c
°IEEE
1996
Presented at Asilomar Conf. on Signals, Systems and Comp., Nov. 1996
Performance of Optimal Digital Page Detection in a
Two-Dimensional ISI/AWGN Channel
Keith M. Chugg∗
Department of Electrical Engineering – Systems
University of Southern California
Los Angeles, CA 90089-2565
chugg@milly.usc.edu
Abstract
The performance of Maximum Likelihood Page Detection (MLPD) of digital pages of data corrupted
by intersymbol interference (ISI) and additive white
Gaussian noise (AWGN) is derived. While the development concentrates on the linear ISI channel with
AWGN, the results characterize the performance of the
ML state estimator for a more general class of Markov
Random Fields. Hence, while page detection in optical
memory systems provides the motivation, the results
are applicable to a broader class of problems including
image de-blurring. While MLPD is infeasible, its performance provides a lower bound for the performance
of practical data detection techniques. This utility is
demonstrated through numerical examples.
1
Introduction
This paper contains a development of performance
bounds and approximations for the detection of twodimensional (2D) digital pages of data which have
been corrupted by a finite-length, two-dimensional
intersymbol interference (ISI) channel and observed
in additive white Gaussian noise (AWGN). The primary motivation for this development is provided by
the detection of two-dimensional binary data pages
in page-oriented memories (POMs) implemented via
volume optical systems (e.g., see [1]). While the noise
and interference environment in practical volume optical memory systems is extremely complex and is a
function of many system parameters, this simple ISIAWGN model is representative of at least some designs and has been adopted elsewhere [2, 3]. Rather
than tie this development to this one application, the
purpose of this paper is to address the issue of the
best possible performance of algorithms designed to
mitigate this 2D ISI-AWGN channel. In fact, with
minor modification the development characterizes the
performance of maximum likelihood estimation of a
discrete-valued Markov Random Field (MRF) from an
observation corrupted by AWGN.
The special 1D case of Maximum Likelihood Sequence Detection (MLSD) can be carried out effi∗ This work supported in part by the National Science Foundation (NCR-9616663).
ciently via the Viterbi Algorithm (VA) [5]. However
in 2D, the two-dimensional nature of the index set and
the ISI pattern complicate the detection problem immensely, primarily due to the lack of a natural order
on the 2D index set. One obvious approach is to raster
the 2D page into a sequence. This process inevitably
leads to a scattering of the ISI dependence from a
small, compact 2D region to a large, sparse region in
one dimension. Conceptually, one may consider the
Maximum Likelihood Page Detection (MLPD) algorithm which compares all possible data pages to find
the best. If each pixel in an (N × N ) data page can
2
take on M values, then there are M N hypotheses to
be searched. No computationally efficient algorithm
for conducting this search is known. Suboptimal algorithms have been considered. For example, an algorithm using the VA across rows with decision feedback
from the above previously detected rows was reported
in [2]. A similar algorithm for quantization of grayscale images was developed in [4]. Algorithms based
on the 2D extensions of linear and decision feedback
equalization were successfully demonstrated in [3].
Despite the fact that the MLPD algorithm may
not be implementable, a characterization of its performance is valuable. It provides the designer of detection algorithms based on ad-hoc performance criteria
and/or constrained structures with a measure of the
degradation suffered relative to the optimal detection
algorithm. The main point of this paper is that, while
the VA may not generalize simply to 2D, the analysis of MLSD does generalize. Thus, the performance
analysis in this paper contains the well-known results
of MLSD performance [5, 6, 7, 8] as a special case.
2
Signal Model and Preliminaries
The signal model assumed is the direct generalization of the standard discrete-time (1D) ISI/AWGN
model
z(i, j) = x(i, j) + w(i, j)
x(i, j) = f (i, j) ∗ a(i, j)
X
f (i − k, j − l)a(k, l)
=
(k,l)∈P̄(i,j)
(1a)
(1b)
(1c)
=
X
f (k, l)a(i − k, j − l).
(1d)
(k,l)∈P(i,j)
This model will be assumed to be real-valued for
simplicity, with independent, identically distributed
(IID) digital data1 A = {a(i, j)} and Gaussian noise
{w(i, j)}. The data is assumed to be uniform over a
finite alphabet A with M = |A|. The support region
of the channel P = P(0, 0) is assumed to be finite and
to contain the point (0, 0). The notation P(i, j) represents a “footprint” of the channel shifted to (i, j) –
i.e., it is the region of the page where x(k, l) is nonzero
due to a nonzero data symbol at (i, j). The region
P̄(i, j) is the set of all points (k, l) for which the value
of a(k, l) can affect the value of x(i, j). Graphically,
P̄ can be obtained from P by reflecting around both
the horizontal and vertical axes. A particularly simple case is the square footprint centered at (i, j) – i.e.,
P(i, j) = P̄(i, j) = {(k, l) : |i − k| ≤ L, |j − l| ≤ L}.
For the development that follows, this square footprint could be assumed without loss of generality, since
any non-square footprint could be inscribed by a sufficiently large square pattern. However, more general
footprints are considered to illustrate the importance
of the various steps. Three cases are illustrated in
Figure 1(a)-(b).
The decision metric to be minimized by an
MLPD algorithm is proportional to the negative loglikelihood functional
X
[z(i, j) − x(i, j; Ã)]2 .
(2)
ΛX (Ã) =
∆Λ[P(i,j)]c (Ã(1) ; Ã(2) ) is not a function of a disagreement at (i, j). Thus, the decision at (i, j) can be made
based solely on {z(k, l)}(k,l)∈P(i,j) . The moat of agreements surrounding the point (i, j) in LP (i, j) corresponds directly to “state agreements” in the trellis for
the special 1D case. Specifically, for the 1D case the
pairwise property states that a pairwise comparison
between sequences between two fixed states can be
made with only a finite segment of the received signal
– this is the basis for the add-compare-select step of
the VA for MLSD. There are several equivalent definitions of the influence region, for example
[
P̄(k, l).
(4)
LP (i, j) =
(k,l)∈P(i,j)
The interpretation of (3) and (4) is the same: LP (i, j)
is the set of indices whose data values can cause noisefree channel outputs which overlap with the noise-free
outputs due to the data at (i, j). The influence region for each of the example channels is illustrated in
Figure 1(c).
A third, equivalent definition of the influence region
is given by
LP (i, j) = {(k, l) : ∃ (m, n) such that (k, l) ∈ P(m, n)
and (i, j) ∈ P(m, n)}.
(5)
This implies that
Pr {x(i, j) = v|X − {x(i, j)}}
(6)
ª
©
= Pr x(i, j) = v|{x(k, l)}(k,l)∈LP (i,j)−{(i,j)} .
(i,j)∈X
The most likely page, denoted by Â, is assumed to be
obtained via the (impractical) exhaustive search with
X taken to be the entire page index range. Also, the
shorthand notation ∆ΛX (Ã(1) ; Ã(2) ) = ΛX (Ã(1) ) −
ΛX (Ã(2) ) will be used.
2.1
The Pairwise Property
The “pairwise property” is a formalization of the
intuitive notion that, due to the finite support region
of the channel, a pairwise decision between two subpages can be made when the two pages agree on a
sufficiently large region. To show this, consider the
“influence region” of a particular point in the page.
The influence region of (i, j) is defined as
n
o
\
(3)
LP (i, j) = (k, l) : P(k, l) P(i, j) 6= ∅ .
Consider two data pages Ã(1) and Ã(2) which agree
at all points in LP (i, j) except for the point (i, j).
Even if these two pages disagree outside of LP (i, j),
a pairwise decision can be made for the data point
a(i, j) based only on a finite sub-page of Z. This
is because ∆ΛP(i,j) (Ã(1) ; Ã(2) ) is not a function
of disagreements outside the influence region and
1A
boldface character will be used to represent the entire 2D
page of the corresponding signal.
Thus, x(i, j) is a Markov Random Field [9] (MRF)
due to the finite support region of the ISI. The influence region, excluding the point (i, j), is often called
the neighborhood in the MRF literature. The received
page z(i, j) is also an MRF, although, unlike x(i, j) it
takes values on the continuum. Some special cases of
influence regions were pointed out in [10, Fig. 3], which
deals with classification of binary MRFs with respect
to their underlying distributions. Therefore, with minor modifications, the performance bounds that follow
characterize the performance of the optimal estimator
of an arbitrary discrete MRF corrupted by AWGN.
For example, if the data itself were a MRF (e.g., an
image model), then X would still be a MRF with a different neighborhood (determined by the neighborhood
of the data and the influence region of the channel).
3
Fundamental Error Patterns
MLPD performance bounds follow from the pairwise property and the concepts used in the MLSD
analysis (although several inconsistencies with the
MLSD literature are pointed out in [11]). Consider an
arbitrary point (i1 , j1 ) in the interior of a very large
2D page. The error probability at a given point is
Ps (i1 , j1 ) = Pr {â(i1 , j1 ) 6= a(i1 , j1 )}
n
o
= Pr  − A ∈ G ,
(7)
(8)
(a)
(b)
(c)
(d)
(e)
(i, j) or (i1 ,j 1)
nonzero element of error pattern
X (E)
Figure 1: Example 2D channels: (a) the footprint P(i, j), (b) P̄(i, j), (c) the influence region LP (i, j), (d) a
fundamental error pattern, and (e) an error pattern from G − F.
where G is defined as the set of difference patterns that
result in an error at (i1 , j1 )
G = {E : e(i, j) ∈ ∆A and e(i1 , j1 ) 6= 0},
(9)
where ∆A is the finite set of possible difference symbols. Define a global error event resulting in error
pattern E by
Ê(E) : E = Â − A.
Thus, we have P (i1 , j1 ) = P (ÊG ), where
[
Ê(E).
ÊG =
(10)
(k,l): e(k,l)6=0
(11)
E∈G
Direct evaluation of P (ÊG ) is difficult because it depends on the global (exhaustive search) decision properties. However, performance bounds can be obtained by considering a specific class of pairwise errors.
Specifically, the set of 2D fundamental error patterns
F is defined as a subset of G with elements from ∆A
and properties similar to the fundamental error pages
of the MLSD analysis [11].2 The weight one error patterns from G are included in F. In addition, E ∈ G is
in F if it has the property
∀ e(i, j) 6= 0 ⇒ ∃ (k, l) ∈ LP (i, j) such that
(k, l) 6= (i, j) and e(k, l) 6= 0. (12)
2 The
term “simple” is used in [7].
This definition means that nonzero elements of a fundamental error pattern cannot be isolated from each
other; they must lie in the influence region of another
nonzero element. Examples of fundamental error patterns and members of G − F are shown in Figure 1(d)(e). The important property of these fundamental error patterns is that a pairwise decision can be made between à and Ã+E based on {z(i, j)} for (i, j) ∈ X (E),
with this composite footprint defined by
[
P(k, l).
(13)
X (E) =
The region X (E) contains all of the noise-free channel outputs which differ for the pairwise comparison.
Another region associated with a fundamental error
event is the composite influence region
[
LP (k, l).
(14)
LP (E) =
(k,l): e(k,l)6=0
Define the event that A + E is more likely than the
correct data A by
E(E) : Λ(A) > Λ(A + E).
(15)
The event EF can now be defined as the event that
there is a fundamental pairwise error
[
E(E).
(16)
EF =
E∈F
4
MLPD Performance Bounds
An upper bound on Ps (i1 , j1 ) is obtained by the
following theorem:3
Theorem 1 ÊG ⊂ EF .
Proof: If Ê(E) has occurred, then there is a corresponding EF ∈ F that coincides with E in LP (EF )
and is zero everywhere else. Any disagreements between A and  located outside of LP (EF ) will not
affect the calculation of ∆Λ(·) on X (EF ). Since A+E
is the best global path, it follows that A + EF is more
likely than A:
(17)
= ∆ΛX (EF ) (A; A + E)
(18)
= ∆ΛX (EF ) (A; A + EF )
(19)
= ∆Λ(A; A + EF ).
(20)
The upper bound on Ps (i1 , j1 ) follows directly from
Theorem 1
X
P (E(E)) (21)
Ps (i1 , j1 ) = P (ÊG ) ≤ P (EF ) ≤
X
X
d∈D
where D is the set of possible values of d2 (E) for E ∈
F 0 and
X
w(E)PC (E),
(27)
KU B (d) =
E∈F 0 (d)
0 < ∆ΛX (EF ) (A; Â)
=
by taking exactly one member of F to represent the
class of size w(E) of fundamental error pages that are
equivalent within a shift. This upper bound can be
compacted further by collecting terms with common
d2 (E)
Ãs
!
X
d2
KU B (d)Q
Ps ≤
,
(26)
2
4σw
E∈F
P (E(E)|A)P (A),
(22)
E∈F A∈C(E)
where C(E) is the set of data pages consistent with
E and P (A) need only be computed over the region
where E is nonzero. It is straightforward to show that
!
Ãs
d2 (E)
,
(23)
P (E(E)|A) = Q
2
4σw
with F 0 (d) representing the set of error pages in F 0
with d2 (E) = d2 . An approximation for large signalto-noise ratio is
Ãs
!
2
d
min
,
(28)
Ps ∼
= KU B (dmin )Q
2
4σw
where dmin is the smallest element of D.
Forney provided a simple, valid lower bound for the
1D case in [6]. This bound can be tightened in a trivial
manner. To summarize, consider the optimal detector
which operates with the side information that either A
or A + E, with E selected at random from F 0 (d), was
sent. The error performance of the optimal detector
which is privy to this side information cannot be worse
than that of MLPD
side information, so that
³q without
´
d2
with
Ps ≥ KLB (d)Q
4σ 2
w

KLB (d) = P 
2
σw
is the variance of w(i), Q(·) is the complewhere
mentary distribution function of a unit variance, zero
mean Gaussian random variable, and
X
[e(i, j) ∗ f (i, j)]2 .
(24)
d2 (E) =
(i,j)
Several steps can be taken to simplify the bound in
(22). Note that as the page size becomes asymptotically large, this upper bound does not depend on
(i1 , j1 ). Thus, if w(E) is the number of nonzero elements in E, then for every E ∈ F, there are w(E)
shifts of E and A resulting in the same P (E(E)). Also,
P (E(E)|A) is not a function of the particular A, only
E. Together, these facts imply that
Ãs
!
X
d2 (E)
w(E)PC (E)Q
,
(25)
Ps ≤
2
4σw
0
E∈F
where PC (E) is the sum of P (A) over all transmitted pages consistent with E. The set F 0 is created
3 This approach is similar to that taken in [8] with a
correction.
[

C(E) .
(29)
E∈F 0 (d)
Forney’s version of this bound is with d = dmin . However, a tighter bound can be obtained by maximizing
over d
!
Ãs
d2min
.
(30)
Ps ≥ sup KLB (d)Q
2
4σw
d∈D
For low noise levels, the Forney lower bound and (30)
will coincide. However, if KLB (dmin ) is small, the new
lower bound in (30) is significantly tighter even at relatively low error rates.
5
Numerical Examples
In this section specific examples are considered with
an alphabet of A = {0, 1} and square channel footprints (L = 1). For this case the error alphabet is
∆A = {−1, 0, +1} with PC (0) = 1 and PC (−1) =
PC (+1) = 1/2, so that PC (E) = 2−w(E) . The symbol
error probability is plotted against the signal-to-noise
ratio
P
2
kf k2
var [x(i)]
(i,j) [f (i, j)]
=
=
. (31)
SNR =
2
2
var [w(i)]
2σw
2σw
0
10
The utility of these performance bounds is demonstrated by considering the fairly simple detector
demonstrated in [3] which is similar to a 2D decision
feedback equalizer. For channel A, this suboptimal detector suffers only a 1 dB degradation in SNR relative
to MLPD at a error rate of 10−4 . However, for channel
B, the degradation relative to MLPD at Ps = 10−4 is
9 dB of SNR. Thus, for a POM accurately modeled by
channel A, there is little motivation (i.e., 1 dB from
no ISI) to improve the algorithm (or space the bits
further apart), which is certainly not the case if the
POM is accurately characterized by channel B.
Probability of Bit Error
-1
10
-2
10
-3
10
-4
10
-5
10
Chan A: Lower/Approx/No ISI
Chan A: Upper bd
Chan B: Forney Lower bd
Chan B: New Lower bd
Chan B: Upper bd
-6
10
-7
10
References
-8
10
0
5
10
SNR (dB)
15
20
Figure 2: The MLPD performance for channels A and
B.
A simple lower bound for this special case is
Ãr
!
SNR
.
Ps ≥ Q
2
(32)
The lower bound in (32) follows from (30) and the
fact that the two weight one error sequences result in
d2 = kf k2 with KLB (kf k2 ) = 1. This lower bound
implies that for any channel, one can never do better
than an equal energy channel without ISI.
The performance expressions were approximated
for two channels which are representative of an optical POM [3]. An L = 1 square footprint with the
symmetry of f (±1, ±1) = c, f (±1, 0) = f (0, ±1) = b,
and f (0, 0) = 1. Channel A is defined by b = 0.181
and c = 0.0327. Channel B is defined by b = 0.352 and
c = 0.0993. The bounds and approximations were approximated by exhaustively searching all ternary fundamental error sequences with nonzero elements limited to a (4 × 4) region. The performance of these
two channels is plotted in Figure 2. As would be expected, the results are qualitatively very similar to
those for MLSD. Note that for channel A, the normalized minimum distance (dmin /kf k) is one, so that
for large SNR there is little degradation suffered due
to ISI if the MLPD algorithm could be implemented.
There are two weight one error patterns with this minimum distance, so the Forney lower bound, (30), and
the approximation of (28) coincide with the No-ISI
curve generated from (32). The more severe channel
B has a normalized minimum distance of 0.76, so that
a 1.2 dB degradation in SNR is suffered due to ISI at
high SNR. Note that the lower bound of (30) is much
tighter than the Forney lower bound over much of the
SNR region plotted. This is because there are two
weight four error patterns with the minimum distance
so that the low SNR characteristics of the lower bound
are dominated by a small value of KLB (d). The high
SNR approximation for channel B is omitted from Figure 2 for the sake of clarity.
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