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2. Author's
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Modeling the bit-level stochastic correlation for turbo decoding
Yi-Nan Lin b
, Wei-Wen Hung a,*, Tsan-Jieh Chen a
, Erl-Huei Lu b
a
Department of Electrical Engineering, Mingchi University of Technology, Taipei, Taiwan
b
Department of Electrical Engineering, Chang Gung University, Tao-Yuan, Taiwan
Received 19 February 2006; received in revised form 26 June 2006; accepted 26 June 2006
Available online 24 July 2006
Abstract
After passing a systematic bit through a turbo encoder, the encoding process will introduce some extent of correlation between a sys-
tematic bit and its associated parity bits. However, this correlation is neglected in the subsequent turbo decoding process so as to reduce
its computational complexity. In this paper, we try to explore the feasibility of modeling the bit-level stochastic correlation for the iter-
ative turbo decoding. By properly adjusting the parameter of the correlation model, we can approximate various degrees of the under-
lying correlation within the received codewords. Reduction in bit error rate (BER) then may benefit from a more accurate capture of the
correlation information at the cost of requiring only a small additional computation complexity. Experimental results indicate that incor-
porating the correlation model into the turbo decoding process can achieve better BER performance than that of conventional turbo
decoders over AWGN channels.
Ó 2006 Elsevier B.V. All rights reserved.
Keywords: Turbo encoder/decoder; Bit-level stochastic correlation model; Bit error rate (BER); Computation complexity; AWGN channel
1. Introduction
In recent years, considerable interest has been devoted
to soft-output decoding schemes that achieve near-Shan-
non limit performance. A parallel concatenated convolu-
tional code (PCCC), named Turbo code, was first
proposed in 1993 by Berrou et al. [1,2]. In the seminal
paper, they used a parallel concatenation of two Recursive
Systematic Convolutional (RSC) encoders interconnected
through an interleaver. Each RSC encoder produces a sys-
tematic output that is equivalent to the original informa-
tion sequence, as well as a stream of parity information.
The two parity sequences can then be punctured before
being transmitted along with the original information
sequence to the decoder module. The underlying interleav-
er is designed to make the two encoded data sequences sta-
tistically independent of each other [3,4]. At the decoder
module, two RSC decoders are employed. Each decoder
receives an a-priory soft input and generates an a-posteriori
soft output, which serves as feedback information to the
other decoder. These soft inputs and outputs indicate not
only whether a received bit was a ‘‘0’’ or a ‘‘1’’, but also
the likelihood that the bit has been correctly decoded.
The RSC decoders operate iteratively and the quality of
the a priori information will improve gradually until some
terminating criterion is met.
The effectiveness of the turbo decoding scheme is based
on iterating the maximum a posteriori (MAP) algorithm [5],
applied to each constituent code. In general, the MAP
algorithm is implemented by means of a soft-in soft-out
(SISO) decoder. This SISO decoder computes the a poste-
riori probability (APP), i.e., a reliability value, for each
received information symbol. However, this computation
is extremely complex owing to the multiplications and
exponential operations required for the forward and
backward recursions in the trellis diagram. In order to
reduce the decoding complexity of the MAP algorithm,
0140-3664/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.comcom.2006.06.023
*
Corresponding author. Tel./fax: +886 02 29061780.
E-mail addresses: ynlin@ccsun.mit.edu.tw (Y.-N. Lin), wwhung@
ccsun.mit.edu.tw (W.-W. Hung), spider@sig.com.tw (T.-J. Chen), lueh@
mail.cgu.edu.tw (E.-H. Lu).
www.elsevier.com/locate/comcom
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3. Author's
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researchers have developed other SISO decoders, which are
less complex and can be used instead of the MAP algo-
rithm. Two of such algorithms are the Max-Log-MAP
and the Log-MAP algorithms.
The Max-Log-MAP algorithm was proposed by Koch
and Baier [6] and Erfanian et al. [7]. This technique simpli-
fied the original MAP version by transferring the recur-
sions into the logarithmic domain and by invoking an
approximation for the sake of dramatically reducing the
associated computation complexity. Due to the approxima-
tion, the performance of the Max-Log-MAP algorithm is
sub-optimal compared to that of the MAP algorithm.
However, Robertson et al. [8] presented the Log-MAP
algorithm, which corrected the approximation used in the
Max-Log-MAP algorithm and hence attained a perfor-
mance virtually identically to that of the MAP algorithm
at a fraction of its complexity.
In this paper, we try to modify the branch metrics
involved in the calculations of the forward and backward
recursions employed in the Max-Log-MAP algorithm. This
modification is intended to reflect various extent of bit-level
stochastic correlation within the received codewords. The
paper is organized as follows. We first give a brief review
of the basic iterative decoding scheme of the Max-Log-
MAP algorithm in the next section. The subsequent section
then explains the formulation of modeling the embedded
correlation for a turbo decoder based on the Max-Log-
MAP algorithm. The modified branch metrics for forward
and backward recursions are also described. Finally, in
section 4, simulation results over AWGN channels are
reported to demonstrate the effectiveness of the
Max-Log-MAP-based turbo decoder when the bit-level
stochastic correlation is properly modeled.
2. Review of the Max-Log-MAP algorithms
In the brief review of the Max-Log-MAP algorithm, we
follow the notations used in [9,10]. Given the received
codeword sequence y, the MAP algorithm computes the
a posteriori Log Likelihood Ratio (LLR) L(ukjy) for each
decoded bit uk, where
LðukjyÞ ¼ ln
Pðuk ¼ þ1jyÞ
Pðuk ¼ À1jyÞ
!
: ð1Þ
Incorporating the code’s trellis, this may be written as
LðukjyÞ ¼ ln
P
ðs0;sÞ;uk¼þ1PðSkÀ1 ¼ s0
^ Sk ¼ s ^ yÞ
P
ðs0;sÞ;uk¼À1PðSkÀ1 ¼ s0 ^ Sk ¼ s ^ yÞ
!
; ð2Þ
where Sk is the state at time k, (s0
,s) is the set of ordered
pairs corresponding to all state transition from the previ-
ous state SkÀ1 = s0
to the present state Sk = s caused by in-
put bit uk at some specific values. Using the Bayes’ rule and
splitting the received codeword sequence y into three sec-
tions, i.e., the codeword associated with the present transi-
tion yk, the codeword sequence prior to the present
transition yj<k and the codeword sequence after the present
transition yj>k, we can thus rewrite the individual term:
PðSkÀ1 ¼ s0
^ Sk ¼ s ^ yÞ
¼ PðSkÀ1 ¼ s0
^ Sk ¼ s ^ yj<k ^ yk ^ yj>kÞ;
¼ Pðyj>kjSk ¼ sÞ Á PðSk ¼ s ^ ykjSkÀ1 ¼ s0
Þ Á PðSkÀ1 ¼ s0
^ yj<kÞ;
¼ bkðsÞ Á ckðs0
; sÞ Á akÀ1ðs0
Þ; ð3Þ
where
ckðs0
; sÞ PðSk ¼ s ^ ykjSkÀ1 ¼ s0
Þ
¼PðukÞ Á PðykjxkÞ ð4Þ
is the branch metric that given the trellis was in state
SkÀ1 = s0
at time k À 1, it moves to state Sk = s and the re-
ceived codeword for this transition is yk. Equivalently, the
transition probability values ck(s0
,s) can be calculated from
the product of the a priori probability P(uk) for the input
bit uk, and the conditional joint probability P(ykjxk) that
given the codeword xk = (xk1, . . . ,xkl, . . . ,xkN) associated
with the transition was transmitted we received the code-
word yk = (yk1, . . . ,ykl, . . . ,ykN), where N denotes the
number of bits in each codeword. The forward path metric
is defined as
akÀ1ðs0
Þ PðSkÀ1 ¼ s0
^ yjkÞ: ð5Þ
It means the probability that the trellis is in state SkÀ1 = s0
at time k À 1 and the received codeword sequence up to
this point is yjk, and can be computed recursively as
akðsÞ ¼
X
all s0
akÀ1ðs0
Þ Á ckðs0
; sÞ: ð6Þ
On the other hand, the definition of the backward path
metric is expressed as
bkðsÞ PðyjkjSk ¼ sÞ: ð7Þ
It gives the probability that given the trellis is in state
Sk = s at time k the future received codeword sequence will
be yjk, and can be computed recursively as
bkÀ1ðs0
Þ ¼
X
all s
bkðsÞ Á ckðs0
; sÞ: ð8Þ
From (2) and (3) we can write for the conditional LLR
L(ukjy) of the decoded uk, given the received codeword yk
LðukjyÞ ¼ ln
P
ðs0;sÞ;uk¼þ1akÀ1ðs0
Þ Á ckðs0
; sÞbkðsÞ
P
ðs0;sÞ;uk¼À1akÀ1ðs0Þ Á ckðs0; sÞbkðsÞ
!
: ð9Þ
The MAP algorithm is extremely complex due to the
exponential and natural logarithm operations required to
calculate L(ukjy) using (9). To simplify the calculation,
the Max-Log-MAP algorithm transfers the forward path
metric ak(s), the backward path metric bk(s) and the branch
metric ck(s0
,s) into the log arithmetic domain and then uses
the approximation
ln
X
i
eXi
% max
i
fXig; ð10Þ
where max
i
ðXiÞ means the maximum value of Xi. Then,
with Ak(s), Bk(s) and Ck(s0
,s) defined as follows:
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4. Author's
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AkðsÞ lnðakðsÞÞ; ð11Þ
BkðsÞ lnðbkðsÞÞ; ð12Þ
and
Ckðs0
; sÞ lnðckðs0
; sÞÞ; ð13Þ
we can change (6) and (8), respectively into
AkðsÞ % max
s0
ðAkÀ1ðs0
Þ þ Ckðs0
; sÞÞ ð14Þ
and
BkÀ1ðs0
Þ % max
s
ðBkðsÞ þ Ckðs0
; sÞÞ: ð15Þ
Finally, from (9) we can write for the a posteriori LLR
L(ukjy) which the Max-Log-MAP algorithm calculates
LðukjyÞ % max
ðs0;sÞ;uk¼þ1
ðAkÀ1ðs0
Þ þ Ckðs0
; sÞ þ BkðsÞÞ
À max
ðs0;sÞ;uk¼À1
ðAkÀ1ðs0
Þ þ Ckðs0
; sÞ þ BkðsÞÞ
: ð16Þ
3. The stochastic correlation model
For simplicity, the correlation among the individual bits
yk1, . . . ,ykl, . . . ,ykN within the received codeword yk is
neglected in the turbo decoding. However, various extent
of correlation is incorporated into each transmitted
codeword xk during the encoding process. Thus,
yk1, . . . ,ykl, . . . ,ykN are correlated. Based on the known
fact, the conditional joint probability P(ykjxk) required to
calculate the branch metric ck(s0
,s) is bounded by [11]
YN
l¼1
PðykljxklÞ 6 PðykjxkÞ 6 min
l
fPðykljxklÞg; ð17Þ
where min
l
ðPðykljxklÞÞ means the minimum value of
P(ykljxkl). Ideally, P(ykjxk) would be calculated directly.
One of the possible adequate joint probabilistic models,
which can simulate both the lower and upper bounds of
inequality (17), is defined as
PðykjxkÞ ¼ exp À
XN
l
½À lnðPðykljxklÞÞŠF
#1
F
0
@
1
A; ð18Þ
where 0 P(ykljxkl) 6 1. It can be shown that (18) reduces
to the lower bound of inequality (17), when the parameter
F = 1, and to the upper bound when F fi 1. However, the
exponential and natural logarithm operations involved in
(18) always make above joint probabilistic model
impractical.
From the inequality (17), we can observe that there are
at least two ways of obtaining an approximate of
P(ykjxk). In the first way using the lower bound, the indi-
vidual bits yk1, . . . ,ykl, . . . ,ykN within the received code-
word yk are assumed to be independent or weakly
correlated. The conditional joint probability P(ykjxk) can
be approximated as
PðykjxkÞLB ¼
YN
l¼1
PðykljxklÞ
¼
YN
l¼1
1
ffiffiffiffiffiffiffiffi
2Áp
p
Ár
Áexp À
Eb
2Ár2
Áðykl ÀxklÞ
2
; ð19Þ
where Eb is the transmitted energy per bit and r2
is the
noise variance. Upon substituting (19) in (4), we have the
branch metric using lower bound:
cLB
k ðs0
; sÞ ¼PðukÞ Á PðykjxkÞLB
¼PðukÞ Á
YN
l¼1
1
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á r
Á exp À
Eb
2 Á r2
Á ðykl À xklÞ2
¼PðukÞ Á
1
ð
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á rÞN Á exp À
Eb
2 Á r2
Á
XN
l¼1
ðy2
kl þ x2
kl À 2 Á ykl Á xklÞ
¼CLB Á PðukÞ Á exp
Eb
r2
Á
XN
l¼1
ðykl Á xklÞ
; ð20Þ
where
CLB ¼
1
ð
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á rÞ
N Á exp À
Eb
2 Á r2
Á
XN
l¼1
ðy2
kl þ x2
klÞ
¼
1
ð
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á rÞ
N Á exp À
Eb
2 Á r2
Á
XN
l¼1
ðy2
klÞ þ N
:
ð21Þ
By means of (21), the log branch metrics CLB
k ðs0
; sÞ using the
lower bound in the recursive formulas of (14) and (15) de-
rived for Ak(s) and BkÀ1(s0
), respectively, can be written as
CLB
k ðs0
; sÞ lnðcLB
k ðs0
; sÞÞ
¼ ln CLB Á PðukÞ Á exp
Eb
r2
Á
XN
l¼1
ðykl Á xklÞ
¼^CLB þ ln½PðukÞŠ þ
Lc
2
Á
XN
l¼1
ðykl Á xklÞ; ð22Þ
where ^CLB ¼ lnðCLBÞ and the channel reliability value
Lc = 2ÆEb/r2
. The term ^CLB does not depend on the sign
of the bit uk or the transmitted codeword xk and so is a
constant in (16) and cancels out. The log branch metrics
CLB
k ðs0
; sÞ derived from the lower correlation bound is wide-
ly used in the turbo decoding. This has the advantage of
low model complexity, but the disadvantage losing the
bit-level stochastic correlation among systematic bits and
the associated parity bits.
The second way is to use the upper bound, which is valid
if there is a strong correlation within a codeword. The min-
imum value of fPðykljxklÞ; 1 6 l 6 Ng will dominate the
calculation of conditional joint probability P(ykjxk) and
PðykjxkÞUB ¼ min
l
ðPðykljxklÞÞ: ð23Þ
Assume the ith term P(ykijxki) has the smallest value among
fPðykljxklÞ; 1 6 l 6 Ng, then
PðykjxkÞUB ¼
1
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á r
Á exp À
Eb
2 Á r2
Á ðyki À xkiÞ
2
ð24Þ
and the corresponding branch metric using upper bound:
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cUB
k ðs0
; sÞ ¼PðukÞ Á PðykjxkÞUB
¼CUB Á PðukÞ Á exp
Eb
r2
Á yki Á xki
; ð25Þ
where
CUB ¼
1
ð
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á rÞ
Á exp À
Eb
2 Á r2
Á ðy2
ki þ x2
kiÞ
¼
1
ð
ffiffiffiffiffiffiffiffiffi
2 Á p
p
Á rÞ
Á exp À
Eb
2 Á r2
Á ðy2
ki þ 1Þ
: ð26Þ
By means of (26), the log branch metrics CUB
k ðs0
; sÞ using
the upper bound can be written as
CUB
k ðs0
; sÞ lnðcUB
k ðs0
; sÞÞ
¼ ln CUB Á PðukÞ Á exp
Eb
r2
Á ðyki Á xkiÞ
¼^CUB þ ln½PðukÞŠ þ
Lc
2
Á ðyki Á xkiÞ; ð27Þ
where ^CUB ¼ lnðCUBÞ. The term ^CUB is also regardless of
the sign of the bit uk or the transmitted codeword xk and
so can be considered a constant in (16) and omitted. How-
ever, there are N correlation components contributed to
the calculation of the log branch metrics Ck(s0
,s) in (22)
whereas only one component in (27). It’s suggested to
incorporate a gain factor N into the correlation term in
(27). Thus, we have
CUB
k ðs0
; sÞ ¼ ^CUB þ ln½PðukÞŠ þ
Lc
2
Á ðN Á yki Á xkiÞ: ð28Þ
To normalize the correlation terms for the log branch
metrics using lower bound and upper bound in (22) and
(28), we have
CLB
k ðs0
; sÞ ¼ ^CLB þ ln½PðukÞŠ þ
Lc
2
Á N
Á
1
N
Á yk1 Á xk1 þ Á Á Á þ
1
N
Á ykN Á xkN
ð29Þ
and
CUB
k ðs0
; sÞ ¼ ^CUB þ ln½PðukÞŠ þ
Lc
2
Á N Á ½yki Á xkiŠ: ð30Þ
Observing the difference between (29) and (30), we can find
that the contribution made by each correlation component
yklÆxkl in the correlation term
PN
l¼1ðykl Á xklÞ can be used to
indicate the degrees of assumed bit-level correlation. There-
fore, we can define the correlative log branch metrics
CCORR
k ðs0
; sÞ, which takes account of the correlation be-
tween each systematic bit and the associated parity bits, as
CCORR
k ðs0
; sÞ ¼ ^CCORR þ ln½PðukÞŠ
þ
Lc
2
Á N Á
XN
l¼1
ðll Á ykl Á xklÞ; ð31Þ
where ^CCORR is a constant, and the weights
{ll, 1 6 l 6 N} are non-negative and satisfy the condition
XN
l¼1
ll ¼ l1 þ Á Á Á þ ll þ Á Á Á þ lN ¼ 1: ð32Þ
Apparently, by properly adjusting the weights, we can
achieve various extent of bit-level correlation. When the
weights are equally distributed with fll ¼ 1
N
; 1 6 l 6 Ng,
then the individual bits within a codeword are assumed to
be weakly correlated or even independent. In this case, the
term CCORR
k ðs0
; sÞ can be formulated as CLB
k ðs0
; sÞ in (22). On
the other hand, neglecting the correlation components with
higher conditional probability values and assigning unity
to the weight corresponding to the correlation component
with the lowest conditional probability value can approxi-
mate the strongest bit-level correlation.
4. The effect of correlative turbo decoder
In this section, we conduct a series of experiments to
illustrate the effectiveness of the stochastic correlation
model we proposed for turbo decoding. The system
architecture for adaptation of the correlative turbo
decoder is shown in Fig. 1. Information bits uk are
grouped into blocks of bits and encoded with a turbo
encoder consisting of a parallel concatenation of two
recursive systematic convolutional (RSC) codes. The
RSC component codes are K = 3 codes with generator
polynomials G0 = 7 and G1 = 5 in octal representation.
These generator polynomials are optimum in terms of
maximizing the minimum free distance of the component
codes [12]. The standard interleaver used between the
two component RSC codes is a 1000-bit random inter-
leaver with odd–even separation [13]. Puncturing of par-
ity bits up1
k and up2
k can be performed to achieve the
required one-third code rate. The coded bits are modu-
lated using binary phase shift keying (BPSK) and white
Gaussian noise n(t) with a double-sided power spectral
density of N0/2 is added to the modulated signal s(t).
At the decoder, only eight iterations are carried out, as
no significant improvement in performance is obtained
with a higher number of iterations. When the received
signal r(t) is demodulated, the correlative decoder first
estimates the SNR value ^cð^c ¼ Eb=2 Á N0Þ, and then deter-
mines the channel reliability value Lc and the correlation
weight {ll, 1 6 l 6 N}.
In our simulation, there are two bits within the received
codeword yk, i.e., N = 2. Assume that
Pðyk1jxk1Þ ¼ min
l
ðPðykljxklÞÞ
¼ minfPðyk1jxk1Þ; Pðyk2jxk2Þg ð33Þ
and according to (32) we have
l1 þ l2 ¼ 1: ð34Þ
If we let l1 = l, then
l2 ¼ 1 À l1 ¼ 1 À l ð35Þ
and the correlative log branch metrics CCORR
k ðs0
; sÞ in (31)
can be rewritten as
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6. Author's
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CCORR
k ðs0
; sÞ ¼ ^CCORR þ ln½PðukÞŠ
þ
Lc
2
Á N Á fl Á yk1 Á xk1 þ ð1 À lÞ Á yk2 Á xk2g:
ð36Þ
It is worth to note that the weight l can be treated as the con-
tribution of a correlation component with the lowest condi-
tional probability value. In the case of l = 1, it implies that
there exists strong bit-level stochastic correlation within each
received codeword. When the weight l is set to a smaller val-
ue, the degree of correlation can be further reduced.
4.1. The BERs versus the correlation weights
Fig. 2 shows the performance of a turbo decoder using
the stochastic correlation model versus different values of
the weight l under predetermined Eb/N0 values at À1, 0
and +1 dB. From this figure, it can be seen that the perfor-
mance of the correlative turbo decoder with +1 dB is poor
at small weights, but improves rapidly as the weight is
increased. After a specific weight, the BER performance
degrades again. The similar phenomena can also be found
in the cases of 0 dB and À1 dB. Apparently, the proposed
stochastic correlation model has the effect on the BER per-
formance of a turbo decoder. Fig. 2 reveals a fact that mul-
tiplying a set of weights to the correlation components
fðykl Á xklÞ; 1 6 l 6 Ng in the log branch metrics Ck(s0
,s)
is highly useful in modeling the underlying correlation
and helpful in achieving better BER performance.
4.2. The relation between Eb/N0 value and the correlation
weight
In our simulation, the modulated signal is corrupted by
white Gaussian noise in which the Eb/N0 value ranges from
À2 dB to +3 dB with 0.2 dB increment. The correlation
weight is also increased from 0 to 1 with increment of
0.05. From the experimental results, the relation between
Eb/N0 value and the optimum correlation weight can be
obtained approximately and is illustrated in Fig. 3.
As shown in Fig. 3, the optimum correlation weight that
achieves the best BER performance is heavily related to the
Eb/N0 value of the underlying environment. In the less
noisy condition (Eb=N0 P 1dB), the optimum weight is
set to 0.5. At this time, the correlative log branch metrics
CCORR
k ðs0
; sÞ is reduced to its lower bound CLB
k ðs0
; sÞ that is
equivalent to conventional log branch metrics. When the
transmission channel becomes noisier, larger weight is used
for the stochastic correlation model in order to obtain opti-
mum BER performance.
ku
Puncturing
BPSK
Demodulator
Interleaver
RSC 1
RSC 2
Determine
Correlation
Weighting factor
)(tn
)(tr )(ts
cL SNR
Interleaver
Deinterleaver
Interleaver
kc ,
p
ku
Parallel-to-serial
BPSK
Modulator
ky
Channel SNR
Estimation
Block
kc yL
Serial-to-Parallel
)( ,,2,1 Nμμ μ
s
kc yL
Decoder1
Decoder2
1p
kc yL
2p
kc yL
int
s
yL
12eL
21eL
ku
k
p2
u
u 1
k
p
Fig. 1. The system architecture for adaptation of the correlative turbo decoder.
Fig. 2. The performance of a turbo decoder using the stochastic
correlation model versus different values of the weight l.
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7. Author's
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4.3. The effectiveness of the correlative turbo decoding
Fig. 4 compares the BER performances of the three
cases, i.e., the uncoded case (denoted as ‘‘Uncoded’’), the
turbo decoder based on the Max-Log-MAP algorithm
(denoted as ‘‘TBC’’) and the ‘‘TBC’’ with stochastic corre-
lation model (denoted as ‘‘CTBC’’). In the implementation
of the ‘‘CTBC’’ case, the optimum correlative weight is
adaptively selected block by block according to the follow-
ing relation obtained from Fig. 3.
lopt ¼
0:50 if Eb=N0 0:8dB;
0:55 if 0:8dB P Eb=N0 À0:6dB;
0:60 if À 0:6dB P Eb=N0 À1:2dB;
0:65 if Eb=N0 À1:2dB:
8
:
ð37Þ
From the Fig. 4, we can find that the ‘‘Uncoded’’ and
‘‘TBC’’ cases have an intersection at about Eb/N0 = À0.8
dB. This phenomenon implies that the turbo decoding
based on the Max-Log-MAP algorithm is no longer effec-
tive when Eb/N0 is less than À0.8 dB. In contrast, the inter-
section of the ‘‘Uncoded’’ and ‘‘CTBC’’ cases is at about
À0.95 dB. This result verifies the effectiveness of the pro-
posed stochastic correlation model.
Just as with a conventional turbo decoder based on the
Max-Log-MAP algorithm, the correlative turbo decoder
needs to calculate the logarithmic forward path metrics
Ak(s), the logarithmic backward path metrics Bk(s), and
the logarithmic branch metrics Ck(s0
,s) to find the a-poste-
riori LLR L(ukjy). In addition, the correlative turbo decod-
er requires N additional multiplication operations to
calculate the correlation term Lc
2
Á N Á
PN
l¼1ðll Á ykl Á xklÞ as
shown in (31). For a turbo decoder with code rate of 1/3
(i.e., N = 2), the proposed correlation model introduces
only a little extra computation complexity.
5. Conclusion
During the encoding process, a turbo encoder always
makes a systematic bit and its associated parity bits cor-
related. In this paper, a stochastic correlation model is
proposed to explore the influence of bit-level correlation
on the BER performance of a turbo decoder. By means
of the correlation model, it provides us a useful frame-
work for modeling the underlying stochastic correlation
in a simple way. Experimental results reveal that the
correlation effect has significant impact on the perfor-
mance of a turbo decoder. By properly adjusting the
correlative weights, we can accurately approximate the
extent of bit-level correlation within each codeword
and by which the robustness of a turbo decoder can
be enhanced.
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Fig. 3. The relation between Eb/N0 value and the optimum correlation
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Fig. 4. Comparison of BER performances for the ‘‘Uncoded’’ case, the
‘‘TBC’’ case and the ‘‘CTBC’’ case.
Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862 3861

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Yi-Nan Lin received his B.S. degree from the
Electrical Engineering Department of National
Taiwan Institute of Technology in 1989, and the
M.S. degree in Computer Science Engineering
from the Yuan Ze University in 2000. He joined
the Department of Electrical Engineering at
Mingchi University of Technology, Taishan,
Taiwan, in 1990. He is now a lecturer in the
Department of Electronic Engineering. He is also
a Ph.D. candidate in the Electrical Engineering
Department of Chang Gung University, Taoyu-
an, Taiwan. His current research interests include error-control coding,
and digital transmission systems.
Wei-Wen Hung received his B.S. degree from the
Electrical Engineering Department of Tatung
Institute of Technology in 1986, and the M.S.
and Ph.D. degrees in electrical engineering from
the National Tsinghua University in 1988 and
2000, respectively. He joined the Department of
Electrical Engineering at Mingchi University of
Technology, Taishan, Taiwan, in 1990. He was
the Vice Dean of Student Affairs from 2000 to
2002. He was also the Chairman of Department
of Electronic Engineering in 2003. He is now a
professor in the Department of Electronic Engineering. His current
research interests include speech signal processing, wireless communica-
tion and embedded system design.
Tsan-Jieh Chen is now an undergraduate,
majored in electrical engineering in Mingchi
University of Technology. His current interests
include signal processing, wireless communica-
tion, VLSI design and coding theory.
Erl-Huei Lu received his B.S. and M.S. degrees in
electrical engineering from the Chung Cheng
Institute of Technology, Taiwan, in 1974 and
1980, respectively, and the Ph.D. degree in elec-
trical engineering from the National Cheng Kung
University, Tainan, Taiwan, in 1988. He is now a
professor in the Department of Electrical Engi-
neering, Chang Gung University, Taoyuan, Tai-
wan. His current research interests include error-
control coding, network security and systolic
architectures.
3862 Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862