recursion of Equation 1 1.21 and 1 1.22, respectively. Using the Jacobian logarithmic relationship of Equation 10.2, M - 1 summations of the ex- ponentials requires 2(M-1) additions/subtractions, (M - 1) maximum search operations and (M - 1) table look-up steps. Together with the M additions necessitated to evaluate the term Fk(s’,s) + Ak-l(s’) and I‘k(S’,s) + B~(s) in Equation 11.21 and 11.22, respectively, the forward and backward recursion requires a total of (6M - 4) additions/subtractions, 2(M- 1) maximum search operations and 2(M-1) table look-up steps. Assuming that the term . ck . lF(ck) in Equation 11.23 is a known weighting coefficient, evaluating the branch metrics given by Equation 1 1.23 requires a total of 2 additions/subtractions, 1 multiplication and 1 division. By considering a trellis having x number of states at each trellis stage and M legitimate transitions leaving each state, there are iMx number of transitions due to the bit ck = +l. Each of these transitions belongs to the set (S’, S) + Ck = +l. Similarly, there will be zMx number of ck = -1 transitions, which belong to the set (S’, S) + Ck = -1. Eval- uating A~(s), Bk-l(s’) and I‘k(s’,s) of Equation 11.21, 11.22 and 11.23, respectively, at each trellis stage IC associated with a total of Mx transitions requires Mx(6M - 2) addi- tions/subtractions, Mx(2M - 2) maximum search operations, Mx(2M - 2) table look-up steps, plus Mx multiplications and Mx divisions. With the terms Ak (S), Bk-1 (S’) and I‘k(s’, S) of Equations 11.21, 11.22 and 11.23 evaluated, computing the LLR Lf(ck) of Equation 11.24 using the Jacobian logarithmic relationship of Equation 10.2 for the sum- mation terms ln(x(s,,s)+ck=+l exp(.)) and ln(C(s,,s)jck=+l exp(.)) requires a total of 4($Mx - 1) + 2Mx + 1 additions/subtractions, Mx - 2 maximum search operations and Mx - 2 table look-up steps. The number of states at each trellis stage is given by x = M L = ns,f/M. Therefore, the total computational complexity associated with gen- erating the a posteriori LLRs using the Jacobian logarithmic relationship for the Log-MAP equaliser is given in Table 1 1.1. 1 For the Jacobian RBF equaliser, the LLR expression of Equation 1 1.8 is rewritten in terms 462 CHAPTER 11. RBF TURBO EQUALIZATION Log-MAP Jacobian RBF subtraction n,~+ ns,f(6M + 2) - 3 multiplication n,,j nsLf division ns>f Mn: - 2 ns,f(2M - 1) - 2 table look-up Mni - 2 ns,f(2M - 1) - 2 max nsLf and addition Mn:(m+2)-4 Table 11.1: Computational complexity of generating the a posteriori LLR L," for the Log-MAP equaliser and the Jacobian RBF equaliser [314]. The RBF equaliser order is denoted by m and the number of RBF centres is ni. The notation n,,f = ML+' indicates the number of trellis states for the Log-MAP equaliser and also the number of scalar channel states for the Jacobian RBF equaliser. of the logarithmic form In (fhBF(vk)) to yield: (1 1.25) The summation of the exponentials in Equation 11.25 requires 2(M-2) additions/subtractions: (M-2) table look-up and (M - 2) maximum search operations. The associated complexity of evaluating the conditional PDF of M symbols in logarithmic form according to Equa- tion 10.4 was given in Table 10.1. Therefore, - similarly to the Log-MAP equaliser - the computational complexity associated with generating the a posteriori LLR L: for the Ja- cobian RBF equaliser is given in Table 11.1. Figure 11.4 compares the number of addi- tions/subtractions per turbo iteration involved in evaluating the a posteriori LLRs C: for the Log-MAP equaliser and Jacobian RBF equaliser according to Table l 1.1. More explicitly, the complexity is evaluated upon with varying the feedforward order m for different values of L, where (L + 1) is the CIR duration under the assumption that the feedback order n = L and the number of RBF centres is n: = Mm+L-n /M. Since the number of multiplications and divisions involved is similar, and by comparison, the number of maximum search and table look-up stages is insignificant, the number of additions/subtractions incurred in Figure 1 1.4 approximates the relative computational complexities involved. Figure 1 1.4 shows signifi- cant computational complexity reduction upon using Jacobian RBF equalisers of relatively low feedforward order, especially for higher-order modulation modes, such as M = 64. The figure also shows an exponential increase of the computational complexity, as the CIR length [...]... higher-dispersion channels and for high-order modulation schemes Throughout the book we have studied a host of adaptive transceiver schemes, which were invoked for mitigating the detrimental effects of the multipath-induced channel quality fluctuations We also argued in thePrologue, namely inChapter l that the same adaptive techniques can be used for combating the effects of the time-variant cochannel interference...11.5 RBF TURBO EQUALISER PERFORMANCE 463 increases Observe in Figure 11.4 that as a rule of thumb, the feedforward order of the Jacobian RBF DFEmust not exceed the CIR length ( L 1)in order to achieve a computational complexity improvement relative to the Log-MAP equaliser, provided that we use... receiver We note, however that fixed-mode modulation based space-time codecs are less efficient in terms of mitigating the effects of the time-variant co-channel interference fluctuations, than their adaptive counterparts 11.8 n r b o Equalization of Convolutional Coded and Concatenated Space Time Trellis Coded Systems using Radial Basis Function Aided Equalizers M S Yee, B L Yeap and L Hanzo 11.8.1 . concatenated arrangement, since it can be modelled with the aid of 4.53 Adaptive Wireless Tranceivers L. Hanzo, C.H. Wong, M.S. Yee Copyright © 2002. in the logarithmic form as follows: which we also used in Section 1 1.3. Thus, we could rewrite Equation 1 1.1 l as: and Equation 1 1.12 as: (11.18)