Towards Fully Optimized BICM Transceivers

Abstract

Bit-interleaved coded modulation (BICM) transceivers often use equally spaced constellations and a random interleaver. In this paper, we propose a new BICM design, which considers hierarchical (nonequally spaced) constellations, a bit-level multiplexer, and multiple interleavers. It is shown that this new scheme increases the degrees of freedom that can be exploited in order to improve its performance. Analytical bounds on the bit error rate (BER) of the system in terms of the constellation parameters and the multiplexing rules are developed for the additive white Gaussian Noise (AWGN) and Nakagami-$m$ fading channels. These bounds are then used to design the BICM transceiver. Numerical results show that, compared to conventional BICM designs, and for a target BER of $10^{-6}$, gains up to 3 dB in the AWGN channel are obtained. For fading channels, the gains depend on the fading parameter, and reach 2 dB for a target BER of $10^{-7}$ and $m=5$. Comment: Submitted to the IEEE Transactions on Communications

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arXiv:1012.1799v1 [cs.IT] 8 Dec 2010
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Towards Fully Optimized BICM Transceivers
Md. Jahangir Hossain†, Alex Alvarado§, Leszek Szczecinski‡
†ITR, University of South Australia, Australia
§Department of Signals and Systems, Communication Systems Group
Chalmers University of Technology, Gothenburg, Sweden
‡INRS-EMT, University of Quebec, Montreal,Canada
jahangir.hossain@unisa.edu.au, alex.alvarado@chalmers.se, leszek@emt.inrs.ca
Abstract
Bit-interleaved coded modulation (BICM) transceivers often use equally spaced constellations and
a random interleaver. In this paper, we propose a new BICM design, which considers hierarchical
(nonequally spaced) constellations, a bit-level multiplexer, and multiple interleavers. It is shown that this
new scheme increases the degrees of freedom that can be exploited in order to improve its performance.
Analytical bounds on the bit error rate (BER) of the system in terms of the constellation parameters and
the multiplexing rules are developed for the additive white Gaussian Noise (AWGN) and Nakagami-m
fading channels. These bounds are then used to design the BICM transceiver. Numerical results show
that, compared to conventional BICM designs, and for a target BER of 10−6, gains up to 3 dB in the
AWGN channel are obtained. For fading channels, the gains depend on the fading parameter, and reach
2 dB for a target BER of 10−7and m = 5.
Index Terms
Bit-interleaved coded modulation, bit error rate, interleaver design, multiple interleavers, L-values,
nonequally spaced constellations, pulse amplitude modulation, quadrature amplitude modulation, trellis
coded modulation.
Research supported by the Australian Research Council (ARC) Linkage Project (research grant #LP 0991370), NSERC, Canada
(research grant #356807-07), by the Swedish Research Council, Sweden (research grant #2006-5599), and by the Department
of Signals and Systems (Solveig and Karl G Eliasson Memorial Fund), Chalmers University of Technology. Parts of this work
have been presented at the International Conference on Communications (ICC) 2010, Cape Town, South Africa, May 2010.
Parts of this work are also under review for possible publication at ICC 2011, Kyoto, Japan.
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I. INTRODUCTION
Bit-interleaved coded modulation (BICM) [1]–[3] is used in most of the existing wireless
communication standards, e.g., HSPA, IEEE 802.11a/g/n, DVB, etc. In BICM, the channel
encoder and the modulator are separated by a bit-level interleaver which allows the designer to
choose the code rate and the constellation independently. BICM maximizes the code diversity,
and therefore, outperforms trellis coded modulation (TCM) in fading channels. Compared to
TCM, BICM is suboptimal for the additive white Gaussian noise (AWGN) channel because it
decreases the minimum Euclidean distance. Nevertheless, its simplicity and flexibility make it
an attractive coded modulation scheme even when fading is not present.
BICM appears as a simple out-of-the box coded modulation scheme, however, its full potential
is achieved only if its design is optimized. It has been shown in [4] that the interleaver and the
code can be jointly designed to exploit the so-called unequal error protection (UEP) caused by
the binary labeling of the equally spaced (ES) constellations. In general, if the channel offers
UEP to the coded bits, gains in terms of bit error rate (BER) can be obtained, cf. [4, Sec. I] and
references therein. UEP can also be intentionally introduced when designing the transceiver. For
example, UEP can be imposed by allowing unequal power allocation for different bits, by deleting
bits using certain patterns (puncturing), by changing the binary labeling of the constellation, or
by using signal shaping, i.e., by using nonequally spaced (NES) constellations or nonequally
likely symbols, known as geometrical and probabilistic shaping, respectively.
In this paper, we propose to control the UEP via geometrical shaping. In particular, we use the
so-called hierarchical constellations [5], which have received a great deal of attention in many
applications where independent data streams with different qualities must be sent at the same
time, e.g., in multi-resolution image transmission [6], [7], and simultaneous voice and multi-class
data transmission [8]. Hierarchical quadrature amplitude modulation (HQAM) constellations are
also used in QUALCOMM’s MediaFLO [9] and have been standardized for the latest digital
video broadcasting-terrestrial (DVB-T2) [10], [11]. In this paper, we use HQAM constellations
in a different context, i.e., to transmit only one data stream with improved error performance.
Although shaping techniques for BICM have received some attention in the literature (e.g., ge-
ometrical in [12]–[14] and probabilistic in [15], [16, Sec. III-F]), they are all based on capacity
maximization, which translates into enhanced performance if capacity-approaching codes (turbo
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or low-density parity-check codes) are used. On the other hand, capacity arguments are less
relevant when convolutionally-encoded BICM is considered. In this paper, we study this simple
and low-complexity BICM configuration, and thus, we adopt a different approach that aims at
the minimization of the BER for a given signal to noise ratio (SNR).
In order to exploit the UEP offered by the constellation, the interleaver must be properly
designed. The most commonly interleaver considered in the literature is the single interleaver
(S-interleaver) of [2], which eliminates the UEP caused by the binary labeling1. Recently, the
the so-called multiple interleavers (M-interleavers) [4] were shown to improve the performance
of the system by exploiting the UEP caused by the modulator. In fact, the use of BICM with
M-interleavers (BICM-M) corresponds to the original BICM configuration proposed by Zehavi
in [1], as well as the original BICM with iterative decoding (BICM-ID) scheme proposed by
Li and Ritcey in [17]. M-interleavers have also been shown to outperform S-interleavers when
BICM-ID is considered [18].
The BICM-M system in [4], [18] uses a random bit-level multiplexing (R-MUX) that connects
the encoder and the M-interleavers, i.e., the M-interleavers assign the coded bits to a particular
bit position in the modulator in a pseudo-random fashion (with predetermined probabilities).
By doing this, the dependency of adjacent coded bits is ignored. In this paper, we propose an
multiplexing/interleaving inspired by the well-known puncturing strategy based on the periodic
elimination of the bits according to a prescribed pattern that matches the temporal structure of
the code, cf. [19]. We show that such a deterministic multiplexing (D-MUX) of the coded bits
(followed by random interleaving) notably outperforms the R-MUX used in [4].
The contributions of this paper can be summarized as follows. We propose and study a BICM
scheme for fading and nonfading channels which considers the use of HQAM constellations
(HQAM-BICM), a periodic (and deterministic) bit-level multiplexer, and M-interleavers. It is
demonstrated that the degrees of freedom of such a scheme can be exploited to notably improve
performance of the system in terms of BER. We use a Gaussian model for the probability
density function (PDF) of the L-values passed to the decoder which consider the nonequal
spacing between the signal points in the constellation, and apply it to develop union bounds
(UB) on the BER of the system for fading and nonfading channels. These UBs are then used
1UEP in BICM was in fact considered an “undesired feature” in [2, Sec. II].
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to optimize the transceiver. Presented numerical examples show that the proposed system offers
gains over previous BICM configurations (ES-QAM and S-interleavers [2] or ES-QAM and M-
interleavers [4]). For the particular cases analyzed in this paper, the gains can be up to 3 dB for
a BER target of 10−6in the AWGN channel, and for the Nakagami-m fading channel, the gains
can reach 2 dB for a target of 10−7and m = 5.
II. PROPOSED BICM TRANSCEIVER
Throughout this paper, we use boldface letters ct= [c1,t,...,cN,t] to denote row vectors and
capital boldface letters C = [cT
1,...,cT
M]Tto denote a matrix of M rows, where (·)Tdenotes
transposition. We denote probability by Pr(·) and the PDF of a random variable X by pX(x). A
Gaussian distribution with mean value µ and variance σ2is denoted by N(µ,σ2), the Gaussian
PDF with the same parameters by ψ(λ;µ,σ) ?
?∞
is l is denoted by Wi(l), where Wi(l) ? {[w1,...,wi] ∈ (Z+)i: w1+ ... + wi= l}.
The HQAM-BICM system model under consideration is shown in Fig. 1. In what follows,
1
√2πσexp(−(λ−µ)2
2σ2 ), and the Q-function by
Q(x) ?
1
√2π
xexp
?
−u2
2
?
du. The combinations of i nonegative integers such that their sum
we describe functionalities of various blocks of such transmission scheme.
A. Encoder, Multiplexing, and Interleaving
The kcvectors of information bits il= [il,1,...,il,Nc] with l = 1,...,kcare encoded by a rate
R = kc/n convolutional encoder (ENC) yielding the vectors of coded bits cp= [cp,1,...,cp,Nc]
with p = 1,...,n. These are then fed to a deterministic multiplexing (D-MUX) unit which
bijectively maps C = [cT
1,...,cT
n]Tonto O = [oT
1,...,oT
q]Twith ok = [ok,1,...,ok,Ns] and
k = 1,...,q. Without loss of generality, we assume Nsq = Ncn. The vector of bits after the
D-MUX are fed to q parallel interleavers πk. The q interleavers are assumed to be independent
and give randomly permuted sequences of the bits, i.e., uk= πk{ok}. Each of the interleavers is
connected to the qth bit positions in the hierarchical M-ary pulse amplitude modulation (HPAM)
constellation, where q = log2M.
In general, the D-MUX can be defined as a one-to-one mapping between the blocks of nNc
and qNsbits, i.e., {0,1}nNc↔ {0,1}qNs. We define it via an n × Ncmatrix˜K, whose (p,t′)th
entry is a pair (k,t) where k ∈ {1,...,q} and t ∈ {1,...,Ns}. The entry (k,t) indicates that
the bit cp,t′ is assigned to the kth D-MUX’s output at time instant t, i.e., ok,t= cp,t′.
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The previous definition of the D-MUX is entirely general but difficult to deal with, and thus,
in this paper we only consider D-MUX configurations that operate periodically over blocks of
nJ bits. We then represent˜ K as a concatenation of Nc/J matrices Kτ, each of dimensions
n × J, i.e.,˜K = [K0,...,KNc/J−1], where the variable J is called the period of the D-MUX.
The entries of Kτare pairs (k,t+τnJ/q) with k ∈ {1,...,q} and t ∈ {1,...,nJ/q}. Without
loss of generality, we assume that (Ncmod J) = 0 and that (nJ mod q) = 0. To clarify these
definitions, consider the following example.
Example 1: Assume kc= 1 and n = 2 (R = 1/2), J = 3, and q = 3 (8-ary constellation).
One possible D-MUX is defined by
Kτ=

(1,1 + 2τ) (2,2 + 2τ) (2,1 + 2τ)
(1,2 + 2τ) (3,2 + 2τ) (3,1 + 2τ)

,
(1)
which results in
˜K =

(1,1) (2,2) (2,1) (1,3) (2,4) (2,3) ...
The mapping between C and O is then
(1,2) (3,2) (3,1) (1,4) (3,4) (3,3) ...

.
C =

c1,1 c1,2 c1,3
Since a matrix Kτ is simply a permutation of the set {1,...,q} × {1,...,nJ/q}, (nJ)!
different matrices Kτ can be generated. However, this number can be reduced since trivial
c1,4 c1,5 c1,6
...
c2,1 c2,2 c2,3
c2,4 c2,5 c2,6
...

⇐⇒ O =



c1,1 c2,1
c1,4 c2,4
...
c1,3 c1,2
c1,6 c1,5
...
c2,3 c2,2
c2,6 c2,5
...


.
(2)
operations that do not affect the performance of the system can be applied to Kτ. For example,
for the matrix in (1) with τ = 0, consider the following two matrices:
K′
0=

(1,2) (2,2) (2,1)
0is obtained by permuting the elements of K0such that the first elements in the
entries of K0are not altered. Because M-interleavers are used after the D-MUX (cf. Fig. 1), the
(1,1) (3,1) (3,2)


K′′
0=

(2,1) (1,1) (2,2)
(3,1) (1,2) (3,2)

.
The matrix K′
temporal structure of the sequences okis randomized, and thus, the second elements of the entries
are not (which determines to which time bit ck,tis assigned) are not relevant. Consequently, the
performance of the system using K0or K′
0will be the same. The matrix K′′
0is obtained by
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