Copyright c 1998 IEEE
Presented at ICC 1998, Atlanta, GA, Paper S27P4
Near-Optimal Data Detection for Two-Dimensional ISI/AWGN Channels
Using Concatenated Modeling and Iterative Algorithms
Xiaopeng Chen and Keith M. Chugg
Abstract | The general problem of performing digital data
detection for a two-dimensional (2D) systems is considered
with the specic application of page-access optical memory
systems in mind. A simple 2D intersymbol interference (ISI)
channel with additive white Gaussian noise (AWGN) is assumed. An equivalent signal model consisting of two onedimensional systems separated by a block interleaver is developed and used to motivate the application of recently developed algorithms for data detection in concatenated systems.
The resulting iterative detection algorithms, which are based
on soft-output algorithms, are demonstrated to achieve nearoptimal performance even for very severe 2D ISI channels.
I. Introduction
Digital data storage systems are key components of modern information systems. Recently, a great deal of attention
has been paid to page-access optical memory (POM) systems
because of their potential for simultaneously achieving high
capacity, fast data transfer and two-dimensional (2D) parallel access. However, the problem of reliable data detection
in POM systems is complicated by imperfect channel characteristics, i.e., various noise sources and limitations from the
optical system itself. Specically, the blur resulting from the
low-pass nature [1] (i.e., nite aperture and aberration effects) of the optical system causes neighboring bits in a data
page to overlap. This results in two-dimensional intersymbol
interference (ISI), as illustrated in Fig. 1. The blurred out1
1
0
0
0 0
c
b
b c
c
0
0
0
0
0
0
(a)
b c
b
0
0
0
0
0
0
(b)
Fig. 1. Eect of 2D ISI: (a) a single `1' on a 2D page without ISI,
(b) observed pattern of the single `1' caused by a 3 3 2D ISI.
put from the 2D ISI channel is also corrupted by observation
noise. In many cases of interest, the observation noise (i.e.,
noise from the detection electronics) may be modeled by additive white Gaussian noise (AWGN). The corresponding 2D
data detection task is therefore a special case of the more general problem: detecting digital page data which have been
corrupted by a nite-size, 2D ISI channel and observed in
AWGN. We address this general problem and note that the
results developed in this paper may be more broadly applicable (e.g., see [2] for applications in image processing).
The recognition of the storage system as a digital communication system [3] encouraged the application of the reThe authors are with Communication Sciences Institute, Electrical
Engineering{Systems Dept., University of Southern California, Los Angeles, CA 90089-2565
The work is supported in part by the National Science Foundation
(NCR-9616663).
lated theoretic principles of signal detection in the digital
communication area to storage problems. The simplest data
detection algorithm is the threshold hard (TH) decision rule,
where a decision on each bit is made based on a single observation pixel. The performance of this straightforward approach, which is optimal for independent, identically distributed (IID) data and no ISI, degradates signicantly when
severe ISI exists. A logical approach to improve performance
when signicant ISI is present is to adapt algorithms that are
eective for one-dimensional (1D) ISI/AWGN channels. In
fact, several such approaches have been suggested. However,
the 2D nature of the data and ISI channel complicates the
detection problem immensely, primarily due to the lack of a
natural order on the 2D index set. As a result, there is no
known method for eciently performing optimal data detection for the 2D ISI/AWGN channel (e.g., minimum symbol
error probability or minimum page error probability). One
approach is to raster the 2D page into a sequence. This leads
to a scattering of the ISI dependence from a small, compact
2D region to a large, sparse region in one dimension. Another is to treat a whole row in the 2D page as a symbol.
This leads to a huge input alphabet. In both cases, the number of states in the Viterbi Algorithm (VA) [4] increases exponentially with the page size. Thus, the application of the
VA is impractical. An attempt to modify the VA for 2D ISI
mitigation was made in [5]. The algorithm developed in [5],
which we refer to as the decision feedback-VA (DF-VA) approach, runs the VA across rows with decisions fedback from
previous rows. The use of row-column iterations of hard decisions and increased state complexity was suggested in [6]
as a modication to the DF-VA. Simpler algorithms based
on 1D linear and decision feedback equalization (DFE) have
also been suggested. Specially, 2D minimum mean-square
error (MMSE) linear equalization with and without decision
feedback was demonstrated in [7].
The approach presented in this paper is based on recent
results in data detection for concatenated systems using softoutput algorithms (SOA) and/or iterative decoding techniques [8], [9]. We present a novel concatenated signal model
for the 2D ISI/AWGN channel whereby the 2D system is precisely modeled as the concatenation of two 1D systems separated by a block interleaver. Based on this concatenated
signal model, we apply SOA-based inner channel processing
with maximum likelihood sequence detection (MLSD) [10]
outer channel processing, as suggested in [11], [8]. The performance of this approach, which is signicantly better than
that of existing approaches, is further improved for severe
channels by using iterative detection techniques. In all cases,
we gauge the performance of these algorithms by using the
performance bounds for optimal maximum likelihood page
detection (MLPD) developed in [12]. The new algorithms de-
veloped approach this ideal performance for ISI channels corresponding to POM's operating near and beyond the Sparrow resolution limit [7].
The paper is organized as follows. The concatenated
model is developed in Section II. In Section III, MLPD
along with two optimal versions of full-record SOA are briey
introduced. The simple concatenated detector structure is
described and its relation to the optimal processing is explained. Finally the detector structure is slightly modied to implement the iterative detection scheme. Two 2D
ISI/AWGN channels are simulated in Section IV. Concluding remarks and a discussion of open issues are contained in
Section V.
II. A Concatenated Model for 2D ISI/AWGN
Channels
x(i; j ) =
L
L
X
X
c
r
m=,Lc n=,Lr
(1)
f (m; n)b(i , m; j , n);
(2)
where fw(i; j )g1 is AWGN with variance 2 , ff (i; j )g : i 2
f,Lc; ,Lc + 1; ; Lcg; j 2 f,Lr ; ,Lr + 1; ; Lr g are the
2D ISI coecients, and fb(i; j )g : i; j 2 f1; 2; ; P g is the
P P matrix of input data drawn from alphabet B.
A simple modeling trick allows the 2D convolution operation to be represented as a concatenation of two 1D nite
state machines (FSM) separated by a block interleaver. Dene the row vector
aj (i) = [b(i; j , Lr ) b(i; j , Lr + 1)
b(i; j + Lr ) ] (3)
as an inner vector symbol. The 2D convolution operation in
(2) may then be rewritten as
P
x(i; j ) = Lmc=,L aj (i , m)(fm )T
(4)
c
where fm = [f (m; Lr ) f (m; Lr , 1) f (m; ,Lr )] and ()T
denotes the transpose operation. This model for the 2D ISI
channel is illustrated in Fig. 2.
The outer FSM simply works as an information lossless
row-wise mapper with memory 2Lr (i.e., analogous to an
error correction code). It maps the original data matrix
fb(i; j )g into a matrix with entries that are the vectors in
(3). When the operation of the inner FSM is executed along
columns as dened in (4), an interleaving operation is implicit. The vector symbols down a given column are statistically independent. The inner FSM may be viewed as
a (2Lc + 1){tap vector ISI channel as dened in (4). This
concatenated signal model is precisely equivalent to the 2D
ISI/AWGN model in (1) and (2). The reason for introducing
this model is that it naturally leads to algorithms analogous
to those used for the decoding of concatenated codes.
1 We use fd(i; j )g to denote a matrix (page) of data
Interleaver
Mapper
Input
Mapper
aj(i)
b(i,j)
Mapper
Inner FSM
ISI
ISI
ISI
Noise
w(i,j)
Observation
z(i,j)
Fig. 2. A concatenated signal model for 2D ISI/AWGN channels.
The 2D ISI/AWGN channel can be modeled as a 2D convolution operation with additive observation noise:
z (i; j ) = x(i; j ) + w(i; j )
Outer FSM
III. Detection Algorithms
A. MLPD and Performance
Extending MLSD to the 2D ISI/AWGN case is conceptually simple. One may consider the MLPD algorithm which
compares all possible data pages to nd the one (f^b(i; j )g)
which maximizes the data page likelihood (i.e., minimizes
kb(i; j ) f (i; j ) , z (i; j )k2). If each pixel in a (2P P ) data
page can take on M values, then there are M P hypotheses
to be compared. Thus, MLPD is not practical for reasonable page sizes. However, it is useful to nd the achievable
optimum performance in the sense of minimum page error
probability. The traditional MLSD performance bounds [10]
have been extended to MLPD in [12]. These bounds provide
a measure of the achievable performance for any 2D detection
algorithm.
B. Concatenated Detector and Soft-Output Algorithm
The concatenated signal model developed in Section II is
exactly the same as the system considered in [8]. This suggests the use of the concatenated detector structure shown
in Fig. 3 directly to 2D data detection. In this concatenated detector, an SOA serves as the inner detector and provides an a-posterior probability (APP) information packet
for each inner symbol aj (i). The outer VA which performs
MLSD utilizing these APP's to detect the outer symbols.
The parallel nature of 2D systems makes the choice of fullrecord (FR) (i.e., type-I in [11]) APP algorithms (e.g., the
BCJR algorithm [13]) over xed-delay (FD) (i.e., type-II)
algorithms [14], [11], [9] reasonable. This is because all
of the observations become available at the receiver at one
time. However, our simulation results [14] suggest that FDAPP algorithms with small smoothing depths perform as
well as the FR-APP algorithms. For purposes of notational
simplicity and to aid in the later development of the iterative detector, the FR-APP is described here briey. Given
an observation sequence 2 zK1 , the FR-APP computes the
likelihood (dk ) = Pr[zK1 jdk ] of an input symbol dk . In
the case where the APP algorithm is applied to the inner channel of the concatenated system in this section, d
2 The notation zk2 is used for the sequence k1 k1 +1
k2 .
k1
fz
;z
;;z
g
A-priori
Information
Pr[a j(i)]
Observation
z(i,j)
Inner Detector
APP
APP
APP
VA
γ [a j(i)]
VA
Decision
^b(i,j)
VA
De-interleaver
Outer Detector
Fig. 3. Simple concatenated detector structure for 2D ISI/AWGN
channels. Note that APP algorithms (inner detectors) could be
replaced by MSM algorithms.
and zK1 corresponds to the vector symbol a and columns
in fz (i; j )g, respectively. For the case where an APP algorithm is used as the outer detector in Section III-D, d
stands for the symbol b. Assuming the memory length of the
FSM is L, we denote its state as sk = fdk,L ; ; dk,1 g and
transition as qk = fsk ; dk g. We also dene the causal likelihood (sk ) = Pr[zk1,1 ; sk ] and the anti-causal likelihood
(sk ) = Pr[zKk jsk ] of the state sk and the \intrinsic likelihood" of transition !(qk ) = Pr[qk ; zk1 ]= Pr[zk jqk ]. Under
the assumption that the input is an IID sequence, the state
sequence constructs a Markov chain. The operation of FRAPP algorithms can be summarized as3 :
P
(sk+1 ) = sk :sk+1 (sk ) Pr[zk jqk ] Pr[dk ]
P
(sk ) = sk+1 :sk (sk+1 ) Pr[zk jqk ] Pr[dk ]
P
(dk ) = qk :dk (sk ) Pr[zk jqk ] (sk+1 )
!(qk ) = (sk ) Pr[dk ] (sk+1 )
(5a)
(5b)
(5c)
(5d)
where Pr[dk ] is the a-priori information. In the FR-APP algorithm, the forward recursion (5a) and backward recursion
(5b) are performed to compute (sk ) and (sk ) recursively.
These two likelihoods are used to complete the soft outputs
via (5c) and (5d). Note that (5d) is only executed at the
outer detector in the iterative structure mentioned later.
As mentioned previously, the sequence faj (i)gPi=1 is IID.
However, if we consider the sequence of aj (i) wrapping
around columns (i.e., as per the interleaver model), this sequence is not IID { e.g., there is clearly correlation between
aj (i) and aj ,1 (i). If the APP for the inner FSM is computed
based only on the associated row (i.e., Pr[fz (i; j )gPj=1jaj (i)])
then this may be completed in parallel for all columns. This
approach, precisely that suggested in [8], [11], is not optimal due to the correlation from column to column. This
fact, which provides the impetus for iterative techniques in
Section III-D, is discussed in detail in [15].
3 The notation a : b means that condition a is consistent with b.
C. Minimum Sequence Metric (MSM) Algorithm
It is well-known that optimal symbol-by-symbol detection
and optimal sequence detection for ISI channels results in
similar performance [16]. Thus, it is reasonable to replace
the likelihood (dk ) by the likelihood of the most likely sequence consistent with the conditional value of dk . This may
be viewed as a generalized likelihood of dk , while (dk ) is
the average likelihood with the nuisance parameter being
fdn gn6=k [14]. The sequence based approach has the advantage that working in the negative-log-likelihood (metric) domain results in simple add-compare-select (ACS) updates in
place of the matrix-vector multiplies in APP algorithms [14].
Thus the term MSM refers to the minimum metric of all
sequences consistent with a conditioned quantity. In the
AWGN case, the transition metric is generated from the corresponding transition probability Pr[zk jqk ] by:
Me[zk jqk ] = ,1 log(Pr[zk jqk ]) + 2 = [zk , xk (qk )]2 (6)
where 1 > 0 and 2 are constants that are selected to simplify the calculations. The execution of the FR-MSM can be
summarized by:
A(sk+1 ) = minsk :sk+1 (A(sk )+Me[zk jqk ]+Me[dk ]) (7a)
B (sk ) = minsk+1 :sk (B (sk+1 )+Me[zk jqk ]+Me[dk ]) (7b)
,(dk ) = minqk :dk (A(sk )+Me[zk jqk ]+ B (sk+1 ))
(7c)
(qk ) = A(sk )+Me[dk ]+ B (sk+1 )
(7d)
All the metric quantities are just the metric version of the
probability quantities in FR-APP (5). Note that the FRMSM actually performs two independent VA's, one forward
(7a) and one backward (7b), with the MSM ,(dk ) and (qk )
computed by adding the survivor metrics of the two VA's (although no survivors need be stored). Therefore, the MSM
algorithm has complexity slightly less than twice that of the
corresponding VA (no trace-back step). MSM algorithms
have been demonstrated to performance closely as APP algorithms for the xed-delay case in [11] and [14] (e.g., the
suboptimal SOA (SSA)).
D. Near-Optimal Iterative Detection for 2D Data
Motivated by the suboptimality of the concatenated detector structure, an iterative structure is developed in this
section. As in any iterative algorithm, some a-posterior information needs to be fedback to improve the performance. In
(5) and (7), the a-priori information Pr[dk ] or Me[dk ] about
the input symbols is used in the forward and backward recursions. The IID assumption was used in the inner and outer
detectors. If more reliable a-priori information for the input
symbol dk can be provided somehow, the performance of the
concatenated detector may be improved. The near-optimal
iterative structure is successfully developed here by pursuing
this concept.
In order to generate the information to be fedback in the
iteration, a soft-input FR-APP or FR-MSM algorithm has to
be used for the outer detector instead of the VA used in Section III-B. The execution of such an algorithm is exactly the
Observation
z(i,j)
Inner
Detector
APP
APP
APP
Refreshed
A-priori
Information
(n)
ω [aj(i)]
ID-1
ID-2
APP
γ (n)[aj(i)]
De-interleaver
APP
γ (n)[b(i,j)]
APP
^b(i,j)
Outer Detector
Decision
Fig. 4. Iterative detector structure for 2D ISI/AWGN channels. Note
that APP algorithms at both inner and outer detectors could be
replaced by MSM algorithms. The decision is made only after the
nal iteration.
the iterative detector is illustrated in Fig. 4. After completing the nth iteration, the obtained !(n) [aj (i)] or (n) [aj (i)]
is used by the (n + 1)th iteration. The total number N of
iterations may be determined by the trade-o between the
computational complexity and the desired performance. At
the termination of the iterative procedure, the soft output
(N )[b(i; j )] or ,(N ) [b(i; j )] is computed and used to make
data decisions by ^b(i; j ) = arg max (N )[b(i; j )] in APP-based
detection or ^b(i; j ) = arg min ,(N ) [b(i; j )] in MSM-based detection. This detection scheme is referred as ID-1 and is as
suggested in [9].
Also, another feedback scheme (referred as ID-2) can be
used with the iterative detection. Alternatively, we redene
the \likelihood" (dk ) = Pr[dk jzK1 ] in the APP algorithms
and the corresponding metric ,(dk ) in the MSM algorithms.
They are computed by
P
(dk ) = s
(sk+1 ) (sk+1 )
(8a)
,(dk ) = minsk+1 :dk (A(sk+1 ) + B (sk+1 ))
(8b)
Again, the soft output [aj (i)] or ,[aj (i)] at the inner ded
k+1 : k
tector is used as the transition cost at the outer detector.
Using the same FR-APP/MSM in (8) at the outer detector,
we obtain a-posterior information (n) [b(i; j )] or ,(n) [b(i; j )],
and the new refreshed a-priori information is generated as:
Q
!(n) [aj (i)] = jk+=Lj,r Lr (n) [b(i; j )]
P
(n) [aj (i)] = jk+=Lj,r Lr ,(n) [b(i; j )]
(9a)
(9b)
The decision rule after the nal iteration is the same as that
of ID-1.
IV. Numerical Examples
In this section, specic examples are considered with a binary signaling of intensity of 0 or 1, and a square 2D ISI pattern with Lr = Lc = 1. This results in an inner trellis with
(23 )2 = 64 states for each column processor (APP/MSM)
algorithm and a 22 = 4{state outer trellis for each row processor (VA/APP/MSM). ThePbit error rate (BER) is plotted
against the SNR dened by (i;j) f 2(i; j )=22 . The detection is simulated for two channels which are representative
of a POM operating near and beyond the Sparrow resolution limit [7]. The 3 3 2D ISI with a symmetric pattern
of f (1; 1) = c, f (1; 0) = f (0; 1) = b, and f (0; 0) = 1
is used. Channel A is dened by b = 0:181 and c = 0:0327,
corresponding to a Gaussian blur with full ll-factor and a
normalized deviation of b = 0:45 [7]. Channel B is dened
by b = 0:352 and c = 0:0993, corresponding to a similar channel with b = 0:623. The page size simulated was P = 128.
For reference, the MLPD bounds4 [12] and the performance
of the threshold detection, DFE [7] (using 5 5 lter design
and 2-stage decision feedback) and DF-VA [5] approaches are
plotted as baselines.
The simulation results for Channel A are plotted in Fig. 5.
The performance of the concatenated detectors is already
10-1
10-2
Channel A
Bit Error Rate
same as (5) and (7), except that the soft output [aj (i)] or
,[aj (i)] from the inner detector is used in the place of transition probabilities or metrics in (5) and (7). These algorithms
are called soft-input soft-output (SISO) algorithms in [9].
Using a soft-input FR-APP/MSM for the outer detector, we
obtain a-posteriori information !(n) [aj (i)] or (n) [aj (i)] from
the nth iteration. When n = 0, it corresponds to the given apriori information (i.e., IID assumption). This a-posteriori
information is more informative than the given IID assumption, and may be plugged back into the inner detector iteratively as the refreshed a-priori information. The structure of
10-3
MLPD Lower Bound
MLPD Upper Bound
TH
DF-VA
DFE
Concatenated-MSM
ID-1 MSM-1
ID-2 MSM-1
Concatenated-APP
ID-1 APP-1
ID-2 APP-1
10-4
10-5
8
10
12
14
16
SNR(dB)
Fig. 5. The performance of the concatenated and iterative detectors
for channels A by using both APP-based and MSM-based versions.
Note that the ` ' in APP- or MSM- is the number of iterations.
l
l
l
quite close to the MLPD bound. Thus, further iterations
do not help signicantly. Also, we note that MSM-based
algorithms work actually as well as APP-based algorithms.
Although the performance of TH is poor, the performance of
the DFE and DF-VA for Channel A is close to the bound. At
a BER of 10,3, with respect to DFE, there is almost no gain
4 The MLPD bounds are based on a nite error pattern search and
are therefore numerical approximations to the analytical bounds.
in SNR obtained by the concatenated detection. With one iteration, both ID-1 and ID-2 push the performance to fall between the lower and upper MLPD bound, with < 0:3 dB difference between them(ID-1 being slightly better). Although
there is a 0:6 dB gain in SNR obtained by one iteration, it
is dicult to justify given the additional complexity.
Simulation results for the more severe Channel B are plotted in Fig. 6. The threshold detection is ineective even at
Acknowledgment
10-1
10-2
Bit Error Rate
tector, respectively. All column-wise or row-wise soft decisions may be executed in parallel. Moreover, the forward and
backward recursions in the soft-output algorithms can be run
simultaneously. Thus, a trade-o between the complexity
and speed can be made to suit the particular application.
We also note that there is still a 1 dB dierence at a BER of
10,3 between the bound and best performance achieved for
Channel B. It may be possible to obtain this additional improvement by further rening the iterative procedure and/or
the concatenated system model.
The authors thank Prof. Mark A. Neied and his research
associates at the University of Arizona, with whom we enjoy
an ongoing collaboration on POM-related research. We also
thank Achilleas Anastasopoulos for many helpful discussions.
Channel B
MLPD Lower Bound
MLPD Upper Bound
TH
DF-VA
DFE
Concatenated-MSM
ID-1 MSM-3
ID-2 MSM-3
Concatenated-APP
ID-1 APP-3
ID-2 APP-3
10-3
10-4
10-5
8
10
12
14
16
18
SNR (dB)
Fig. 6. Simulation results for Channel B.
high SNR. Also, the DFE and DF-VA algorithms perform
6.5 dB away from the MLPD bound at a BER of 10,3. The
concatenated detector provide a 2.5 dB improvement relative
to the DF-VA and DFE, which represent the best previouslyknown methods. However, this approach is still 4 dB worse
than the MLPD bound. Iterative detection yields performance that is only 11:7 dB away from the MLPD bound
at a BER of 10,3 with 3 iterations. Additional iteration do
not signicantly improve the performance. While the BER
changes slightly at relatively low SNR when increasing the
number of iterations, this improvement is much more significant at high SNR. Also, we note that the MSM-based and
APP-based algorithms have the same performance, which
is consistent with all of our simulation results. Our experience suggests that MSM-based algorithms can be used in the
place of APP-based algorithms for signicant complexity reduction with almost no degradation in performance. Again,
ID-1 and ID-2 perform closely. Specically, ID-1 is slightly
better at low SNR, while ID-2 is slightly better at high SNR.
V. Concluding Remarks
We have presented a new concatenated signal model and
two iterative detection schemes for 2D data detection. Compared to the theoretical MLPD bounds, the two iterative
detection schemes are nearly optimal with complexity that
grows only linearly with the page size. Specically, in the
case with P P page size and L L ISI size, approximately
P 2 and 2LP 2 memory units are necessary to store the soft
information generated by the outer detector and inner de-
References
[1] J. W. Goodman, Introduction to Fourier Optics. McGraw-Hill,
Inc., 1968.
[2] K. M. Chugg, X. Chen, A. Ortega, and C.-W. Chang, \An iterative
algorithm for two-dimensional digital least metric problems with
applications to digital image compression," submitted to IEEE
ICIP-98, Jan. 1998.
[3] H. Kobayashi, \A survey of coding schemes for transmission
or recording of digital data," IEEE Trans. Commun. Technol.,
vol. COM-19, pp. 1087{1100, Dec. 1971.
[4] G. D. Forney, Jr., \The Viterbi algorithm," Proc. IEEE, vol. 61,
pp. 268{278, Mar. 1973.
[5] J. F. Heanue, K. Gurkan, and L. Hesselink, \Signal detection for
page-access optical memories with intersymbol interference," Appl.
Opt., vol. 35, pp. 2431{2438, May 1996.
[6] C. Miller, B. R. Hunt, M. A. Neifeld, and M. W. Marcellin, \Binary
image reconstruction via 2-D Viterbi search," in IEEE ICIP-97,
(Santa Barbara, CA), Oct. 1997.
[7] M. A. Neifeld, K. M. Chugg, and B. M. King, \Parallel data
detection in page-oriented optical memory," Opt. Lett., vol. 21,
pp. 1481{1483, Sept. 1996.
[8] U. Hansson and T. M. Aulin, \Soft information transfer for sequence detection with concatenated receivers," IEEE Trans. Commun., vol. 44, pp. 1086{1095, Sept. 1996.
[9] S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, \A softinput soft-output maximum a posteriori (MAP) module to decode
parallel and serial concatenated codes," tech. rep., TDA Progress
Report 42-127, Nov. 1996.
[10] G. D. Forney, Jr., \Maximum-likelihood sequence estimation of
digital sequences in the presence of intersymbol interference,"
IEEE Trans. Inform. Theory, vol. IT-18, pp. 363{378, May 1972.
[11] Y. Li, B. Vucetic, and Y. Sato, \Optimum soft-output detection
for channels with intersymbol interference," IEEE Trans. Inform.
Theory, vol. 41, pp. 704{713, May 1995.
[12] K. M. Chugg, \Performance of optimal digital page detection in
a two-dimensional ISI/AWGN channel," Presented at the 30th
Asilomar Conf. on Signal, Systems and Comp., Nov. 1996.
[13] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, \Optimal decoding
of linear codes for minimizing symbol error rate," IEEE Trans.
Inform. Theory, vol. IT-20, pp. 284{287, Mar. 1974.
[14] K. M. Chugg and X. Chen, \Ecient architectures for soft output
algorithms," in Proc. IEEE ICC'98, (Atlanta, GA), June 1998;
Also see \Ecient a-posteriori probability (APP) and minimum
sequence metric (MSM) algorithms," submitted to IEEE Trans.
Commun., Aug. 1997.
[15] X. Chen, A. Anastasopoulos, and K. M. Chugg, \On optimal data
detection in serially concatenated systems," submitted to IEEE
ISIT'98, Nov. 1997.
[16] J. F. Hayes, T. M. Cover, and J. B. Riera, \Optimal sequence
detection and optimal symbol-by-symbol detection: Similar algorithms," IEEE Trans. Commun., vol. COM-30, pp. 152{157, Jan.
1982.