Multi-dimensional Mapping for Bit-Interleaved Coded ... - CiteSeerX

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Multi-dimensional Mapping for Bit-Interleaved Coded Modulation with BPSK/QPSK Signaling Frederik Simoens, Student Member, IEEE, Henk Wymeersch, Student Member, IEEE, Herwig Bruneel, and Marc Moeneclaey, Fellow, IEEE

Abstract— In recent years, it has been recognized that bitinterleaved coded modulation with iterative decoding (BICMID) achieves excellent performance on virtually any channel, provided the signal mapping is carefully designed. In this paper, we introduce multi-dimensional mapping for BICM-ID, where a group of bits is mapped to a vector of symbols, rather than a single symbol. This allows for more flexibility and potential performance improvements. Our analysis shows that multidimensional mapping leads to an increase in Euclidean distance, thus boosting the performance compared to conventional mapping schemes. We derive a design criterion for optimal mappings, and we provide such optimal mappings for BPSK and QPSK constellations.

I. I NTRODUCTION In Ungerboeck’s trellis-coded modulation (TCM) [1], convolutional codes are designed to match the specific signal constellation. On some channels, however, combining modulation and encoding does not yield the best performance. BitInterleaved Coded Modulation (BICM), suggested by Zehavi [2] and profoundly analyzed by Caire et al. [3], increases the time diversity of coded modulation compared with TCM, thus improving performance over fast-fading channels. This idea was later extended to BICM with iterative decoding (BICM-ID), whereby the decoder and the demapper exchange information in order to improve the error performance [4]. It was recognized that the choice of mapping (i.e., how a group of bits is mapped to a complex constellation symbol) is a crucial design parameter for BICM-ID [5]–[7]. Multi-dimensional mapping schemes, with a number of coded bits being mapped to a vector of symbols (instead of a single symbol), were proposed in [8] for TCM. Flexibility and higher code rates were the main motivations for their development. In this contribution we extend the idea of multidimensional mapping to a BICM-ID context. The concept of multi-dimensional mapping can also be exploited in spacetime BICM configurations [9]. Simulation results verify that the proposed scheme has a significant performance gain compared to conventional one-dimensional mappings. Manuscript received September 27, 2004. The associate editor coordinating the review of this letter and approving it for publication was Dr. Cha’o-Ming Chang. This work has been supported by the Interuniversity Attraction Poles Program - Belgian Science Policy. The authors are with the Department of Telecommunications and Information Processing, Ghent University, Gent B-9000, Belgium (e-mail: fsimoens,hwymeers,hb,mm @telin.ugent.be).




II. S YSTEM

MODEL

A. Transmitter We consider the following BICM-ID system. A sequence of information bits is convolutionally encoded and bit-interleaved by a random interleaver. The resulting coded bits are grouped in blocks of bits; the -th block is denoted . Then, is mapped to a vector having elements in the -ary signal set , using a bijective (multi-dimensional) time-invariant mapping function , with , and denoting the -ary vector signal set. We write

 

        "!  # %$& '( ) * + ,.- #/01325476 8* 2 )9 5: 1 6 ) 2 (1) ';& % ?@< 23ACB > /EDF + ?G!( In our multi-dimensional model, each symbol < H> JIK* (L3

 MNO ) is a function of all  coded bits within ? . This is the main difference with a conventional mapping scheme, which assumes that each symbol < H>  depends on  coded bits only. When P Q , the multi-dimensional mapping reduces to the conventional mapping. The symbols contained in the R$S symbol vector 'T are

transmitted over an AWGN channel in consecutive symbol intervals. The channel model has the form

U

U & ';WVXC  ?Y ZV[\!]^_ 

(2)

where corresponds to the vector of signals received at time , and is a a vector of instants iid zero-mean complex-valued AWGN samples, with variance for both the real and imaginary part.

X.



Z=/`a:

B. Receiver The iterative receiver operates according to the turboprinciple, by exchanging so-called extrinsic probabilities between the demapper and decoder [4]. The extrinsic probabilities computed in the demapper (and to be forwarded to the detector) are given by [4]

bdcfe&g    hij lk!m onp U 3q  hi" rks! (3) 

d b ‚ g & e c U

onut p 3q '(|!|~} € ~  hiƒ„j! (4)  v/w/xy?z{ ~… where n is a normalizing constant and 6 denotes the subset of all symbol '† for which the - h -th bit of the inverse + ACvectors kOI #/0 ). The extrinsic equals mapping b‡B cfe&g   k hij! (with probabilities are deinterleaved and applied to the decoder. The decoder, operating according to the BCJR

2

algorithm [10], calculates extrinsic probabilities on the coded bits. These are interleaved and fed back to the demapper etc. III. PAIRWISE

ERROR PROBABILITY ANALYSIS

In this section, we derive a criterion for mapping optimization, based on the pairwise error probability (PEP). It is obvious that the overall error performance is dominated by the PEPs of the symbolvectors with a small Euclidean distance. Hence the PEP corresponding to codewords which differ only in a small number of bit positions (and therefore only differing in a small number of symbol vectors) will dominate the total error rate. Assume that two codewords and differ in bits ( is . Thanks to the presence the Hamming distance), with of the interleaver, these bits that are different among the two codewords will most likely end up in distinct transmit . In other words, the vectors with indices transmitted signals corresponding to and differ in distinct vectors; we denote these vectors corresponding to and by and respectively. This is referred to as perfect interleaving. According to this principle, the PEP, conditioned on the perfect interleaver, can be upper bounded by

Š

ˆ ŒŠ ‹

ŽI -  B   aG0 ˆ

ˆ

‰ˆ

 '( @'( ‘ 

‰ˆ

Š

Š

 ';‰  '(‰ ‘

‰ˆ

Š

 1 EH Ä Ã 'T à 1£ EH Ä ! 1£ H„Ä à £ H„Ä 1 à £ H„Ä 1

'

where denotes the number of neighbors of at distance with an inverse mapping differing in only one bit-position. Maximizing the minimal squared Euclidean distance and minimizing the number of terms in (6) (or, equivalently, ) yields the best corresponding to asymptotic performance. IV. BPSK/QPSK

 1 „H Ä

MAPPING SCHEME

In this section, we present a mapping scheme for a BPSK and QPSK signaling constellation, optimized for the multidimensional set-up outlined above. Remember that an optimal mapping, assuming perfect interleaving and ML decoding, should maximize and minimize the corresponding . : Let us define the bijective mapping-functions and : as follows:

à £ H„Ä 1  1 H„Ä -  0\2 4 ° ÏÑÐ  - la/\02 -  ÉË0 £ Ê|2 ÌCÍ?Î 4 r É C Ò C Ì ? Í Î ° : ÏÓÐ  - laa/ÔLM@LÕ0 2 Š (9) ÉlÊCÌCͨΠ#!Õ ŒÖ ž@Ù¨ÏÑÚÕÛÐ  :a&×   ÉlÒCÌCͨΠ#!Õ rØ Ö ÏÑÐ  :a?ܼ×\!mVSÝ :a?ÞF\!ß (10) -  02 and ?Þ5I - #/0\2 denote the bits of where #ÜOI I #/0 £ 2 at even position and odd positions respectively and ÏÑÐ denotes the energy per coded bit. This É -mapping

 corresponds to a conventional Gray-mapping, i.e. each symbol bQ’ ˆQ4 “‰ˆ q ”•– —™› ˜ œš  t ŸŸ '(¡ ¢ '(‰ ¡  ŸŸ£ ¥¤ is obtained by Gray-mapping only one bit (BPSK) or two bits œ :Z= ž  (QPSK). For any two binary  and  £ , with ' B B B RÉ  B ! , «  , we obtain ' [

É  ! £  ŸŸ ';    '(   ŸŸ £­ (5) £ ¦ }  ¬ ‰ ' B F' £ ¾ £ ¬ ÏÑÐÕà3á  B F £ ! (11) Z= ž ¾ ¨ B  § ? © ª "® !¯ \`¨° :±Z²dµ ³|´ à3á ! denotes the Hamming weight and  B K £ ! is with — W· £ `a:a¸|ŠG· . The average where + b the modulo-2 difference between  and  £ . This means that Š@6¹ ! for codewords§©?ªfwith PEP ¶ Hamming distance Š the Hamming distance between twoB binary vectors determines is obtained by averaging (5) over all possible weight Š errorthe Euclidean distance of their É -mapped symbols. patterns (i.e., all ˆ and ‰ˆ that differ in Š bits), and all possible We point out à that the conventional Gray-mapping or É perfect interleavers. Following a line of reasoning similar to ¬ ÏÑÐ for both BPSK and QPSK. As mapping yields £ H„Ä

[3], [6], we obtain: - Ä 1 each point  in the space  /\0 has exactly â neighbors b Š@6¹ + ! at Hamming distance â“ã and exactly one neighbor à £ HEÄ forat  Hamming distance , the minimal squared distance â «  1 ¦›º»  13t 2 t… B t ­¢ÀÁ (6) any bijective BPSK or QPSK mapping, will never exceed £  ¬ ¿ '  ' ‰ ¼: 132 H   = vaxy?½{ Z=Z¾ ¾ à £äßåæ 5ç ¬ ÏÑÐ ¬ â×\!èâFé[ §©?ª B (12) âê Ž Ó Ï Ð where ' ‰ is uniquely defined as the symbol vector correspond… H ing to the same bit-vector as 'IÂ6 except for the L -th bit where âê  for BPSK and âê “: for QPSK. This means which is inverted. Note that for ^ Ž we obtain the criterion that all BPSK or QPSK (multi-dimensional) mappings will found in [3], [6] for one-dimensional mappings. For all  , have a squared Euclidean distance à £ H„Ä such that ¬ ÏÑÐ ¦ 1 we observe that the PEP is dominated by the terms with the à £ H„Ä ¦ à £äåæ . 1 We will now show that optimized mappings with à £ H„Ä

smallest squared Euclidean distance, which we will denote as à £äåæ and minimal  HEÄ can be constructed. Define 1  ƒ

à £ H„Ä Å–Æ„Ç … '¿ ' £  (7)  = Ä ! so that1 Q  ƒ  Ä ! . Let us precode the ‰ vaxy?½{ > È/H> ¾ ¾ 1 A£ A|B à The number of terms in (6) corresponding to £ HEÄ can bit-sequence  in the following way: equivalently be expressed, up to a scaling factor, as the1 average  ƒ ;?! Ä _ and àÓá  ƒ !¹ even number of nearest neighbors: ð  ƒ | ¸ Ä A|B _ and à3á  ƒ !‡ odd ë # &ì #! îíï  ƒ |/¸ Ä ACB Ž and àÓá  ƒ !¹ even (13)  1 H„Äf :  vat xy  1 HEÄ 'm à 1£ H„Ä ! (8) ¶  ƒ ;?! Ä A|B Ž and à3á  ƒ !‡ odd 132 ïñ ¶ ACB

3

Zò ŽWF . The final mapping is given by + #!‡ óÉ #ë !¹ [É ì #!M! (14) where É ! was defined in (9) and (10) for BPSK and QPSK, respectively. I, this mapping scheme à As shown à £äßåæ inandappendix yields £ HEÄ

a minimal number of nearest an asymptotic neighbors1  1 H„Ä . Taking (12) into account, performance gain of at least dô„õGö âó÷/! dB w.r.t. the conventional É -mapping can be expected (12).

0

where

10

Gray BPSK/QPSK Set−Partition QPSK Opt. Multi−Dim. 3*BPSK Opt. Multi−Dim. 2*QPSK/4*BPSK

−1

10

−2

first iteration

BER

10

−3

10

genie bounds −4

10

V. S IMULATION

10 iterations

RESULTS −5

10

In this section, we will illustrate the performance of the proposed multi-dimensional mapping. We will make comparisons with conventional one-dimensional mapping strategies, and with the performance of a so-called genie receiver [11], where the demapper acquires the correct extrinsic information from the decoder. Or equivalently, in the computation of the extrinsic probability of a bit , the demapper knows all the other relevant bits. It is clear that the genie performance corresponds to a lower bound on the actual error rate. Furthermore, we have verified that (6), which was obtained under the assumption of perfect interleaving, is also an upper bound for the error-rate of a genie receiver. We consider a 16-state rate recursive convolutional encoder with a random interleaver of 1200 bits. Fig. 1 shows the BER-performance of the new multi-dimensional mapping scheme (13) compared to conventional (single-dimensional) Gray- and Set Partitioning schemes [1]. Note that conventional Gray mapped BPSK or QPSK schemes do not benefit from iterating between demapper and decoder. Note also that the Set-Partitioning mapping corresponds to the optimal mapping (13-14) for . In the first iteration, the multi-dimensional schemes perform a lot worse then Gray and Set Partitioning. However, after 10 iterations and above a certain SNR threshold, the new schemes outperform Gray mapping (for dB) and Set-Partitioning (for dB). The genie bounds illustrate the potential gain of the new schemes at high SNR (compared to Gray mapping: 3.5dB gain for BPSK and 4.75dB for QPSK or BPSK ).

bdg‚e&c  s  hi"!

 hE

b¹c¿eøg    hij!

\`a:

P Q

… Ï ¡` Z=Kéù:

P ¬

… Ï ¡` Z=Kéù:#ûú

u Pü

ý P:

VI. C ONCLUSIONS A multi-dimensional mapping scheme for bit-interleaved coded modulation with iterative decoding over an AWGN channel was presented. Analysis based on the pairwise error probability shows that multi-dimensional mapping is able to achieve a significant performance gain. Based on a simple design criterion, optimal BPSK and QPSK mapping strategies are derived. Simulation results verify the improved performance compared to conventional mapping strategies. Optimal mapping schemes for higher order constellations can be found through heuristics (e.g. binary switching algorithm [6]). This remains a topic for future research.

−6

10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

E /N (dB) b

0

Fig. 1. BER-performance for different mappings AWGN channel. (   over denotes the energy per information bit,
)

þ.ÿ

þ

þdÿ

VII. A PPENDIX I

à £ „H Ä Ã ä£ åæ  1 1 HEÄ Ã £ HEÄ ¬ ÏÑÐ è à P£äßå æ

 éQ › à £ H„Ä ¬ ÏÑÐ 1â×\! ê â 

 âê “:\ 1 ì  £ !M! ×â¿  B 3à á ì -  B !.K Ä  £ I  0  B ®  £  B  ƒ  ® !  £

 ƒ ò ! àÓá ì  B !d ì  £ !M!3

â ¬ ÏlÐ â B   B £  ƒB  ® !  £  ƒ£  ® ! à3á  ƒB !‡ à3á  ƒ£ !  à3á ì  B !. ì  £ !Õ!Œ âFó ¬ B £ ÏÑÐ â\! ÃB +  B ! + â Ž  £! H„Ä 1 ì  C ! K  ì  M ! ¿ ! 

â   àÓá £ B + ƒ #! àÑá ì  B !. ì  £ !M!J Râ à H„Ä â× ¬ 1 ÏlÐ â ' W' à HEÄ 1

In this appendix we prove that the proposed mapping scheme (13)-(14) satisfies with a minimal number of nearest neighbors . Euclidean distance: For , it is readily seen that (13)-(14) yields . For , we prove , with for BPSK and that for QPSK. Taking into account (11), it suffices to prove that  for any two bit-vectors and that differ only in one bit position. There are two possible scenarios:  and differ only in the last bit: and . From (13), we see that . The squared Euclidean distance between the corresponding symbol vectors equals .  and differ in one bit, but not in the last bit: and . Noticing that  , it is readily seen from (13) that . The squared Euclidean distance between and equals . Number of nearest neighbors: From the previous paragraph, we see that for each binary vector , has exactly neighbors at distance (i.e., for ) and one neighbor at which larger distance (i.e., for which ). A different mapping yielding a minimum Euclidean distance with less than nearest neighbors should have at least two neighbors at distance . However, since each vector has only one such neighbor (i.e., vector ), the mapping (13)-(14) yields the smallest possible number of neighbors at distance . ACKNOWLEDGMENT The first author gratefully acknowledges the support from the Fund for Scientific Research in Flanders (FWO).

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R EFERENCES [1] G. Ungerboeck, “Channel Coding with Multilevel/Phase Signals,” IEEE Trans. Inform. Theory, vol. 1, no. 28, pp. 55–67, Jan. 1982. [2] E. Zehavi, “8-PSK Trellis Codes for a Rayleigh Channel,” IEEE Trans. Comm., vol. 40, no. 5, pp. 873–884, May 1992. [3] G. Caire, G. Taricco and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 44, no. 3, pp. 927–946, May 1998. [4] X. Li and J.A. Ritcey, “Trellis-coded modulation with bit interleaving and iterative decoding,” IEEE Select. Areas Comm., vol. 17, no. 4, Apr. 1999. [5] S.Y. Le Goff, “Signal Constellations for Bit-Interleaved Coded Modulation,” IEEE Trans. Inform. Theory, vol. 49, no. 1, pp. 307–313, Jan. 2003. [6] F. Schreckenbach, N. G¨ortz, J. Hagenauer and G. Bauch, “Optimization of Symbol Mappings for Bit-Interleaved Coded Modulation with Iterative Decoding,” IEEE Comm. Letters, vol. 7, no. 12, pp. 593–595, Dec. 2003. [7] S. ten Brink, J. Speidel and R.-H. Yan, “Iterative demapping for QPSK modulation,” IEE Electron. Letters, vol. 34, no. 15, pp. 1459–1460, July 1998. [8] S.S. Pietrobon, R.H. Deng, A. Lafanechere, G. Ungerboeck and D.J. Costello, “Trellis-Coded Multidimensional Phase Modulation,” IEEE Trans. Inform. Theory, vol. 36, no. 1, pp. 63–89, Jan. 1990. [9] F. Simoens, H. Wymeersch and M. Moeneclaey, “Spatial Mapping for MIMO systems,” in IEEE Information Theory Workshop proceedings, San Antonio, Oct. 2004. [10] L.R. Bahl, J. Cocke, F. Jelinek and J. Raviv, “Optimal decoding of linear codes for minimising symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. pp. 284–287, Mar. 1974. [11] J. Boutros, N. Gresset and L. Brunel, “Turbo Coding and Decoding for Multiple Antenna Channels,” in Int. Symp. on Turbo Codes and Related Topics, Brest, Sept. 2003.