Signalling constellations for power-eff icient bit4 nterleaved coded modulation schemes

S.Y. Le Goff

Abstract: Bit-interleaved coded modulation (BICM) is a bandwidth-efficient coding technique consisting of serial concatenation of binary errckorrecting coding, bit-by-bit interleaving, and high-order modulation. The author addresses the problem of finding the signal sets that are most suitable for designing power-efficient BICM schemes over an additive white Gaussian noise channel. To this end, the expression of the BICM capacity limit is exploited and evaluated for several 8- and 16-ary constellations. It is also shown that the bit error probability curves of the modnbation schemes without coding can be used to detemiine the most attractive constellations. Finally, the bit error rate performance of some BICM schemes made up of turbo codes and various signal sets is investigated to illustrate the theoretical results.

1 Introduction

Bit-interleaved coded modulation (BICM) is a handwidth- efficient coding technique based on serial concatenation of binary error-correcting coding, bit-by-bit interleaving, and high-order Mary modulation [I]. Despite its simplicity. BlCM has proved to be a very power-efficient approach provided that state-of-the-art codes, such as turbo codes [ 2 4 ] or low-density parity-check (LDPC) codes [7, 81, are employed.

It is possible to design BICM schemes by employing any two-dimensional signal constellation. However, it has been shown that the choice of the signal set may have a strong influence on the error performance of the system. In Fact, some recent studies carried out on BlCM systems combin- ing turbo codes and 16-ary modulations have indicated that the use of a rectangular 16-QAM signal set leads, in general, to optimal BER performance at the turbo decoder output, on an additive white Gaussian noise (AWGN) channel [9]. This surprising result was obtained by using a specific approach that relies on some properties of the 16-QAM constellation, and thus cannot easily be generalised to other types of BICMs.

In this paper, the goal is to investigate the influence of the signal set on BICM performance by following a more general and theoretical approach, valid for any code and any type of two-dimensional constellation. An expression for the BICM capacity is exploited so as to determine some hasic rules for designing BICMs. The capacity limit of BICM is evaluated for various 8- and 16-ary constellations, and, among these signal sets, those being potentially the most suitable ones for the dcsign of power-efficient BlCM schemes are determined. The idea of evaluating capacity limits to find the best signal sets comes from the fact that power-efficient BlCMs usually employ state-of-the-art codes displaying near-capacity performance. As a result,

ID IEE. 2003 IEE Proceeding> online no. ?CO30230 doi: IO. 1049/ip-cam:20030230 Paper h t received lOlh April ZW2 and in revised form 7th Januilly 2003 The author is with Etisalai College of Engineefing. Emirales Telccommunica- tionr Corporation. PO Box 980, Sharfah, United Arab Emirates

IEE Pmc. C o m m n V d 150, N o 3. Julzp 2003

the actual error performance of these BICMs can he predicted by evaluating their capacity limits. It is shown how the error probability curves of the modulation schemes without coding can be exploited to find the most attractive constellations. The main advantage of this alternative method over the previous one based on capacity limit computations is that it does not require the assumption that codes displaying near-capacity performance are used. In other words, it allows for determination of the best signal sets in practical cases where the code cannot perform close to the capacity due to design constraints such as coding/ decoding complexity, transmission delay etc. To illustrate the theoretical study, the BER performance of some BICM systems using turbo codes and various signal sets i s also investigated by computer simulations.

2 constellations

In this Section, the capacity limit of BlCM is determined for different 8- and 16-ary signal sets_ when the com- munication channel is an AWGN channel. To this end, some results given by Caire rf U!. [ I ] are exploited; they derived a general expression for the BICM capacity valid for a large number of channels and N-dimensional signal sets.

2. I System model

Capacity of BlCM using various 8- and W a r y

The BICM model is represented by the block diagram of Fig. 1. The BICM transmitter is made up of a binary encoder (ENC) followed by a bit-by-bit interleaver E and a modulator modelled by a signal set S composed of M = 2”’ complex signals. The interleaved sequence of coded bits is broken into subsequences of m hits each that are mapped onto signals in S according to Gray labelling. If this type of labelling is impossible due to the structure of the constellation, then a quasiGray mapping i s used which is as close to Gray mapping as possible. At time k, the output of the modulator is a complex signal .q&S, labelled by a set {qI, q L ,_.., q,,,} of m bits. The BICM receiver acquires, at time k, a complex signal yk. We consider here an AWGN channel characterised by a transition probability density

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info

OEC LLR computation

Block dikgrmn ofgeneric BlCM scheme Fig. 1

function pbrl.qJ given bq

where dy,, designates the Euclidean distance between the complex signals xk and yk, and 2 is the variance of complex zero-mean Gaussian noise.

From signal yk , the logarithm of likelihood ratio (LLR) A&;) associated with each bit Q;, is{ I, _.., m}, is computed and used, after de-interleaving, as a soft decision by the binary decoder (DEC). Over an AWGN channel, the LLRs A ( c , ~ ) are obtained using the relation, for iE{ l , ..., in}:

IES,,“

where K is a constant, and S,, designates the subset of all the signals s of S whose labels have the value j E { O , I} in position i. Finally. the de-interleaver f I , associated with the interleaver T[ used at the transmitter side, suppresses correlation between samples A(cJ, and thus ensures an efficient decoding.

2.2 BlCM channel capacity Let c = {cl, czI.. ., c,,,}e{O, I)’” be a set of m bits at the modulator input, and y the corresponding received signal (we drop the time index k to simplify the notations). In [I J an expression for the BlCM capacity limit C was derived, valid under the constraint of uniform input distribution, which is as follows:

where ECJ. denotes expectation with respect to c and y, and &, designates the subset of all the signals stS whose labels have the value c , E { O , I} in position i. Capacity is here expressed in information hits per channel use (bit/channel use), where each channel use corresponds to the transmis- sion of a complex signal xsS. Assuming an AWGN channel and performing some minor modifications, we finally obtain

where dv.,v is the Euclidean distance between signals y and s. Equation (4) can provide some important indications on how to define suitable signal sets for the design of powerful BICM schemes. In fact, to maximise the capacity C for a

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given signal-to-noise ratio (SNR), it appears that the term

should be minimised, while the term

has to he maximised. Minimising A consists in ensuring that, when a complex

signal y is received, the 2”’ Euclidean distances between y and the signals s t $ other than the transmitted one, are maximal. In other words; the signal set should he chosen so that the signal points are as ‘spread as possible. This is actually the characteristic of a constellation having, ideally, the largest minimum Euclidean distance hetween signal points and the smallest average number of nearest neighbour signal points. In practice, it means that any signal set displaying symbol error rate (SER) performance .sufficiently close’ to the optimum over the entire SNR range could be suitable [1&13].

At the same time, it is necessary to maximise the term B. Each time a hit c; is transmitted in position i in the label of the signals of S, a signal x E S;.,, is actually emitted. When receiving the corresponding signal y, it is possible to maximise B by making sure that the 2‘”’-’) Euclidean distances between y and signals s E S i , , are minimal. Practically, this can he achieved, without contradicting the previous condition, leading to a minimal value of A, by using a mapping technique that globally keeps all signals s of a same subset Si.<, as close to each other as possible. The most efficient technique for doing so is the well known Gray labelling. There are many constellations of practical interest for which the complex signals cannot he labeled using Gray mapping. For these signal sets, the most appropriate technique is then to employ a quasiGray mapping.

At this point, it is convenient to introduce the parameter N,,i,, which is here defined as the average number of hits that diffcr between two closest signal points in the constellation. It is computed as follows:

where N., is the number of nearest neighhour signal points of siLTal s, and d&,s’) is the Hamming distance between labels of signals s and s’. S,, denotes the subset of all signals that are nearest neighbours o f s (IS,vl = IVJ. We recall that s and s’ are nearest neighhours if ds.,, =do, where do designates the minimum Euclidean distance between signal points. The value of N,,i, depends both on the mapping and the shape of the constellation, and is always greater than or equal to I . The case N,,,, = I can be achieved by using Gray mapping to label signal points. Therefore, for a given signal set, the value of Nmi, indicates how close the mapping is to the Gray mapping.

To summarise this discussion, one can state that determining the most attractive constellations for the design of power-efficient BICMs basically consists of searching for the modulation schemes satisfying the two following requirements:

Conditiufi I : the SER performance of the modulation scheme should be as close to the optimum as possible over the entire SNR range.

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Coridition 2: The parameter N,,i, of this modulation scheme should be minimum.

These two conditions provide basic design rules for obtaining powerful BICM systems. It is iriteresting to compute the BICM capacity for various 8- and 16-ary signal sets using (4) so as to illustrate these design rules. Note that nUmenca1 integration via the Monte Carlo method was used to compute the expectation in (4).

2.3 Example 1: 8-aty signal constellations The X-ary Constellations considered here are 8-PSK, optimum, (1. 7), and cross [1&-13]. They are depicted in Fig. 2. The BICM capacity for other signal sets, such as triangle, rectangular, and (4, 4) constellations was also evaluated. However, to be able to display results clearly using a reasonable amount of space, it was decided to consider, in this paper, only constellations offering the best performance. Additional results are available in another contribution [14]. Note that the optimum constellation was given this nime since it performs optimally at high SNR over an AWGN channel [ I I]. The 'best' mappings found (i.e. as close to Gray mapping as possible) as well as the resulting values of N,,,, are also indicated for each constellation in Fig. 2.

01 0 ,.' #m

a b

c d

Fig. 2 U 8-PSK. h",,, = I h OptinwnL N,,,$,,= 1.36 (: ( I , 71, IV,~,~,, = I .27

Siruciure ofthe 8-nry signal seis consi&rcd in this puper

d Cross. N""" zz I .22

Fig. 3 shows BICM capacity Cagainst SNR for the four 8-ary constellations over an AWGN channel. The SNR is defined as the ratio &/No, where 4 is the average energy per signal and No is the one-sided power spectral density of white Gaussian noise. I t is seen that, at low-to-medium SNRs, the highest capacity is achieved by the 8-PSK signal set. As the SNR is increased, constellation 8-PSK is progressively outperformed by the three other signal sets. In the medium-to-high SNR region, the best performance is obtained with the cross constellation. The capacity curves of the cross and 8-PSK signal sets have a crossover point at

IEE Pmr: Gx,s,iri.. Yo/. /XI, A?>. 3. Jum 2W3

&/No: 5.3 dB. Finally, at higher SNRs corresponding to capacities greater than 2.5 bit/channel use, it is observed that constellation ( I , 7) tums out to be the most attractive signal set. Froin an error performance point of view, BICMs using 8-ary modulations are, however, not of interest when the desired capacity is close to 3 bit/channel use because, in such a case, a much better alternative would consist of employing 16-ary modulation [ I , 151.

2.8

2.6

2.4 0

3

0 - 2 2.2 c P 6

0 8

0

2.0 = a

1.8

1.6

1.4 3 4 5 6 7 8 9 1 0

SNR. dB

Fig. 3 seis ouer the A WCN channel, j h SNR rariyinqjrom 3 Io IOdB

BICM cupacily upinst SNR j b r jour dfj'erenr 8-UrY Sign<//

All these results indicate that NnIi, constitutes the key parameter when optimal performance at low-to-medium SNRs is targeted. But, as the SNR is increased, SER performance becomes progressively a more and more crucial parameter. Eventually. at very high SNRs (for capacities close to 3bit/channel use), BICM capacity is essentially determined by the SER performance of the modulation scheme. This explains the results obtained here. In the low-to-medium SNR range, constellation 8-PSK is the most attractive signal set due to its unit N,,,.. At medium SNRs, the highest capacity is obtained with the cross signal set which actually achieves an excellent trade-off 'optimal SER performance-minimum Nmi,,'; indeed it has a relatively small N,j, (: 1.22) and displays SER perfor- mance that is vety C l O S to the optimum as will be illustrated in Section 3. On the other hand, constellations ( I , 7) and optimum present SER performance advantages over the cross signal set that are not sufficient, in this particular SNR region, to compensate for their larger Nmi, (% 1.27 and 1.36, respectively). For higher SNRs, where N,,,,, is no longer a critical parameter, the ( I , 7) signal set outperforms the cross constellation owing to its better SER performance. If a capacity almost equal to 3 bit/channel use was the objective. the best choice would be to use the optimum signal set.

2.4 Example 2: Wary signal constellations Consider the following 16-ary constellations depicted in Fig. 4 rectangular 16-QAM, optimum, (1, 5, IO), and (5, I I ) . These signal sets were selected from the existing literature because they have been found to be the four most

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power-efficient constellations over an AWGN channel [ I & 131. I t will be seen later that no other 16-ary signal sets need actually to be considered in this work. For constellations ( I , 5_ IO) and ( 5 , I I), the values of the ring ratio /l (defined as the ratio between the radii of the outer and inner circles) leading to optimal performance at high SNRs are approximately 1.902 and 2.175, respectively [13]. The optimum constellation was given this name since it performs optimally at high SNR over an AWGN channel [ I I]. As for 8-ary signal sets, Fig. 4 indicates the 'best' mappings found as well as the resulting values of N,+ Fig. 5 shows BICM capacity C against SNR E,INo for the four 16-ary constellations over an AWGN channel. It is seen that the best performance is clearly achieved by the rectangular 16-QAM signal set over the entire SNR range considered here.

a b

0 10 1 11 I 10 1 10 1111 og,'.~ ..... ~+ 1 0 & I & a

c d

Fig. 4 a Optimum. 1.35 b (I. 5. IO). "!J,;,,= 1.46 for /j= 1.902 c Regular 16-QAM. h',,,= I d ( 5 . l l ) , i V , i n ~ l . 4 1 forp=2.175

Smcrure of the l6-0ry signal XIS coti.sidwd in 1lri.s pctper

The superiority of the rectangular 16-QAM modulation scheme is due to the excellent trade-off 'optimal SER performance-minimum that it achieves. In fact, 16-QAM performs very close to the optimum on the AWGN channel and its parameter Nmi, is equal to I . It is worthwhile mentioning that, to the best of the author's knowledge, optimum. ( I _ 5, IO), and (5, I I ) are the only known constellations presenting better SER performance than 16-QAM. In other words, they are the only ones that could have been more attractive than 16-QAM for the design of powerful BICMs.

3 Alternative approach to the design of power- efficient BlCMs

In this Section, an alternative method is proposed for finding the most suitable signal sets for the design of BICMs. This method is based on the analysis of the error probability curves of the uncoded modulation schemes rather than on the theoretical concept of capacity. As a result, it has the advantage of providing a more practical insight of the issue addressed throughout this paper. In

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addition. it will he seen that this method does not actually require the assumption that codes displaying near-capacity performance are used. This point is important for practical applications where the design constraints (coding/decoding complexity, transmission delay.. .) are such that the code is not always capable of performing very close to the capacity limit.

3. I Discussion Before starting the search for signal sets that optimise error performance of BICM, it is useful to think about the codes that, ultimately, are expected to be used in practical designs of BICM schemes. At the time of writing, there are basically two classes of binary error-correcting codes that are of great interest for the design of near-capacity perfomlance coding systems: turbo codes [2. 31 and low-dcnsity parity-check (LDPC) codes [7, 81. These state-of-the-art codes have both been shown to perform withiii a few tenths of dB away from capacity, when a moderate BER (around 10-3 is taken as a reference. Turbo and LDPC codes have some similar characteristics. One of them is that the decoding process is performed using an iterative technique. Typically, the BER curve at a turbo or LDPC decoder output exhibits a very sudden change in its slope at the SNR where the iterative decoding algorithm starts operating in an efficient way, i.e. the error rate starts decreasing from one iteration to the other. At this particular SNR, called 'conuergence rAreshold' [16, 171, it becomes theoretically possible to achieve an arbitrarily small probability of error by simply increasing the nuinher of decoding iterations. Therefore, optimisation of the code performance consists, at least in theory, of minimising the convergence threshold of the iterative algorithm. In practice, therc are obviously other parameters that should be taken into account. such as the well known error-floor of turbo codes [3] . However, in the context of this discussion, it can be assumed that the code performance is simply determined by the convergence threshold, and measured by the parameter ASNR defined as the difference between this convergence threshold and the capacity limit.

~ E E P, , ,~ . conu,iu,7.. i+i. ISIJ. N,,. 3, zinu

At this stage, we need to introduce a new parameter, called 'threshold BER, which is closely related to the convergence threshold. This parameter indicates the lcvel of reliability that the sequence of channel samples at the decoder input must reach in order to make the iterative decoding converge. For any rate-R code with a gwen ASNR, an accurate value of the threshold BER can be determined as follows. First, the capacity limit over an AWGN channel is evaluated when such code is combined with a binary modulation scheme. such as QPSK. The threshold BER is then the BER obtained at the binary demodulator output for an SNR equal to this capacity limit plus ASNR. For instance, the threshold BER is about equal to 1.54 x IO- ' for a rate-112 code having n convergence threshold equal to capacity limit (AsNR=OdB). If the coding rate of the same code is increased to 213 and to 1/4, the value of the threshold BER falls to approximately 9.7 x and 7.1 x IO->, respectively. In Table I, the threshold BERs obtained for different values of the coding rate R are reported for a code having a convergence threshold equal to either capacity limit (ASNR= OdB), or capacity limit plus 2dB (ASNR=2dB), or capacity limit plus 4dB (ASNR = 4dB).

Table 1: Threshold BER of a code as a function of its coding rate Rand its performance parameter AsNn

Coding rate R AsNn=OdB A s ~ n = 2 d B A s N R = ~ ~ B

112 1.54 x lo-' 9.9 x 5.2 x IO-' 213 9.7 x 10.' 5.0 x 1.9 x

314 7.1 x IO-* 3.2 x 10.' 9.8 x 415 5.5 10-2 2.2 10-2 5.7 10-3 516 4.6 x lo-' 1.7 x lo-' 3.8 x lo-' 617 4.0 x lo-' 1.3 x lo-' 2.5 x

718 3.4 10-2 1.0 10-2 1.9 10-3

When M-ary modulation is combined with turbo or LDPC coding according to the BICM approach, it is possible to minimise the convergence threshold of the code by ensuring that the threshold BER of this code is obtained at the smallest SNR at the demodulator output. Hence, the issue is simply to determine the modulation scheme that performs optimally at this particular BER. In practice, the value of the threshold BER is high provided that the code is not too weak and/or its rate is not excessively close to 1, as can be seen from Table 1. One can thus conclude that, for a given value of M. the most attractive Mary modulation scheme for designing power-efficient BlCMs is, generally, the one achieving the best BER perfonnance in the low SNR region.

3.2 BER performance of uncoded modulation schemes At this point, it is interesting to determine the main parameter that affects BER performance of uncoded modulation systems at low SNRs. Over AWGN channels, it is well known that an excellent approximation of the bit error probability P,i, of an M-ary modulation scheme is given by

where P,,,y designates the symbol error probability, and IV,,,~, is the average number of bits that differ hetween two closest

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signal points as previously defined. Although (8) is generally considered as being valid essentially at high SNRs, we have checked that it is also a relatively accurate approximation at the SNR values dealt with in this paper (it becomes inaccurate only at very low SNRs).

For a given value of M, it is known that SER curves of different M-ary signal sets tend to coincide as the SNR decreases. In particular, the SER performance diflerence between the most power-efficient constellations is always very small at low SNR. This will be illustrated later by several simulation results. Hence, it is clear from (8) that, among these most power-efficient constellations, the signal set with the smallest Nmi,, will display the best BER performance at low SNR. This confirms what was found in Section 2 by analysis of the general expression for BICM capacity: a modulation scheme is optimal for the design of powerful BICM systems if it displays SER performance sufficiently close to the optimum and has a minimum N,,,im

3.3 Example 7: 8-ary signal constellations Fig. 6 shows the error probability (SER and BER) against EdNo, obtained by computer simulations, for constellations 8-PSK, ( I , 7), optimum, and cross over the AWGN channel. These results can be exploited to find the most suitable constellation for the design of any BICM system with a given spectral efficiency. As an example, consider a BICM system based on a code having a convergence threshold equal to the capacity limit (ASNR=O dB). Fig. 6 shows that constellation 8-PSK is the most attractive signal set when a threshold BER greater than about 1.1 x IO-' is targeted. One can see from Table I that it corresponds to a code whose rate is lower than about 0.6; i.e. a BICM scheme with a spectral efficiency less than 1.8bit/s/Hz. The cross signal set turns out to he the hest constellatio: for threshold BER values ranging from 4.6 x IO-- to 1.1 x IO- ' , which corresponds to the use of a code with a rate falling between 0.6 and 5/6. In other words, the cross

-1 0 1 2 3 4 5 6 7 Eb/No, dB

Fig. 6 Prr/orrnwce ronipari.son over A WGN clrunnel between r w r a l X-ury .sipuI e ts , in low SNR reyion

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constellation is the most suitable signal set for designing BICM systems whose spectral efficiencies range from 1.8 to 2.5 bit/s/Hz. Finally, it is seen in Fig. 6 that the ( I , 7) signal set achieves the hest performance for BERs lower than 4.6 x IO-’ at the demodulator output, and consequently offers the highest coding gain when used to design a BlCM system whose spectral efficiency is higher than 2.5 bit/s/Hr. However: Fig. 6 does not show that constellation (1, 7) remains i n fact the best constellation up to an SNR of 1 I dB [II, 131; above which it is outperformed by the optimum signal set. As a result. the optimum constellation is actually optimal for designing BlCM systems if the desired spectral efficiency is almost equal to 3 bit/s/Hr. All these results agree with those obtained in Section 2.

The method described here can be applied to find the most suitable signal sets, even in cases where the code does not perform close to the capacity. This method is therefore more general than the one described in Section 2, based on capacity limit calculations, which gives relevant results only when a code displaying near-capacity performance is used. To illustrate this, consider the example of a 2-hit/s/Hz BlCM scheme made up of 8-ary modulation and rate-2/3 code. Assume that the convergence threshold of the code is equal to capacity limit plus 4dB (ASNR=4dB), i.e. relatively far away from this capacity limit. From Table I , it is seen that such a code has a threshold BER approximately equal to 1.9 x IO-’, which means that the best choice consists of using the ( I , 7) signal set in the design of this particular BlCM scheme. This contradicts the results in Fig. 3, indicating that the best performance is achieved with a 2-bit/s/Hz BICM when using the cross constellation, and thus shows that a search procedure based on capacity limit computations is effective only if a code displaying near-capacity performance is employed. In conclusion. the method proposed in this Section is particularly interesting from a pi-actical point of view since it allows for precise determination of the best constellation in any application for which the code is not capable of perfonning near capacity due to some stringent design constraints.

3.4 Example 2: 76-ary signal constellations Fig. 7 shows the error probability (SER and BER) against EJN,. obtained by computer simulations. for constellations 16-QAM, ( I . 5 ; IO), optimum, and ( 5 , 11) over the AWGN channel. It is observed that the best BER performance is achieved by the 16-QAM signal set over the entire SNR range considered here, which indicates that the use of this signal set in the design of BICM schemes leads to the highest coding gains for all spectral efficiencies of practical interest. For EdNo greater than I 1 dB, i.e. threshold BERs lower than about 4.0 x I r 3 , one could, however, show that the 16-QAM signal set is outperformed by the optimum constellation. It is thus preferable to employ the optimum signal set in place of the 16-QAM constellation to design BlCM systems with spectral efficiencies almost equal to 4bit/s/Hz. As mentioned earlier. this kind of systems is, however, not of interest from an error performance point of view since a much better alternative would consist of using a hgher-order modulation scheme in such a case [ I , 151. Once again, note that these results agree with those obtained in Section 2.

4 codes and several 8- and 16-ary signal sets

To illustrate the study carried out so far, we investigate hereafter the BER performance of some BlCM systems using turbo codes and the signal sets previously considered.

Performance of BlCM schemes combining turbo

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t

4. I Simulation parameters All computer simulations consider the same rate-1/3 mother turbo code built from two 16-state recursive and systematic convolutional (RSC) codes with polynomials (23, 31). The pseudorandom interleaving separating these RSC codes has a size of 16384 bits. The desired rate-R turbo code is obtained after periodic puncturing of the mother turbo encoder. The MAP algorithm is used for the decoding of each RSC code [2], and turbo decoding is performed in eight iterations. To achieve a Fair comparison between all signal sets, the log-likelihood ratios A(c ,3 are computed from the received signal yk by applying (2) without any simplification. However, it is worth noting that. for constellations having a simple structure such as 8-PSK and I6-QAM, it is possible to simplify (2) tremendously [4, 181 without significantly degrading the error performance of the system. Finally. in all cases_ Gray and quasi-Gray mappings are defined so that information bits are the most protected bits. This technique has been shown to maximise the coding gains obtained at medium BERs (around lo-’) [4].

4.2 Simulation results Fig. 8 shows graphs of BER against EJN, for several 2-bit/ s/Hz and 3-bit/s/Hz BICMs. The 2-bit/s/Hz BICMs combine a rate-2/3 turbo code and the four 8-ary signal sets depicted in Fig. 2, while the 3-bit/s/Hz BICMs are made up of a rate-3/4 turbo code and three different 16-ary constellations, namely I6-QAM, optimum, and ( I , 5, IO). The BER curve obtained with the 2-bit/s/Hz 8-PSK turbo trellis-coded modulation (turbo TCM) proposed in [I91 is also plotted for reference purposes. This coding scheme is made up of two parallel-concatenated 16-state TCMs. and uses 16 384-bit interleaving and eight decoding iterations. The BER graph of the 3-bit/s/Hz turbo TCM proposed in [20] is also shown. This system is built from parallel

IEE Pluc. Commun., Vol. 180, No. 3, June 2003

-cross - 2 biWdHz -(1,7)-2biWdHr - --D - turbo TCM - 2 bitidHz t 16-QAM ~ 3 bitidHz +--optimum-3biVslHz d - ( l . 5 , lO)-3biVslHr

1

l O - B l . . . , 1 " " " " " " " " " " " " " 3.0 3.5 4.0 4.5 5.0 5.5 6.0

EblNO. dB

Fig. 8 Perfmi?unce cori!puri.son otiw A WGh' d m n d hc~tween severul 2-bitlslffz und 3-bitl.sIHz BICM .scIime.s ruiny turbo (.o~/e.r conihined with diflrrrenr 8 oid 16-nrj sigilul seis The performance of tmm turbo TCMs is also indicated for reference PUrpOSCS

BlCM schemes: 16-state codes, 16384bit interleaving. and 8 iterations; 2-bit/r/Hr turbo TCM: R-PSK. 16-state codes. 16384-bit interleaving. and 8 iterations: i-bit/s/Hz turho TCM: I6-QAM. &state codes. 15000-bit interleaving. and 8 iterations

concatenation of two 8-state TCMs using IGQAM, 15000- bit interleaving, and eight iterations.

From Fig. E. it is seen that the highest coding gains at medium BERs are obtained using 8-PSK, cross, and 16-QAM constellations. On the other hand, note that the signal sets that have been shown to display excellent BER performance in the absence of coding, such as 8 and 16-ary optimum constellations. are not of interest for the design of BICM schemes. These results agree with those reported in Sections 2 and 3. It is worth noting that the BICM system using the 8-PSK constellation displays error performance at medium BERs almost identical to that of turbo TCM based on the same signal set. In fact, this result is not surprising since it has been shown iii [ I ] that BlCM schemes can achieve coding gains very close to those obtained with bandwidth-efficient coding systems designed according to TCM principles [IS], provided that very powerful codes are employed. If the coding rate of the code is high enough, the performance gap between the two approaches is reduced to zero. Also note that the BICM scheme using 16-QAM slightly outperforms the 3-bit/s/Hz 16-QAM turbo TCM taken as a reference. although BlCM and TCM capacities have been found to be identical in such a case. This can be explained by the fact that this 3-bit/s/Hz turbo TCM is made Up of constituent RSC codes that have only eight states.

5 Conclusions

Determining the signal sets that perform optimally when combined with error-correcting codes is a fundamental

IEE Pwc. Conimioz., Voi. 150, No. 3, June 2M;

issue. This paper has addressed this problem by considering several 8- and 16-ary constellations that have been reported to be the most power-efficient ones in the existing literature. The main goal was to find, among these signal sets, those which are the most suitable ones for the design of powerful BICMs. To this end, two radically different approaches have been used. The first approach consists of evaluating the BICM capacity limit for each constellation, while the second one is based on the analysis of BER curves of these signal sets without coding. I t has been shown that the two approaches lead to identical results provided that codes displaying near-capacity performance are used. However, when the code is not capable of performing close to capacity, the second approach is the only one to provide relevant results.

It has been found that the best 8-ary constellation is the 8-PSK, or cross. or ( I , 7) depending on the desired spectral efficiency and the convergence threshold of the code. As for the case of 16-ary constellations, the most attractive signal set is rectangular 16-QAM for all spectral efficiencies of practical interest, as long as the code performance is not excessively poor. The study carried out in this paper does not formally prove the optimality of these particular signal sets for designing power-efficient BICM schemes. To the hest of the author's knowledge, however, the signal sets considered in this work are those that present the best SER performance and/or the lowest N,,,, among all possible 8 and 16-ary constellations. Therefore, it is believed that there is no constellation that significantly outperforms 8-PSK, cross, (1, 7), or 16-QAM when combined with an error- correcting code in a BlCM system.

An important conclusion of this work is that signal sets having a simple structure, such as 8-PSK and 16-QAM, are particularly attractive. This remarkable and rather surpris- ing result strongly suggests that some other higher-order constellations such as cross 32-QAM, rectangular 64-QAM, cross 128-QAM, 256-QAM etc. are also very much of interest for the design of power-efficient BICM schemes.

6 Acknowledgments

The author wishes to thank the reviewers for their valuable and constructive comments that improved the presentation of this paper. The author would also like to express his special gratitude to R. Benlamri, C. Berrou. A. Glavieux, E. Moore, .I. Ortiz-Odejar, L. Tnnogga: and D. Vernon for their encouragement.

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