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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) L 11 I RBF Turbo Equalization This chapter presents novel a turbo equalization scheme,which employs a RBF equaliserinstead 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 areprovided. A novel element of our design isthat 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 conceptof turbo equalization. 11.1 Introductionto Turbo equalization In the conventionalRBF DFEbased systems discussedin Chapter 10 equalization and channel decoding ensued independently. However, it is possible to improve the receiver’s performance, 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 11.1, which will be detailed during our forthcoming discourse. Gertsmanand Lodge [308] extendedthis 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], demonstratingan increased performance gain due to turbo coding as well as with advent of enhanced IS1 mitigation achieved by turbo equalization. The principlesof iterative turbo decoding[ 1521 were modified appropriately for the coded M - Q A M system of Figure 1 l .2. The channel encoder fed is 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 isviewed as an ’inner encoder’ of a serially concatenated arrangement, since it can be modelled with the aid of 4.53 CHAPTER 11. RBF TURBO EOUALIZATION 454 Channel Decoder output Figure 11.1: Iterative turbo equalization schematic Interleaver Channel 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 thea priori information associatedwith the codedbits ck transmitted over the channeland - in conjunction with the channel output values V k -computes the aposteriori information concerning the coded bits. The soft values of the channel coded bits ck are typically quantifiedin the formof the log-likelihood ratiodefined 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 informationof the source bits only, while in turbo equalization the a posteriori information concerning all thecoded bits is required. In our descriptionof the turbo equaliser depicted in Figure 1 1.1, we have used the notation L E and C D 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 extrinsicLLR, 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," ( c k ) of the coded bits, and generates the a posteriori LLR values Cf(ck) of the interleaved coded bitsc k seen in Figure 11.2. Before passing theabove a posteriori LLRs generated by the SISO equaliser to the SISO decoder of Figure 11.1, the contribution of the decoder - in the formof the a priori informationC," ( c k ) -from theprevious 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 reliabilityestimation characterizing the coded bits, i.e. the extrinsic information. Thecombined channel 455 11.2. RBF ASSISTED TURBO EQUALIZATION and extrinsic LLR values are channel-deinterleaved - as seen in Figure 11.1 - 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 constitutedby the extrinsic LLR : C (c,) and the channel-deinterleaved combined channel and extrinsic LLR C?(C,) extracted from the equaliser’s a posteriori LLR ( 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 thea posteriori LLR values :C (c,) at the channeldecoder’s output, as seen alsoin Figure 11.1, yielding: C -: C,D(c,) = C,D(c,) - Cf(c,). (11.1) The extrinsic information: C (c,) of the coded bits isthen interleaved in Figure 11.1, in order to yield C:(ck), which is fed back in the required bit-orderto 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 importantthat only the channel-interleaved extrinsicpart - i.e. C:(ck) of) , . ( : C - is fed back to the equaliser, since the interdependence between the a priori information Cf(ck) = : C ( c k ) 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 forthe 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 subtracted from the a posteriori LLR value : C ( c k ) at the output of the equaliser in Figure 11.1, before passing the LLR values to the channel decoder. In the final iteration, the a posteriori 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 equalizationresearch has implemented theSISO equaliser using the SoftOutput Viterbi Algorithm (SOVA) [ 1531, the optimal MAP algorithm [3 1l ] 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 schemeof [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 11.1, 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 Equation 8.80, describing the conditional probability density function (PDF) of the ith symbol, CHAPTER 11. RBF TURBO EQUALIZATION 456 i = 1,. . . , M , associated with the ith subnet of the M-ary RBFequaliser: nf (11.2) i = l ,. . . j = l ,. . . ,n: ,M, 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 highlighted 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 statesp; = P(.;) and the RBF width introduced in Section 8.7.l is given thevalue of 20; where 0; is the channelnoise variance. The actual numberof channel states n: is determined by the specific designof the algorithm invoked, reducing the number of channel states from the optimum number of Mm+L-l,where m is the equaliser feedforward order and L l is the CIR duration [246,286,287]. The probabilitypf of the channel states rf , and therefore the weightsof the RBF equalisercan be derived from the LLR values of the transmitted bits, as estimated by the channeldecoder. Expounding further from Equation8.2 and 8.10, the channel output canbe defined as + rj = Fsj, (11.3) where F is the CIR matrix defined in Equation 8.11 and sj is the jth possible combinationof T the ( L + m ) transmittedsymbolsequence, sj = [ sjl . . . s j p . . . 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: P(rj) = = P(sj) p(sjl n . . . slP n . . . s ~ ( ~ + ~ , ) j (11.4) = 1,. 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 c j p l rc j P 2 , .. . ,clpm. Therefore, P(sjp) = P(cjPln . . . cjPqn . . . c j p r n ) = nP(cj,,) m j = 1, . . . , n t ? p = l , . .. , L + m . (11.5) q= 1 We have to map the bits cjps representing the M-ary symbol s j P to the corresponding bit { c k } . Note that the probabilityP(rj)of the channel output statesand therefore alsothe RBF weights defined in Equation 11.2 are time-variant, since the values of C p ( c k )are time-variant. 11.3. COMPARISON OF THE RBF MAPAND EOUALISER 457 Based on the definition of the bit LLR of Equation 10.6, the probabilityof bit c k having the value of + l or -I can be obtained after a few steps from the a priori information l F ( c k ) provided by the channel decoderof Figure 1 1.1, according to: (11.6) Hence, referring toEquation 11.4, 1 1.5and 11.6, the probability P(rj)of the received channel output vector can be represented in terms of the bit LLRs C F ( c j p q )as follows: P(rj) = P ( q ) L+m p= I p = l q=1 L+m m ,. yr:I where the constant = n;=1 l+exp(-L,E(cjpq)) e x p ( - L , E ( c 3 p q ) ’ 2 ) is independent of the bit cjpq. Therefore, we have demonstrated how the soft output C f ( C k ) of the channel decoder of Figure 1 1. l can be utilized by the RBF equaliser. Anotherway of viewing this process is that the RBF equaliser is trainedby the information generatedby the channel decoder. The RBF equaliser provides thea posteriori LLR values of the bits ck according to (11.8) where f k B F ( v k ) was defined by Equation 11.2 and the received sequence v k 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 [ 3131. 11.3 Comparison of the RBF and MAP Equaliser c ‘: The a posteriori LLR value of the coded bit ure 11.2, canbe calculated accordingto [ 3 1l]: ck, given the received sequence V k of Fig- (11.9) 458 - - - - - - - -> kk -- 3l CHAPTER 11. RBF TURBO EQUALIZATION -1 +l k-2 k k+l Figure 11.3: Example of a binary ( M = 2) system’s trellis structure where S’ and S denote the statesof the trellis seenin Figure 11.3 at trellis stagesIC - 1 and IC, respectively. The joint probabilityp(s’, S,v k ) is the product of three factors [31l]: d s ’ , S , Vk) = d s ’ , U j < k ) . . P(SlS’) P(UkIS’, S ) . P ( q > k l s ) , p ak-l(S‘) (11.10) Plc ( 3 ) Yk(S‘,S) where the term c q - 1 ( S ’ ) and /3k ( S ) are the so-called forward-and backward oriented transition 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 equalization, this corresponds to the length 3of the received sequence or the transmissionburst. 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 probabilityp ( 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 isgiven by: (11.14) and the a priori probability of bit c k being a logical where 6 k is the noiseless channel output, 1 or a logical 0 can be expressed in terms of its LLR values according to Equation 11.6. Since the term1 in the transition probability expression of Equation 11.14 and the term &Gq exp(-L:(ck)’2) in the a priori probability formula of Equation 11.6 are constant over the l+exp(-LE(ck)) summation in the numerator and denominator of Equation 11.9, they cancel out. Hence, the transition probability is calculated according to [3 1l]: yk(S’, S) = w k ’ ?;(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 describedby 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 complexityof both the MAPand the RBF equaliserscan 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 I k are made in possession of the information related to all the received symbols of a transmission burst. In the MAP algorithm the recursive relationships of the forward and backward transition probabilitiesof Equation 1 1.1 l and 1 1.12,respectively, allow us to avoid processing the entire received sequence v k everytime the a posteriori LLR , C f ( c k ) is evaluated from the joint probability p(s’, S , v k ) according to Equation 11.9. This approach is different from that of the RBF based equaliser having a feedforward orderof m, where the received sequence V k 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 forwardand backward recursively calculated transition probabilities c t k ( S) and P k ( S ) , before the LLR values Cf ( Q ) can be calculated fromEquation 1 1.9. The equaliser’s delay facilitates invoking information from the ’future’ samples u k , . . . , u~lc-~+lin the detection of the transmitted symbol I k - 7 . In other words, the delayed decision of the MAP equaliser provides the necessary information concerning the ’future’ samples u j > k - relative 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 increased Euclidean distance between channel states, as willitbe demonstrated in Section 1 1.5. However, this is not so for the more practical RBFDFE 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 probabilityin the logarithmic formas follows: (11.18) (11.19) (1 1.20) which we also used in Section 1 1.3. Thus,we could rewrite Equation 1 1.1 l as: (11.21) and Equation 1 1.12 as: (11.22) LFrom Equation l 1.21 and 1 1.22, the logarithmic-domain forwardand backward recursion can be evaluated, once I‘k(s’, S ) was obtained. In order to evaluate the logarithmic-domain branch metric r k (S’, S ) , Equations 11.15-1 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 P k ( S ) into the logarithmic domain in the Log-MAP algorithm, the expression for the LLR, :C (Ck) in Equation 11.9 is alsomodified to yield: In the trellis of Figure 11.3 there are M possible transitions from stateS’ to all possible states S or to state S from all possible statesS’. Hence, there areM - 1summations of the exponentials in the forward and backward recursion of Equation 11.21 and 11.22, respectively. Using the Jacobian logarithmic relationshipof Equation 10.2, M - 1 summations of the exponentials requires2(M-1) additions/subtractions, (M - 1)maximum search operations and (M - 1)table look-up steps.Together with the M additions necessitatedto evaluate theterm Fk(s’,s) Ak-l(s’) and I ‘ k ( S ’ , s ) B ~ ( sin) Equation 11.21 and 11.22, respectively, the forward and backward recursion requires a total of ( 6 M - 4) additions/subtractions, 2(M1) maximum search operations and 2(M-1) table look-up steps. Assuming that the term . c k . l F ( c k ) in Equation 11.23 is a known weighting coefficient, evaluating the branch metrics given by Equation 11.23 requires a totalof 2 additions/subtractions, 1 multiplication and 1 division. By considering a trellishaving x number of states at each trellis stageand M legitimate transitions leaving each state, there arei M x number of transitions due to thebit ck = +l. Each of these transitions belongs to the set (S’, S ) + Ck = +l. Similarly, there will be 1 zMx number of ck = -1 transitions, which belong to the set ( S ’ , S ) + Ck = -1. Evaluating 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 M x ( 6 M - 2) additions/subtractions, M x ( 2 M - 2) maximum search operations,M x ( 2 M - 2) table look-up steps, plus Mx multiplications and Mx divisions. With the terms Ak ( S ) , B k - 1 ( S ’ ) and I‘k(s’, S ) of Equations 11.21, 11.22and 11.23 evaluated, computing the LLR Lf(ck) of Equation 11.24 using the Jacobian logarithmic relationship of Equation 10.2 for the summation terms ln(x(s,,s)+ck=+l exp(.))and ln(C(s,,s)jck=+l exp(.)) requires a total of 4 ( $ M x - 1) 2 M x 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 = n s , f / M . Therefore, the total computational complexity associated with generating the a posteriori LLRs using the Jacobian logarithmic relationship for the Log-MAP equaliser isgiven in Table 1 1.1. For the Jacobian RBF equaliser, the LLR expression of Equation 11.8 is rewritten in terms + + + + CHAPTER 11. RBF TURBO EQUALIZATION 462 Log-MAP subtraction and addition multiplication division max table look-up n s , f ( 6 M+ 2 ) - 3 n,,j nsLf n s , f ( 2 M- 1) - 2 n s , f ( 2 M- 1) - 2 Jacobian RBF n,~+ Mn:(m+2)-4 ns>f nsLf Mni - 2 Mn: - 2 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 formIn (fhBF(vk)) to yield: (1 1.25) The summationof the exponentialsin 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 Equation 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 Jacobian RBF equaliser is given in Table 11.1. Figure 11.4 compares the number of additions/subtractions per turbo iteration involved in evaluating the a posteriori LLRs :C for the Log-MAP equaliserand Jacobian RBF equaliser accordingto Table l 1.1. More explicitly, the complexity is evaluatedupon with varying the feedforward orderm for differentvalues of L, where ( L 1)is the CIR duration under the assumptionthat the feedback order n = L and the number of RBF centres is n: = Mm+L-n/ M . Since the numberof 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 significant 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 showsan exponential increaseof the computationalcomplexity, as the CIR length +