Convergence Analysis of Iterative Detectors for Narrow-Band Multiple Access 1
2
Fredrik Br¨annstr¨om1,† , Tor Aulin1 , Lars Rasmussen1,2 and Alex Grant2
Chalmers University of Technology, Department of Computer Engineering, SE-412 96 G¨ oteborg, Sweden Institute for Telecommunications Research, University of South Australia, Mawson Lakes SA 5095, Australia
Abstract– Convergence analysis of iterative detectors for narrow-band multiple access is performed by using extrinsic information transfer charts. The system has no bandwidth expansion, so K users are using the same bandwidth as one single user. The load (the number of bits per channel use) of the system is therefore higher than the load in, for example, conventional CDMA systems. The analysis provide useful guidelines how to combine outer codes with the chosen mapper for different loads of the system. Both memoryless mappers and differential mappers are evaluated. The limitations of the system measured in the maximum number of users and the required SNR for different scenarios are presented.
I. Introduction In [1], [2] constellation-constrained capacity calculations were used to find the maximum theoretically achievable information rate together with the minimum required signal-to-noise ratio (SNR) in a multiple access (MA) system known as trellis code multiple access (TCMA) [1], [3], [4]. These calculations were based on the superimposed constellations of all the users, disregarding the outer code of each user. The calculations could therefore not predict what outer code for the single user (SU) that could be combined with the chosen mapper to achieve reliable communication. Unfortunately, the minimum required SNR in terms of Eb /N0 given from these calculations were several dB from the actual simulated threshold for the chosen combination of code and mapper [1]. Eb is the energy used to transmit one information bit and N0 /2 is the double sided power spectral density of the noise. Extrinsic information transfer (EXIT) charts were introduced in [5], [6] as a tool to design mappers to be used together with demappers and outer codes. Demappers use a priori information on the input bits to the mapper together with the received observable to update the extrinsic information on the same bits. EXIT charts have also been used to design both serially and parallel concatenated codes [7], [8]. Similar methods used for convergence analysis are based on density evolution, [9]. In this paper EXIT charts are applied to narrow-band MA systems, extending the ideas in [5] to include a MA † Corresponding
author, E-mail:
[email protected] This research was conducted while Fredrik Br¨ annstr¨ om visited Dr. Alex Grant at the Institute for Telecommunications Research, University of South Australia.
demapper. In contrast to the constellation-constrained capacity calculations [1], [2], the outer code for the SU can now be included in the convergence analysis. II. System Model The MA system in Figure 1 consists of K users transmitting statistically independent and identically distributed binary data xk [1], [3], [4]. Each user’s data is passed through an outer encoder Ck of rate Rc . The coded bits yk are then fed to a user unique random bit interleaver Πk , permuting blocks of L bits. The interleaved bits zk are mapped with Mk onto an Mk -ary twodimensional symbol sk , referred to as the user symbol. In contrast to [1], [3], [4], Mk can here be a memoryless mapper or a mapper with memory. All users synchronously modulate their user symbol, sk , onto a continuous-time waveform s(t). Note that there is no spreading in this system besides the outer code. Each waveform (or channel use) will therefore carry bk = Rc log2 (Mk ) information bits. These waveforms are superimposed on the AWGN channel, where s is a symbol from the joint constellation of all the user symbols [1], [3], [4]. The dashed box in Figure 1 indicates that the K mappers can be identified as a singe mapper M, mapping the K bitstreams zk in parallel onto a symbol s. A Gaussian noise vector n is then added to s in the vector channel
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model, Figure 1. The variance of each component in this vector is σ 2 which is also the double sided noise power spectral density of the continuous time system, N0 /2. To be able to tell what block of information bits that corresponds to which user, each user must have a unique combination of the outer code Ck , the interleaver Πk and the mapper Mk . They can be combined in different ways as long as the combination is unique among the users [1], [3], [4]. In this narrow-band MA system some of the signal points in the joint constellation can be ambiguous, [1]. If the mappers Mk are memoryless, the MA demapper works in principle as described in [10]. If the individual mappers have memory the MA demapper will work on a trellis, where the number of states and edges grows exponentially with the number of users. That is usually not a problem here, since only a few users are considered. The iterative detector in Figure 1 consists of a MA demapper followed by K SU decodes. Ayk , Eyk , Azk and Ezk denote the a priori and the extrinsic information on yk and zk respectively. The MA demapper M−1 uses the received observable r together with the a priori values Ayk delivered from all SU decoders to calculate the extrinsic information on zk for all users. Ayk¯ in Figure 1 denotes the set of a priori information from all users except user k, Ayk¯ = {Ay1 , . . . , Ayk−1 , Ayk+1 , . . . , AyK }. The same notation is used for the extrinsic information, Ezk¯ . The extrinsic information Ezk is passed through a deinterleaver Π−1 k and becomes the a priori information Ayk fed to the a posteriori probability (APP) calculator of user k, Ck−1 . This module makes decisions on the information bits x ˆk and calculates the extrinsic information of the outer coded bits yk . The APP calculator is an extension of the BCJR algorithm [11] and a detailed description can be found in [12]. After the interleaver, the extrinsic information Eyk becomes the a priori information Azk fed to the MA demapper. The next iteration of the MA demapper can start when all users have delivered their updated Azk . III. Convergence Analysis and Performance The mutual information between the coded bits from the outer code and the log-likelihood ratio of the a priori knowledge IAy = I(yk ; Ayk ) is calculated according to [7]. Mutual information is also used to measure the information content of the extrinsic output from the APP calculators IEy = I(yk ; Eyk ). IEy depends only on the a priori knowledge and can therefore directly be plotted versus IAy in an EXIT chart [5], [6]. Three Rc = 1/2 outer codes are chosen for analysis, a repetition code and two feed forward convolutional codes (CC). CC(2,3) denotes the CC with code polynomials
(2,3) given in octal. CC(2,3) has two states and CC(5,7) has four states. Any half rate code will have IEy = 0.5 when IAy = 0.5 [5]. The extrinsic characteristic of the input bits to the mapper IEz depends on both the a priori knowledge IAz and the SNR. For brevity, the analysis of the inner code is here restricted to QPSK with Gray mapping, where all users have the same phase-offset. The Gray mapping will give a higher initial value of IEz than, for example, QPSK with natural mapping [5]. This scenario, where all users have the same phase-offset, is the worst case scenario in terms of achievable information rate, since some of the signal points in the superimposed constellation can be ambiguous [1]. The results can easily be extended to include higher order constellations Mk > 4, random phase-offset and other mappings than Gray mapping. For a SU with a memoryless QPSK Gray mapper, IEz remains constant when increasing IAz , i.e., the level only depends on the SNR [5]. Therefore, nothing can in this case be gained by iterating between the outer code and the demapper. The EXIT chart for two users with memoryless QPSK is shown in Figure 2a together with the SU case. Thick lines corresponds to the outer code and thin lines to the inner MA demapper. In the two-user case, IEz has a slope greater than zero and something can be gained by iterating between the outer SU code and the inner MA demapper. When there is more than one user in the system, IEz will initially start at a lower level than in the SU case, but it will for all K finally end up at the same level as in the SU case when IAz = 1. For turbo codes, there is a waterfall at a critical SNR, [7]. This effect will only appear if the curve for the inner code combined with the curve for the outer code will open a tunnel at a certain SNR such that the tunnel ends in IEz = IAy = 1. When that happens, the a priori knowledge of the coded bits yk will be perfectly known and the APP calculator of the outer code will have no problem to make reliable decisions on the information bits x ˆk . The performance turns from high error rate to zero error rate within fractions of a dB. In other words, the performance is related to how much a priori knowledge the outer code has on the coded bits [8], i.e., how high IAy = IEz is. Figures 2a–c show that IEz for the multi user case is never higher than IEz for the SU case. This means that the SU performance is the lower limit on the performance of the individual users in the multi-user case. SU performance can therefore only be achieved in the point where IEz in both the SU case and the multi-user case intersect at the same SNR and this will only happen when IAz = IEy = 1. For two users together with CC(5,7) this seems to be around 4–5 dB and that corresponds well
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with the simulation of this scenario, Figure 3a. In Figures 3a–b, ’K=2 #6’ denotes the performance of K = 2 users after 6 iterations. For CC(2,3) SU performance is achieved at 7 dB, Figure 2a and Figure 3a. In the two-user case with CC(5,7), the tunnel in Figure 2a is opened somewhere between 2 and 3 dB, but the curves will not end up in IEz = 1 and therefore no sharp waterfall in the performance will appear as shown in Figure 3a. The minimum required SNR according to the constellation-constrained capacity for this scenario with Rc = 1/2 and memoryless QPSK is close to 2 dB [1]. That is why IEz at 2 dB is around 0.5 when IAz = 0.5 in Figure 2a. Based on the interleaver sizes used in [1], it was not possible for three users with QPSK Gray mappers together with CC(5,7) to achieve reliable communication. Figure 2b shows that this is indeed possible if large bitinterleavers and many iterations are applied. The threshold is observed to be close to 10 dB and many iterations are necessary. At this SNR the tunnel will be opened and IEz ≈ 1, i.e., a waterfall region will appear. If the SNR is increased even more, the curve for the inner code will still not differ from the 10 dB curve, i.e., is not going to open up a tunnel for stronger outer codes. With three users and CC(2,3) SU performance is, as in the two-user case, expected at around 7 dB according to Figure 2b. This is also indicated in the performance shown in Figure 3a. In addition, Figure 3a shows that using the simple repetition code as the outer code does not provide as good performance as the other codes. Figure 2a shows that two users with the repetition code is not expected to have SU performance until the SNR is higher than 10
dB and after many iterations. For three users SU performance is probably never going to occur even for very high SNR as indicated in Figure 2b and Figure 3a. The maximum number of users with QPSK mappers together with any half rate code is four according to the constellation-constrained capacity [1], [2]. Even for high SNR, IEz will not exceed the crucial 0.5 point when IAz = 0.5 as illustrated in Figure 2c. This is necessary for the inner code to be combined with any half rate outer code. According to the constellation-constrained capacity it will eventually just exceed this point when the SNR and the block size go towards infinity [1], [2]. To make the inner code recursive, a differential QPSK mapper [13] is suggested as the mapper for each user. This recursive code is a four state, rate one, code where the transmitted symbol at time index l depends on the symbol at time index l −1. The trellis in the MA demapper for this implementation is a fully connected trellis with 4K states and 16K edges per trellis section, where K is the number of users. The inner code will now be recursive and IEz = 1 when IAz = 1. This means that if a tunnel is opened, the performance will have a very steep slope at this SNR value, at least after many iterations. Figure 2d shows that a tunnel is opened around 1.0 dB in the SU case together with CC(5,7) and just below 1 dB with CC(2,3). That is also confirmed in Figure 3b, where the performance is shown for both cases. The EXIT chart for two users with differential QPSK is shown in Figure 2e. The tunnel is not opened before approximately 10 dB with CC(5,7). When the tunnel is passed, only a few more iterations are required for con-
vergence. If a weaker outer code is used, like the repetition code or CC(2,3), the EXIT chart indicates that the performance is expected to be much better at a lower SNR. Figure 3b shows that two users with repetition codes have SU performance at 4 dB. With CC(2,3) the tunnel is opened at around 3.5 dB, Figure 2e and Figure 3b. These two figures also show that two users with the repetition code have better performance than two users with CC(2,3) for SNR less than 4 dB. The EXIT chart for the three-user case is shown in Figure 2f. The tunnel is only opened if a repetition code is used as the outer code and it will then require an SNR higher than 8 dB, Figure 3b. EXIT charts can also be used to estimate the BER in MA system after an arbitrary number of iterations using methods similar to [7], [8]. It can also predict what the dB loss is compared to the SU case. For example, Figure 2a shows that around six iterations is required at 2 dB in the two-user case with CC(5,7) to get up to the SU 0 dB level. This means that after six iterations in the two-user case at 2 dB, the dB loss compared to the SU case is 2 dB. That can be confirmed by inspection of Figure 3a.
References [1]
[2]
[3] [4]
[5] [6] [7]
[8]
[9]
IV. Conclusions The narrow-band system described here has no bandwidth expansion compared to a SU with no spreading. The number of users can therefore not be as high as in MA schemes with bandwidth expansion. This fact is shown here using convergence analysis of the MA channel. EXIT charts are used as a tool to choose the SU codes to be combined with the chosen mappers. The EXIT charts for the MA system can predict when and also at what SNR SU performance can be accomplished. For example, three users with a two-state CC and memoryless QPSK mappers can have SU performance at 7 dB. This is also confirmed by simulations. Instead of using memoryless mappers, differential mappers can be used, making the inner code recursive. The EXIT charts show that the performance in this case is only limited by the outer code and perfect knowledge of the coded bits is achievable. Unfortunately, three users is the maximum number of users in the worst case scenario described here. Convergence is then only accomplished if the users have a repetition code as the outer code together with high SNR. The EXIT chart predicts how many users that can have reliable communication for a given scenario. It can also predict if a threshold is expected in the performance and in that case at what SNR. These fundamental system limits provide useful guidelines when designing MA systems with high spectral efficiency.
[10]
[11]
[12]
[13]
F. Br¨ annstr¨ om, Trellis Code Multiple Access (TCMA) – Detectors and Capacity Considerations. Lic. Eng. thesis, Chalmers Univ. of Techn., G¨ oteborg, Sweden, Nov. 2000. Available: http://www.ce.chalmers.se/staff/fredrikb. F. N. Br¨ annstr¨ om, T. M. Aulin, and L. K. Rasmussen, “Constellation-constrained capacity for trellis code multiple access systems,” in Proc. IEEE Global Commun. Conf. (GLOBECOM’2001), San Antonio, TX, Nov. 2001, vol. 2, pp. 791–795. F. N. Br¨ annstr¨ om, T. M. Aulin, and L. K. Rasmussen, “Iterative detectors for trellis code multiple access,” to appear in IEEE Trans. Commun., 2002. F. N. Br¨ annstr¨ om, T. M. Aulin, and L. K. Rasmussen, “Iterative multi-user detection of trellis code multiple access using a posteriori probabilities,” in Proc. IEEE Int. Conf. Commun. (ICC’2001), Helsinki, Finland, June 2001, vol. 1, pp. 11–15. S. ten Brink, “Convergence of iterative decoding,” IEE Electron. Lett., vol. 35, pp. 806–808, May 1999. S. ten Brink, “Iterative decoding for multicode CDMA,” in Proc. IEEE Vehicular Technol. Conf. (VTC’99), May 1999, pp. 1876–1880. S. ten Brink, “Design of serially concatenated codes based on iterative decoding convergence,” in Proc. Int. Symp. on Turbo Codes and Rel. Topics, Brest, France, Sept. 2000, pp. 319–322. S. ten Brink, “Iterative decoding trajectories of parallel concatenated codes,” in Proc. 3rd IEEE/ITG Conference on Source and Channel Coding, Munich, Germany, Jan. 2000, pp. 75–80. D. Divsalar, S. Dolinar, and F.Pollara, “Iterative turbo decoder analysis based on density evolution,” IEEE J. Selected Areas Commun., vol. 19, pp. 891–907, May 2001. S. ten Brink, J. Speidel, and R.-H. Yan, “Iterative demapping and decoding for multilevel modulation,” in Proc. IEEE Global Commun. Conf. (GLOBECOM’98), Sydney, Australia, Nov. 1998, pp. 579–584. L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding,” IEEE Trans. Inform. Theory, vol. IT-44, pp. 909–926, May 1998. P. Hoeher and J. Lodge, “”turbo DPSK”: iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. COM-47, pp. 837–843, June 1999.
