SIGNAL MAPPINGS OF 8-ARY CONSTELLATIONS FOR
BICM-ID SYSTEMS OVER A RAYLEIGH FADING CHANNEL
Nghi H. Tran and H a H . Nguyen
Department of Electrical Engineering
University of Saskatchewan
Saskatoon, Saskatchewan, Canada S7N 5A9
hut461 @engr.usask.ca,hanguyen @engr.usask.ca
Encoder Jnterleaver Modulator
It has been known that in a bit-interleaved coded-modulation
with iterative decoding (BICM-ID), signal constellation
and mapping have a strong influence to the system’s error priori p r o , The a posteriori probability
perjormance. This paper presents good mappings of various Interleaver
8-ary constellations for BICM-ID systems operating over
a frequency non-selective block Rayleigh fading channel.
Simulation results for the error pe$onnance o different
constellations/mappings are also provided and discussed.
Figure 1: The block diagram of a BICM-ID system.
Keywords: Bit-interleaved coded-modulation, iterative
decoding, signal constellation, signal mapping.
The paper also introduces a new constellatiotdmapping that
1. INTRODUCTION performs very well at high signal-to-noise ratio (SNR). The
error performance and its convergence behavior of different
Coded modulation that jointly optimizes coding and modu-
constellations/mappings are also studied.
lation is a powerful technique to improve the performance of
digital transmission schemes. This technique, independently
developed by Ungerboeck [l] and Imai and Hirakawa , 2. BICM-ID SYSTEM MODEL
optimizes the code in the Euclidean space rather than deal-
ing with the Hamming distance. Ungerboeck’s approach, Figure 1 shows the block diagram of a BICM-ID system.
called trellis-coded modulation (TCM), is based on map- The transmitter is built from a serial concatenation of the
ping by set partitioning that maximizes the minimum intra- convolutional encoder, the bit interleaver and the memory-
subset Euclidean distance [ 13. To improve the performance less modulator. Since this paper concentrates on signal con-
of TCM over Rayleigh fading channels, a scheme called bit- stellation and mapping, a rate-213 convolutional code is al-
interleaved coded-modulation (BICM) was suggested by Ze- was assumed. This code together with an 8-ary constellation
havi . This scheme increases the time diversity of the yields a spectral efficiency of 2 bits/s/Hz. Denote the input
coded modulation at the expense of reducing the free squared bits of the encoder during the nth symbol interval by U(.) =
Euclidean distance (FED), leading to a degradation over ad- [ul(n), u2(n)]and its corresponding three output bits (which
ditive white Gaussian noise (AWGN) channels [3,4]. )
make up a code symbol) by ~ ( n= [vl(n),vz(n),w3(n)],
Since the invention of turbo codes , interleaving and it- where ui(n) or q ( n ) is the ith bit, taking values in ( 0 , l )
erative decoding/demodulation have been applied to coded with equal probabilities. After the interleaver, every group
modulation systems. It was shown in [6-111 that BICM with of three binary bits, c ( n ) = [cl(n), ] mapped
c2(n), ( n ) is ,
iterative decoding (BICM-ID) can in fact be used to provide to a complex channel symbol chosen from an 8-ary constel-
excellent error performance over both AWGN and fading lation I? according to some mapping rule. For a frequency
channels. References  and  present good mappings of non-selective Rayleigh fading channel, the discrete-time re-
8-PSK and square 16-QAM constellations for Rayleigh fad- ceived signal can be represented as follows:
ing channels. Motivated by the results in [8,12], this paper
obtains good mappings of other popular 8-ary constellations. r(n) = gs(n) w(n) + (1)
CCECE 2004-CCGEI 2004, Niagra Falls, May/mai 2004
0-7803-8253-6/04/$17 @ 2004 IEEE
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where s ( n ) is one of M complex-valued transmitted sym- systems operating over a Rayleigh fading channel, the code
bols, ~ ( nis )a complex white Gaussian noise with two- and the mapping have independent impacts on the asymp-
sided power spectral density No/2. The scalar g is a Rayleigh totic performance as can be seen from the following expres-
random variable representing the fading amplitude of the re- sion [8,12]:
ceived signal and E [ g 2 ] = 1. Also assume a coherent detec-
tion with a perfect channel state information (CSI), i.e., g can
be perfectly estimated at the receiver.
Due to the presence of bit-based interleaver, the true maxi- (4)
mum likelihood decoding of BICM requires joint demodula- where Pb is the probability of bit error, d H ( C ) is the mini-
tion and convolutionaldecoding and is therefore too complex mum Hamming distance of the code C , R is the information
to implement in practice. In , Zehavi suggested a subopti- rate. Therefore, with a fixed convolutional code, the general
mal method using two separate steps, namely bit metric gen- :
rule is to choose a mapping that maximizes 2 while still hav-
eration and Viterbi decoding. Although BICM performs well ing a sufficient large dz to make the first iteration work well
over fading channels because of an increase in diversity or- and the iterative receiver can reach its ideal performance after
der, its performance is degraded over Gaussian channels due a few iterations.
to the “random modulation” caused by bit interleaving . In , different mappings for 8-PSK constellations were
This makes the conventional BICM less efficient than TCM studied. It was shown that a BICM-ID system using semi-set
over Gaussian channels. partitioning (SSP) mapping offers the best performance over
Iterative processing (with soft feed-back) studied in  other popular 8-PSK mappings provided that a large enough
shows that with perfect knowledge of other two bits, an 8- block length of the interleaver is used. It can be verified that
ary constellation is translated to binary modulation selected zz
SSP mapping has the biggest among all the mappings of 8-
from four possible sets of binary constellations. It then illus- PSK constellations. The next section presents mappings that
trates that iterative decoding of BICM not only increases the maximizes d i for other popular 8-ary constellations.
inter-subset Euclidean distance, but also reduces the number
of nearest neighbors. This leads to a significant improvement 3. SIGNAL MAPPINGS OF 8-ARY
over both AWGN and fading channels. The receiver shown CONSTELLATIONS
in Fig. 1 uses a suboptimal, iterative method, that is based on
individually optimal, but separate demodulation and convo- Three popular constellations considered in this paper for
lutional decoding. BICM-ID systems include (1,7), cross 8-ary, and the opti-
Signal mapping has a critical influence to the performance mum 8-ary constellations. They are shown in Fig. 2 together
of BICM-ID systems. That influence can be quantified by the with the 8-PSK constellation. The exact locations of the sig-
two Euclidean distances, namely the harmonic mean distance
(dz) and the harmonic mean distance with perfect knowledge 010 111
of the other bits (@) [8,12]. These distance parameters can 100 WO looo OW 001
be calculated for any M-ary constellation as follows:
(a) SSP &PSK
010 100 001 010
110 0 Y
101 011 110
0 . 0 0
ow 0 Om
In (2) and (3), I: denotes the subset of I7 that contains all 0 111 011
the symbols whose labels have the value b E ( 0 , l ) in the
position i. The symbol j: in (2) belongs to I’t (where 6 is the
complement of b) and it is the nearest neighbor of s. The
symbol S in (3) is in I? whose label has the same binary bit Figure 2: Various 8-ary constellations/mappings.
values as those of s, except at the ith position.
The distance d i affects to the asymptotic performance of nal points in the optimum 8-ary constellation are provided
BICM while the distance zi dominates the asymptotic per- in . This constellation is optimum in the sense that it per-
formance of BICM-ID. It should be noted that for BICM-ID forms optimally at high SNR over an AWGN channel. Those
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constellations were considered in  for BICM and lately tiodmapping shown in Fig. 2-(e), which we refer to as an
in [ 151 for BICM-ID systems operating over an AWGN chan- asymmetric 8-PSK constellation. In each odd or even sub-
nel. set of this constellatiodmapping, the angel between any two
nearest signal points is ~ / 6 The harmonic mean d: and
For each constellation, the mapping of interest to us is the .
one that maximizes 2:. Such mappings were found for the ;
2 of this constellatiodmapping are shown in Table 1. It
above constellations' and they are also presented in Fig. 2-
(a), (b), (c), and (d), respectively. These mappings are the
is clear that the distance :
of asymmetric 8-PSK constella-
tiodmapping is much higher than that of SSP 8-PSK. This
best mappings for each constellation as far as the asymptotic
constellatiodmapping is therefore expected to have a very
performance is concerned. Table 1 lists the distances d; andgood asymptotic performance. However, it should be men-
2 of the proposed mappings where it is assumed that the tioned that due to the small value of d:, it might need a long
average energy per symbol is unity. It is interesting to ob-interleaver and a large number of iterations to achieve the
asymptotic performance of this asymmetric 8-PSK constel-
latiodmapping. Again, this prediction is confirmed by simu-
Table 1: The harmonic mean distances d: and 2; for the pro- lation results in Section 4.
posed mappings (maximized-di mappings)
4. SIMULATION RESULTS
I Constellation I d? I d? I
Asymmetric 8-PSK 0.3052 3.5241 This section provides the simulation results to verify the
8-PSK 0.5858 2.8766 mapping designs introduced in Section 3. As the investi-
8-cross 0.8348 2.5946 gation concentrates on signal mapping, a rate-2/3, 8-state
Optimum 8-ary 0.9163 2.3420 convolutional code (CC) with the generator sequences g1 =
(1.7) 0.8880 2.3045 (4,2,6) and 92 = (1,4,7) is always assumed. The bit-wise
interleavers used in all simulations are designed according to
the rules outlined in [8, 161. Each point in the BER curves
serve that d: of SSP 8-PSK is smallest. This suggests that is obtained by simulating the systems with lo7 to lo8 coded
if iterations work well with SSP 8-PSK, they also work well bits. Also observe from Table 1 that the 8-cross, (1,7) and op-
with other contellations/mappings. Even with a short inter- timum 8-ary constellations/mappings have almost the same
leaver length that makes the performance of BICM-ID sys- values of dz and 22. Thus it can be expected that these con-
tem with SSP-8PSK mapping degraded, the BICM-ID sys- stellations/mappings perform very closely.
tems employing other constellations/mappingsmight still de- At first, a long interleaver with a length of 12000 coded
liver a good performance. In contrast with d:, SSP 8-PSK bits are used. Figures 3, 4 and 5 present the performances
has the biggest value of 2.
This means that the asymptotic with 1 to 8 iterations of the BICM-ID systems employing the
performance of BICM-ID using SSP 8-PSK mapping outper- asymmetric 8-PSK, 8-PSK, 8-cross and the proposed map-
forms other constellations/mappings. The above discussion pings (see Fig. 2), respectively. Also shown in each figure
is confirmed by simulation results in Section 4. is the asymptotic performance obtained by assuming the per-
As far as the asymptotic performance is concerned, it is in- fect knowledge of the other two bits in one symbol. It is
teresting to find a constellation/mapping that performs better clear from these figures that the iterations work well with
than SSP 8-PSK, i.e., having a larger than that of SSP 8- all the constellations/mappings and they also converge to the
PSK. This can be accomplished by a simple modification of asymptotic performances. Note, however, that the conver-
SSP 8-PSK as follows. Observe that the signal points in SSP gence behavior of each constellation/mapping is very differ-
8-PSK are divided into even and odd subsets. The even sub- ent. This is of course due to the difference in the parameters
set includes all the signal points whose labels have even Ham- :
d: and 2 and can be summarized as follows. For a con-
ming weights. Similarly, the odd subset includes all the signal stellatiodmapping with a small d: and a large 2; (such as
points whose labels have odd Hamming weights. Now if the asymmetric 8-PSK), it requires a higher value of SNR for the
signal points in each subset move closer to each other, 2 will iteration to work, but it converges to a lower asymptotic per-
increase. It is easy to verify that when all four signals in each formance. For a constellatiodmapping with a large d: and a
subset collapse to one point, the distance 2 is biggest. How- small J; (such as 8-cross), it is the other way around: The it-
ever, the use of this ambiguity constellation/mapping will re- erations start working at a lower SNR region but it converges
sult in a very poor error performance at any practical S N R to a higher asymptotic performance.
level. Figure 6 compares the BER performance of different con-
A compromise solution proposed here is a constella- stellations/mappings after 8 iterations. Having a bigger value
of 2 makes the asymptotic performance of the asymmet-
'For 8-PSK, it is the SSP mapping presented in . ric 8-PSK constellation better than that of both the 8-cross
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*" 6 6.5 7 1.5 8 8.5 9 9.5
Figure 3: BER performance of a BICM-ID system: Asym- Figure 5: BER performance of a BICM-ID system: 8-
metric 8-PSWproposed mapping, rate-2/3, %state CC, cross/proposed mapping, rate-213, &state CC, 12000-bit in-
12000-bit interleaver, 8 iterations. terleaver, 8 iterations.
1 systems employing different constellations/mappings,a short
interleaver and when only a few iterations can be afforded.
This is important when the receiver complexity andor delay
are of primary concern. Figure 7 presents the BER perfor-
mance of BICM-ID systems using an interleaver with length
of 1200 coded bits and after 5 iterations. As can be ex-
pected, the performance of the asymmetric 8-PSK constel-
latiodmapping is very poor compared to that of other con-
stellations/mappings. Obviously, the iterations do not work
with this asymmetric 8-PSK constellation because of its small
value o dz and a short interleaver. With bigger values o d;,
the advantage of the %cross, (1,7) and the optimum 8-ary
. - - BER floor constellations is clear from Figure 7: They outperform both
6 6.5 1 1.5 8 8.5 9 9.5 the asymmetric 8-PSK and 8-PSK constellations at practical
EJNo (a) BER level from to
Figure 4: BER performance of a BICM-ID system: 8- 5. CONCLUSIONS
PSWSSP mapping, rate-2/3, M a t e CC, 12000-bit inter-
In this paper, the best mappings in terms of asymptotic per-
leaver, 8 iterations.
formance were presented for various 8-ary constellations in
BICM-ID systems operating over a frequency non-selective
and 8-PSK constellations. On the other hand the asymptotic Rayleigh fading channel. The BER performance as well as
performance of the 8-PSK is only slightly better than that of its convergence behavior of BICM-ID employing the pro-
the %cross. As the convergence behaviors of these constella- posed constellations/mappings are discussed and verified by
tions/mapping are different, the appropriate choice of signal computer simulation. In particular, it is demonstrated that,
constellatiodmapping depends on the desired BER level. For for a fixed convolutional code, the most suitable constella-
example, Fig. 6 suggests that the 8-cross, optimum 8-ary and tiodmapping depends on the length of the interleaver, the
(1,7) are the best choices if the targeted BER level is number of iterations and the desired BER level.
The 8-PSK performs the best at BER levels in the range from
to Finally, to achieve a BER of the asym- Acknowledgements
metric 8-PSK is the most efficient constellation (about 1dB is
This work was supported in part by a Scholarship from
gained compared to other constellations). TRlabsSaskatoon and an NSERC Discovery Grant.
It is also of interest to study the performance of BICM-ID
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