Detection in the Presence of Additive Noise and Unknown Offset

Transcription

1 Detectio i the Presece of Additive Noise ad Ukow Offset Kees A. Schouhamer Immik ad Jos H. Weber Abstract - The error performace of optical storage ad No-Volatile Memory (Flash) is susceptible to ukow offset of the retrieved sigal. Balaced codes offer immuity agaist ukow offset at the cost of a sigificat code redudacy, while miimum Pearso distace detectio offers immuity with low-redudat codes at the price of lesseed oise margi. We will preset a hybrid detectio method, the distace measure is a weighted sum of the Euclidea ad Pearso distace, so that the system desiger may trade oise margi versus amout of immuity to ukow offset. Key words: Costat compositio code, permutatio code, flash memory, digital data storage, No-Volatile Memory, NVM, mismatch, fadig, Pearso distace, Euclidea distace, parity check. I. INTRODUCTION The receiver of a trasmissio or storage system is ofte igorat of the exact value of the offset, also kow as drift or baselie wader, of the received sigal. I optical data storage, both the amplitude ad offset of the retrieved sigal deped o the reflective idex of the disc surface ad the dimesios of the writte features []. I ovolatile memories, such as floatig gate memories, the data is represeted by stored charge, which ca leak away. The amout of leakage depeds o various physical parameters ad, clearly, o the time elapsed betwee writig ad readig the data. I such a case, the covetioal approach of combattig the effects of chael mismatch by usig a automatic offset or gai cotrol that estimates the mismatch from the previously received data ca ot be applied, ad other methods have to be developed. I optical disc storage devices ad o-volatile memories, costraied codes, specifically dc-free or balaced codes, have bee used ad/or proposed to couter the effects of offset mismatch []. Jiag et al. [] addressed a q- ary balaced codig techique, called rak modulatio, for circumvetig the difficulties with flash memories havig agig offset levels. Zhou et al. [], Sala et al. [], ad Immik [6] ivestigated the usage of balaced codes for Kees A. Schouhamer Immik is with Turig Machies Ic, Willemskade b-d, 06 DK Rotterdam, The Netherlads. immik@turigmachies.com. Jos Weber is with Delft Uiversity of Techology, Delft, The Netherlads. j.h.weber@tudelft.l. eablig dyamic readig thresholds i o-volatile memories. Alterative detectio methods, such as miimum Pearso distace detectio, that are immue to offset ad gai mismatch, have bee preseted i [7]. Miimum Pearso distace detectio offers immuity to both gai ad offset mismatch, which is bought at the price of a lesseed oise margi. We will ivestigate the properties of a hybrid detector that weighs miimum Pearso ad miimum Euclidea distace measures, the desiger has the optio of tradig the amout of mismatch immuity o the oe had versus loss of oise margi o the other had. We start, i Sectio II, with a descriptio of hybrid miimum Pearso ad Euclidea distace detectio. I Sectio III, we preset a aalysis of the error performace of the hybrid detector, ad a mismatch sesitivity aalysis. I Sectio IV, we will show that a simple rate / parity-check code may improve the oise margi by slightly less tha db. I Sectio V, we will discuss implemetatio issues, ad i Sectio VI, we will describe our coclusios. II. DESCRIPTION OF HYBRID DETECTION We cosider a commuicatio codebook, S, of chose q- ary codewords x = (x, x,..., x ) over the q-ary alphabet Q = {0,,..., q }, q,, the legth of x, is a positive iteger. Suppose the set codeword, x, is received as the vector r = x + ν + b, r i R, b R is a ukow (dc-)offset, ad ν = (ν,..., ν ), ν i R are oise samples with distributio N(0, σ ). We use the shorthad otatio x + b = (x + b, x + b,..., x + b). Before we proceed with a descriptio of the ew detector, we will defie T -costraied codes. T -costraied codes, preseted i [6] for eablig simple dyamic threshold detectio of q-ary codewords, cosist of q-ary -legth codewords, T, 0 < T q, preferred or referece symbols must appear at least oce i a codeword. The code, deoted by S, comprises codewords the symbol 0 appears at least oce. The size of S equals [6] S = q (q ), q >. () For the biary case, q =, we simply fid S =, the all- word is deleted. We will assume that the code used is S = S. Immik & Weber [7] advocated the usage of the Pearso distace for chaels with both gai ad offset mismatch. 08

2 For chaels with offset mismatch oly, they itroduced the modified Pearso distace δ p (r, ˆx) = (r i ˆx i + ˆx), () i= betwee r ad ˆx, x are ˆx are take from S, ad ˆx = ˆx i. () i= The receiver outputs the codeword that is closest to r, or x o = arg mi ˆx S δ p (r, ˆx). () It has bee show i [7] that the outcome () is itrisically resistat to the ukow offset, b, at the cost of oise margi. We defie a ew hybrid distace measure by δ(r, ˆx) = γδ e (r, ˆx) + ( γ)δ p (r, ˆx), () δ e (r, ˆx) = (r i ˆx i ) (6) i= is the (squared) Euclidea distace betwee the received sigal vector r ad the codeword ˆx S. The parameter γ, 0 γ, ca be judiciously chose for tradig the sesitivity to offset mismatch versus oise margi loss. Clearly, for γ = we obtai a pure miimum Euclidea distace detector, which offers optimal oise margi, but it is sesitive to chael mismatch. O the other had, for γ = 0 we obtai a pure miimum Pearso distace detector, which is immue agaist offset mismatch, but it comes at a lesseed oise margi. Other values of γ will lead to a detector that balaces the properties of the two types of detectors. The hybrid detector outputs the codeword x o = arg mi ˆx S δ(r, ˆx). (7) A direct implemetatio of (7) requires aroud q computatios of (), which is prohibitive for larger values of. I Sectio V, we will show that the computatio of (7) ca be accomplished with much less computatioal load, for example, for q =, oly istead of evaluatios of () are eeded. I the ext sectio, we will compute the error performace of the ew detector. III. ERROR PERFORMANCE ANALYSIS The detector errs if there is at least oe codeword ˆx x, ˆx S, such that or δ(r, ˆx) < δ(r, x) (8) γ(δ e (r, ˆx) δ e (r, x)) + ( γ)(δ p (r, ˆx) δ p (r, x)) < 0. (9) We have ad δ e (r, ˆx) δ e (r, x) = δ p (r, ˆx) δ p (r, x) = e i ν i + i= e i (e i + b) i= (e i e)ν i + i= (e i e), i=, for clerical coveiece, we itroduce the short-had otatio e i = x i ˆx i (0) ad e = x ˆx. () The, we may rewrite the above iequality (8) by ζ 0 + ζ i ν i < 0, () ad ζ 0 = γ i= i= e i (e i + b) + ( γ) (e i e), () i= ζ i = γe i + ( γ)(e i e), i. () The oise samples ν i are draw from N(0, σ), so that the probability that a error is made, equals ( ) d(x, ˆx) P r(δ(r, ˆx) < δ(r, x)) = Q, σ the Q-fuctio is defied by Q(x) = e u du. π The oise distace, d(x, ˆx), betwee the codewords x ad ˆx is give by d(x, ˆx) = α β, () ad i= β = x α = ζ 0 (6) ζi. (7) i= Workig out yields α = γ e i (e i + b) + ( γ) (e i e) ad = ( γ)e + γbe + β = i= e i (8) i= [e i ( γ)e] i= = ( γ )e + e i. (9) i= 09

3 γ= Fig.. Word error rate () of the ew detector as a fuctio of the sigal-to-oise ratio (SNR) for q =, = 8 ad γ = 0, γ = 0., ad γ =. No offset, b = 0. The diagram shows the error performace computed by upperboud (0) ad computer simulatios (dotted lies). 0 0 γ=0 γ= performace of the ew detector improves upo that of the traditioal Euclidea distace detector. At asymptotically high sigal-to-oise ratios, the word error rate () is overbouded by [8] < (q ) Q q ( dmi σ ), () the miimum distace betwee ay pair of codewords i S, d mi, is defied by d mi = We simply fid mi d(x, ˆx) = x, ˆx S ˆx x mi x, ˆx S x ˆx α β. () d mi = γ γ b. () γ Note that d mi is idepedet of the alphabet size q. By varyig the value of γ, we select a error performace that lies betwee that of a miimum Euclidea ad a miimum Pearso distace detector. For the ideal case, b = 0, we fid that the asymptotic codig gai, Γ(γ), equals 0 0 γ= Γ(γ) = γ. () γ 0 0 γ=0 γ=0. The term γ b i () shows how the offset mismatch affects the error performace. The receiver will completely fail if d mi Fig.. Word error rate () of the ew detector as a fuctio of the sigal-to-oise ratio (SNR) for q =, = 8 ad γ = 0, γ = 0., ad γ =. Offset b = 0.. The diagram shows results by computer simulatios. Note that the curve for γ = 0 (Pearso detectio) is the same as show i Figure. The word error rate () over all coded sequeces x is upperbouded by < ( ) d(x, ˆx) Q. (0) S σ x S ˆx x Figure shows results of computer simulatios (dotted lies) ad computatios ivokig (0) for q =, = 8, ad γ = 0, γ = 0., ad γ = for the matched case b = 0. The sigal-to-oise-ratio (SNR), is defied by SNR(dB) = 0 log 0 σ. () Figure shows results of computer simulatios (dotted lies) for the same parameters as used i Figure, but ow with a offset b = 0.. We otice that i this case the error or (b) worst (γ) γ ( γ ), (6) (b) worst (γ) deotes the maximum offset that the receiver ca tolerate just before collapsig. Clearly the maximum allowable offset (b) worst (γ) varies betwee 0. for Euclidea distace detectio (γ = ) ad ifiity for Pearso distace detectio (γ = 0). Usig () ad (6), we plotted i Figure the parametric equatio of the maximum offset that the receiver ca tolerate, (b) worst (γ), versus codig gai, 0 log Γ(γ) (db), for = 6, 8 ad 0. The curves show the crucial trade-off betwee codig gai ad offset tolerace. For example, for = 8, the desiger may buy for a ivestmet of 0. db codig loss, a maximum offset tolerace improvemet by a factor of aroud six with respect to regular Euclidea distace detectio. The receiver is completely immue to offset mismatch for the price of a codig loss of 0 log(7/8) 0.8 db. IV. PARITY CHECK CODES For certai applicatios it may be desirable to improve the oise margi at the expese of code redudacy. A parity check T -costraied code, deoted by S p, cosists 00

4 8 0 (b) worst o parity = =6 =8 = parity =6 =6 = logΓ(dB) Fig.. Parametric equatio of the maximum offset, (b) worst (γ), that the receiver ca tolerate just before failure versus codig gai 0 log Γ(γ) (db) for = 6, 8, ad 0. of codewords, x, take from S that additioally satisfy the parity check ( x i ) mod q = a, i= a Q is a iteger chose by the code desiger, for example, to maximize the code size. Below we will cocetrate o the biary case, q =, for a = ( + ) mod, the umber of codewords equals, so that the rate of these codes is exactly /. We easily compute that the miimum distace, d p,mi, betwee ay pair of codewords equals d p,mi = γ γ b. (7) γ The codig gai, Γ p (γ), uder matched coditios, b = 0, equals Γ p (γ) = γ γ. (8) Figure shows a compariso of the error performace of offset-resistat detectio with ad without extra parity check. For asymptotically large SNR, we may, for = ad γ = 0, expect to gai a factor Γ p (0)/Γ(0) = 0/ (=.6 db) i oise margi with respect to the o-parity case. We deduce from the diagram that the gai is aroud.9 db i the SNR regio show. The receiver will fail if (b) p,worst (γ) γ ( γ ). (9) We coclude that the parity check receiver is slightly more susceptible to offset mismatch tha a stadard receiver, see (6). We plotted i Figure the parametric equatio of the maximum offset that the receiver ca tolerate, (b) p,worst (γ), versus codig gai, 0 log Γ p (γ), for = 6, 8, ad 0. Fig.. Word error rate () of offset-resistat detectio, γ = 0, as a fuctio of the sigal-to-oise ratio (SNR) for q =, = 6 ad q =, =, with ad without parity check computed usig upperboud (0). (b) worst 7 6 =6 =8 = logΓp(dB) Fig.. Parametric equatio of the maximum offset, (b) p,worst (γ), that the parity-check receiver ca tolerate just before failure versus codig gai 0 log Γ p(γ) for = 6, 8, ad 0. V. IMPLEMENTATION ISSUES The proposed distace measure () is a weighted sum of two distace measures, ad, clearly, is more complicated to evaluate tha a sigle distace. After combiig (), (), (), ad (6), we may rewrite the distace as δ(r, ˆx) = ri r iˆx i + i= i= + ( γ)( ˆx + ˆx i= ˆx i r i ). (0) i= The two right-had terms are the extra load for the receiver with respect to traditioal Euclidea detectio, which is embodied by the three left-had terms. Before we further discuss the complexity issue of the distace measure, we will first discuss how may evaluatios of the distace are eeded per received word. We tacitly assumed that the 0

5 receiver computes the distace () for all codewords i S before it will give its verdict o the codeword set. This would imply a expoetial icrease i the computatioal load with codeword legth. However, it was show i [7] that the codebook S cosists of K costat compositio codes. A costat compositio code is a set of codewords the umbers of occurreces of the symbols withi a codeword is the same for each codeword [9]. These specialize to costat weight codes i the biary case. It was show i [7] that we may sigificatly reduce the umber of distace computatios () to the umber of costat compositio codes, K, that costitute the full set S. For q =, sice S cosists of K = costat weight codes, istead of for a full set oly computatios are required. For a biary parity-check code the umber of costat weight subsets equals ( + )/ for odd ad / for eve, which reduces the umber of computatios eve more. Let a costat compositio code be deoted by Sw i, i K. Each costat compositio code, Sw i, is characterized by oe codeword of the code, called pivot word, deoted by x pi, i K. The pivot word x pi is, by defiitio, the largest word i the lexicographical orderig (the largest symbols first) of the codewords i Sw i. Thus, for a pivot word we have x pi Sw i ad x pi = (x pi,, x pi,,..., x pi,), x pi, x pi, x pi,... x pi, x pi,, i K. The remaiig codewords i S cosist of all distict sequeces that ca be formed by permutig the order of the symbols of each pivot word x pi, i K. Detectio of the received vector, r, is accomplished by ivokig the followig two-step procedure: ) The symbols of the received word, r, are sorted, largest to smallest, i the same way as taught i Slepia s prior art [0]. ) Compute the distace, usig (0), betwee the sorted received word ad all K pivot words. The receiver decides that a word take from Sw k, k K, was set i case the pivot word x pk is at miimum distace to the received vector r. Now that we kow that the word set is a member of Sw k, we ca decode the received word by Slepia s procedure for sigle permutatio codes. I a embodimet of the secod part of the above procedure, usig a micro-processor, for example, the receiver has tabulated ad stored i memory the K pivot words. I additio, the receiver has, for each pivot word, pre-computed the terms, see (), A j = x pj,i, j K ad, see (0), B j = i= x p j,i ( γ)a j, j K, i= which are idepedet of r. The receiver computes for each received codeword C = i= ad for each of the K pivot words, see (0), D j = << r, x pj >> +B j + CA j, j K, << u, v >> deotes the ier product of the vectors u ad v, ad r deotes the sorted received vector. Note that we do ot eed to compute the term r i, sice it is irrelevat. As we may observe from the above, the mai computatioal load is the computatio of the ier product. This is also the case for Euclidea detectio, ad we coclude that the hybrid detectio is oly slightly more ivolved tha Euclidea distace detectio. r i VI. CONCLUSIONS We have preseted a hybrid detectio method, the distace measure is a weighted sum of the Euclidea ad Pearso distace, which offers the desiger the optio of tradig oise margi versus amout of immuity agaist ukow offset. We have computed the error performace of the hybrid detectio method i the presece of ukow offset ad additive oise. We have computed the maximum allowable offset, (b) worst, ad the pealty i oise margi. We have itroduced parity-check T -costraied codes, ad have show that with a code rate /, we may gai aroud - db i oise margi with respect to regular T - costraied codes. We have discussed issues related to the computatioal load of the hybrid detector. REFERENCES [] G. Bouwhuis, J. Braat, A. Huijser, J. Pasma, G. va Rosmale, ad K.A.S. Immik, Priciples of Optical Disc Systems, Adam Hilger Ltd, Bristol ad Bosto, 98. 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Forey Jr., Maximum-Likelihood Sequece Estimatio of Digital Sequeces i the Presece of Itersymbol Iterferece, IEEE Tras. Iform. Theory, vol. IT-8, pp. 6-78, May. 97. [9] W. Chu, C.J. Colbour, ad P. Dukes, O Costat Compositio Codes, Discrete Applied Mathematics, Volume, Issue 6, pp. 9-99, April 006. [0] D. Slepia, Permutatio Modulatio, Proc. IEEE, vol., pp. 8-6, March 96. 0