AbstractâThe goal of this paper is to assess the performance of turbo-equalization (TE), considering at the transmitter bit- interleaved coded modulation (BICM) ...
The complete list of Xavier Jaspar’s publications is currently (2005) available at: http://www.tele.ucl.ac.be/digicom/jaspar/publications/
Turbo-equalization considering bit-interleaved turbo-coded modulation: performance bounds Antoine Dejonghe∗ ,Xavier Jaspar† , Xavier Wautelet† and Luc Vandendorpe† ∗ Interuniversitair
Micro-Electronica Centrum (IMEC) Kapeldreef 75, B-3001 Leuven, Belgium † Communications and Remote Sensing Lab., Université catholique de Louvain Place du Levant 2, B-1348 Louvain-la-Neuve, Belgium
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A. Dejonghe and X. Jaspar thank the Belgian NSF for its financial support. This work is partly funded by the Federal Office for Scientific, Technical and Cultural Affairs, Belgium, through IAP contract No P5/11.
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I. I NTRODUCTION Bit-interleaved coded modulation (BICM) is a flexible solution for bandwidth-efficient coded transmission over memoryless channels [1]. It besides enables turbo processing at the receiver (i.e., iterative joint demodulation and decoding), which leads to very attractive performance. This latter approach is often referred to as turbo-demodulation (TD) [2], [3]. As an extension, it is possible to obtain more general BICMbased communication systems with iterative detection and decoding at the receiver. For instance, the extension of TD to frequency-selective channels leads to bit-interleaved turboequalization (TE) [4], [5]. The receiver then performs iterative joint equalization, demodulation and decoding. Capitalizing on [1] and [3], we have recently proposed a framework for bounding the error-floor of such BICMbased TE schemes [6]. This enables in particular to accurately predict asymptotic performance, and, to emphasize the dominant parameters. The goal of this paper is to extend these bounding techniques to the subset of the BICM-based TE schemes which have the specific feature of using a turbocode as error correcting code, considering as a case study transmission over static FS channels. These turbo-systems are indeed very attractive: they simultaneously offer a high level of flexibility (thanks to the use of BICM, the code and constellation mapping can be picked up independently), and very attractive performance (thanks to the use of a turbo-code as error correcting code, a strong coding gain can be expected). Note finally that bounding techniques are especially useful for this kind of communication systems due to their very low bit error rates (BER) which cannot be measured by simulations within a reasonable time.
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Abstract— The goal of this paper is to assess the performance of turbo-equalization (TE), considering at the transmitter bitinterleaved coded modulation (BICM) with a turbo-code as error correcting code. The bounding techniques that we have recently proposed in [6] are extended to this particular problem. Simulation results show the relevance of the proposed bounds. By the way, this results in analytical tools for the understanding and optimization of such turbo-systems.
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Receiver: three SISO modules turbo-receiver
II. T RANSMITTER A BICM transmitter which makes use of a turbo-code as error correcting code is considered. As represented in Fig. 1, it consists in the serial concatenation of a binary turbo-code, of a random bit interleaver Π2 , and of a memoryless modulator. The considered turbo-code classically consists in the parallel concatenation of two recursive systematic convolutional (RSC) codes separated by an interleaver Π1 . The memoryless modulator is characterized by a complex symbol constellation χ and a labeling map µ, which associates groups of m bits to one of the 2m symbols in χ. Let u, c, x and r denote the sequences of information bits, coded bits, transmitted symbols and received symbols, respectively. The channel is assumed static (i.e., non fading) and may be frequency-selective. III. I TERATIVE RECEIVER The resulting iterative receiver is illustrated in Fig. 2. It logically corresponds to a bit-interleaved TE scheme [5], whose outer SISO stage is the conventional turbo-decoder of the considered parallel concatenated turbo-code. By the way, note that it consists in the association of three SISO modules. For the sake of tractability, the inner SISO stage is implemented using low-complexity MMSE-based solutions for the required equalization process [4], [5]. When considering transmission over a memoryless channel, note that the considered iterative receiver reduces to a turbo-coded TD scheme.
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Fig. 3. Factor graph of the proposed scheme. Variable nodes are represented by circles, local function nodes by black squares. The nodes u i are the information bits, zi1 and zi2 the parity bits of the turbo-codes, xi the symbols (after mapping) sent through the channel. Π1 and Π2 are the two interleavers. The nodes t1i (resp. t2i ) are the states of the first (resp. second) RSC, qi the states of the ISI channel modeled as a hidden Markov chain. The functions f j1 , fj2 , dj and gj express the conditional probabilities between their neighbors for the first RSC, the second RSC, the mapper and the channel respectively. For example, gj (qj−1 , qj , xj ) = P (rj |qj−1 , xj ) I{qj ← (qj−1 , sj )} where the indicating function (I(a) = 1 if a is true, 0 otherwise) is used because the transition from (qj−1 , sj ) to qj is deterministic. At last, the indicating functions in the graph specify the start or the end of the chains in the zero state. Note that the last bits of u are used for the trellis termination of the first RSC.
A. A view based on factor graph theory The receiver goal is the bit error rate (BER) minimization given the received sequence r. This is achieved by the following maximum a posteriori (MAP) detection rule: argmaxui {P (ui |r)}.
(1)
In the considered case, this implies to factorize the joint probability P (u, z1 , z2 , x|r), and to compute the marginal P (ui |r) for all i. It is nowadays well-known that the required probabilities can be efficiently computed by message-passing algorithms like the sum-product algorithm (SPA) on factor graphs [10]. Fig. 3 shows the factor graph of the proposed scheme given in Fig. 1. The two RSCs and the ISI channel are modeled as Markov chains with states t1i , t2i and qi respectively. These chains are separated by the interleavers Π1 and Π2 as in Fig. 1. This factor graph does contain cycles, introduced by the interleavers. As a consequence, the SPA becomes iterative and approximate. Concerning the implementation, the order in which the messages are exchanged in the graph during the iterations can be chosen freely. The most intuitive order is to apply the SPA on each Markov chain separately, one at a time, and to exchange messages between them through the interleavers. As expected, this leads to the considered three SISO modules turbo-receiver (Fig. 2). Note that, for a fixed number of iterations and with some assumptions verified by the proposed scheme [11], the SPA is ensured to converge towards the correct probabilities required in (1) if, in our case, the interleavers lengths tend towards infinity. By the way, the considered turbo-receiver is guaranteed to provide the MAP detected bits for sufficiently long interleavers. IV. B OUNDING THE ERROR - FLOOR As an extension of [1] and [3], we have recently proposed a framework for bounding the error-floor of BICM-based TE schemes in quasi-static fading conditions [6]. This errorfloor is reached above a given SNR-threshold by the iterative
process, and corresponds to a perfect a priori information (PP) situation. The proposed bounds lead to drastic simplifications w.r.t. existing results and often to analytical expressions. The results of [6] which are relevant in the current context (i.e., static frequency-selective channel) will first be summarized here, and then properly extended to the particular problem of interest (turbo-code as error-correcting code). A. The error-floor of TD over memoryless channels Let us first consider the problem of bounding the errorfloor of TD over a memoryless channel. The basic building block of these bounds is the pairwise error probability (PEP) ˆ with between two possible coded binary sequences c and c ˆ when Hamming distance d (i.e., the probability to detect c c was transmitted). This PEP is computed assuming a PP situation (which corresponds to the asymptotic error-floor), and for given constellation χ and labelling µ. It is denoted (pp) ˆ). The associated union bound (UB) on the bit P(d,µ,χ) (c → c error probability Pb , considering a rate-1/n binary code, is: ∞ X (b) (pp) ˆ), Pb ≤ Ad P(d,µ,χ) (c → c (2) d=df
where df is the free Hamming distance of the considered (b) code, and where the coefficients Ad classically denote its bit-spectrum or weighted spectrum (see subsection IV-C). In the sequel, let χib denote the subset of all the symbols x ∈ χ whose label has the value b ∈ {0, 1} in position i ∈ {1, . . . , m}, and let ¯b denote the complement of b. Assuming a memoryless and stationary channel, let us also introduce the metric difference ∆(x, z¯) for given symbols x and z¯ in χ, such that P (x → z¯) = P (∆(x, z¯) < 0). The Laplace transform of the metric difference pdf is denoted Φ∆(x,¯z) (s). Capitalizing on the mathematical framework introduced in [8], the considered PEP can then be expressed as [1], [3]: Z c+j∞ h id ds 1 (pp) ˆ) = P(d,µ,χ) (c → c Ψ(pp) (s) , (3) 2πj c−j∞ s
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Fig. 4. Complementary pairs differing in bit i (i = 1, 2, 3, from left to right), for 8PSK modulation and GM.
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where for a given symbol x ∈ χib , the associated symbol z¯ ∈ χ¯ib has a label only differing in position i w.r.t. x. In other words, the computation of the PEP using (4) implies to consider all the pairs of symbols, whose labels only differ in one bit. These pairs will be referred to as ordered complementary pairs. As an example, they are represented in Fig. 4 in the particular case of 8PSK modulation and Gray mapping. The metric difference ∆(x, z¯) being a function of the intersymbol distance between x and z¯, drastic simplifications are possible. Note first that, thanks to symmetry, is is sufficient to consider unordered (instead of ordered) complementary pairs, which reduces the number of terms to take into account by a factor 2. Note then that, depending on χ and µ, one can identify, in the set of the unordered complementary pairs, a given number τ of subsets containing pairs having the same intersymbol distance. Let us assume in the sequel that the subset denoted t ∈ {1, . . . , τ } contains nt pairs having the same distance dt . The parameters τ , nt and dt (t ∈ {1, . . . , τ }) obviously depend on the constellation χ and the labelling µ. As an example, in the case of 8PSK modulation with Gray mapping (GM), there is τ = 2 subsets; n1 = 8 pairs have distance d1 and n2 = 4 pairs have distance d2 , as shown in Fig. 4. Similarly, there is τ = 3 subsets for the set partitioning (SP) and for the so-called EFD optimal [9] mappings. Taking the above into account, the averaging required in (4) for the computation of the PEP using (3) can thus be performed by considering these τ subsets: Ψ(pp) (s) =
τ X 1 nt Φ∆(dt ) (s), m2m−1 t=1
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where the Laplace transform Φ∆(dt ) (s) is related to a pair of symbols having distance dt (i.e., one of the nt pairs belonging to the subset t). Note that the number of terms required is reduced to τ in (5) instead of m2m in (4), which often enables tractable analytical solutions. TD over an AWGN channel: In the particular case of an AWGN channel, it can be shown that Φ∆(dt ) (s) = exp s(N0 s − 1)Es d2t [6]. Equation (5) writes thus: Ψ(pp) (s) =
τ X 1 nt exp s(N0 s − 1)Es d2t . m−1 m2 t=1
(6)
The low values taken by the parameter τ then often enable a binomial expansion in (3). This leads to simple analytical expressions of the PEP, depending on the parameter τ of the considered constellation χ and labelling µ. As an example, for τ = 2 (e.g., considering 8PSK modulation with GM), the following solution can be put forward: ! d X d d 1 1 (pp) P(d,µ,χ) (c →ˆ c) = 2 m2m−1 l l=0 s (7) 2 l + d2 (d − l)) E (d s 1 2 nl1 n2d−l erfc 4 N0 Similarly, for τ = 3 (e.g., considering 8PSK modulation with SP or EFD mapping), it comes: ! ! d X l d X l d 1 1 (pp) ˆ) = P(d,µ,χ) (c → c 2 m2m−1 j l l=0 j=0 s 2 j + d2 (l − j) + d2 (d − l)) E (d s j l−j d−l 1 2 3 n1 n2 n3 erfc 4 N0
(8)
Note that a more general (i.e., valuable for every τ ) but also more complex (complexity evolving exponentially instead of linearly with d as in (7) and (8)) analytical solution can be obtained if the channel stationarity is not taken into account [6]. This solution enables to point out that in all cases the PEP on the error-floor of TD over an AWGN channel evolves asymptotically as [6]: s h n id 1 2 dd E min s (pp) min (9) ˆ) ≈ P(d,µ,χ) (c → c erfc m2m−1 2 4 N0 where nmin is the number of unordered complementary pairs having the minimum intersymbol distance dmin , for the considered constellation χ and labelling µ. By the way, this enables to emphasize that the asymptotically dominant parameters in the present context are dmin and nmin . Finally, the UB (2) provides an upper-bound on the errorfloor of TD over an AWGN channel. Considering only the single term corresponding to d = df , (2) leads to a lower bound. In the present case, these upper and lower bounds merge for increasing SNRs. The asymptotic behavior of the error-floor can thus be explained using (9).
B. The error-floor of TE over frequency-selective channels As shown intuitively in [5], and proved rigorously in [6], the error-floor of a bit-interleaved TE scheme over a static (i.e., non fading) frequency-selective channel is the same as that of the associated TD scheme over an AWGN channel1 . However, this error-floor is reached at a higher SNR in the case of TE, due to the intersymbol interference introduced by the frequency-selective channel. The same bounds as those provided in subsection IV-A can thus be used in this case. 1 Note that this affirmation does not hold in fading conditions, where a multipath diversity gain is possible. We refer the interested reader to [6].
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Turbo−equalization, iteration 20. Turbo−demodulation,iteration 20. Measured error−floor (PP situation). Analytical upper and lower bounds. Analytical asymptotic approximations.
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Turbo−equalization, iteration 20. Turbo−demodulation, iteration 20 Measured error−floor (PP situation). Analytical upper and lower bounds. Analytical asymptotic approximations.
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Fig. 6. BER measurements and analytical bounds, considering SP mapping.
C. Spectrum of punctured turbo-codes
plotted in dashed in both figures: it is obtained considering perfect a priori information at the equalizer/demapper and letting the turbo-decoder perform 20 iterations. Accordingly to what is mentioned above, it can be seen in Fig. 5 that this error-floor is reached by both the TE and TD schemes. This is possible at the cost of a SNR-gap for the TE scheme, due to the presence of the frequency-selective channel (similarly to what is observed with classical convolutional codes [6]). In Fig. 6, the convergence could only be emphasized for the TD scheme, due to the very low error-floor in this case. The above BER measurements are now compared with the theoretical bounds introduced in the current paper. The upper bound (2) is provided in Fig. 5, based on the expression (7) of the PEP (GM). The same is done in Fig. 6, based on the expression (8) of the PEP (SP). The corresponding lower bounds, obtained considering the single term d = df in (2), are plotted in both cases. In both figures, it can then be seen that the upper bound becomes rapidly and remarkably tight, which enables to accurately predict the error-floor of the considered turbo-receivers. This is especially useful in the current context, in order to avoid heavy BER measurement simulations. Note also that the lower bounds are relatively loose in this case (on the contrary to what is observed with classical convolutional codes [6]) due to the low value observed for the first element (b) Adf of the bit spectrum of a turbo-code. Finally, based on the asymptotic expression of the PEP (9), simple asymptotic approximations of the upper and lower bounds are plotted in dotted in Fig. 5 and 6. These approximations are very useful for a theoretical asymptotic analysis on the one hand, and for a practical optimization of the error-floor on the other hand. In Fig. 7, the results related to TD over an AWGN channel which are provided in Fig. 5 (GM) and Fig. 6 (SP) are combined in a single graph, with the corresponding upper bounds. The first iterations are also added, for the sake of comparison. This enables to emphasize the constellation mapping impact on the performance of the proposed scheme, in analogy to what is observed with conventional TD schemes [2]. Note thus that this conclusion holds with a turbo-code as
The turbo-code bit-spectrum is required so as to compute the union bound provided in (2). In the case of an unpunctured systematic parallel turbo-code, the latter spectrum can be computed following [12], based on the concept of uniform interleaver. An extension to punctured parallel turbo-codes can be found in [13]. We refer the reader to these references, for the sake of conciseness. V. S IMULATION RESULTS In order to validate the proposed bounds, let us consider the following bit-interleaved turbo-coded modulation scheme. The turbo-code consists in the parallel concatenation of two identical RSC codes, having memory 2 (i.e., 4 states) and octal generators (5, 7)8 . Trellis termination is classically performed for the first RSC code. The global coding rate is raised to 1/2 by using a regular puncturing pattern. Both interleavers Π1 (turbo-code) and Π2 (BICM) of the transmission scheme are selected randomly and vary independently from one frame to the other. The length of interleaver Π1 (resp. Π2 ) is fixed to 1998 (resp. 3996). The chosen modulation is 8PSK. Simulation results are provided for two different labellings: GM in Fig. 5 and SP mapping in Fig. 6. In both figures, BER measurements are provided considering transmission over an AWGN channel on the one hand (the proposed turbo-receiver reduces in this case to a TD scheme), and transmission over a frequency-selective channel on the other hand (the proposed turbo-receiver is in this case a TE scheme). In the latter case, the length-10 frequencyselective channel with discrete-time channel impulse response (0.04, −0.05, 0.07, −0.21, −0.5, 0.72, 0.36, 0, 0.21, 0.03, 0.07) is considered. For the sake of tractability, a filter-based lowcomplexity MMSE implementation of the required equalizer is used (see [5] and references therein for further details). The performance of the TD and TE schemes (corresponding to the AWGN and frequency-selective channels, respectively) are given at iteration 20. In addition to these results, the simulated error-floor (i.e. obtained through measurements) is
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VII. C ONCLUSION
TD, constellation mapping impact.
The performance of BICM-based TE schemes having the specific feature of using a turbo-code as error correcting code has been analyzed. To do so, new performance bounds have been developed by extending the bounding techniques recently proposed by the authors in [6]. The bounds are remarkably tight as the simulation results show. To conclude, this paper provides analytical tools for the understanding and optimization of such BICM-based turbo-systems.
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Taking as reference curve the first curve of Fig. 8 — where M is the memory of the RSC encoders —, the correcting performance of the proposed scheme can be improved by increasing the interleaver length N (second curve, interleaver gain in N −1 , N = 19980), by using more powerful RSC codes (third curve, M = 4) and/or by applying another mapping (fourth curve, optimal mapping EFD [9]). However, all these asymptotic improvements come at a price: there is an obvious trade-off between them and, respectively, the decoding delay, the decoding complexity and the turbo-convergence at low SNR (predicted by EXIT charts). At last, the fifth curve shows a BICM scheme whose code is a robust turbo JSC code (see [7]). This emphasizes the wide range of applications of the proposed bounding technique.
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error-correcting code. The same trade-off between asymptotic performance and SNR-threshold is encountered here: a lower error-floor results in a higher SNR-threshold, which offers room for optimization. The proposed bounds are very useful for this purpose. Note finally that the same conclusion holds for the results obtained for the proposed TE scheme over frequency-selective channels, similarly to what is observed in [5] with simple convolutional codes. VI. F UTURE PROSPECTS The most obvious prospect of this work is the joint optimization, at high SNR, of the code and the mapping of the BICM scheme. This can be combined, at low SNR, with convergence analysis tools such as the EXIT chart technique. In this optics, in Fig. 8, a few application examples of the proposed bounds are illustrated, which underline possible optimizations. We here focus on parallel turbo-codes, but note that any code for which a feasible SISO decoder exists can also be used, e.g. a serial turbo-code or a joint source-channel (JSC) turbo-code (as proposed in [7]).
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