Performance of Optimal Digital Page Detection in a Two-Dimensional ISI/AWGN Channel

Transcription

1 c IEEE 1996 Presented at Asilomar Conf. on Signals, Systems and Comp., Nov 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 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 (D 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 ISI- AWGN model is representative of at least some designs and has been adopted elsewhere [, 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 D 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 ciently via the Viterbi Algorithm (VA [5]. However in D, 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 D index set. One obvious approach is to raster the D page into a sequence. This process inevitably leads to a scattering of the ISI dependence from a small, compact D 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 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 []. A similar algorithm for quantization of grayscale images was developed in [4]. Algorithms based on the D 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 D, 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. 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 = f(i k, j la(k, l (k,l P(i,j (1a (1b (1c

2 = (k,l P(i,j f(k, la(i k, j l. (1d This model will be assumed to be real-valued for simplicity, with independent, identically distributed (IID digital data 1 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 P(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, P 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 = P(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 Λ X (Ã = [z(i, j x(i, j; Ã]. ( (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 Λ X (Ã(1 ; Ã( = Λ X (Ã(1 Λ X (Ã( will be used..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 { L P (i, j = (k, l :P(k, l } P(i, j. (3 Consider two data pages Ã(1 and Ã( which agree at all points in L P (i, j except for the point (i, j. Even if these two pages disagree outside of L P (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 Λ P(i,j (Ã(1 ; Ã( is not a function of disagreements outside the influence region and 1 A boldface character will be used to represent the entire D page of the corresponding signal. Λ [P(i,j] c(ã(1 ; Ã( is not a function of a disagreementat(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 inl P (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 L P (i, j = P(k, l. (4 (k,l P(i,j The interpretation of (3 and (4 is the same: L P (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 L P (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 = Pr { x(i, j =v {x(k, l} (k,l LP (i,j {(i,j}}. 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 (i 1,j 1 in the interior of a very large D page. The error probability at a given point is P s (i 1,j 1 =Pr {â(i 1,j 1 a(i 1,j 1 } (7 } = Pr {Â A G, (8

3 (a (b (c (d (e (i, j or(i 1,j 1 nonzero element of error pattern X (E Figure 1: Example D channels: (a the footprint P(i, j, (b P(i, j, (c the influence region L P (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 (i 1,j 1 G = {E : e(i, j A and e(i 1,j 1 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 =  A. (10 Thus, we have P (i 1,j 1 =P (ÊG, where Ê G = E G Ê(E. (11 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 D 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]. The weight one error patterns from G are included in F. In addition, E Gis in F if it has the property e(i, j 0 (k, l L P (i, j such that (k, l (i, j and e(k, l 0. (1 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 X (E = P(k, l. (13 (k,l: e(k,l 0 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 L P (E = L P (k, l. (14 (k,l: e(k,l 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 E F can now be defined as the event that there is a fundamental pairwise error E F = E(E. (16 E F

4 4 MLPD Performance Bounds An upper bound on P s (i 1,j 1 is obtained by the following theorem: 3 Theorem 1 ÊG E F. Proof: If Ê(E has occurred, then there is a corresponding E F Fthat coincides with E in L P (E F and is zero everywhere else. Any disagreements between A and  located outside of L P(E F will not affect the calculation of Λ( onx (E F. Since A+E is the best global path, it follows that A + E F is more likely than A: 0 < Λ X (EF (A;  (17 = Λ X (EF (A; A + E (18 = Λ X (EF (A; A + E F (19 = Λ(A; A + E F. (0 The upper bound on P s (i 1,j 1 follows directly from Theorem 1 P s (i 1,j 1 =P (ÊG P (E F E F P (E(E (1 = E F A C(E P (E(E AP (A, ( 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 P (E(E A =Q, (3 where σ w is the variance of w(i, Q( is the complementary distribution function of a unit variance, zero mean Gaussian random variable, and = (i,j[e(i, j f(i, j]. (4 Several steps can be taken to simplify the bound in (. Note that as the page size becomes asymptotically large, this upper bound does not depend on (i 1,j 1. 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 P s E F w(ep C (EQ, (5 where P C (E is the sum of P (A over all transmitted pages consistent with E. The set F is created 3 This approach is similar to that taken in [8] with a correction. 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 P s d K UB (dq, (6 d D where D is the set of possible values of for E F and K UB (d = w(ep C (E, (7 E F (d with F (d representing the set of error pages in F with =d. An approximation for large signalto-noise ratio is P s = KUB (d min Q d min, (8 where d min 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 (d, was sent. The error performance of the optimal detector which is privy to this side information cannot be worse than that of MLPD without side information, so that P s K LB (dq d with K LB (d =P E F (d C(E. (9 Forney s version of this bound is with d = d min.however, a tighter bound can be obtained by maximizing over d d min P s sup K LB (dq d D 4σw. (30 For low noise levels, the Forney lower bound and (30 will coincide. However, if K LB (d min 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 P C (0 = 1 and P C ( 1 = P C (+1 = 1/, so that P C (E = w(e. The symbol error probability is plotted against the signal-to-noise ratio SNR = var [x(i] (i,j var [w(i] = [f(i, j] σw = f σw. (31

5 Probability of Bit Error Chan A: Lower/Approx/No ISI Chan A: Upper bd Chan B: Forney Lower bd Chan B: New Lower bd Chan B: Upper bd SNR (db The utility of these performance bounds is demonstrated by considering the fairly simple detector demonstrated in [3] which is similar to a D decision feedback equalizer. For channel A, this suboptimal detector suffers only a1dbdegradation in SNR relative to MLPD at a error rate of However, for channel B, the degradation relative to MLPD at P s =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. References [1] D. Psaltis, Parallel Optics Memories, Byte, vol. 17, pp , 199. Figure : The MLPD performance for channels A and B. A simple lower bound for this special case is SNR P s Q. (3 The lower bound in (3 follows from (30 and the fact that the two weight one error sequences result in d = f with K LB ( f = 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 = Channel B is defined by b =0.35 and c = 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. As would be expected, the results are qualitatively very similar to those for MLSD. Note that for channel A, the normalized minimum distance (d min / f 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 (8 coincide with the No-ISI curve generated from (3. The more severe channel B has a normalized minimum distance of 0.76, so that a 1. 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 K LB (d. The high SNR approximation for channel B is omitted from Figure for the sake of clarity. [] J. Heanue, M. Bashaw, and L. Hesselink, Decision Feedback Viterbi Detection for Page-Access Optical Memories, J. of the Optical Society of America A, vol. 1, pg. 43, [3] M.A. Neifeld, K. M. Chugg and B. M. King, Parallel Data Detection in Page Oriented Optical Memory, Optical Letters, vol. 1, No. 18, Sept. 15, 1996, pp [4] L. Ke and M. W. Marcellin, Near-lossless image compression: Minimum-entropy, constrained-error DPCM, submitted to IEEE Trans Image Proc. [5] G. D. Forney, Maximum-Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference, IEEE Trans. Information Theory, vol. IT-18, May 197, pp [6] G. D. Forney, Lower Bounds on Error Probability in the Presence of Large Intersymbol Interference, IEEE Trans. Communications, vol. 0, Feb. 197, pp [7] S. Verdú, Maximum Likelihood Sequence Detection for Intersymbol Interference Channels: A New Upper Bound on Error Probability, IEEE Trans. Information Theory, vol. 33, Jan. 1987, pp [8] G. L. Stüber, Principles of Mobile Communication, Kluwer Academic Press, [9] R. Chellapa and A. Jain (eds. Markov Random Fields Theory and Applications, Academic Press, [10] K. Abend, T. J. Hartley, and L.N. Kanal, Classification of Binary Random Patterns, IEEE Trans. Information Theory, vol. 11, Oct. 1965, pp [11] K. M. Chugg, The Performance of Maximum Likelihood Page Detection in the Presence of Intersymbol Interference, IEEE Trans. Information Theory, (submitted, May 1996.