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
Abstract
TRANSMlTTER
15
Infor. bits
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
perjormance. This paper presents good mappings of various
8-ary constellations for BICM-ID systems operating over
a frequency non-selective block Rayleigh fading channel.
Simulation results for the error pe$onnance of different
constellations/mappings are also provided and discussed.
Keywords: Bit-interleaved coded-modulation,
decoding, signal constellation, signal mapping.
Encoder
r----------------I
priori p r o ,
Demodulator
Received signal
iterative
Jnterleaver
m
Interleaver
Modulator
The a posteriori probability
Deinterleaver
Decoder
EEElYEi
lL‘
,
Decoded bits
Figure 1: The block diagram of a BICM-ID system.
The paper also introduces a new constellatiotdmapping that
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 moduconstellations/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],
2. BICM-ID SYSTEM MODEL
optimizes the code in the Euclidean space rather than dealing 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 memorysubset Euclidean distance [ 13. To improve the performance less modulator. Since this paper concentrates on signal conof TCM over Rayleigh fading channels, a scheme called bit- stellation and mapping, a rate-213 convolutional code is alinterleaved coded-modulation (BICM) was suggested by Ze- was assumed. This code together with an 8-ary constellation
havi [3]. 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 [5], 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),
c2(n),~ ( n ) is
] ,mapped
iterative decoding (BICM-ID) can in fact be used to provide to a complex channel symbol chosen from an 8-ary constelexcellent error performance over both AWGN and fading lation I? according to some mapping rule. For a frequency
channels. References [8] and [12] present good mappings of non-selective Rayleigh fading channel, the discrete-time re8-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.
(1)
r(n) = gs(n) w(n)
1. INTRODUCTION
+
CCECE 2004-CCGEI 2004, Niagra Falls, May/mai 2004
0-7803-8253-6/04/$17 @ 2004 IEEE
- 1809 -
where s ( n ) is one of M complex-valued transmitted symbols, ~ ( nis )a complex white Gaussian noise with twosided power spectral density No/2. The scalar g is a Rayleigh
random variable representing the fading amplitude of the received signal and E [ g 2 ] = 1. Also assume a coherent detection 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 maximum likelihood decoding of BICM requires joint demodulation and convolutionaldecoding and is therefore too complex
to implement in practice. In [3], Zehavi suggested a suboptimal method using two separate steps, namely bit metric generation and Viterbi decoding. Although BICM performs well
over fading channels because of an increase in diversity order, its performance is degraded over Gaussian channels due
to the “random modulation” caused by bit interleaving [3].
This makes the conventional BICM less efficient than TCM
over Gaussian channels.
Iterative processing (with soft feed-back) studied in [8]
shows that with perfect knowledge of other two bits, an 8ary constellation is translated to binary modulation selected
from four possible sets of binary constellations. It then illustrates that iterative decoding of BICM not only increases the
inter-subset Euclidean distance, but also reduces the number
of nearest neighbors. This leads to a significant improvement
over both AWGN and fading channels. The receiver shown
in Fig. 1 uses a suboptimal, iterative method, that is based on
individually optimal, but separate demodulation and convolutional decoding.
Signal mapping has a critical influence to the performance
of BICM-ID systems. That influence can be quantified by the
two Euclidean distances, namely the harmonic mean distance
(dz) and the harmonic mean distance with perfect knowledge
of the other bits (@) [8,12]. These distance parameters can
be calculated for any M-ary constellation as follows:
systems operating over a Rayleigh fading channel, the code
and the mapping have independent impacts on the asymptotic performance as can be seen from the following expression [8,12]:
(4)
where Pb is the probability of bit error, d H ( C ) is the minimum Hamming distance of the code C , R is the information
rate. Therefore, with a fixed convolutional code, the general
: while still havrule is to choose a mapping that maximizes 2
ing a sufficient large dz to make the first iteration work well
and the iterative receiver can reach its ideal performance after
a few iterations.
In [8], different mappings for 8-PSK constellations were
studied. It was shown that a BICM-ID system using semi-set
partitioning (SSP) mapping offers the best performance over
other popular 8-PSK mappings provided that a large enough
block length of the interleaver is used. It can be verified that
SSP mapping has the biggest among all the mappings of 8PSK constellations.
The next section presents mappings that
maximizes d i for other popular 8-ary constellations.
zz
3. SIGNAL MAPPINGS OF 8-ARY
CONSTELLATIONS
Three popular constellations considered in this paper for
BICM-ID systems include (1,7), cross 8-ary, and the optimum 8-ary constellations. They are shown in Fig. 2 together
with the 8-PSK constellation. The exact locations of the sig010
111
looo
WO
100
OW
001
011
011
110
110
(a)
10 I
SSP &PSK
I11
0
and
010
100
’%
010
001
0
100.
110
101
0
011
.
110
0
0
Y
0
w1
0
Om
110
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 [13]. This constellation is optimum in the sense that it performance of BICM-ID. It should be noted that for BICM-ID forms optimally at high SNR over an AWGN channel. Those
ow
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constellations were considered in [14] for BICM and lately
in [ 151 for BICM-ID systems operating over an AWGN channel.
For each constellation, the mapping of interest to us is the
one that maximizes 2:. Such mappings were found for the
above constellations' and they are also presented in Fig. 2(a), (b), (c), and (d), respectively. These mappings are the
best mappings for each constellation as far as the asymptotic
performance is concerned. Table 1 lists the distances d; and
of the proposed mappings where it is assumed that the
average energy per symbol is unity. It is interesting to ob-
tiodmapping shown in Fig. 2-(e), which we refer to as an
asymmetric 8-PSK constellation. In each odd or even subset of this constellatiodmapping, the angel between any two
.
harmonic mean d: and
nearest signal points is ~ / 6 The
2; of this constellatiodmapping are shown in Table 1. It
is clear that the distance
of asymmetric 8-PSK constellatiodmapping is much higher than that of SSP 8-PSK. This
constellatiodmapping is therefore expected to have a very
good asymptotic performance. However, it should be mentioned that due to the small value of d:, it might need a long
interleaver and a large number of iterations to achieve the
asymptotic performance of this asymmetric 8-PSK constellatiodmapping. Again, this prediction is confirmed by simuTable 1: The harmonic mean distances d: and 2; for the pro- lation results in Section 4.
posed mappings (maximized-di mappings)
z:
2
-
I
Constellation
Asymmetric 8-PSK
8-PSK
8-cross
Optimum 8-ary
(1.7)
I
d?
0.3052
0.5858
0.8348
0.9163
0.8880
I
d?
4. SIMULATION RESULTS
I
3.5241
2.8766
2.5946
2.3420
2.3045
serve that d: of SSP 8-PSK is smallest. This suggests that
if iterations work well with SSP 8-PSK, they also work well
with other contellations/mappings. Even with a short interleaver length that makes the performance of BICM-ID system with SSP-8PSK mapping degraded, the BICM-ID systems employing other constellations/mappingsmight still deliver a good performance. In contrast with d:, SSP 8-PSK
has the biggest value of
This means that the asymptotic
performance of BICM-ID using SSP 8-PSK mapping outperforms other constellations/mappings. The above discussion
is confirmed by simulation results in Section 4.
As far as the asymptotic performance is concerned, it is interesting to find a constellation/mapping that performs better
than SSP 8-PSK, i.e., having a larger
than that of SSP 8PSK. This can be accomplished by a simple modification of
SSP 8-PSK as follows. Observe that the signal points in SSP
8-PSK are divided into even and odd subsets. The even subset includes all the signal points whose labels have even Hamming weights. Similarly, the odd subset includes all the signal
points whose labels have odd Hamming weights. Now if the
signal points in each subset move closer to each other, 2: will
increase. It is easy to verify that when all four signals in each
subset collapse to one point, the distance 2: is biggest. However, the use of this ambiguity constellation/mapping will result in a very poor error performance at any practical S N R
level.
A compromise solution proposed here is a constella-
2.
'For 8-PSK, it is the SSP mapping presented in [8].
This section provides the simulation results to verify the
mapping designs introduced in Section 3. As the investigation concentrates on signal mapping, a rate-2/3, 8-state
convolutional code (CC) with the generator sequences g1 =
(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
is obtained by simulating the systems with lo7 to lo8 coded
bits. Also observe from Table 1 that the 8-cross, (1,7) and optimum 8-ary constellations/mappings have almost the same
values of dz and 22. Thus it can be expected that these constellations/mappings perform very closely.
At first, a long interleaver with a length of 12000 coded
bits are used. Figures 3, 4 and 5 present the performances
with 1 to 8 iterations of the BICM-ID systems employing the
asymmetric 8-PSK, 8-PSK, 8-cross and the proposed mappings (see Fig. 2), respectively. Also shown in each figure
is the asymptotic performance obtained by assuming the perfect knowledge of the other two bits in one symbol. It is
clear from these figures that the iterations work well with
all the constellations/mappings and they also converge to the
asymptotic performances. Note, however, that the convergence behavior of each constellation/mapping is very different. This is of course due to the difference in the parameters
d: and 2: and can be summarized as follows. For a constellatiodmapping with a small d: and a large 2; (such as
asymmetric 8-PSK), it requires a higher value of SNR for the
iteration to work, but it converges to a lower asymptotic performance. For a constellatiodmapping with a large d: and a
small J; (such as 8-cross), it is the other way around: The iterations start working at a lower SNR region but it converges
to a higher asymptotic performance.
Figure 6 compares the BER performance of different constellations/mappings after 8 iterations. Having a bigger value
of 2: makes the asymptotic performance of the asymmetric 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
EJNo (dB)
Figure 3: BER performance of a BICM-ID system: Asym- Figure 5: BER performance of a BICM-ID system: 8metric 8-PSWproposed mapping, rate-2/3, %state CC, cross/proposed mapping, rate-213, &state CC, 12000-bit in12000-bit interleaver, 8 iterations.
terleaver, 8 iterations.
loo
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 performance of BICM-ID systems using an interleaver with length
of 1200 coded bits and after 5 iterations. As can be expected, the performance of the asymmetric 8-PSK constellatiodmapping is very poor compared to that of other constellations/mappings. Obviously, the iterations do not work
with this asymmetric 8-PSK constellation because of its small
value of dz and a short interleaver. With bigger values of d;,
.
- - BER floor
6
6.5
1
1.5
8
EJNo (a)
8.5
9
9.5
Figure 4: BER performance of a BICM-ID system: 8PSWSSP mapping, rate-2/3, M a t e CC, 12000-bit interleaver, 8 iterations.
the advantage of the %cross, (1,7) and the optimum 8-ary
constellations is clear from Figure 7: They outperform both
the asymmetric 8-PSK and 8-PSK constellations at practical
BER level from
to
5. CONCLUSIONS
In this paper, the best mappings in terms of asymptotic performance were presented for various 8-ary constellations in
BICM-ID systems operating over a frequency non-selective
Rayleigh fading channel. The BER performance as well as
its convergence behavior of BICM-ID employing the proposed constellations/mappings are discussed and verified by
computer simulation. In particular, it is demonstrated that,
for a fixed convolutional code, the most suitable constellatiodmapping depends on the length of the interleaver, the
number of iterations and the desired BER level.
and 8-PSK constellations. On the other hand the asymptotic
performance of the 8-PSK is only slightly better than that of
the %cross. As the convergence behaviors of these constellations/mapping are different, the appropriate choice of signal
constellatiodmapping depends on the desired BER level. For
example, Fig. 6 suggests that the 8-cross, optimum 8-ary and
(1,7) are the best choices if the targeted BER level is
The 8-PSK performs the best at BER levels in the range from
Acknowledgements
to
Finally, to achieve a BER of
the asymmetric 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
- 1812 -
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lo46
6.5
I
1.5
E,,”
8.5
8
9
9.5
(dB)
Figure 6: BER performance of BICM-ID systems: Different constellations/mappings, rate-2/3, 8-state CC, 12000-bit
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1
loo t
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6.5
I
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