Iterative Row-Column Soft-Decision FeedbackAlgorithm Using Joint Extrinsic Information for

Two-dimensional Intersymbol InterferenceYiming Chen

School of Electrical Engineeringand Computer Science

Washington State UniversityPullman, WA, 99164-2752

Email: ychen@eecs.wsu.edu

Benjamin J. BelzerSchool of Electrical Engineering

and Computer ScienceWashington State UniversityPullman, WA, 99164-2752

Email: belzer@eecs.wsu.edu

Krishnamoorthy SivakumarSchool of Electrical Engineering

and Computer ScienceWashington State UniversityPullman, WA, 99164-2752Email: siva@eecs.wsu.edu

Abstract-In this paper, we redesign the previous iterative row­column soft decision feedback algorithm (Cheng et al., 2007)for a two-dimensional intersymbol interference channel withadditive white Gaussian noise; a 2 x 2 averaging mask and two3 x 3 masks are considered. In particular, we consider the jointstatistics for a block of pixels involved in the exchange of extrinsicinformation between the detectors; previously, these pixels wereconsidered to be statistically independent. The new algorithm,referred to as the block (BLK) algorithm, provides more than 1dB SNR gain; the resulting performance is within 0.3 dB from themaximum likelihood bound for the masks considered. To addressthe increased computational and storage complexity introducedby the joint statistics, we have developed a simplified version ofthe block (SBLK) algorithm. Experimental results demonstratethat the SBLK algorithm performs almost as well as the BLKalgorithm.

Index Terms-Two-dimensional intersymbol interference, Softdecision feedback, Row-column algorithm, Joint statistics

I. INTRODUCTION

We consider the detection of an M x N binary equiprobableindependent and identically distributed (i.i.d.) image f withelements f (k, l) E {-I, I} from received image r withelements:

r(m, n) == L L h(m - k, n - l)f(k, l) + w(m, n). (1)k l

In the above equation, h(k, l) are the elements of a two­dimensional (2D) finite impulse response blurring mask handthe w(m, n) are zero mean i.i.d. Gaussian random variables(rv). The model in (1) applies, e.g., to 2D data storage systems,which suffer from 2D intersymbol interference (lSI) at highstorage densities. Such systems are under active developmentby industry for next-generation optical disk storage (e.g., [1]),and holographic data storage (e.g., [2]).

In an earlier work [3], we have developed an iterativerow-column soft-decision feedback (IRCSDF) algorithm anddemonstrated its good performance; by concatenating theIRCSDF detector with the zigzag iterative soft-decision feed­back (ZZISDF) detector [4], we obtain further improvementwith overall performance close to the corresponding maximum

978-1-4244-7417-2/10/$26.00 ©2010 IEEE

likelihood (ML) bound. However, these algorithms have havetwo main limitations: the pixels involved in the extrinsicinformation exchanged between the constituent decoders areassumed to be independent in [3]. This was mainly a sim­plifying assumption and is not necessarily true. Moreover,the extrinsic information between detectors is weighted (by afactor less than 1) to account for imperfect information. Thisleads to a time consuming problem of designing an appropriateweight schedule; as shown in [4], proper design of the weightschedule results in significant performance improvement.

In this paper, we redesign the IRCSDF algorithm by drop­ping the independence assumption on the extrinsic informa­tion exchanged between the row and column decoders; wecall the resulting algorithm the block (BLK) algorithm. Weestimate and exchange joint statistics for the pixels involvedin the extrinsic information exchange. To address the increasedcomputational and storage complexity introduced by the jointstatistics, we have developed a simplified version of theblock (SBLK) algorithm. Experimental results demonstratethat the SBLK algorithm performs almost as well as the BLKalgorithm.

One additional advantage of exchanging joint statistics isthat the new algorithms (both BLK and SBLK) require a muchsimpler weight scheme - a single parameter (constant weight)to choose. This weight can be designed by simple simulation;in particular, no EXIT chart method is needed here (comparedto [4]). The new algorithm offers us more than 1 dB gain forboth 2 x 2 and 3 x 3 masks, and brings us to within 0.3 dBof the corresponding ML bound.

This paper is organized as follows: In section II, we presentthe required modifications to our original IRCSDF algorithm[3] (which was based on the BCJR algorithm) to estimate andexchange joint extrinsic information. Section III discusses theissue of subtraction of input extrinsic information from theoutput log-likelihood ratio (LLR), before it is passed on tothe next decoder. A simplified version of the BLK algorithm,referred to as SBLK algorithm, is presented in section IV.Simulation results are presented in section V and section VI

II. BLOCK ALGORITHM

provides concluding remarks.

P(s Is') == P(U == i) == P(io) x P(i 1 ) x P(i2) (4)

(6)

(7)

(8)

(9)

(10)

shown below, respectively.

p' (Yk I U == i, Sk == S, Sk-l == s') ==

P(Yk2 I io, il, iz, S, Sf) X [ L P(Ol, O2)fh,02

P(Yk1 I io,i1,s,sf,02)P(ykO I i o,S,Sf,01,02)],

P(s Is') == P(U == i) == P(io,i1,i2),

We now address the problem of obtaining the joint prob­abilities in equations (7) and (8). In the BCJR algorithm wecompute Qk(S) == P(Sk == s, Yk), (3k(S) == P(Yk+l I Sk == s)and ri(Yk,S,S') == P(U == i,Sk == S,Yk I Sk-l == s') bya forward-backward procedure. We then compute A~ (s) ==Ls'Qk-l(S'){3k(S)ri(Yk, s, s'), which gives the unnormal­ized probability P(U == i); corresponding log likelihood ratio(LLR) is given by (in the binary case, there is one LLR perpixel, the "other LLR" being 0):

L(k) == 10g(Ls A~O=+I(s)).Ls A10=-I(S)

In our block (BLK) algorithm, we compute the joint prob­ability of the block of pixels that constitute the state andinput (block of four pixels for 2 x 2 mask and block ofnine pixels for 3 x 3 mask) as shown in Figures 1 (a)and 2 (a). We first compute the same Q, {3; r is computedbased on (2), (6)-(8). For computing A, we do not sum

'.. \i ( ') - P(U - • S - S -' )over s , I.e, /\k s, s - - 1, k - S, k-l - S ,Yk ,which is the joint block probability. Instead of computingtwo LLRs (one of them being equal to zero) for each pixel,we now have 24 == 16 for 2 x 2 mask, and 29 == 512for 3 x 3 mask LLRs, as shown in equation (10). Here, for2 x 2 mask i == (i o, i 1 ) , i0 == (-1, -1), S == (so, S 1 ) ,

So == (-1, -1), S' == (s~, s~), Sb == (-1, -1). For 3 x 3 case,i == (io, iI, i 2), iO == (-1, -1, -1), S == (so, SI, S2, S3, S4, ss),So == (-1, -1, -1, -1, -1, -1), S' == (s~, s~ , s~, s~, s~, s~),

and Sb == (-1, -1, -1, -1, -1, -1).

(e ) ( A~(S, S') )Lk 1, S = log >.~=iO(S = So, Sf = Sb) .

In other words, we compute a 16-valued LLR at each pixel(512-valued LLR for 3 x 3 mask), instead of a single LLR ateach pixel, when we used the independence assumption.

Before passing this information to the next decoder (inan iterative scheme), the input extrinsic LLR is subtractedfrom the LLR computed in (10). It is customary to alsomultiply the extrinsic LLR by a weight factor less than one,since the extrinsic information is not reliable, at least in theinitial iterations. Designing an optimal weight schedule (asa function of iteration number) has been discussed in [4],under the independence assumption. Finally, note that, thejoint probability of the four/nine-pixel block can be suitablymarginalized to obtain the joint input probability and joint

(5)

(2)

P(OI) == P(wo) x P(Wl) x P(W2)

P(02) == P(W3) x P(W4) x P(ws).

ri(Yk,S',S) ==p'(Yk I U == i,Sk == S,Sk-l == s')x

P(U == i I s, s') x P(s Is'),

where

and for feedback pixel block:

Here k represents the trellis stage, Y» == (YkO, Ykl, Yk2) isthe received vector, i == (i o, iI, i 2) is the input vector, s, s'are the current and previous states, and 0 1 , O2 representsthe two rows of feedback pixels. In [3], [4], for simplicity,we assumed that the pixels in the input/feedback block werestatistically independent. Consequently, marginal probabilities(from the extrinsic information) were multiplied to obtain therequired joint probabilities. In other words, we set (for inputpixel block):

p' (Yk I U == i, Sk == S, Sk-l == s')

== P(Yk2 I i o, iI, i 2, s, s')

x [L P(02)P(Yk1 I io, i 1 , S, Sf, O2 ) (3)O2

x L P(Ol)P(Yko I io, S, Sf, 0 1 , O2 )l0 1

We follow the notations and derivation of the IRCSDFalgorithm (assuming independence of pixels in extrinsic in­formation) from [3], [4]. For lack of space, we will presentonly the most relevant equations from [3], which get modifiedin the BLK algorithm. Figures 1 (a) (2 x 2 mask) and 2 (a)(3 x 3 mask) show the state (labeled s), input (labeled i),and feedback (labeled w) pixels for the row decoder. Figures1 (d) and 2 (d) show similar information for the columndecoder. The main modification is in the computation of rin the traditional BCJR algorithm [5]. Computation of r inthe traditional BCJR algorithm [3] (see equations (2), and (3)below) requires apriori probability of the input block (twopixels for 2 x 2 mask and three pixels for 3 x 3 mask)and apriori probability of the feedback block (two pixels for2 x 2 mask and six pixels for 3 x 3 mask), from the extrinsicinformation obtained from the previous decoder in the overalliterative scheme. In the sequel, we present these equations forthe 3 x 3 mask; obvious changes lead to expressions for the2 x 2 case.

The independence assumption in (4) and (5) is strictlyspeaking not valid in practice; this was verified based on thejoint probabilities obtained in our block algorithm. The keyidea is to replace equations (3)-(5) with equations (6)-(8)

Fig. 2. Structure definition of the BLK and SBLK algorithm for 3 x 3mask, (a) state and input pixels definition of row decoder (b) joint block fromthe row decoder and the method to apply to the column decoder for BLKalgorithm (c) jo int pairs from the row decoder and the method to apply to thecolumn decoder for SBLK algorithm (d) state and input pixels definition ofcolumn decoder (e) joint block from the column decoder and the method toapply to the row decoder for BLK algorithm (f) joint pairs from the columndecoder and the method to apply to the row decoder for SBLK algorithm. (In(b) and (e), 'X ' indicates the pixels which are marginalized out)

Fig. I . Structure definition of the BLK and SBLK algorithm for 2 x 2mask, (a) state and input pixels definition of row decoder (b) joint block fromthe row decoder and the method to apply to the column decoder for BLKalgorithm (c) joint pairs from the row decoder and the method to apply to thecolumn decoder for SBLK algorithm (d) state and input pixels definition ofcolumn decoder (e) joint block from the column decoder and the method toapply to the row decoder for BLK algorithm (f) joint pairs from the columndecoder and the method to apply to the row decoder for SBLK algorithm. (In(b) and (e), ' X' indicates the pixels which are marginalized out)

(II)

III. SUBTRACTIO N OF INPUT LLR

weight scheme design [4].

As discussed in [6], we subtract the input LLR, whichcorresponds to an apriori probability (APP), before passingit to the next decoder. Based on the modified LLR in (10), theA computation formula and the input APP terms in our BLKalgorithm , we derived a similar subtraction term for our BLKalgorithm . However, with the subtraction of (joint) input APP,the overall performance of the algorithm, although better thanthat with independence assumption, was not as good as theone without such subtraction. This observation was initiallypuzzling but can be explained based on the factor graph ofour BLK algorithm and that of the original algorithm.

Figure s 3 and 4 present the factor graph of original IRCSDFalgorithm [3] and the BLK algorithm. By studying the factorgraph, we find that there are two kinds of loops in ouralgorithm : input APP loops (input loops) and feedback APPloops (fdbk loops) . We also identify two types of edges inthe factor graph: edges of the BCJR forward and backwardprocess within a decoder (called "B edges" in the sequel)and the edges resulting from exchange of extrinsic informationbetween decoders (called regular or "R edges" in the sequel).Note that these edges are not labeled in Figures 3 and 4.

As seen from Figure 3 (f) , in our previous algorithm withindependence assumption [3], the input loops contain only theR edges (same holds for the 3x3 case although we do not showit in Figure 4), whereas the fdbk loops contain both Band Redges. However, in our new block algorithm (and simplifiedblock algorithm to be discussed in section IV), both the inputloops and fdbk loops contain both Band R edges.

In our previous algorithm, the input APP subtraction wasbased on loops consisting of only R edges (input loops) andnot for fdbk loops, which contain both Rand B edges . Inthe current BLK (and SBLK) algorithm, based on how theblock extrinsic information is used (after marginalization) ­see Figures I (b, e) and 2 (b, e) - the input and fdbk loopsboth consist of Rand B edges. This argues for eliminating theinput LLR subtraction step.

Another argument supporting the elimination of the sub­traction step is that in the BLK (and SBLK to be discussedlater) algorithm, the LLR (or APP) after the row decoder isindependent of the LLR after the column decoder in the sameiteration . This is unlike the case in our previous algorithm [3).In other words, we can experimentally verify that

50 5 3 io io i, i2 X /w~ ~V;~~/~/

i, X X X X ///./ ;W05 , 54 »'f; //"~ (b)

i2 X X X X .~////.

52 55// / / ~"ff;

,.---~r'////53 lo ; 0; ~W3

f----- - ~~54 i, ~w~ ~::

f-- - ~~ (c)

55 i,//<~ //.//:W2' /wr;

~- ~~

(~~ ~W{; 50 5 , S, 50 5 , 52 io X X X X X//' / /// /

~{" ///~i, X X '///. ;o/f; ;Wf"Jf4/ 53 54 55 53 54 55 /ytOj (e)//// / ///

:V4 (~~ io i, i, lo i, i2 i, X X ~0 ///~ ///,1

//, / / / // »» /)tj,4; Yls;

1 5 31 541 551 It~IW:r//~1(d) ~'X~ ~/?: /Yfi;

1 io 1 i, I i,I VVf~ I(<<~I%~I,/// 3/ // / ;'1/ ///~/

(I)

~ ~53 I,

io i,

(~~ ~«j ~~/// '/ // '/ //' /

~tc///~ :Vf/'

/ / / / ~4/ ~//~

50 53 lo

5 , 54 i,

52 55 i2

[iliJ ~ ffiB ~////. ////

,)'fo; ,}If;; S1 11 10 I, i, X50 io

X //// a5 , i, ~VjO,(e)

X YJi;(d)

(b) Yf~: ~Vj/(a)

G:EJ~~ bB[; (I) 10 I,r ?f~

W~ [IE]~I, (c) Wo' ,W{// /; '/,,' ////

(a)

where i R is the pixel i after row decoder, whereas P is thesame pixel i after column decoder. We do that by verifying

feedback probability required in equation s (7) and (8). Thismarginalization is shown in Figures I (b, e) and 2 (b, e).

For the current BLK algorithm, we argue next that (a)it is not necessary to subtract the input extrinsic LLR be­fore passing it on to the next decoder and (b) a constantweight scheme (single parameter) results in good performance,thereby avoiding the need for a sophisticated EXIT chart based

P( i R IR+)P(ic IC+) = P(iR, i C I R+,C+)

P(iR I R+)P(ic I C-) = P(iR,P I R+,C-)

P( i R IR-)P(i c IC+) = P( i R,P IR- ,C+ )

P( i R I R-)P(i c I C-) = P( i R , i C I R- ,C-)

(12)

(13)

(15)

and

P(R+)P(C+) = P(R+,C+)

P(R+)P(C-) = P(R+, C-)

P(R-)P(C+) = P(R-, C+)

P(R-)P(C-) = P(R- ,C-) .

Note that if (12) and (13) hold, then

P(iR)P(P) = [P (i R IR+)P(R+) + P(iR IR-)P(R-) ]

x [P (p IC+)P(C+) + P(ic IC-)P(C-) ]

= P(iR I R+)P(ic I C+)P(R+)P(C+)

+ P(iR IR+)P(ic IC-)P(R+)P(C-)

+ P(iR IR-)P(ic IC+)P(R-)P(C+)

+ P(iR I R-)P(ic I C-)P(R-)P(C-)

= P(iR, i C I R+, C+)P(R+)P(C+)

+ P(iR,ic IR+,C-)P(R+)P(C-)

+ P(iR, i C IR- ,C+)P(R-)P(C+)

+ P(iR, i C IR- ,C-)P(R-)P(C-)

= P(iR,ic IR+,C+)P(R+,C+)

+ P(i R, i c IR+,C-)P(R+,C-)

+ P(iR, i C I R- ,C+)P(R- ,C+)

+ P(iR, i C IR- ,C-)P(R- ,C-) = P(i R , i C ) ,

(14)

which establishes (11). In equations (12), (13), and (14),R+ means "this pixel is decoded as +1 after row decoder,"whereas R - means "this pixel is decoded as - 1 after rowdecoder ;" similarly for C+ and C - .

We verified (12) by plotting the corresponding histograms(for right-hand and left-hand sides) based on Monte Carlosimulation. Equation (13) was verified using counts of thefour possible configurations (again, based on Monte Carlosimulation) in both left-hand and right-hand sides of theequation. The norm of the difference vector (between the twosides of the equation) was less than 0.1%.

The above observations and our simulation results show thatno subtraction of input LLR (APP) is requited for our blockalgorithm. In the following section, we address the complexityof the block algorithm and present a simplified block (SBLK)algorithm.

IV. SIMPLIFIED BLOCK ALGORITHM

Note that the block algorithm requires the exchange of16-valued (512-valued) LLR for each pixel for the 2 x 2(3 x 3) mask. This involves significantly more storage andcomputation compared to our original IRCSDF algorithm,where there was a single LLR for each pixel. We now presenta simplified version of the block algorithm, which we call theSBLK algorithm .

The key idea is to store and exchange LLRs only for thejoint pairs that we need in the next decoder. This is illustratedin Figures 1 (c, f) and 2 (c, f). For the 2 x 2 mask, we onlyneed to store two pixel pairs which is 22 x 2 = 8 LLRs -

-1 -1 -1 -1 -1 -1 ~-1 e e f f 9 9 h h-1

-1 a b c d -1

-1 e f 9 h -1 ~-1 i i j j k k I 1-1-1 i j k I -1

-1 m n 0 p -1

~-1 -1 -1 -1 -1 -1-1 m m n noD p p-1

(a)

~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(d)

~~abe f i j m n -1 -1

~~be f 9 j k no -1 -1

~~

(b)

cd 9 h kiD P -1 -1

~~

~d -1 h -1 1-1 P -1 -1 -1

(e)

~ ~

~ 4illr~C If9lC~j k ~ LJ..I:.J i j (~

(c)

Fig. 3. Factor graph and loops for 2 x 2 mask. (a) Sample image size 4 x 4with one-pixel boundary (b) factor graph of row decoder of RC algorithm in[31 (c) factor graph of column decoder of RC algorithm in [31 (d) factor graphof row decoder of BLKJSBLK algorithm (e) factor graph of column decoderof BLKJSBLK algorithm (f) sample loops for RC algorithm [31 (on pixel g)and for BLKJSBLK algorithm (on block [f g; j k]) in (a). (The index ' R' in(f) means from row decoder and 'C ' means from column decoder)

half the 16 LLRs required in the BLK algorithm. For the 3 x 3mask, we only need 26 +23 = 72 LLRs - almost one seventhof the 512 LLRs required in the BLK algorithm. Since we donot need subtraction of the input LLR, the performance of thesimplified algorithm is almost as good as that of the blockalgorithm (see Figures 6, 7 for details).

V. SIMULATION RESULTS

In this section, we present Monte Carlo simulation resultsfor the BLK and SBLK algorithms, and compare their perfor­mance with the previous RC algorithm (based on independenceassumption) and also the ML bound. All simulations employ arandom 64 x 64 binary image f (m , n) with pixel values chosenfrom the alphabet {-I, +I}. The plots presented below showthe bit error rate (BER) of the estimated binary input image,versus SNR. The SNR is defined as in [7] :

(Var[f *h])

SNR = 10 10glO (J~ ,

where * denotes the 2D convolution in (I) and (J~ is thevariance of the Gaussian rv w(m,n) in (1). To computethe received image r(m,n), we assume a boundary of - 1pixels around f(m,n); the receiver uses this known boundarycondition to simplify the trellis near edge pixels.

Both the BLK algorithm and the SBLK algorithm givesus significant improvement over our previous algorithm, espe-

-1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1-1 -1 abc d -1 -1-1 -1 e f 9 h -1 -1-1 -1 i j k I -1 -1-1 -1 m n 0 p -1 -1-1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1

(d)

(b) (e)

Fig. 4. Factor graph and loops for 3 x 3 mask. (a) Sample image size 4 x 4 with two-pixel boundary (b) factor graph of row decoder of BLKlSBLKalgorithm (c) factor graph of column decoder of BLKlSBLK algorithm (d) sample loops for BLKlSBLK algorithm (on block [e f g; i j k; m n oj) in (a).(The index 'R' in (d) means from row decoder and 'C' means from column decoder)

Fig. 5. Structure of 3-rows version of SBLK algorithm for 2 x 2 mask. (a)state and input pixels definition of 3-row version row decoder (b) joint pairsfrom the row decoder, s~ comes from previous step (c) the method to applyto the column decoder for BLK algorithm (d) state and input pixels definitionof 3-row version column decoder

the original RC algorithm ; their performance is only 0.3 dBaway from the ML bound. A constant weight was was used(0.3 for both BLK and SBLK) with ten iterations in each case.Results for an easier 3 x 3 mask (called Channel B [8]) is alsoshown in Figure 7, with some performance improvement. Thegains are relatively modest, since we are already quite closeto the ML bound.

VI. CONCLUSION A ND F UTUR E WORK

This paper has demonstrated a new row-column iterativesoft-decision feedback algorithm using joint extrinsic infor­mation and a much simpler weight scheme. At high SNR,the new algorithm achieves more than I dB gain compared tothe previous row-column algorithm based on an independenceassumption, for both the 2 x 2 and 3 x 3 masks; larger gainswere observed at lower SNRs. A simplified version of the jointalgorithm was presented , which reduces both computationaland storage complexity.

The next natural step is to concatenate the BLK/SBLKdecoder with the zigzag decoder (which has good low SNRperformance) ; based on our earlier work with concatenatedsystems , we expect to see overall gain from a concatenated

~ 5 0 51 5 2

~ io i1 i2

(d)

tfj I50 lo 11

51 i1

5 2 i2 ~~(a) (b) (e)

cially for the 3 x 3 mask; the results are even better than ourconcatenated system with zigzag decoder [4].

For the 2 x 2 mask, the BLK and SBLK algorithm (2-rowsversion) gives us almost I dB gain over the corresponding 2­rows version with independence assumption (see Figure 6). Weapplied a constant weight (over all iterations) to the extrinsicLLR (0.5 for BLK and 0.6 for SBLK), with ten iterations ineach case. The weights were obtained after a semi-exhaustivesearch. This is in contrast to the original RC algorithm, whichrequired an iteration-dependent weight schedule optimizedbased on EXIT charts.

A 3-rows and a 4-rows version of the original RC algorithmprovided some improvement (at the cost of added trellis com­plexity) ; however, the BLK/SBLK algorithm still outperformsthem. Performance of the 3-rows version is shown in Figure 6;the 4-rows version has a very similar performance and is notshown. In Figure 6, we also compare our results with a modi­fied version of the original algorithm in [3]. The modificationis that for the feedback pixels, extrinsic information is usedfrom that provided by the previous decoder (ColumnlRow)instead of from the current decoder (Row/Column) as done in[3). All of the 2-rows, 3-rows and 4-rows version provide someimprovement due to this modification; the 2-rows version sawthe most gain.

A 3-rows/4-rows version of the BLK/SBLK algorithm didnot provide any improvement. This is because the joint block isnot square anymore but is rectangular. Therefore, it is not easyto obtain the required joint input and feedback probabilities.This is illustrated in Figure 5 for the SBLK algorithm (3-rowsversion), where some partial independence assumption wouldbe required. For example, in Figure 5 (b), although we canobtain joint extrinsic information for ( s~ , so), we will have toassume that io is independent of s~ and so.

For the 3 x 3 averaging mask, Figure 7 shows that both theBLK and SBLK algorithms provide almost 1.2 dB gain over

10°r -r-- --r- --= 2 =- ---r r--TIl

10

Fig. 6. Monte Carlo simulations with 2 x 2 averaging mask for the newBLK and SBLK algorithms , the original RC algorithm with independenceassumption in [3) and ML bound.

161514

~0-3r •••••••••••••••••••••••••••·••••" " *', !:~s~~~\~ , l ~~1{ 1en

-4

10 -a-- ML bound avg. mask-e- avg. mask in [3)-+- BLK avg . mask

- 510 -+- SBLK avg. mask

- 8 - ML bound channel B- 8- channel B in [3)

10-6 - +- BLKchannel B- ~- SBLK channel B- *- ZR channel B in [4)

-7 --ZR avg. mask in [4)10 8 9 10 11 12 13

SNR (dB)

Fig. 7. Monte Carlo simulations for the new BLK and SBLK algorithm, theoriginal RC algorithm with independence assumption in [3), the concatenatedsystem in [4) and ML bound. Results for 3 x 3 averaging mask (solid line)and channel B from [8) (dashed line) are shown.

131211

0::llJCD

-+- ML bound10-4 --e- 2 rows version in [3]

-e- 3 rows version in [3]--v- BLK 2 rows version-4- SBLK 2 rows version

-5 --modified version of [3]

10 6 7 8 9 10SNR (dB)

10-3F : : : : : : : : : : : : :

10-2F :: :: : : : : : : : : :

system. Since the zigzag algorithm only provides independentextrinsic information (pixel-by-pixel) based on its specialstructure, whereas the BLKlSBLK algorithm requires jointextrinsic LLR, we will have to carefully design the interfacebetween the two decoders. We are actively pursuing this idea.A longer term goal would be to develop a version of the zigzagalgorithm that provides joint extrinsic information .

[7) C. L. Miller, B. R. Hunt, M. W. Marcellin, and M. A. Neifeld, "Imagerestoration using the Viterbi algorithm ," Journal of the Optical Societyof America A, vol. 16, pp. 265-274, February 2000.

[8) X. Chen and K. M. Chugg, "Near-optimal data detection for two­dimensional ISIIAWGN channels using concatenated modeling and it­erative algorithms ," in Proc. IEEE International Conference on Commu­nications, ICC'98, 1998, pp. 952-956.

ACKNOWLEDGM ENT

This work was partially supported by NSF grants CCR­0098357 and CCF-0635390.

R EF ER ENC ES

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[2) G. T. Huang, "Holographic memory," MTf Technology Review, vol. 9,Sept. 2005.

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[4) Y. Chen, T. Cheng, P. Njeim, B. Belzer, and K. Sivakumar, "Iterativesoft decision feedback zig-zag equalizer for 2D intersymbol interferencechannels ," IEEE Journal on Selected Areas in Communications , vol. 28,no. 2, February 2010.

[5) L. R. Bahl, J. Cocke, F. Jelinek , and J. Raviv, "Optimal decoding oflinear codes for minimizing symbol error rate," IEEE Transactions onInformation Theory, vol. 20, pp. 284-287, March 1974.

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