IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 7, JULY 2007 433

Row-Column Soft-Decision Feedback Algorithm forTwo-Dimensional Intersymbol Interference

Taikun Cheng, Benjamin J. Belzer, and Krishnamoorthy Sivakumar

Abstract—We present a novel iterative row-column soft decisionfeedback algorithm (IRCSDFA) for detection of binary imagescorrupted by 2-D intersymbol interference and additive whiteGaussian noise. The algorithm exchanges weighted soft infor-mation between row and column maximum a posteriori (MAP)detectors. Each MAP detector exploits soft-decision feedbackfrom previously processed rows or columns. The new algorithmgains about 0.3 dB over the previously best published results forthe 2 2 averaging mask. For a non-separable 3 3 mask, theIRCSDFA gains 0.8 dB over a previous soft-input/soft-outputiterative algorithm which decomposes the 2-D convolution into1-D row and column operations.

Index Terms—Iterative algorithm, soft decision feedback, 2-D in-tersymbol interference.

I. INTRODUCTION

CONSIDER the detection of an binary-equiprob-able 2-D independent and identically distributed (i.i.d.)

image (with elements ) from re-ceived image

(1)

where is a finite-impulse-response 2-D blurring mask,the are zero mean i.i.d. Gaussian random variables(r.v.s) with variance , and the double sum is computed overthe mask support region . It isassumed that a boundary of elements surrounds the image

. This system model applies, e.g., to 2-D image storagesystems, which suffer from 2-D ISI at high storage densities.Direct maximum likelihood (ML) detection of from

requires comparison of with candidateimages, and is therefore impractical for typical image dimen-sions. Standard Wiener filtering is significantly inferior to MLdetection, especially at high SNR [1]. Hence, it is desirable todevelop a low-complexity 2-D ISI detection algorithm that ap-proximates the performance of 2-D ML detection. For 1-D sig-nals, the Viterbi algorithm (VA) provides efficient ML detection

Manuscript received July 6, 2006; revised October 18, 2006. This workwas supported in part by the National Science Foundation under GrantCCR-0098357. This work was previously presented at the 39th Conferenceon Information Sciences and Systems (CISS’05), Johns-Hopkins University,Baltimore, MD, March 2005. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Jitendra K. Tugnait.

The authors are with the School of Electrical Engineering and Computer Sci-ence, Washington State University, Pullman, WA 99164-2752 USA (e-mail:tcheng@eecs.wsu.edu; belzer@eecs.wsu.edu; siva@eecs.wsu.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LSP.2006.891329

of ISI-corrupted data [2], but the VA does not generalize to twoor higher dimensions. Union bounds on the performance of 2-DML detection are described in [3]; these bounds are tight at highSNR, and are useful in assessing the performance of 2-D detec-tion algorithms.

To our knowledge, [1] and [4] employed the first iterativealgorithm for 2-D ISI reduction; a 2-D decision-feedback VA(DFVA) was run on rows and columns, and bits which agreedin both directions were fixed for subsequent iterations. Subse-quent work has employed the turbo principle (after [5]). In [6],the 2-D convolution operation is decomposed into two 1-D oper-ations, and an iterative decoding algorithm exchanges soft infor-mation between 1-D soft-input/soft-output (SISO) detectors. In[7], mask separability is exploited to construct an iterative row-column detector for low-density-parity-check (LDPC) coded bi-nary images, in which extrinsic information is exchanged be-tween a non-binary column SISO decoder, a binary row SISOdecoder, and a LDPC decoder. In [8] and [9], soft information isexchanged between MAP detectors operating on multiple rowsand multiple columns; this scheme handles nonseparable masks.

The primary contribution of this paper is a new iterative soft-decision feedback (SDF) MAP detection algorithm for reduc-tion (or elimination) of 2-D ISI. Our scheme, while similar tothat of [8], was developed independently, and has several keydifferences. First, we make decisions one row at a time and useSDF, rather than making decisions two or more rows at a timeand using “feed-forward” [8]. Second, we weigh the extrinsic in-formation passed between SISOs, and increase the weights witheach iteration; the weight schedule significantly improves thealgorithm’s performance. Third, we achieve additional gains byadding rows (respectively, columns) to the state and input pixelblocks of the row (column) SISOs. And fourth, we demonstrateperformance with both 2 2 and 3 3 masks on 128 128 and64 64 images, whereas the maximum source image size con-sidered in [8], [9] is 5 5. The IRCSDFA achieves about 1.5 dBof SNR improvement over the hard-decision iterative algorithmof [1], for the 3 3 averaging mask. For a more rapidly decaying3 3 mask, the IRCSDFA achieves about 0.8 dB gain over [6].For the 2 2 averaging mask, the IRCSDFA gains about 0.3 dBover the separable algorithm of [7] (without coding), the previ-ously best published result for that mask.

II. TRELLIS DEFINITION

Two-dimensional convolution can be viewed as the innerproduct of image with inverted mask ,with mask coefficient at pixel position . Theinverted mask raster-scans through the image row-by-row orcolumn-by-column.

For the row-by-row case, we use Miller et al.’s method [1] todefine the IRCSDFA trellis states and inputs as in Fig. 1. Trellis

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434 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 7, JULY 2007

Fig. 1. State, input, and feedback pixels for the 2� 2 mask and 3� 3 mask.

generation for the 3 3 mask on the th image row is initi-ated by placing the input marked in Fig. 1 at the leftend of the row, where the initial values of the six state pixelsare due to the boundary conditions, and the vector of threeinput pixels can take eight different values. The entire state/inputblock is then shifted right to pick up the next three input pixels,and the previous three input pixels become the middle threestate pixels. The trellises for each row are terminated at the rightend of the row by extra shifts into the boundary pixels. For the3 3 mask, the 64-state trellis has eight branches entering andleaving each state, with no parallel branches. At each position

, the trellis branch output vector consists of three 3 3inner products between the inverted mask and the pixel valuesdefined by the trellis; the upper inner product uses two feedbackrows, the middle uses one feedback row, and the lower uses re-ceived pixels only. The branch metric is the squared Euclideandistance between the branch output and the received pixel vector

.The 2 2 mask has a fully connected trellis with four states

and four branches per state. We can improve system perfor-mance by extending the state and input blocks shown in Fig. 1by one or more rows. For the 2 2 mask, adding one morerow gives an 8-state trellis with eight branches per state, andthree (rather than two) inner products per branch metric. Thecolumn-by-column case is similar to the row-by-row case.

As the image pixels are i.i.d., the above-described trellis con-structions impose the Markov condition that, given the currenttrellis state, subsequent states and branch outputs are indepen-dent of past states or outputs. This Markov condition allows theuse of a modified BCJR [10] algorithm for detection.

III. ITERATIVE ROW-COLUMN

SOFT-DECISION FEEDBACK ALGORITHM

Fig. 2 shows a block diagram of the algorithm. The basic el-ement is a soft decision feedback, soft-input soft-output (SDF-SISO) detector. Each SDF-SISO processes received image ,which is corrupted by 2-D-ISI and by additive white Gaussiannoise (AWGN). The SDF-SISOs use a modified BCJR [10] al-gorithm, in which soft estimates of branch outputs from earliertrellis stages are used as SDF to aide computation of the currentpixel’s log-likelihood ratio (LLR).

The row and column SDF-SISOs exchange weighted softinformation. The SDF-SISOs assume that their decision

Fig. 2. IRCSDFA block diagram.

Fig. 3. (a) Source image. (b) Source with 2-D ISI and noise. (c) Error imagefrom hard decoding of image (b) (white pixels are errors, black are correct). (d)Error image from IRCSDFA algorithm.

feedback is correct, but in fact it contains errors. Decisionfeedback errors cause error propagation; an example is shownin Fig. 3(d), which shows the error map from a Monte-Carlosimulation of the IRCSDFA on the i.i.d. binary source imageshown in Fig. 3(a). When the information weights are setto one (the nonweighted case), the algorithm converges intwo to three iterations. By employing the weight schedule

, where , , is theiteration number, we slowed the convergence of the algorithmto six iterations, at all SNRs tested. (SNR gains from twoadditional iterations were limited to at most 0.01 dB, and theyoccurred at low SNR.) For the 2 2 averaging mask at highSNR, we observed that this weight schedule gave us an SNRgain of about 0.5 dB over the nonweighted case, at a bit errorrate (BER) of 2 . (This particular weight schedule wasarrived at experimentally after a nonexhaustive search, andmust therefore be considered suboptimal.)

A theoretical analysis of the weight schedule and its optimiza-tion is the subject of future work. Based on observations, we hy-pothesize that decision feedback errors make the output LLRslarger than their true values, and that the weights move the LLRscloser to their true reliabilities, thereby preventing the algorithmfrom converging too quickly.

The SDF branch output computation computes LLRsfor inner products between the mask and candidate binaryestimates of the image pixels. To illustrate theSDF LLR calculation on row scans, assume the 3 3 aver-aging mask is used to compute the convolution

. For pixelat the th trellis stage, , the correspondingreceived pixel vector is ,and the input vector is ,as shown in Fig. 1. To simplify, let ,

CHENG et al.: ROW-COLUMN SOFT-DECISION FEEDBACK ALGORITHM FOR 2-D ISI 435

and . The LLR is, where

is the extrinsic estimate of passed to decoder , ,from the other decoder. Decoder ’s extrinsic LLR inputis ,and the extrinsic LLR output to the next decoder is

, where .By using the input extrinsic information, we can compute theconditional probability of the input pixel

(2)

Given trellis state , input vector , and received vector, define , where

, , and is the number of inputbits per trellis stage. We can then compute the a posteriori prob-ability (APP) . As in [10],by setting ,and , wehave . The SDF outputLLRs can be incorporated into the pixel transition probabilities

. The modified is the product of a modified con-ditional channel probability density function (pdf) , trellistransition probabilities, and extrinsic information from the otherdecoder

(3)

For the given states and input , is 0 or 1and is based on the trellis. Theextrinsic information can be computed as

(4)

where comes from (2), and .The modified channel pdf sums over the values of inner prod-

ucts associated with state transition that are affectedby past decisions

(5)

where denotes feedback rows, inner productdepends on the feedback pixels, and the row probabilities

, where arefeedback pixel probabilities. For the 3 3 averaging-maskchannel, inner products and arenine-pixel averages of the pixels labeled “inner product 1”and “inner product 2” in the 3 3 mask of Fig. 1. The

are computed from feedback LLRs from previ-ously processed rows (or columns) during the current it-eration. Since the original image is subject to AWGN, the

are normal pdfswith means and variances .

To estimate the pixel located on from the vector s,we sum the s over and

(6)

The pixel LLR is computed as

(7)

If , we decide that pixel is ; otherwise, it isdetected as .

IV. SIMULATION RESULTS

We present Monte Carlo simulations for the IRCSDFA on therandom binary image with pixel alphabet .The plots below show the BER of the estimated binary inputimage, versus SNR. The SNR is defined as in [1]:

(8)

where denotes 2-D convolution, and is the variance of r.v.sin (1). When computing (1), we assume a boundary

of pixels around ; the receiver uses this knownboundary condition to simplify the trellis at image edge pixels.

Fig. 4 shows IRCSDFA simulation results on a random128 128 binary image blurred by the 2 2 averaging maskand AWGN. We plot results using two, three, and four rows inthe 2 2 trellis state and input block of Fig. 1, which showsthe basic two-row case. The maximum likelihood estimator(MLE) upper bound of [3] is also plotted. In addition, we plotresults for the row-by-row MAP with SDF (but without columnextrinsic information), and iterative row-column MAP withextrinsic LLR exchange but with hard-decision feedback (HDF)on past rows (IRCHDF); these additional results are based onthe basic (2 row) trellis definition. Plots marked “Opt. Weights”use the six-iteration weight schedule given in Section III; plotsmarked “Unit Weights” fix the weights at 1.0.

Table I shows the SNR loss relative to the ML bound at BER2.0 for selected curves from Fig. 4; all results in Table Irefer to weight-optimized IRCSDFA, except where noted.Three-row IRCSDFA gains about 0.4 dB over the two-row ver-sion, and performs within 0.6 dB of the ML bound. Four-rowIRCSDFA performs as well as the three-row at high SNR, andgains up to 0.3 dB at lower SNRs. Additional state/input rowsallow the algorithm to correct larger error patterns, which occurmore frequently at low SNR.

For comparison, we plot simulation results for the separablealgorithm of [7]. (Because [7] reports results only for one it-eration of the separable equalization algorithm without LDPCcoding, we implemented the equalization algorithm and testedits multi-iteration performance. We found that two iterationsof the separable algorithm achieve almost all available perfor-mance gain, so two iterations are used in all separable-algorithmresults presented here.) In Table I, 3-row IRCSDFA gains about0.3 dB over the separable algorithm. The IRCSDFA works forgeneral 2-D masks, whereas the separable algorithm must use

436 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 7, JULY 2007

Fig. 4. Simulation results for 128� 128 binary input image and 2� 2 aver-aging mask.

TABLE IDISTANCE FROM ML BOUND (IN DB) AT BER 2�10 FOR CURVES FROM

FIG. 4. RESULTS ARE FOR IRCSDFA, EXCEPT WHERE NOTED

the closest separable approximation to a nonseparable mask,which leads to an error floor in many cases.

The Marrow-Wolf (M-W) algorithm of [8], [9] also employsiterative row/column MAP decoding, and their trellis (like ourtwo-row version in Fig. 1) is fully connected with four statesand four branches/state. Based on simulation results performedunder identical conditions to those in [8], [9], we believe thetwo-row version of our algorithm with unit weights is essentiallyequivalent to the 2 2 mask M-W algorithm. Hence, based onTable I, we believe our algorithm’s performance is about 1 dBbetter than the M-W algorithm, for the 2 2 averaging mask.

The improved performance of the three- and four-row IRCS-DFAs comes at a complexity cost relative to the two-row versionand the separable algorithm. The number of operations per pixelfor the two, three, and four row IRCSDFAs, and the separablealgorithm, are as follows: add/subtract, 423, 1935, 8735, and480; multiply/divide, 742, 3398, 15366, and 943; exp/log, 87,391, 1543, and 75. We note that the two-row IRCSDFA com-plexity is roughly equal to that of the separable algorithm.

Fig. 5 shows IRCSDFA simulation results on a random64 64 binary image blurred by the 3 3 averaging mask( , ) and AWGN; here we use thethree-row trellis definition shown in Fig. 1. The MLE upperbound for this mask is also plotted. We also plot simulationsof the IRCHDFA. The IRCHDF and IRCSDF results shownin Fig. 5 were run with the weight schedule described inSection III. At high SNR, the IRCSDFA requires about 1.2dB more SNR than the MLE. By comparison, the iterative

Fig. 5. Simulation results for two 3� 3 masks.

algorithm by Miller et al. [1] is about 3 dB away from the MLbound.

We also simulate a random 128 128 image blurred by the3 3 mask (named Channel B) defined by Chen and Chugg in[6], and show the results in Fig. 5. (Chen and Chugg’s SNR

is based on i.i.d. equiprobable pixels withalphabet {0, 1}; when that alphabet is used in (8), it gives

, i.e., our SNR definition is 3 dB to the leftof Chen and Chugg’s. Hence, we left-shifted the original curvein [6] by 3 dB in Fig. 5.) The IRCSDFA gains about 0.8 dBcompared to [6] at BER , and is within 0.3 dB of the MLbound at high SNR; also, Chen-Chugg’s curve diverges fromthe ML bound, whereas the IRCSDFA is parallel to it.

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