Turbo Multiuser Detection for Coded DMT VDSL Systems * Huaiyu Dai, Student Member, IEEE, and H. Vincent Poor, Fellow, IEEE Department of Electrical Engineering, Princeton University Princeton, NJ 08540 Tel: (609)258-4634 Fax: (609)258-1560 Email: [email protected], [email protected]ABSTRACT In recent years, iterative processing techniques with soft-in/soft-out (SISO) components have received considerable attention. Such techniques, based on the so-called turbo principle, are exemplified through turbo decoding, turbo equalization and turbo multiuser detection. In this paper, turbo multiuser detection is applied to a discrete multitone (DMT) very-high-rate digital subscriber line (VDSL) system to combat crosstalk signals and to obtain substantial coding gain. The proposed iterative DMT receiver is shown to achieve an overall 7.0 dB gain over the uncoded optimum receiver at a bit error rate of 7 10 - for a channel with severe intersymbol interference and additive white Gaussian noise and with one dominant crosstalk signal. Impulse noise is detrimental to the proposed scheme but can be overcome through erasure decoding techniques, as is shown by example. Index Terms Coded DMT, Crosstalk, Gray coding, Impulse noise, Multiuser detection, Turbo decoding, VDSL. * Manuscript received Jan. 5, 2001; revised May 23 and June 28, 2001. This research was supported by the National Science Foundation under Grant CCR-99-80590
Transcript
Turbo Multiuser Detection for Coded DMT VDSL Systems∗ Huaiyu Dai, Student Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Department of Electrical Engineering, Princeton University Princeton, NJ 08540
∗ Manuscript received Jan. 5, 2001; revised May 23 and June 28, 2001. This research was supported by the National Science Foundation under Grant CCR-99-80590
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I. Introduction*
* This paper was partly presented at the 2001 Conference on Information Sciences and Systems, held at The Johns Hopkins University, Baltimore, MD, Mar. 21-23, 2001.
Digital Subscriber Line (DSL) technology provides transport of high-bit-rate digital information over telephone
subscriber lines. Various DSL techniques (Basic Rate ISDN, HDSL, ADSL, and VDSL) involving sophisticated
digital transmission schemes and extensive signal processing have recently become practical due to advances in
microelectronics. The latest in DSL technology is very-high-rate DSL (VDSL), which provides tens of megabits per
second to those customers who desire broadband entertainment or data services. At such high rates, signals on
twisted pairs can be reliably transmitted at most to a few thousand feet. Thus, VDSL will primarily be used for loops
fed from an optical network unit (ONU) or a central office (CO) to a customer premises, i.e., it addresses the so-
called “ last mile” problem. The modulation scheme for VDSL can either be multicarrier-based or single carrier-
based, typically discrete multitone (DMT) and carrierless amplitude/phase modulation (CAP)/quadrature amplitude
modulation (QAM). The duplexing methods can be either time-division duplexing (TDD) or frequency-division
duplexing (FDD) [9], [20].
Intersymbol interference (ISI) is one of the major obstacles to high-data-rate, bandwidth-efficient communications.
Multicarrier modulation (MCM), following Shannon’s optimum transmission suggestion, achieves the highest
performance in channels with ISI. DMT is a particular form of MCM that has been found to be well suited for DSL
application and is adopted in ANSI T1.413 ADSL standards. With this approach, a channel is divided into many
independent ISI-free subchannels in the frequency domain, and power and bits are allocated adaptively according to
the channel characteristics [8], [20]. The advantages of using DMT for VDSL include optimality for data
transmission, adaptivity to changing environments and flexibility in bandwidth management.
Normally VDSL signals occupy the band 300 KHz to 30 MHz within the twisted-pair bandwidth, and are separated
from POTS/ISDN signals by splitter devices. Noise on a phone line usually occurs because of imperfect balance of
the twisted pair. There are many types of noises that couple through imperfect balance into the phone line, the most
common of which are crosstalk noise, radio noise and impulse noise. While the radio noise problem can be solved or
at least alleviated by restricting VDSL transmission within radio bands, crosstalk and impulse noise are two
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principal sources of degradation in VDSL transmission systems. The traditional single-user detector (SUD) for such
systems merges crosstalk into the background noise, which is assumed to be white and Gaussian. Actually, crosstalk
is the result of the sum of several filtered discrete data signals. Its distribution deviates from Gaussian, and its power
spectral density (PSD) is significantly greater than that of background additive white Gaussian noise (AWGN).
Recent research has explored the nature of crosstalk signals and has shown the potential benefits of robust multiuser
detection to jointly mitigate crosstalk and impulsive noise for contaminated VDSL signals [7], [10].
Coding is a common way to reduce the gap in channel capacity experienced by uncoded systems. A concatenated
coding scheme consisting of an inner trellis code (a 4-D Wei’s code) and an outer Reed-Solomon (RS) code was
proposed for ADSL DMT systems to provide a 5 dB coding gain at bit error rate (BER) 10-7 without bandwidth
expansion [26], [27]. There are two problems with this approach. First, since the constellation size varies from tone
to tone, a time-varying trellis-coded modulation (TCM) encoder is required. Second, further improvement is very
difficult from a practical implementation perspective because of the complexity of Viterbi decoding for
multidimensional TCM. More recently, powerful turbo coding has been proposed for DMT systems [2], [13], [16],
[22]. One typical turbo code is used to code across all subchannels. The coded bits are then interleaved and allocated
to various tones for quadrature amplitude modulation. Thus, a single standard binary decoder can be employed at the
receiver and further improvement in turbo coding is easily incorporated. A coding gain of 6.0 dB for bandwidth
efficiency of 2 bits/s/Hz and 4.1 dB for 3 bits/s/Hz was reported for a channel with severe ISI at BER 10-5 [16].
In recent years, iterative processing techniques with soft-in/soft-out (SISO) components have received considerable
attention. The basic idea is to break up optimum joint signal processing, e.g. concatenated decoding, joint
equalization and decoding, or joint decoding and multiuser detection (MUD), which are typically very complex and
require large amounts of memory, into separate components, iterating between them with the exchange of
probabilities or “soft” information. This approach typically results in almost no loss of information. This so-called
turbo principle is exemplified through turbo decoding [15], turbo equalization [11] and turbo multiuser detection
[18], [25]. An iterative decoding technique, called soft cancellation, was used to mitigate the effect of home-LANs
on uncoded VDSL systems in [6]. In this paper, we consider the application of turbo multiuser detection in a coded
DMT VDSL system to combat crosstalk and to obtain substantial coding gain. We also consider the effects of
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impulse noise, which has been found to greatly impact the performance of our proposed receiver, and an erasure
decoding technique is proposed as a remedy.
The paper is organized as follows. In Section II a signal model for the DMT VDSL communication system is
described, together with the iterative receiver structure for demodulation (with multiuser detection) and channel
decoding. In Section III, we describe MUD-based schemes for DMT VDSL signal detection, while in Section IV
details of the turbo decoding process are given. Simulation results are given in Section V, and Section VI concludes
the paper.
II. System Description We consider a convolutionally encoded DMT system with crosstalk as shown in Fig. 1. The information bits d are
first encoded into coded bits b with a standard binary convolutional encoder with code rate R. A code-bit interleaver
is used to decorrelate the noise on the coded bits at the input of the channel decoder. The interleaved bits are
optimally allocated to N subchannels and mapped to QAM signals of various constellation sizes. Then the
conjugate-symmetric vector of length NN 2= is transformed using the inverse fast Fourier transform (IFFT) to get
a real time-domain vector. After parallel-to-serial and digital-to-analog conversion, the DMT VDSL signal )(tx is
transmitted into the channel, where it is corrupted by additive coupled crosstalk signals and background noise. At
the receiver end, after analog-to-digital and serial-to-parallel conversion, the received signal )(tr is transformed
back to the frequency domain using an FFT, where it can be written as
i
M
mmimiiii ECFXHY +∑ ⋅+⋅=
=2,, , Ni ,...,1= , (1)
where for the ith subchannel, iH is the channel gain, iX is the transmitted (complex) DMT symbol, miC , is the
mth crosstalk signal, Mm ,,2 �= , miF , is the corresponding crosstalk coupling function, and iE is the
background noise. In this paper, we will primarily assume that the noise iE in (1) is Gaussian in order to focus on
the issue of crosstalk reduction. However, in Section V, we will consider a non-Gaussian model briefly in order to
treat the effects of impulse noise. Output values of the FFT are fed into the demodulator and decoder for further
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processing. Note that when the VDSL signal and crosstalk signals are asynchronous (almost always the case), the
crosstalk coupling functions can vary in time. However, our model considers only one DMT block at a time so we
omit the time index for simplicity. In practice, the crosstalk coupling functions can be estimated for each block to
allow the application of our models.
Figure 2 shows the turbo structure for iterative demodulation and decoding. It consists of two stages: a soft metric
calculator (the demodulation stage) and a SISO channel decoder (the decoding stage). The two stages are separated
by an interleaver and a de-interleaver. The crosstalk signals are first detected via a multiuser detection technique,
discussed in Section III. Then a channel log-likelihood ratio (LLR) for the kth bit carried by the ith subchannel
symbol is calculated as follows:
)})({|1(
)})({|1(log)(
,
,,1 trbP
trbPb
ik
ikik −=
==Λ , (2)
where { r(t)} is the received waveform as shown in Fig. 1(c). Using Bayes’ formula and discarding the common term
)})(({ trp , (2) can be written as
�� ��� ������ ����� ��)(
,
,
)(
,
,,1
,2,1
)1(
)1(log
)1|)}(({
)1|)}(({log)(
ikp
ik b
ik
ik
b
ik
ikik bP
bP
btrp
btrpb
λλ
−==
+−=
==Λ , (3)
where the second term )( ,2 ikp bλ represents the a priori LLR delivered from the decoding stage in the previous
iteration. For the first iteration, this term is set to zero if we assume equally likely coded bits. The first term )( ,1 ikbλ ,
denoting the extrinsic information obtained from the demodulation stage about the bit ikb , , is then de-interleaved
and sent to the channel decoder as its a priori information. Similarly, the SISO channel decoder computes the a
posteriori LLR of each code bit and then excludes the influence of a priori knowledge to get extrinsic information
from the decoding stage about the bit jb as follows:
)(decoding)|1(
decoding)|1(log)()()( 1122 j
p
j
jj
pjj b
bP
bPbbb λλλ −
−==
=−Λ= , (4)
where jb is the de-interleaved version of ikb , , alternatively the coded bits before the interleaver in Fig. 1(a). The
above factorization is derived in Section IV. Again, this extrinsic information is interleaved and fed back to the
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demodulation stage as a priori knowledge for the next iteration. At the last iteration, the SISO decoder also
computes the a posteriori LLRs for information bits, which are used to make final decisions. More details on this
turbo decoding process will be given in Section IV.
III. Mitigation of Crosstalk via Multiuser Detection As we mentioned in Section I, it is possible to apply multiuser detection to jointly detect the VDSL signal and the
crosstalk signals and thereby to greatly improve the system performance. According to the system model given in
Fig. 1, the optimal maximum likelihood multiuser detector (ML-MUD) for Gaussian noise is one that estimates the
VDSL input and crosstalker inputs in unison so as to minimize the distance between the channel output received
signal and all the possible discrete waveform outcomes. Although it is possible that the crosstalk signals are
incorrectly estimated, the probability of erroneous selection of the desired VDSL signal will be lower for such a
detector than if the crosstalk signals are merely absorbed into the background Gaussian noise for detector design.
We would expect a greater improvement in performance using multiuser detection when the difference between the
power spectral density level of the crosstalk signals and that of background noise is larger. There are two types of
crosstalk, the so-called near-end crosstalk (NEXT) caused by signals traveling in the opposite direction as the signal
of interest, and far-end crosstalk (FEXT) caused by signals traveling in the same direction as the signal of interest.
Generally speaking, crosstalk strength increases with frequency: NEXT with 5.1f and FEXT with 2f . Fortunately,
FEXT experiences the same line attenuation as the desired signal; but unfortunately, NEXT does not. For VDSL
systems, high-frequency NEXT is the most detrimental type of crosstalk, and consequently is also the most
promising for reduction via MUD. A typical background noise level in VDSL transmission is –140dbm, while the
typical NEXT is –90 ~ –110dbm; thus we can expect substantial gain from multiuser detection relative to traditional
single user detection in this situation. Moreover, in DMT VDSL subchannels where there are substantially stronger
crosstalk signals (typically in the high frequency bands on long loops), the so-called "near-far" problem of wireless
code-division multiple-access (CDMA) systems, single-user detection (SUD) will fail to work properly while
optimal MUD will essentially achieve the single-user lower bound. Note that the crosstalk signals in DSL
transmission are of various types and cannot be represented under a uniform framework, to the best of the authors’
knowledge. In our application of MUD to signal detection in DSL systems, we deal mainly with NEXT of other
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types (in contrast to the self NEXT coming from the phone lines carrying the same VDSL service) for the following
reason. FEXT experiences the same line attenuation as the desired signal while NEXT does not, which makes NEXT
the most detrimental type of interference, especially at high frequencies. Self NEXT can be largely alleviated by
duplexing methods that separate the upstream and downstream data in time or frequency. Therefore, the other-type
NEXT provides the best opportunity for performance gain. Nevertheless, although we consider other-type NEXT,
multiuser detection is a valid technique for mitigation of crosstalk of all types, albeit modifications of the techniques
proposed here may be necessary for each specific situation.
Let us consider the detection problem for the data model given in (1) for the case of Gaussian ambient noise. The
traditional single user detector demodulates QAM symbols tone-by-tone independently. On the other hand, joint
maximum-likelihood detection of both VDSL and crosstalk signals selects a set of N inputs { }iX and the crosstalk
sequences { })(,
)(,2
)(,1
)( ,...,, lmN
lm
lm
lm CCC=C , Mm ,,2 �= , to satisfy
},minarg{1
2
2
)(,,
}{},{ )(,
∑ ∑ ⋅−⋅−== =
N
i
M
m
lmimiiii
CXi CFXHYX
lmii
Ni ,...,1= , (5)
where the minimization is taken over the DMT signal alphabet and all possible crosstalk sequences
[ ] ,,,2,,...,1,)( Mml ml
mm �=== CCC that can occur within the VDSL symbol period of interest. The size
,,,2, Mmm �=C of the set of all possible crosstalk sequences can be large but is always finite when all the
crosstalkers are digital signals or are derived from digital signals.
Just as its counterpart in wireless CDMA does, the maximum-likelihood multiuser detector achieves optimum
performance but suffers from very high complexity. A full search in the input domain requires approximately
N |C||M| squared-error computations, where N is the number of subchannels, ∏=
=M
mm
2|||| CC is the number of
possible crosstalk sequences, and |M| is the average size of the transmitted alphabet. In practice N and especially |C|
can be very large, introducing prohibitive computational complexity. The large number of possible crosstalk
sequences also means an exponentially greater number of states, making dynamic programming inappropriate.
Therefore, we need to consider a simplified receiver structure that maintains satisfactory performance while
7
requiring far less computational complexity. As we mentioned before, the crosstalk signals in DSL transmission are
of various types and cannot be represented under a uniform framework. The type we examine here is the dominant
near-end QAM-like crosstalk (e. g. [5]).
One lower-complexity approach to MUD is to employ a linear multiuser detection technique, such as decorrelating
(zero forcing) or MMSE multiuser detection. However, unlike CDMA or space-division multiple-access (SDMA)
where linear detection has been effective, there is no identifying signature such as the spreading code for CDMA or
the steering vector for SDMA, to aid linear detection in VDSL. Moreover, the desired signals and crosstalk signals
are often of different data format. An alternative approach that is better suited to this situation is to employ
interference cancellation (IC) [21], i.e., to attempt excision of the crosstalk from the received signal before applying
traditional DMT VDSL signal detection. We adopt this approach here.
The interference cancellation approach is based on a natural idea: if decisions have been made about an interfering
user’s information bits, the interfering signal can be reconstructed and subtracted at the receiver. Of course, this will
achieve perfect interference elimination if the decisions were correct; however, with incorrect decisions things could
be worse than without the canceller. Nevertheless, in VDSL applications, incorrect detection of the crosstalk
sequence may not be as bad as one might think. Dominant NEXTs from other communication systems [4], [5], [6]
usually are detected in the time domain and a DMT symbol interval will contain more than one crosstalk bit. Often
distortion effects in the frequency (resp. time) domain will diffuse in the time (resp. frequency) domain, in which
further gain can be obtained by making hard decisions in the high-SNR scenario. These phenomena make IC a good
choice in this application.
Our interference cancellation multiuser detection (IC-MUD) scheme is described as follows:
1) A hard decision is made on the VDSL signal in the frequency domain. This decision can be obtained
through the received signal TNYY ),,( 1 �=Y directly (for the first iteration) or through soft LLR
values 2 (see (4)) from a SISO decoder.
2) The DMT symbols TNXX ),,(ˆ
1 =X are reconstructed based on these detected bits.
8
3) The desired signal is subtracted and the known crosstalk coupling function is applied to get a
frequency domain estimate of the entire crosstalk sequence TmNmm CC )
~,,
~(
~,,1
�=C via
)ˆˆ(~ 1 ∑−−=
<
−
miiimm CFXHYFC ��� , (6)
where TNHH ),,( 1 =H , T
mNmm FF ),,( ,,1 �=F , TmNmm FF )/1,,/1( ,,1
1 �=−F , iC are formerly
detected and recreated crosstalk signals, and “ � ” denotes Kronecker (elementwise) product. The order
of crosstalk detection and cancellation can be decided according to the relative strength of the different
signals.
4) A time-domain sequence is obtained through the IFFT, )~
IFFT(~mm Cc = , after which hard decisions
are made on the crosstalk symbols and the crosstalk signal is recreated and transformed to the
frequency domain to get TmNmm CC )ˆ,,ˆ(ˆ,,1
����=C .
5) The above process is repeated until all crosstalk signals [ ]Mmm ,...,2,ˆˆ == CC are estimated and
reconstructed.
6) Finally, SUD is used for DMT signal detection, i.e.,
}ˆminarg{2
2,,
}{∑−⋅−==
M
mmimiiii
Xi CFXHYX
i
, Ni ,...,1= . (7)
IV. Turbo Decoding for Coded DMT System In contrast to trellis-coded modulation where coding and modulation are considered jointly to obtain an optimal
signal constellation in the sense of maximum Euclidean distance in the signal space, the iterative decoding process
we propose separates demodulation and channel decoding stages for ease of implementation. The greatest benefits
of this turbo process are the very large increase in code memory due to the interleaver, and almost no loss of
optimality due to the iterative exchange of extrinsic soft information.
Assuming the independence of the background noise between subchannels and between the two dimensions of each
subchannel, the soft metric delivered by the demodulation stage (see (3)) can be expressed as
9
)1()1,(
)1()1,(log
)1|(
)1|(log
)1|)}(({
)1|)}(({log)(
,,
,,
,
,
,
,,1
=−=−==
=
−==
=
−==
=
ikiki
ikiki
iki
iki
ik
ikik
bPbYp
bPbYp
bYp
bYp
btrp
btrpbλ
.)()}ˆ{,|(
)()}ˆ{,|(log
)1()()|(
)1()()|(log
)1(),(
)1(),(log
1:,2,
1:,2,.
1:,
1:,
,1:
,1:
,
,
,
,
,
,
∑ ∏
∑ ∏=
∑ =⋅
∑ −=⋅=
=⋅∑
−=⋅∑=
−= ≠=
= ≠=
−=
=
−=
=
iki
iki
iki
iki
iki
iki
bX klil
Mmmiii
bX klil
Mmmiii
bikiii
bikiii
ikb
ii
ikb
ii
bPCXYp
bPCXYp
bPPYp
bPPYp
bPYp
bPYp
b
b
b
b
bb
bb
b
b
(8)
Here the second equality holds as TNYY ),,( 1 �=Y is a sufficient statistic for )}({ tr and has independent
components. The third equality comes from Bayes’ formula. The iL different bits },,{ ,,1 iLii ibb �=b assigned to a
symbol iX can be modeled as being independent due to the interleaving effect, which leads to the last equality,
where the summations are over all possible DMT symbol realizations with the indicated conditions. Note that, to be
exact, the marginal distribution for the crosstalk signals should be incorporated in the last equality of (8). But this
unrealistic approach would hinder the implementation of the proposed algorithm in practice. Instead, we use the
MUD schemes discussed in Section III to detect the crosstalk signals first and subtract them from the received
signal. For ML-MUD, the crosstalk signals are detected and subtracted from the received signal before iteration
begins. As we can see, ML-MUD makes a thorough search over the product space spanned by the desired VDSL
DMT signal and the crosstalk signals, so the refined estimates from the decoding stage for the data bits would not
help here. However, for IC-MUD, after each iteration, the LLRs from the SISO decoder are used to help better
detect the crosstalk signals, and ultimately obtain a better estimate of the code bits at the soft metric calculator (see
step 1 of the IC-MUD scheme in Section III). For the last expression in (8), we have the formula
))](tanh(1[2
1)( 2, j
pjik bbbP λ+= , (9)
10
where jb is the de-interleaved version of ikb , , and )(2 jp bλ represents the corresponding a priori LLR delivered
from the channel decoding stage in the previous iteration. Invoking the Gaussian assumption on the background
noise, we can also write
2
2
2,,
2
ˆ
22, 2
1)}ˆ{,|( σ
πσ
∑=
=
−−
−
=
M
mmimiiii CFXHY
Mmmiii eCXYp , (10)
where 2σ is the noise variance per dimension. Due to the independence of the in-phase and quadrature data, the
above formulas (8) through (10) can be treated for each dimension separately.
For the channel decoding stage we consider a binary 1/n convolutional code with constraint length υ . The BCJR
decoding algorithm [1] is known to yield the optimal symbol estimate with minimum symbol-error-rate. What is
more important for turbo processing is the BCJR algorithm’s ability to yield soft information in the form of a
posteriori LLRs for coded and information bits. Using the same notation as in [1] and [25], for stage t of the code
trellis transiting from state sSt ′=−1 to sSt = associated with input td and output ),,( 1 nttt bb �=b (where
}{}{ ktj bb ↔ with kntj +−= )1( for a rate- n/1 convolutional code), we have
∑ ′′
∑ ′′=Λ
−
+
−
−
k
k
Sttt
Sttt
kt ssss
ssssb
)(),()(
)(),()(log)(
1
1
βγα
βγα, (11)
where +kS is the set of state pairs ),( ss′ such that the kth coded bit at stage t is 1 and −
kS is the corresponding set
for -1; and
∑ ′′
∑ ′′=Λ
−
+
−
−
k
k
Uttt
Uttt
t ssss
ssssd
)(),()(
)(),()(log)(
1
1
βγα
βγα, (12)
where +kU is the set of state pairs ),( ss′ such that the information bit at stage t is 1 and −
kU is the corresponding
set for -1. The )(stα , )(stβ and ),( sst ′γ terms are defined as follows. ),( sst ′γ denotes the transition probability
for the branch ss →′ for the stage t of the code trellis. As no channel outputs are available for the outer code (the
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convolutional code is treated as the outer code while the QAM modulation on DMT tones is treated as the inner
code for this concatenated system) of a concatenated coding system, we have
))](tanh(1[2
1)()|(),( ,1,
1 11 ik
pik
n
l
n
l
ltttt bbbPsSsSPss λγ +∏ ∏==′===′
= =− , (13)
where ikb , is the interleaved version of ltb , and )( ,1 ik
p bλ represents the corresponding a priori LLR delivered
from the demodulation stage that is related to the state transition of ss →′ . The tα and tβ terms are defined with
forward and backward recursions as
∑ ′′=′
−s
ttt ssss ),()()( 1 γαα , τ,,2,1 �=t (14)
and
∑ ′′=′
++s
ttt ssss ),()()( 11 γββ , 0,,2,1 �−−= ττt (15)
with boundary conditions
,1)0(0 =α and ,0)(0 =sα for 0≠s , (16)
and
,1)0( =τβ and ,0)( =sτβ for 0≠s , (17)
where τ denotes the frame length of the information bits. The summations of (14) and (15) are over all states s′ for
which the transition ss ↔′ is possible. So the extrinsic information (4) produced by the SISO channel decoder
can be written as
)()()()()(
)()()(log)( 12
,11
,11
2kt
pkt
S
n
kll
lttt
S
n
kll
lttt
kt bb
bPss
bPssb
k
k λβα
βαλ −Λ=
∑ ∏′
∑ ∏′=
−
+
≠=−
≠=−
, (18)
where again }{}{ ktj bb ↔ with kntj +−= )1( for a rate- n/1 convolutional code.
The BCJR algorithm is known to have numerical problems associated with the representations of probabilities due
to the large dynamic range of tα and tβ . Thus it is preferable that operations be processed in the logarithmic
domain [19], [23]. With the substitution of )log( tta α= , )log( ttb β= and )log( ttc γ= , (11) can be rewritten as
12
∑ ++−
∑ ++=Λ
−+−−
kk Sttt
Sttt
kt bcabcab )exp(log)exp(log)( 11 (19)
where
∑ ∑+
==′= =
n
l
n
llt
plt
lt
pltl
tt bb
bbbPssc
1 1 1
1 )))(exp(1
))(exp(log())(log(),(
λλ
, (20)
∑ ′+′=′
−s
ttt sscsasa )),()(exp(log)( 1 , (21)
with
,0)0(0 =a and ,)(0 −∞=sa for 0≠s , (22)
and
∑ ′+′=′
++s
ttt sscsbsb )),()(exp(log)( 11 , (23)
with
,0)0( =τb and ,)( −∞=sbτ for 0≠s . (24)
Using the approximation
iii
x xe i max)log( ≈∑ , (25)
Equations (19), (21) and (23) become
[ ] [ ])(max)(max)( 11 tttS
tttS
kt bcabcab
kk
++−++=Λ −− −+, (26)
[ ]),()(max)( 1 sscsasa tts
t ′+′= −′
, (27)
and
[ ]),()(max)( 11 sscsbsb tts
t ′+′= ++′
, (28)
where (27) and (28) can be readily recognized as forward and backward Viterbi algorithms. This simplified
maximum a posteriori probability (MAP) algorithm is the Max-Log-MAP algorithm.
13
Another approximation of the MAP algorithm is the soft-output Viterbi algorithm (SOVA) [14]. In traditional
convolutional decoding, the Viterbi algorithm (VA) is known to be optimal for bit sequence detection and to
perform almost as well as the MAP algorithm but with much lower complexity. One of the drawbacks of the VA is
its inability to produce soft output information, which hurts the performance of systems in a concatenated form. The
SOVA corrects this problem by not only finding the most likely path sequence in a finite state Markov-chain but
also delivering additionally a reliability value for each coded and information bit. The fundamental information for
allocating a reliability value to the choice of one path, the survivor, of the two paths which merge in a node of a
binary code trellis, is the difference between the two accumulated metrics associated with this node. This reliability
