A systematic bit-wise decomposition of M-ary symbol metric

Abstract

In this paper, we present a systematic recursive formula for bit-wise decomposition of M-ary symbol metric. The decomposed bit metrics can be applied to improve the performance of a system where the information sequence is binary-coded and interleaved before M-ary modulated. A traditional receiver designed for certain system is to de-map the received M-ary symbol into its binary isomorphism so as to facilitate the subsequent bit-based manipulation, such as hard-decision decoding. With a bit-wise decomposition of M-ary symbol metric, a soft-decision decoder can be used to achieve a better system performance. The idea behind the systematic formula is to decompose the symbol-based maximum-likelihood (ML) metric by equating a number of specific equations that are drawn from squared-error criterion. It interestingly yields a systematic recursive formula that can be applied to some previous work derived from different standpoint. Simulation results based on IEEE 802.11a/g standard show that at bit-error-rate of 10^-5 , the proposed bit-wise decomposed metric can provide 3.0 dB, 3.9 dB and 5.1 dB improvement over the concatenation of binary-demapper, deinterleaver and hard-decision decoder respectively for 16QAM, 64QAM and 256QAM symbols, in which the in-phase and quadrature components in a complex M²-QAM symbol are independently treated as two real M-PAM symbols. Further empirical study on system imperfection implies that the proposed bit-wise decomposed metric also improves the system robustness against gain mismatch and phase imperfection. In the end, a realization structure that avails the recursive nature of the proposed bit-decomposed metric formula is addressed

Full text

Page 1

2742IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
A Systematic Bit-Wise Decomposition
of M-ary Symbol Metric
Chia-Wei Chang, Member, IEEE, Po-Ning Chen, Senior Member, IEEE,
and Yunghsiang S. Han, Member, IEEE
Abstract—In this paper, we present a systematic recursive
formula for bit-wise decomposition of M-ary symbol metric. The
decomposed bit metrics can be applied to improve the perfor-
mance of a system where the information sequence is binary-
coded and interleaved before M-ary modulated. A traditional
receiver designed for certain system is to de-map the received
M-ary symbol into its binary isomorphism so as to facilitate
the subsequent bit-based manipulation, such as hard-decision
decoding. With a bit-wise decomposition of M-ary symbol metric,
a soft-decision decoder can be used to achieve a better system
performance.
The idea behind the systematic formula is to decompose the
symbol-based maximum-likelihood (ML) metric by equating a
number of specific equations that are drawn from squared-
error criterion. It interestingly yields a systematic recursive
formula that can be applied to some previous work derived
from different standpoint. Simulation results based on IEEE
802.11a/g standard show that at bit-error-rate of 10−5, the
proposed bit-wise decomposed metric can provide 3.0 dB, 3.9
dB and 5.1 dB improvement over the concatenation of binary-
demapper, deinterleaver and hard-decision decoder respectively
for 16QAM, 64QAM and 256QAM symbols, in which the in-
phase and quadrature components in a complex M2-QAM
symbol are independently treated as two real M-PAM symbols.
Further empirical study on system imperfection implies that the
proposed bit-wise decomposed metric also improves the system
robustness against gain mismatch and phase imperfection. In the
end, a realization structure that avails the recursive nature of
the proposed bit-decomposed metric formula is addressed.
Index Terms—Maximum-likelihood decoding, QAM modula-
tion, soft-decision decoding, Viterbi decoder.
I. INTRODUCTION
T
OFDM to achieve a high data rate. In order to make a
better use of the error correcting capability of the adopted
(2,1,6) convolutional code, the standard specified a two-step
bit interleaver, where the first step maps adjacent code bits
onto non-adjacent sub-carriers, and the second step permutes
HE state-of-the-art wireless transmission technique of
IEEE 802.11a/g [8], [9] incorporated high QAM into
Manuscript received May 14, 2004; revised June 6, 2005 and April 17,
2006; accepted July 17, 2006. The editor coordinating the review of this paper
and approving it for publication is L. Vandendorpe. This work was supported
by the National Science Council, Taiwan, R.O.C., under Grant NSC 92-2213-
E- 009-118, 93-2213-E-009-050 and 94-2213-E-009-017.
C.-W. Chang is with the Dept. of ECE, University of California, San Diego,
CA 92093 USA (e-mail: sekwan.cm87@nctu.edu.tw).
P.-N. Chen is with the Dept. of Communications Eng., National Chiao Tung
Univ., Taiwan, R.O.C. (e-mail:qponing@mail.nctu.edu.tw).
Y. S. Han was with the Dept. of CSIE, National Chi Nan University,
Taiwan. Now he is with the Graduate Institute of Communication Engineering,
National Taipei University, Taiwan, R.O.C. (e-mail: yshan@mail.ntpu.edu.tw).
Digital Object Identifier 10.1109/TWC.2006.04322
the code bits alternately onto less and more significant bits of
the QAM constellation. Such an interleaver design, although
straight and simple in concept, may restrict the potential
structures of a receiver in practice.
For a system where the binary information sequence is
encoded and bit-interleaved before M-ary modulated, a tra-
ditional receiver design is to de-map the received M-ary
symbol into its binary isomorphism so as to facilitate the
subsequent bit-based manipulation as shown in Fig. 1. Take
IEEE std. 802.11a as an example. After de-interleaving, the
two code bits, c1 and c2, that decide a trellis branch of
(2,1,6) convolutional code will respectively come from the
least significant bit of 16QAM quadrature component r1and
the most significant bit of 16QAM quadrature component r7
[8]. The dependence of one single trellis branch on two time-
inconsecutive QAM quadrature symbols somehow suggests
that the received M-ary symbol should be hard-demapped
before decoding, since other receiver structure such as one
with soft-decision decoding may require a branch metric that
can be determined by two distant and non-orderly M-ary
symbols. Such a soft-decision-based receiver will become
more involved, when a receiver structure that also supports
multi-rate transmission through code punctuation is further
considered.
In order to design a general receiver for use of M-ary
symbol transmission of coded and interleaved information se-
quence, researchers have proposed several heuristic methods to
perform bit-wise decomposition of M-ary symbol metric [10],
[12], [15]. With a bit-wise decomposition of M-ary symbol
metric, a better system performance can possibly be achieved
by adopting a soft-decision decoder, even if code punctuation
for dynamic rate transmission is incorporated. Even though
some of them perform well in practice, there is still lack of a
systematic method for bit-wise decomposition of M-ary sym-
bol metric. Thus, we propose in this work a general recursive
formula for bit-wise decomposition of M-ary symbol met-
ric[4]. The proposed approach is to approximate the symbol-
based maximum-likelihood (ML) metric by equating a number
of specific equations that are drawn from square error criterion.
Notably, the square error between symbol-based ML metric
and its bit-decomposed approximation reduces to zero when
all the listed equations can be simultaneously satisfied, which
means that the performance of the symbol-based ML metric
can be achieved by taking the proposed bit-wise decomposed
metrics. This optimistic result of zero square error however
can not be obtained in general due to the bit-wise interleaving.
1536-1276/06$20.00 c ? 2006 IEEE

Page 2

CHANG et al.: A SYSTEMATIC BIT-WISE DECOMPOSITION OF M-ARY SYMBOL METRIC2743
M-ary
Symbol Demapping
Bitwise
Deinterleaving
Convolutional Code
(Viterbi) Decoding
Fig. 1.An exemplified receiver design.
A suboptimal bit-decomposition metric in terms of maximal
subset of simultaneously satisfiable equations is then proposed.
Exemplified study on QAM modulation interestingly yields
a systematic recursive bit metric formula. By re-examining
their metrics, we interestingly found that the bit reliability
metrics of some previous work also have similar recursive
forms. Details are given in Section II-C.
Empirical study under the system setting of IEEE 802.11a/g
and the additive white Gaussian noise (AWGN) channel
showed that at bit error rate (BER) = 10−5, the proposed bit-
decomposed metric has 3.0 dB, 3.9 dB and 5.1 dB gains over
the hard decision system for 16QAM, 64QAM and 256QAM,
respectively. Also, only 0.13 dB performance degradation
is resulted by introducing 32-level quantization for 16QAM
signals. The quantization impact for 64QAM signals under 64-
level uniform quantization can even be reduced to 0.07 dB.
No further performance degradation, in addition to that due
to quantization, can be observed, when mismatch of AGC
gain is limited to be within ±40%. The robustness of the
proposed bit-decomposed metric against phase imperfection
is also examined. When the phase drift increases up to ±6o,
the BER due to our bit-decomposed metric will increase from
10−5to around 4 × 10−5at Eb/N0 = 6.7 dB for 16QAM
modulation. This phase drift tolerance reduces to ±4oat
Eb/N0 = 9.7 dB for 64QAM modulation, where Eb/N0 is
chosen such that the no-phase-drift BER is approximately
10−5.
Some previous works on bit reliability study are summa-
rized below. In [15], Zehavi proposed a decoding scheme
that consists of a sub-optimal Log-Likelihood-Ratio (LLR)
bit demapping for subsequent use of path metric computation
of Viterbi decoder. Later, Pyndiah et al. [10] provided a
pragmatic algorithm based on LLR to turbo codes associated
with high QAM modulation, and showed that the block turbo-
coded QAM modulation outperforms Trellis-Coded Modu-
lation (TCM) scheme by at least 1 dB at BER = 10−5.
Then, Caire et al. [3] presented a maximum-likelihood bit
demapping for bit-interleaved coded modulation (BICM) and
gave guidelines for its design. Tosato and Bisaglia [12] adapted
the Pyndiah’s algorithm to COFDM system, and proposed
a simplified bit reliability decomposition for 16QAM and
64QAM constellations. Their simulations showed that for
64QAM constellation, adopting their bit reliability decompo-
sition results in 8.5 dB gain at BER = 10−4over a hard-
decision-based receiver under HIPERLAN/2 system model.1
This paper is organized in the following fashion. Section
II provides the analysis of the proposed bit-decomposed
metric, followed by its complexity comparison with other soft-
demapping schemes. Section III summarizes the simulation re-
1Similar to IEEE std 802.11a/g, the scrambled input sequence of HIPER-
LAN/2 [5] is convolutionally encoded with rate 1/2 and constraint length 7
before bit-wise interleaving and QAM modulation.
sults over AWGN channels, and examines the robustness of the
proposed bit-decomposed metric against system imperfection.
Section IV provides a realization structure for our proposed
bit-wise decomposition of M-ary symbol metric. Concluding
remarks appear in Section V.
II. SYSTEMATIC BIT-WISE DECOMPOSITION OF M-ARY
SYMBOL METRIC
Denote by r = (r1,r2,...,rK) the real-valued received
vector when M-ary symbols s = (s1, s2, ..., sK) that are
mapped from an interleaved version of encoding output c =
(c1,c2,...,cN) ∈ {0,1}Nare transmitted. Assume that the
M-ary symbol transmission suffers additive white Gaussian
noise (AWGN), n1,n2,n3,...,nK, with single-sided noise
power per hertz N0. The received vector r then satisfies
ri= si+ ni
for 1 ≤ i ≤ K. For the AWGN channel, the maximum-
likelihood decision upon the receipt of r is given by:
dML(r)=argmax
s∈SPr{r1,...,rK|s1,...,sK}
1
(πN0)K/2exp
=argmax
s∈S
?
−
K
?
i=0
(ri− si)2
N0
?
=argmin
s∈S
K
?
i=1
(ri− si)2,
(1)
where S represents the set of all possible mappings from the
convolutional codeword in C to its respective symbol word.
Due to the non-linear (e.g., interleaving) relation between
codeword c and transmitted symbol s, Eq. (1) cannot be
equivalently transformed to the sum of code bit metrics. An
approximation is therefore necessary to perform soft-decision
decoding. Our goal then becomes to find a sequence of
function η = (η1,η2,...,ηN) such that the sum of all code
symbol metrics,?K
vector r, where functions η1,η2,...,ηN can be distinct bit-
metric functions.
For clarity and simplicity, we use the 16QAM modulation as
an example for the presentation of our subsequent derivation
in this section. The general results for 64QAM and 256QAM
will be given at the end. Let s(·,·) be the real quadrature
4PAM mapping from the code bit to the transmitted symbol,
and denote by σ(·) the bit-interleaving function mapping.
Then, received vector component ri is given by the sum of
transmitted symbol s(cσ(2i−1),cσ(2i)) and noise ni. Here, the
in-phase and quadrature components are independently treated
as received scalars due to the transmission of real 4PAM
symbols although our system presumes complex 16QAM
constellation.
As ri is only a function of code bits cσ(2i−1)and cσ(2i),
its contribution to the summation in (1) is equal to [ri−
s(cσ(2i−1),cσ(2i))]2. Because r1,...,ri−1,ri+1,··· ,rK are
nothing to do with cσ(2i−1)and cσ(2i), we can simplify our
goal to the finding of functions η2i−1 and η2i such that
η2i−1(cσ(2i−1),ri) + η2i(cσ(2i),ri) well-approximates [ri−
s(cσ(2i−1),cσ(2i))]2for all legal cσ(2i−1) and cσ(2i) in the
i=1(ri−si)2, can be well-approximated by
i=1ηi(ci,r) for every mapping pair (c, s) and the received
?N

Page 3

2744 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
codebook. Since function pairs (η2i−1,η2i) are determined
according to independent and identical statistical structure, it
is reasonable to presume that they are universal for all received
scalars, and hence, we can set , η2i−1= f1and η2i= f2for
some functions f1and f2for 1 ≤ i ≤ K.2
We next address the criterion of “well-approximation”
adopted in this paper.
A. Equating the Coefficients in Squared Error
The criterion we adopt is the minimization of average square
error, namely,
min
f1,f2E
?
?f1(c,r) + f2(¯ c,r) − [r − s(c,¯ c)]2
?2
?
,
(2)
where the expectation is taken over the statistics of c, ¯ c and r. Without
loss of generality, we can re-write f1 and f2 as:3
f1(c,r) =1
2r2+a1,c,rr+b1,c,r
f2(¯ c,r) =1
2r2+a2,¯ c,rr+b2,¯ c,r.
This transforms (2) into minimization of
E
?
?(a1,c,r+ a2,¯ c,r+ 2s(c,¯ c))r +
?b1,c,r+ b2,¯ c,r− s2(c,¯ c)
??2
?
(3)
subjected to a1,c,r, a2,¯ c,r, b1,c,r, b2,¯ c,r.
It is neither practical nor analytically tractable to consider
general coefficient functions, a1,c,r, a2,¯ c,r, b1,c,rand b2,¯ c,r, for
continuous r. Instead, we consider a piece-wise simplification
of them by setting a1,c,r, a2,¯ c,r, b1,c,r and b2,¯ c,r equal to a
constant for r ∈ Iρ = (λρ−1,λρ], where 1 ≤ ρ ≤ q and
−∞ = λ0 ≤ λ1 ≤ ··· ≤ λq = ∞. Apparently, there will
be totally 4q constants and (q − 1) interval thresholds to be
determined. For notational convenience, we use a1,c,ρ, a2,¯ c,ρ,
b1,c,ρ and b2,¯ c,ρ to denote the constant values specified for
each interval. This reduces (3) to the minimization of
q
?
?b1,c,ρ+ b2,¯ c,ρ− s2(c,¯ c)??2???r ∈ Iρ
a2,¯ c,ρ+2s(c,¯ c) and b1,c,ρ+b2,¯ c,ρ−s2(c,¯ c), can be made zero
by some specific {(a1,c,ρ,a2,¯ c,ρ)}q
for some finite q, Eq. (4) will be exactly zero. As a result, we
list the coefficients in (4) according to the value of s(c,¯ c),
and let them be zero.
ρ=1
Pr{r ∈ Iρ}E [[(a1,c,ρ+ a2,¯ c,ρ+ 2s(c,¯ c))r+
?
.
(4)
Intuitively from (4) , if all coefficients of r, namely, a1,c,ρ+
ρ=1and {(b1,c,ρ,b2,¯ c,ρ)}q
ρ=1
a1,0,ρ+ a2,0,ρ+ 2 · s(0,0) = 0, b1,0,ρ+ b2,0,ρ− s2(0,0) = 0
a1,0,ρ+ a2,1,ρ+ 2 · s(0,1) = 0, b1,0,ρ+ b2,1,ρ− s2(0,1) = 0
a1,1,ρ+ a2,1,ρ+ 2 · s(1,1) = 0, b1,1,ρ+ b2,1,ρ− s2(1,1) = 0
a1,1,ρ+ a2,0,ρ+ 2 · s(1,0) = 0, b1,1,ρ+ b2,0,ρ− s2(1,0) = 0
It can be seen that the solution should be a function of the
mapping s(·,·) adopted.
2As a consequence, the number of distinct functions among η1,η2,...,ηN
is m = log2(M) for real M-PAM modulation, which will be denoted by
f1,f2,...,fm in the sequel. These bit metric functions, f1,f2,...,fm,
will be applied to decompose the symbol metric for every QAM quadrature
component symbol received.
3The optimal f∗
(f∗
(f∗
of (1/2)r2+ a1,c,rr + b1,c,r does cover the optimal solution.
1(c,r) can apparently be re-expressed as (1/2)r2+
1(c,r)/r − r/2)r + 0. Therefore, as long as the coefficient a1,c,r =
1(c,r)/r − r/2) and b1,c,r= 0 are allowed to be r-dependent, the form
A common mapping used in practice for PAM modulation
is the Gray code mapping. For 16QAM quadrature component,
we can number the equations according to ascending value of
s(·,·)4and yield:
a1,0,ρ+ a2,0,ρ− 6 = 0,
a1,0,ρ+ a2,1,ρ− 2 = 0,
a1,1,ρ+ a2,1,ρ+ 2 = 0,
a1,1,ρ+ a2,0,ρ+ 6 = 0,
b1,0,ρ+ b2,0,ρ− 9 = 0
b1,0,ρ+ b2,1,ρ− 1 = 0
b1,1,ρ+ b2,1,ρ− 1 = 0
b1,1,ρ+ b2,0,ρ− 9 = 0
(5a)
(5b)
(5c)
(5d)
It can be easily verified that at most three of the four
equations (5a)–(5d) can be made valid simultaneously. In
addition, it is not necessary to equate coefficient equations
with non-contiguous equation numbers (e.g., (5a), (5c) and
(5d)) because the squared error in (4) can be further reduced by
replacing the non-contiguous numbered equation (e.g., (5a))
with one having contiguous equation number (e.g., (5b)). This
suggests that among those cases we tried, the best choice
that minimizes (4) for Gray code mapping is to take q = 2,
for which coefficient constants for ρ = 1 corresponds to
the validity of (5a)–(5c), and those for ρ = 2 are selected
to validate (5b)–(5d). By experimenting over 64QAM and
256QAM with Gray code mappings, we found that the same
observation is also applied.
Based on the above finding, we propose a systematic ap-
proach to define a bit-decomposition of M-ary symbol metric
specifically for Gray code mapping as follows.
Initialization: List and number the coefficient equations in
ascending value of Gray code mapping. (Suppose
there are L of them, which are numbered with
1,2,...,L.)
Step 1. For each 1 ≤ ρ ≤ L, find the largest set Aρ of
equations
• that contains equation ρ (namely, the equation
whose index equals ρ),
• that can be simultaneously made valid, and
• that are contiguous in their equation numbers.
Step 2. Delete all duplicate sets among A1,A2,...,AL.
(Let q be the number of sets remained.)
Step 3. Output the remaining q distinct equation sets.
Step 4. Determine the optimal λ = (λ1,...,λq) that mini-
mizes the average squared error in (4).
Notably, the solution obtained from the above procedure is
only suboptimal in the minimization of average squared error
over all legitimate f1(c,r) and f2(¯ c,r). However, simulations
introduced later show that at BER = 10−5, the piece-wise
simplification of a1,c,r, a2,¯ c,r, b1,c,rand b2,¯ c,ris only 0.83 dB
4For 16 QAM, s(0,0) = −3, s(0,1) = −1, s(1,1) = +1, s(1,0) = +3.

Page 4

CHANG et al.: A SYSTEMATIC BIT-WISE DECOMPOSITION OF M-ARY SYMBOL METRIC2745
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
BER
Eb/No
64QAM
Symbol-ML
ML-bit
ML-bit simplified
Soft-proposed
Soft-TB
Hard
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
4 6 8 10 12
Eb/No
14 16 18 20
BER
256QAM
Symbol-ML
ML-bit
ML-bit simplified
Soft-proposed
Soft-TB
Hard
Fig. 2.System performances under AWGN channels.
and 1.4 dB inferior to the non-achievable5Symbol-ML lower
bounds under 64QAM modulation and 256QAM modulation,
respectively (cf. Fig. 2). The results indicate that even if the
optimal coefficient functions for continuous r are adopted, the
performance improvement is very limited. This empirically
justifies the use of piece-wise simplifications of the coefficient
functions.
B. Bit-Decompositions of 16QAM, 64QAM and 256QAM
Symbol Metrics
Applying Step 1 of the proposed algorithm to (5a)–(5d)
results:
A1
={(5a), (5b), (5c)}
A2
={(5a), (5b), (5c)}
A3
={(5b), (5c), (5d)}
A4
= {(5b), (5c), (5d)}.
5As mentioned earlier, with a nonlinear mapping between codeword c and
symbol word s, a receiver can no longer implement Eq. (1). However, by
turning off the interleaver, which in turn linearizes the relation between c
and s, a M-ary modulated (2,1,6) convolutional codeword can be decoded
by tracing over the code trellis in which each branch metric is determined by
exactly one received symbol. Notably, the performance of the symbol-based
ML decision with disabled interleaver is an apparent performance lower bound
to any bit-demapped schemes under AWGN channels. Thus, by means of the
symbol-ML performance curve, one can tell that the improvement margin
for a bit-demapped scheme is limited, if it already performs close to the
symbol-based ML decision. As a disabled interleaver is not a valid system
option for IEEE 802.11a/g system, we therefore term this performance bound
non-achievable, and only illustrate it for the sake of comparison.
Notably, A1, A2, A3 and A4 must respectively contain
equation (5a), (5b), (5c) and (5d) according to the algorithm.
Since A1 = A2 and A3 = A4, Step 2 yields q = 2. The
coefficients that validate the equations in sets A1and A3are
respectively:
⎧
⎪
⎪
⎪
Accordingly, with uniform distributed c and ¯ c, the average
squared error becomes:
??λ1
?λ2
16
√πN0
−∞
?∞
Taking the derivative of (6) with respective to λ1 then
concludes that the optimal λ1 that minimizes W(λ) is zero,
and the average squared error is reduced to
?
We summarize the derivation above as follows.
1
2r2+ a1,c,ρr + b1,c,ρ
=
2r2+ [2(1 − 2c) − a2,1,ρ]r + (1 − b2,1,ρ)
f2(c,r)=
2r2+ a2,c,ρr + b2,c,ρ
=
2r2+ [a2,1,ρ− 4(1 − c) · sgn(r)]r
+[b2,1,ρ+ 8(1 − c)],
where
⎧
⎩
we can remove the constants in?N
that depend on c), and since a universal scaling on all bit-
decomposed metric functions also preserves the decoding re-
sult, the bit-decomposed metric functions can be equivalently
reduced to:
A1(ρ = 1)
:
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
⎩
⎨
⎧
a1,0,1= 2 − a2,1,1
a1,1,1= −2 − a2,1,1
a2,0,1= 4 + a2,1,1
b1,0,1= b1,1,1= 1 − b2,1,1
b2,0,1= 8 + b2,1,1
a1,0,2= 2 − a2,1,2
a1,1,2= −2 − a2,1,2
a2,0,2= −4 + a2,1,2
b1,0,2= b1,1,2= 1 − b2,1,2
b2,0,2= 8 + b2,1,2
A3(ρ = 2)
:
W(λ)=
1
4√πN0
λ0
(8r)2e−(r−3)2/N0dr+
λ1
(−8r)2e−(r+3)2/N0dr
??λ1
r2e−(r+3)2/N0dr
?
=
r2e−(r−3)2/N0dr+
λ1
?
.
(6)
8(18 + N0)erfc
3
√N0
?
− 48
?
N0
π
exp
?
−9
N0
?
.
f1(c,r)=
1
1
1
sgn(r) =
⎨
1,
0,
−1,
if r > 0;
if r = 0;
if r < 0
and
ρ =
?
1,
2,
if r ≤ 0;
if r > 0,
and a2,1,1,a2,1,2,b2,1,1,b2,1,2 can be any fixed values. Since
i=1ηi(ci,r) without affect-
ing the decoding result (namely, we can keep only those terms
f16QAM
1
f16QAM
2
(c,r)
(c,r)
=
=
c|r| · sgn(−r)
c(|r| − 2).

Page 5

2746 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 10, OCTOBER 2006
Consequently, we replace [ri − s(cσ(2i−1),cσ(2i))]2
f16QAM
1
(ri,cσ(2i−1)) + f16QAM
sgn(−ri) + cσ(2i)(|ri| − 2) for cσ(2i−1),cσ(2i−1)∈ {0,1} for
which a decoding algorithm like Viterbi decoding becomes
applicable.
We can similarly obtain the equivalent bit-decomposed
metric functions for 64QAM (where the images of quadrature
component symbol mapping include −7, −5, −3, −1, 1,
3, 5 and 7) and 256QAM (where the images of quadrature
component symbol mapping include −15, −13, ..., 13 and
15) under the IEEE 802.11a system setting6as:
by
2
(ri,cσ(2i)) = cσ(2i−1)|ri| ·
?
?
?
and
f64QAM
1
f64QAM
2
f64QAM
3
(c,r)
(c,r)
(c,r)
=
=
=
c(|r − 4| + |r| + |r + 4| − 8) · sgn(−r)
f16QAM
1
(c,4 − |r|)
f16QAM
2
(c,|r| − 4)
?
?
?
?
?
?
?
?
?
We end this subsection by noting that the above result suggests
a recursive bit-metric decomposition formula. Specifically, for
M-ary PAM modulation (or equivalently, M2-QAM) with
amplitude spacing u,7
?
?
f256QAM
1
(c,r)=c(|r − 4| + |r| + |r + 4| − 8) · sgn(−r)
+c(|r − 8| + |r + 8| − 16) · sgn(−r)
f64QAM
1
(c,8 − |r|)
f64QAM
2
(c,|r| − 8)
f64QAM
3
(c,8 − |r|)
f256QAM
2
f256QAM
3
f256QAM
4
(c,r)
(c,r)
(c,r)
=
=
=
f(m)
1
(c,r)=
c · sgn(−r)
m−2
?
i=−(m−2)
?c,(−1)j?2m−2u − |r|??,
(|r + 2ui| − 2ui)
f(m)
j
(c,r)=
f(m−1)
j−1
where m = log2(M) ≥ 2, 2 ≤ j ≤ m and f(1)
−cr. Note that m is respectively 2, 3 and 4 for 16QAM,
64QAM and 256QAM. Consequently, after specifying the
first-bit metric f1, the subsequent bit-metrics are alternately
the left-shift mirror f(m−1)
j−1
f(m−1)
j−1
Such bit-metric assignment somehow balance the bit reliability
for decoding.
1(c,r) =
?c,2m−2u − |r|?
and right-shift
?c,|r| − 2m−2u?
of the bit-metrics for m less one.
C. Bit Metrics Recursively Generated from Other First-Bit
Metric
In this subsection, we briefly describe some existing struc-
tures of receiver design for M-ary modulated interleaved code.
In 2002, Tosato and Bisaglia [12] has proposed and ex-
amined a simplified soft-output demapper for binary in-
terleaved COFDM with application to HIPERLAN/2 [5].8
We interestingly found that their proposed bit metrics
6The IEEE 802.11a standard did not specify the interleaver for 256QAM
transmission. Here, we simply extend its design philosophy for 16QAM
and 64QAM transmission to obtain an extension interleaver for use of
256QAM transmission. To be specific, 96 256QAM quadrature components
are tabularized in the same fashion as 16QAM, where each component is now
comprised of four bits instead of two bits, and circular shift (from bottom to
top) is repeated (i−1) times for those bits belonging to the same quadrature
component that locates at ith column.
7The amplitude spacing, u, is 2 for all QAM considered in this work.
8Their simplified soft-output demapper has been appeared in a book
published in 1997 [14]. In the book, the soft-output demapper is heuristically
obtained through a direct derivation, as opposed to the simplified-from-LLR-
decision approach taken by Tosato and Bisaglia.
{g(m)
expressed in terms of our recursive formula as:
j
(·,·)}1≤j≤m,m≥1 for 22m-QAM can be equivalently
g(m)
1
g(m)
j
(c,r)
(c,r)
=
=
c(−r)
g(m−1)
j−1
(·,·)}1≤j≤m, m≥1 are actually sim-
(·,·)}1≤j≤m, m≥1 derived from the bit-based log-
likelihood ratio (LLR) decision [12], which can also be re-
expressed using our recursive formula as:
?c,(−1)j(2m−2u − |r|)?.
The bit metrics {g(m)
plified from the bit metrics
{¯ g(m)
j
j
¯ g(m)
1
(c,r)=
1
2· c · sgn(−r)
m−1
?
i=0
(|r + ui| + |r − ui| − 2ui)
¯ g(m)
j
(c,r)=¯ g(m−1)
j−1
?c,(−1)j(2m−2u − |r|)
?.
Note that Tosato and Bisaglia’s simplified formulas (or the
original LLR formulas) are different from our bit metrics only
on the initial functions. This suggests that by varying the first-
bit metric, variants of bit-decomposed metrics can be resulted
through the recursive formula we established. As anticipated,
with f(m)
1
= g(m)
1
for m = 1,2, our bit decomposed metric
coincides with that proposed in [12] for 16QAM, but is
different for 64QAM and 256QAM modulations.
Next, we briefly describe the so-called ML bit demapper [3,
Formula (7)], which operates according to the maximum like-
lihood criterion. It follows the convention that decoding in bit-
interleaved coded modulation is performed by applying an ML
bit demapper followed by de-interleaving and Viterbi decod-
ing. A suboptimal simplified branch metric that is obtained by
the log-sum approximation, i.e., log(?
(SNR), is also given in [3, Formula (9)]. We found that
the ML bit demapper metric {¯ q(m)
simplification {q(m)
be expressed in recursive forms as follows.
jZj) ? maxjlog(Zj),
as typically occurs in channels with high signal-to-noise ratio
j
(·,·)}1≤j≤m,m≥1 and its
j
(·,·)}1≤j≤m,m≥1for 22m-QAM can also
¯ q(m)
1
(c,r)=
2m−1
?
¯ q(m−1)
j−1
+¯ q(m−1)
j−1
2m−1−1
k=1
exp
?
−(r − (2k − 1)(2c − 1))2
2σ2
?c,2m−2u + r?
?
¯ q(m)
j
(c,r)=
?c,(−1)j?2m−2u − r??
(|r − 2(2c − 1)k| − 2k)
−2m(2c − 1)r + 1
q(m−1)
j−1
−2m|r| + 22(m−1),
q(m)
1
(c,r)=
−2
?
k=1
q(m)
j
(c,r)=
?c,(−1)j(2m−2u − |r|)?
where σ2is the variance of the additive white Gaussian noise
and m = log2(M) ≥ 2, 2 ≤ j ≤ m.
D. Complexity Comparison of the Metrics Introduced Previ-
ously
All the metrics introduced in the previous subsection are
piece-wise linear functions of r for given c except the ML bit
demapper {¯ q(m)
linear function in r. For this reason, it is indicated in [3]
j
(·,·)}1≤j≤m,m≥1, which is an apparent non-