In this paper we show that APP decoding for a linear code C is optimum not for C, but for a minimum-distance-2 code which contains C as a subcode when the codewords of are transmitted with equal probability. However, APP decoding is shown to be a symptotically optimum for C for high SNR when C is a binary one-step orthogonolizable code with equiprobable codewords transmitted over the AWGN channel

Hartmann, Carlos R.P.

Mehrotra, Kishan

Rudolph, Luther D.

Snedeker, Guy J.

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Electrical Engineering and Computer Science - Technical Reports College of Engineering and Computer Science 6-1981 On APP Decoding Carlos R.P. Hartmann Syracuse University, chartman@syr.edu Luther D. Rudolph Syracuse University Kishan Mehrotra Syracuse University, mehrotra@syr.edu Guy J. Snedeker Syracuse University Follow this and additional works at: https://surface.syr.edu/eecs_techreports  Part of the Computer Sciences Commons Recommended Citation Hartmann, Carlos R.P.; Rudolph, Luther D.; Mehrotra, Kishan; and Snedeker, Guy J., "On APP Decoding" (1981). Electrical Engineering and Computer Science - Technical Reports. 38. https://surface.syr.edu/eecs_techreports/38 This Report is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in Electrical Engineering and Computer Science - Technical Reports by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. 5-81ON APP DECODINGCarlos R.P. HartmannLuther D. RudolphKishan G. MehrotraGuy J. SnedekerJune 81SCHOOL OF COMPUTERAND INFORMATION SCIENCEO d · +n APP Deco 1ngCarlos R.P. Hartmann*Luther D. Rudolph*Kishan G. Mehrotra**Guy J. Snedeker**+This work was partially supported by the National Science Foundationunder Grant ECS 80-19484.*Communication Studies Laboratory, School of Computer and InformationScience, Syracuse University, Syracuse, New York, 13210, (315) 423-2368.**School of Computer and Information Science, Syracuse University,Syracuse, New York, 13210, (315) 423-2368AbstractIn this paper we show that APP decoding for a linear code Cis optimum not for C, but for a minimum-distance-2 code C whichcontains C as a subcode when the codewords of C are transmittedwith equal probability. However, APP decoding is shown to be a-symptotically optimum for C for high SNR when C is a binary one-step orthogonolizable code with equiprobable codewords transmittedover the AWGN channel.-1-1. IntroductionIn recent years there has been a growing interest in soft de-cision decoding schemes for error-correcting codes. The intent isto avoid, in part or in whole, the degradation of communication sys-tem performance which results when symbol-by-symbol "hard decision"quantization precedes decoding. In most cases the additional infor-mation provided by using soft decisions is worth about 2 dB of ad-ditional coding gain.Among the soft decision decoding techniques which seek to mini-mize the probability of symbol error, Massey's APP (a postetioriprobability) decoding [1J is one of the most important. In thispaper we show that APP decoding for a linear code C is optimum notfor C,but for a minirnurn-distance-2 code ~ which contains C as asubcode when the codewords of C are transmitted with equal probability.However, APP decoding is shown to be asymptotically optimum for Cfor high SNR when C is a binary one-step orthogonolizable code withequiprobable codewords transmitted over the AWGN channel.-2-2. Main ResultIn this section we first show that the complexity of the optimumerror symbol decoding rule is independent of the number of paritychecks used and is prohibitve for almost any code. Based on thisoptimum decoding rule we then define a new error symbol decoding rulewhich is analogous to maximum-likelihood word decoding. We then showthat for a particular case this new decoding rule becomes APP decoding.Let H be the parity check matrix for an (n,k,d) linear code Cover GF(p), P a prime. Also let B be a Jxn matrix whose rows corre-spond to J parity checks of C. A codeword £ = (cO' c l ' ... , c 1)n-is transmitted over a time-discrete memoryless channel with outputalphabet~. The received word is denoted by £ = (rO' r 1 , ... , r n - 1 ) ,r j E h and its quantized version by ! = (ro ' r1 , ••• , rn - 1 ), rj € GF(p).- T ~ ~Define § = rB , and let~denote addition over GF(p) andC'x theadditive inverse of x in GF(p). Now let y = (vO' VI' ... , v n - 1 ) and~ = (wO' WI' .•• , wn- 1 ) be arbitrary vector of Vn , the n-dimentionalvector space over GF(p). Finally, let ~ = (eO' e l , •.. , en-I)'e. € GF (p), be the error vector, that is, ~ = £ e ~.JThe decoding problem is: given that a codeword from C was trans-mitted, and given d B · Ar an , compute an est1rnate em of the roth error"symbol e in such a way that the probability that e equals e ism m mmaximized. Now we state the optimum error symbol decoding rule.Decoding Rule 1: 1\ -Set e = rea, where a € GF (p) maximizesrn mPr(em-3-Pr(vlr),""J ""J( 1)whereE(a) = {v, v = r 0 a yB Tf"W m m '.-We now show that this decoding rule is -equivalent to the op-timum symbol-by-symbo1 decoding rule, which is:a € GF(p) maximizesASet c = a, wheremPr(crn = aIr) = Lf"'o..I~€G (a)(2)Awhere c m is the estimate of em andWe demonstrate this equivalence by showing that (1) and (2) areequal. Define ~ = r - ~ for ~ € Vn andwhereU (a) = {~f~ W € C(a)},~We now prove two technical lemmas.Lemma 1: L~EC (a)Pr(~I;E) = EJ:!€U(a)Pr(ulr).,.." """'"Proof: Since r is determined by knowing [, we may writePr(f8 ~I!:) =Pr(~I!:). Sincel3=r0~, Pr(M1r) =Pr(!tlr). Now,since for each element ~EC(a) there is a corresponding element-4-1:! = ~0)t € U(a) and vice versa, we may conclude thatL:~€C(a)Pr(~1 £) = L Pr(2Ir).1d EU (a)Q.E.D.Lerruna 2: 1:~€U (a)Pr(ulr) =I"0oI I"oJ1: Pr(vlr).yEE(a) I"oJ I"oJProof: We prove Lemma 2 by proving that U(a)=E(a) • t\Te first assumethat ~EU(a), that f 8 x, where yEC (a) • Thus and T - T §.so ~ = Ym=a ?SB = ;(B =I"oJThen ?,Se:E(a) • Now that ~€E(a). Then z =r 0a and T §.we assume ~B =m mDefine w = r 0 z .'" I"oJ I"0oITThen w =a and wB =0.m I"oJ I"oJ So ~EC(a) and ~EU(a).Q.E.D.As a consequence of Lemmasl and 2 we have the following theorem:Theorem 1: L: Pr(~lf) =yEE(a)Now (3) may be written asL~EC(a)( 3)L)lEE (a)Pr(ylr) = L~e:C(a)Pr(rlw)Pr(w)/Pr(r) .I"oJ f'O,J I"oJ '"(4 )Since only codewords of C are transmitte~we may write (4) asLYEE(a)Pr(yl!:) = L~€G (a)Pr (wi r) •I"oJ ,...,So we may conclude that for any given matrix B we havePr (e = r t::\ at r) =In m\.../ I"oJ L~€G (a)An alternative way to see this result is to observe that-5-Pr (yl!:) = Pr (~ = r e Xl [) =0 whenever ~ = ~ e ~ I. c. This also showsthat the complexity of calculating Pr(em = rmE)a1r) is O(min(pk,pn-k»[2]and it is independent of the given matrix B. This complexity of de-coding is prohibitive for almost any code. We now modify DecodingRule 1 in order to decrease its complexity. From the proof of Lemmas1 and 2 we know that we may write Pr(~1 r) = P(~ = rSyl r) wherey€E(a) and ~ = £e X € C(a). Since the channel is memoryless we haven-lPr(xlr) = 'ITi=On-l= iTi=OPr(r.1 w.) (Pr(w)/Pr(r»1 1. /'OoJ /'OoJ(Pr (w. I r . ) Pr (r . ) /Pr (w. ) ) (Pr (w) /Pr (r) )111 1 ~ ~where ~ = [ep)l. We note that Pr(w.lr.) = Pr(r. t:'w.lr.) = Pr(v.lr.).1 ~ 1 ~ 1 1 ~ 1Thus (1) may be written as:orPr (e =r Cl al r) =m m\:lf'Oo.I t"o.Jn-lL (iT Pr(v.1 r. )Pr(r. )/Pr(r.C\f7.» (Pr(~)/Pr(r»1 1 1 i\Jv 1 ~ f'Oo.I ~y€E(a) i=On-lPr(e =r Gal r)=R(r) L ('If Pr(v.lr.)/Pr(r.<9v.»Pr(rev) (5)m m ~ - yEE(a) i=O 1 1 1 1 ~-wheren-lR(!:) = 1Tj=OPr (r · ) /Pr (r) ,J ~and if the codewords of Care equiprobable and B=H, (5) may bewritten as-6-Pr(e =r ~ a1 r)m m"-/ - Lv€E(a)~n-l11"i=OPr(v.1 r.).1. 1.(6)Based on (6) we define the following decoding rule:Decoding Rule 2: 1\ - eSet e =r - a, where a € GF(p) maximizesm rnL~EE(a)n-l11'i=OPr (v. I r. ) •1. ~(7)The reader should note the analogy between Decoding Rule 2 andmaximum-likelihood word decoding. Using arguments similar to thoseused in the proof of Theorem 1 we can prove the following:Theorem 2:n-1E 7TYEE(a) i=OPr(v.' r.) =1. ~n-1L 7Twe:C(a) i=O,....,Pr(w.1 r .. ).1. ~(8)Now let C be the linear (n,k,d) code over GF(p) defined by thematrix B, that is, ~ € C if and only if !BT=Q. Theorem 2 shows thatthe performance of a decoder which implements Decoding Rule 2 is equalto the performance of a decoder which implements the optimum syrnbol-by-symbol decoding rule for C whenever the codewords of C are transmittedwith equal probability. We note that C is a subcode of C and thusd ~ d and k ~ k. Since the complexity of calculating (7) is O(min(pk,pn-k)and d ~ d, it appears that Decoding Rule 2 may be more complex than,and yields an inferior performance to, Decoding Rule 1. However,we will show that when the parity check corresponding to the rows ofthB are orthogonal on the m position, Decoding Rule 2 is the same as-7-Massey's original APP decoding rule (APPDR) which has complexityO(Jn} [1J. To show this we let Q.={ jl' j2' ••• , jp. }, lsjsJ,J Jbe the set of positionschecked by the jth row of B, excluding theroth position. Since the parity checks corresponding to the rows ofB are orthogonal in the roth position, we know that Q.nQ. = ~ forJ ~i~j, l:5i,j~J. Now letandS.=JF.JEi=lb. r. G)r ,J. J.. m1. 1.b. E: GF (p) ,JiM.= {x= (x. ,J "J ) 1... ,F.Jx . ) I x. ~ GF (p), A b. x .J F . Ji i=l Ji J i)=s·0 r 0aLJ mWe now define APPDR in terms of our formulation.APPDR: Set ~ = r 1"::\ a where a E GF (p) maximizesrn rn \:,.J ,Pr (x =r e a I r )m m InJ1f (j=lF.Jr "ITxEM. i=l"J JPr (x. f r. ).J · J ·1. 1.(9)Since Q.nQ.=~ for i~j, it is easily seen that (7) and (9) are equal.J 1When the rows of B correspond to parity checks orthogonal on theroth position,then d=2. Thus the performance of an APP decoder is thesame as the performance of the optimum symbol-by-syrnbol decoder forC, with d=2, whenever the codewords of Care equiprobable. However,when a binary code C is transmitted over the AWGN channel and J=d-l,the APP decoder is asymptotically optimum for high SNR. This isobtained as follows: from (8) and [3J we may conclude that PAPp 4SNR+OO),-8-the asymptotic probability of error of the mth bit for APPDR, isgiven bywhere em is the minimum Hamming weight of vectors in C(l), N(Gro ) thenumber of codewords of weight em in C(l), R=k/n, Yb=Eb/No the SNR pertransmitted digit of information andIt is easily seen that 0 =J+l. Thus for J=d-l, e =d andm mwhich is proportional to the asymptotic behavior of the probabilityof error of the mth bit for the optimum symbol-by-syrnbol decoding rulefor C whenever its codewords are transmitted over the AWGN channelwith equal probability. [3J~9-3. Simulation ResultsIn this section we present performance curves for the (21,11)binary projective geometry code transmitted over the AWGN channel forAPP decoding and maximum-radius decoding [4J, both using five ortho-gonal parity checks, their iterative extensions [4J, majority logicdecoding and optimum symbol-by-symbol decoding. The APP decoder and themaximum radius decoder with the demodulation function[1,p(x} = COS (1TX),-1,~~oo<x<l,l~x [4Jare asymptotically optimum for high SNR. Figures 1 and 2 show thebit-error-rate and the word-error-rate performance respectively. Thedotted curve in Figure 2 shows the asymptotic behavior of optimumdecoding [5J.The APP decoding rule is a function of the channel SNR. However,as Figures 1 and 2 show for the (21,11) PG code transmitted over theAWGN channel, maximum radius decoding, which is fixed and thus indepen-dent of the channel SNR, gives almost the same performance as APP de-coding when both decoders use the same five orthogonal parity checks.This make maximum-radius decoding very attractive for practical ap-plications.-10-References[lJ J.L. Massey, Threshold Decoding. Cambridge, MA.: MIT Press,1963.[2J C.R.P. Hartmann and L.D. Rudolph, "An optimum symbol-by-symboldecoding rule for linear codes," IEEE Trans. Inform. Theory,vol. IT-22, pp. 514-517, September, 1976.[3J C.R.-P. Hartmann, L.D. RUdolph and K.G. Mehrotra, "Asymptoticperformance of optimum bit-by-bit decoding for the whiteGaussian channel," IEEE Trans. Inform. Theory, vol. IT-23,pp. 520-522, July, 1977.[4J L.D. Rudolph, C.R.P. Hartmann, T.-Y. Hwang and N.Q. Duc,"Algebraic analog decoding of linear binary codes," IEEETrans. Inform. Theory, vol. IT-25, pp. 430-440, July, 1979.[5J G.D. Forney, Jr., Concatenated Codes, Cambridge, MA.: MITPress, 1966, p.52.[6J CNR, Incorporated, "Demod/decoder intergration," Rome AirDevelopment Center, Griffiss AFB, NY, final technical re-port RADC-TR-78-70 on Contract F30602-76-C-0361.-11-Figure 1:Bit Error Rate of the(21,11) P.G. Code over the AWGN Channel(Antipodal Signalling)8.(APP DECODING(APPD)MRD (5 ITERATIONS)MAXIMUM-RADIUS DECODING(MRD)UNCODED DATADIGITAL MAJORITY DECODINGOPTIMUM BIT-BY-BITDECODING [6]m,..-12-Figure 2:Word Error Rate of the(21,11) P.G. Code over the AWGN Channel(Antipodal Signalling)UNCODEO DATAMRD (5 ITERATIONS)MAXIMUM-RADIUS DECODING (MRO)DIGITAL MAJORITY DECODINGIO-~ .............---"'------"'--.......-.....-.----........--'"---...--....-~-.-.---'1.0 2.0 3.0 6.0 8.'..o..~lAJ 10--eo~o~•-0...o~-13-

On APP Decoding

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On APP Decoding

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