Spatially-Coupled MacKay-Neal Codes Universally Achieve the Symmetric Information Rate of Arbitrary Generalized Erasure Channels with Memory

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1 Spatially-Coupled MacKay-Neal Codes Universally Achieve the Symmetric Information Rate of Arbitrary Generalized Erasure Channels with Memory arxiv: v1 [csit] 27 Jan 215 Abstract This paper investigates the belief propagation decoding of spatially-coupled MacKay-Neal SC-MN codes over erasure channels with memory We show that SC-MN codes with bounded degree universally achieve the symmetric information rate SIR of arbitrary erasure channels with memory We mean by universality the following sense: the sender does not need to know the whole channel statistics but needs to know only the SIR, while the receiver estimates the transmitted codewords from channel statistics and received words The proof is based on potential function Index Terms spatially-coupled codes, MacKay-Neal codes, LDPC codes, potential function I INTRODUCTION Felström and Zigangirov introduced convolutional LDPC codes [1] Later the codes are called spatially-coupled SC codes Lentmaier et al confirmed that regular SC LDPC codes achieve excellent BP decoding [2] Kudekar et al proved that SC codes achieve MAP threshold by BP decoding on the binary erasure channel [3] and binary memoryless symmetric channels [4] In [5], Takeuchi et al introduced potential function to understand how threshold saturation phenomenon happen With some modifications, Yedla et al proved threshold saturation of spatially-coupled LDPC codes over in [6], [7] Kasai et al introduced SC MacKay-Neal MN codes, and showed that these codes with finite maximum degree achieve capacity of by numerical experiment [8] Obata et al [9] and Okazaki et al [1] proved respectively,,d r = 2,d r = 2 SC-MN codes and,d r = 3,d g = 3 SC-MN codes achieve the capacity of for any > d r, where each of,d r,d g describes the degree of nodes in the Tanner graph of the codes Reliable transmissions over channels with memory are practically important, eg, magnetic recording with ISI and Rayleigh fading channel with memory [11] Pfister and Siegel proved that carefully designed irregular LDPC codes can achieve the symmetric information rate SIR of dicode erasure channels under joint iterative decoding [12] In [13] and [14], it was empirically observed that the BP threshold Masaru Fukushima, Takuya Okazaki and Kenta Kasai Department of Communications and Computer Engineering, Tokyo Institution of Technology fukushima,osa,kenta}@commcetitechacjp of spatially-couple regular codes achieve the SIR of and channels, respectively, by increasing node degree In this paper, we show that SC-MN codes with bounded degree universally achieve the symmetric information rate of wide variety of erasure channels with memory We mean by universality the following sense: the sender does not need to know the whole channel statistics py x but needs to know only the SIR, while the receiver estimates the transmitted codewords from channel statistics and received words The proof is based on the powerful potential function method [5], [6], [7] II BACKGROUND A Generalized Erasure Channel GEC Denote a channel input and output as x =,,x n,1} n and y = y 1,,y n Y n, respectively We assume that x,1} n is uniformly distributed Let us assume that by introducing some appropriate state nodes σ = σ 1,,σ n such that px y = σ px,σ y, we can factorize px,σ y so that its factor graph is a tree By Bayes rule, we have px, σ y py σ, xpσ xpx Consider the APP detector implemented by sum-product algorithm to calculate pxj = y,px j = 1 y =: µ j,µ j 1 for j = 1,2,,n We say the channel py x is a generalized erasure channel GEC if µ j,µ j 1 is one of 1,,, 1,1/2, 1/2} The corresponding LLR values are +, and Some authors use, 1 and?, instead B LDPC codes over GEC Next, consider x is uniformly distributed in an LDPC code C In precise, 1/#C x C px = x / C

2 Let us denote the factor graph of py σ,xpσ xpx by G We divide G into two subgraphs One is the factor graph of py σ,xpσ x and the other is the factor graph of px They are corresponding to APP detector and LDPC decoder, respectively Since the message from APP detector to the bit nodes is one of M, the messages used in the LDPC decoder also takes value in M Consider the density evolution of the BP decoding on G in the limit of code length We define φx;ǫ as a function which maps the erasure probability of messages from LDPC decoder to APP detector via bit nodes x, to the erasure probability of messages from APP detector to from LDPC decoder φx; ǫ, where ǫ is a parameter which defines the channel From the definition, it follows that φx;ǫ is nondecreasing in x [,1] In this paper, we further assume that φx; ǫ is twice continuously differentiable with x and strictly increasing with ǫ [,1] In this setting, the density evolution for,d r regular LDPC codes over GEC with φx;ǫ is written as follows x = 1, x t+1 = φ 1 1 x t dr 1 ;ǫ 1 1 x t dr 1 1, where x t is the erasure probability of messages from bit nodes to parity-check nodes at the t-th decoding round C Symmetric Information Rate SIR Iǫ is the mutual information is defined as follows 1 Iǫ := lim n n IX;Yǫ, under the existence of limit, where capital letters represent random variables In [15], [16], it is shown that the SIR is calculated via φx; ǫ as follows Iǫ = 1 1 φx;ǫdx = 1 Φ1;ǫ, where Φx;ǫ := x φx ;ǫdx Let R be the coding rate Define SIR limit as ǫ such that Iǫ = R, and denote it by ǫ SIR R Uniqueness of ǫ SIR R is again due to the assumption that φx; ǫ is increasing in ǫ D MacKay-Neal Codes,d r,d g MN codes are multi-edge type MET LDPC used over GEC with φx;ǫ codes [17] defined by pair of multi-variables degree distributions µ, ν listed below νx;φx;ǫ = d r x 1 +Φxdg 2 ;ǫ, µx = xdr 1 xdg 2 Here, we slightly extended the definition of degree distribution in such a way that the bits corresponding to the term Φx dg 2 ;ǫ are transmitted through the GEC with φ ǫ In the case of ǫ, Φx, ǫ = ǫx We define the erasure probability message sent from bit nodes along edges of type j at the t-th decoding round by x t j The recursion of density evolution of MET-LDPC codes on is given by j = ν jy t ;Φx;ǫ, y t+1 j = 1 µ j1 x t, ν j 1;id R µ j 1 x t where ν j x;φx;ǫ := x j νx;φx;ǫ, µ j x := x j µx and id R is the identity function id R x = x Then, the density evolution of,d r,d g MN codes is where x = 1, x t+1 = fgx t ;ǫ, fx;ǫ := x 1 1,φx dg 2 ;ǫ 1 gx := 1 1 dr 1 1 x 2 dg, 1 1 dr 1 x 2 dg 1 2 E Spatially-Coupled MacKay-Neal Codes SC-MN codes of chain length L and of coupling width w are defined by the Tanner graph constructed by the following process First, at each section i Z, place rm/l bit nodes of type 1 and M bits nodes of type 2 Bit nodes of type 1 and 2 are of degree and d g, respectively Next, at each section i Z, place M check nodes of degree d r +d g Then, connect edges uniformly at random so that bit nodes of type 1 at section i are connected with check nodes at each section i [i,,i+w 1] with d r M/w edges, and bit nodes of type 2 at section i are connected with check nodes at each section i [i,,i + w 1] with d g M/w edges Bits at section i / [,L 1] are shortened Bits of type 1 and 2 at section i [,L 1] are punctured and transmitted, respectively The rate of SC-MN codes R MN,d r,d g,l,w is given by R MN,d r,d g,l,w = d r 1+w 2 w i= + 1 i w dr+dg L = d r L 3 F Vector Admissible System and Potential Function In this section, we define vector admissible systems and potential functions Definition 1 Define X [,1] d, and F : X [,1] R and G : X R as functional satisfying G = Let D be a d d positive diagonal matrix Consider a general recursion defined by x l+1 = fgx l ;ǫ 4 where f : X [,1] X and g : X X are defined by F x;ǫ = fx;ǫd and G x = gxd, where F x;ǫ Fx,, Fx x n Then the pair f,g defines a vector admissible system if 1 f, g are twice continuously differentiable, 2 fx;ǫ and gx are non-decreasing in x and ǫ with respect to 1, 1 We say x y if x i y i for all 1 i d

3 3 fg;ǫ = and Fg;ǫ = We say x is a fixed point if x = fgx;ǫ Definition 2 [18, Def 2] We define the potential function Ux;ǫ of a vector admissible system f,g by Ux;ǫ gxdx T Gx Fgx;ǫ Definition 3 [18, Def 7] Let Fǫ x X \} x = fgx;ǫ} be a set of non-zero fixed points for ǫ [,1] The potential threshold ǫ is defined by ǫ supǫ [,1] min x Fǫ Ux;ǫ > } Let ǫ s be threshold of uncoupled system defined in [18, Def 6] For ǫ such that ǫ s < ǫ < ǫ, we define energy gap Eǫ as Eǫ max inf ǫ [ǫ,1] x Fǫ Ux;ǫ Definition 4 [18, Def 9] For a vector admissible system f, g, we define the SC system of chain length L and coupling width w as x t+1 i ǫ i = = 1 w w 1 k= f w 1 1 w j= gx t i+j k ;ǫ i k ǫ, i,,l 1},, i /,,L 1} If we define f,g as the density evolution for,d r,d g MN codes in II-D and II-D, the SC system gives the density evolution of SC-MN codes with chain length L and coupling width w Next theorem asserts that if ǫ < ǫ then fixed points of SC vector system converge towards for sufficiently large w Theorem 1 [18, Thm 1] Consider the constant K f,g defined in [18, Lem 11] This constant value depends only on f,g If ǫ < ǫ and w > dk f,g /2 Eǫ, then the SC system of f,g with chain length L and coupling width w has a unique fixed point It can be seen that the density evolutionf,g of,d r,d g MN codes over GECφx; ǫ is a vector admissible system by choosing F x;ǫ,gx and D as below, since this system f,g satisfies the condition in Definition 1 Fx;ǫ = d r x 1 d +Φxdg 2 ;ǫ, l Gx = d r +d g x 2 +1 dr 1 x 2 dg 1, dr D = d g From Definition 2, the potential function U,x 2,ǫ of,d r,d g MN codes is given by U,x 2 ;ǫ 5 = 1 Φ1 1 dr 1 x 2 dg 1 } dg ;ǫ 1 1 dr 1 1 x 2 dg } 1 dr 1 x 2 dg + d gx x 2, III PROOF OF ACHIEVING SIR In this section, we will prove that,d r = 2,d g = 2 and,d r = 3,d g = 3 SC-MN codes, for any > d r, achieve the SIR of any GEC First, we calculate the potential threshold ǫ,d r,d g of MN codes, and show that the potential threshold equals to the SIR limit ǫ SIR d r /d r Then we apply Theorem 1 which proves that density evolution of SC-MN code has a unique fixed point for ǫ smaller than SIR limit ǫ SIR d r /d r A Potential Function at Trivial Fixed Point Recall the definition of potential threshold ǫ,d r,d g in Definition 3 We need to investigate the structure of Fǫ to calculate the potential threshold ǫ,d r,d g The density evolution 1 of,d r,d g MN codes at fixed point,x 2 ;φx;ǫ can be rewritten as =1 1 dr 1 1 x 2 dg 1, 6 x 2 =φ1 1 dr 1 x 2 dg 1 } dg ;ǫ 1 1 dr 1 x 2 dg 1 } dg 1 7 First, observe that = 1,x 2 = φ1;ǫ Fǫ for ǫ [,1] We call these fixed points trivial From II-F and the definition of SIR limit, the next lemma asserts that the sign of U 1,φ1;ǫ;ǫ changes at the SIR limit ǫ SIR d r Lemma 1 U 1,φ1;ǫ;ǫ = 1 Φ1;ǫ, d l U 1,φ1;ǫ;ǫ > if ǫ < ǫ SIR d r, = if ǫ = ǫ SIR d r, 8 < if ǫ > ǫ SIR d r Proof: The first part is straightforward from II-F The second part is obvious from the definition of SIR limit B Potential Function at Non-Trivial Fixed Point Next, for given,1, solve III-A in terms of x 2 Denote this by x 2 [ ] x 2 [ ] = 1 1 d 1 x l 1 1 dg 1 1 dr 1 Note that for some, x 2 [ ] may fall outside [,1] as one can see from Fig 2 Such points are excluded from the set of fixed points Then it follows that,x 2 [ ] Fǫ[ ] iff x 2 [ ],ǫ[ ] [,1] 2, 9 where ǫ[ ] is the unique solution of the equation III-A with x 2 = x 2 [ ] holds In precise, the equation is as follows x 2 [ ] =φ1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ[ ] 1 1 dr 1 x 2 [ ] dg 1 } dg 1 Uniqueness is due to the assumption that φx;ǫ is strictly increasing in ǫ We call such fixed points III-B non-trivial

4 Ux5 1,x 2 [ ];ǫ[ ] U,x 2 [ ];ǫ[ ] Fig 1 Potential functions U,x 2 [ ];ǫ[ ] at non-trivial fixed points of 4,2,2 MN codes and 6,3,3 MN codes for 3 types of generalized erasure channels ǫ[ ], ǫ[ ], ǫ[ ] and PR3ǫ[ ] For example of ǫ, ǫ [ ] can be written in an explicit form x 2 [ ] ǫ [ ] := 1 1 dr 1 x 2 [ ] dg 1 } dg 11 Obviously, trivial and non-trivial fixed-points cover the set Fǫ, in precise, Fǫ = F t ǫ F n ǫ We denote the set of trivial and non-trivial fixed-points respectively by F t ǫ and F n ǫ F t ǫ := 1,φ1;ǫ } F n ǫ :=,x 2 [ ] ǫ[ ] = ǫ,,1 } F t ǫ F n ǫ is an empty set Let φ[ ] be the unique solution of III-A in terms of φ for given,x 2 = x 2 [ ] and ǫ = ǫ[ ] φ[ ] := Equivalently, we have where x 2 [ ] 1 1 x1 dr 1 x 2 [ ] dg 1 d g 1 11 φ[ ] = φψ[ ];ǫ[ ], 12 ψ[ ] := 1 1 dr 1 x 2 [ ] dg 1 d g 13 Note thatx 2 [ ],φ[ ] andψ[ ] do not depend on the channel GECφ ;ǫ, whileǫ[ ] does From III-B and III-B, it can be seen that ǫ [ ] = φ[ ] 14 Substituting,x 2 [ ] and φ[ ] into II-F, at non-trivial fixed points,x 2 [ ] Fǫ[ ], we have the potential function as U,x 2 [ ];ǫ[ ] 15 = 1 Φψ[ ];ǫ[ ] 1 1 dr 1 1 x 2 [ ] dg } 1 dr 1 x 2 [ ] dg 1 1 x 2 [ ] From [9, Lemma 4] and [1, Lemma 1, Lemma2], we have the following lemma Lemma 2 Consider transmissions over ǫ, ie, we have φx = φ x = ǫ and φ[ ] = ǫ[ ] Let U,x 2 [ ];ǫ [ ] be the potential function of,d r,d g MN codes at the non-trivial fixed points For any > d r it holds that U,x 2 [ ];ǫ [ ] > for,1 for d r,d g = 2,2 and 3,3 Lemma 3 For any GECφx;ǫ, let U,x 2 [ ];ǫ[ ] be the potential function at the non-trivial fixed point,x 2 [ ] Fǫ[ ] for,1 as defined in III-B Then, it holds that U,x 2 [ ];ǫ[ ] U,x 2 [ ];ǫ [ ] Proof: From III-B, we have x 2 [ ],ǫ[ ] [,1] 2 From III-B, we have ψ[ ] [,1] From this and using III-B and III-B, we obtain ǫ [ ] [,1] This justifies that U,x 2 [ ];ǫ [ ] is well-defined since,x 2 [ ],ǫ [ ] [,1] We have U,x 2 [ ];ǫ[ ] U,x 2 [ ];ǫ [ ] a =ǫ [ ] 1 1 dr 1 x 2 [ ] dg 1 dg Φ 1 1 dr 1 x 2 [ ] dg 1 dg ;ǫ[ ] b =φ[ ] ψ[ ] Φ ψ[ ];ǫ[ ] =φ[ ] ψ[ ] ψ[x1] c = ψ[x1] d = e ψ[x1] φx ;ǫ[ ]dx φ[ ] φx ;ǫ[ ]dx φψ[ ];ǫ[ ] φx ;ǫ[ ]dx The equality a is due to III-B and 1 The equality b is due to III-B In c, we used the fact that ψ[ ] does

5 not depend on channel GECφ ;ǫ The equality d is due to III-B In e, we used the fact that φψ[ ];ǫ[ ] φx ;ǫ[ ] for x [,ψ[ ] ] since φx;ǫ is non-decreasing in x The equality is attained with φx;ǫ = φ x;ǫ = ǫ In Fig 2, we plotted the potential functions at non-trivial fixed points of4,2,2 MN codes and6,3,3 MN codes for 3 types of generalized erasure channels ǫ[ ], ǫ[ ] and ǫ[ ] One can see that the potential function for is lower than other curves for any,1 as claimed in Lemma 3 C Potential Threshold Equals to SIR Limit Next theorem shows the potential threshold of some MN codes is equal to the SIR limit Theorem 2 For any GECφx;ǫ, the potential threshold ǫ of,d r,d g MN codes is equal to ǫ SIR dr for any > d r, d r,d g = 2,2 and 3,3 Proof: From Lemma 2 and Lemma 3, we have that for any,1 such that ǫ = ǫ[ ] [,ǫ SIR dr, It follows that ǫ = supǫ [,1] U,x 2 [ ];ǫ[ ] > 16 min U,x 2 ;ǫ > },x 2 Fǫ = supǫ [,1] min,x 2 F tǫ F nǫ U,x 2 ;ǫ > } a = supǫ [,ǫ SIR dr b = ǫ SIR d r min U,x 2 ;ǫ > },x 2 F tǫ F nǫ In a, we used 1 In b, we used 1 and III-C Define the BP threshold ǫ BP,d r,dg,l,w as the spremum value of ǫ such that the SC system with chain length L and coupling width w of,d r,d g MN codes over GECǫ converges to zero In precise, ǫ BP,d r,dg,l,w is supǫ [,1] lim x t t i = for i =,1,,L 1} Then, from II-E and Theorem 3 and 1 we have the following theorem Theorem 3 For any GECφx;ǫ, the potential threshold ǫ of,d r,d g MN codes is equal to ǫ SIR dr for any > d r, d r,d g = 2,2 and 3,3 lim lim w L ǫbp,d r,dg,l,w = ǫ SIR d r /, lim lim w L RMN,d r,d g,l,w = d r In words, some SC-MN codes universally achieve the SIR limit of any GECφx;ǫ in the limit of large L and w IV CONCLUSION AND FUTURE WORK We have shown that some SC-MN codes achieve the SIR limit of any GECφx; ǫ via potential function The future works include an extension erasure multi-acess channels [19] and to more general channels, eg, channels with Gaussian noise REFERENCES [1] A J Felström and K S Zigangirov, Time-varying periodic convolutional codes with low-density parity-check matrix, IEEE Trans Inf Theory, vol 45, no 6, pp , June 1999 [2] M Lentmaier, D V Truhachev, and K S Zigangirov, To the theory of low-density convolutional codes II, Probl Inf Transm, no 4, pp , 21 [3] S Kudekar, T Richardson, and R Urbanke, Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the, IEEE Trans Inf Theory, vol 57, no 2, pp , Feb 211 [4], Spatially coupled ensembles universally achieve capacity under belief propagation, IEEE Trans Inf Theory, vol 59, no 12, pp , 213 [5] K Takeuchi, T Tanaka, and T Kawabata, A phenomenological study on threshold improvement via spatial coupling IEICE Transactions, vol 95-A, no 5, pp , 212 [6] A Yedla, Y-Y Jian, P Nguyen, and H Pfister, A simple proof of threshold saturation for coupled vector recursions, in Proc 7th Int Symp on Turbo Codes and Related Topics, Sept 212, pp [7], A simple proof of maxwell saturation for coupled scalar recursions, IEEE Trans Inf Theory, vol 6, no 11, pp , Nov 214 [8] K Kasai and K Sakaniwa, Spatially-coupled MacKay-Neal codes and Hsu-Anastasopoulos codes, IEICE Trans Fundamentals, vol E94-A, no 11, pp , Nov 211 [9] N Obata, Y-Y Jian, K Kasai, and H D Pfister, Spatially-coupled multi-edge type LDPC codes with bounded degrees that achieve capacity on the under BP decoding, in Proc 213 IEEE Int Symp Inf Theory ISIT, July 213, pp [1] T Okazaki and K Kasai, Spatially-coupled MacKay-Neal codes with no bit nodes of degree two achieve the capacity of, in Proc 214 IEEE Int Symp Inf Theory ISIT, June 214, pp [11] A P Worthen and W Stark, Unified design of iterative receivers using factor graphs, IEEE Trans Inf Theory, vol 47, no 2, pp , Feb 21 [12] B Kurkoski, P Siegel, and J Wolf, Joint message-passing decoding of LDPC codes and partial-response channels, IEEE Trans Inf Theory, vol 48, no 6, pp , June 22 [13] S Kudekar and K Kasai, Threshold saturation on channels with memory via spatial coupling, in Proc 211 IEEE Int Symp Inf Theory ISIT, Aug 211, pp [14] S Sekido, K Kasai, and K Sakaniwa, Threshold saturation on erasure channel via spatial coupling, in Proc Symp on Inf Theory and its Applications SITA211, Dec 211 [15] H Pfister and P Siegel, Joint iterative decoding of ldpc codes for channels with memory and erasure noise, IEEE J Sel Area Commun, vol 26, no 2, pp , February 28 [16] A Ashikhmin, G Kramer, and S ten Brink, Extrinsic information transfer functions: model and erasure channel properties, IEEE Trans Inf Theory, vol 5, no 11, pp , Nov 24 [17] T Richardson and R Urbanke, Modern Coding Theory Cambridge University Press, Mar 28 [18] A Yedla, Y-Y Jian, P Nguyen, and H Pfister, A simple proof of threshold saturation for coupled vector recursions, in Proc 212 IEEE Information Thoery Workshop ITW, Sept 212, pp [19] S Kudekar and K Kasai, Spatially coupled codes over the multiple access channel, in Proc 211 IEEE Int Symp Inf Theory ISIT, Aug 211, pp APPENDIX In Appendix, we give three GECs as example

6 ǫ[ ] 4 ǫ[ ] Fig 2 Plot of ǫ[ ] at non-trivial fixed points of 4,2,2 MN codes and 6,3,3 MN codes for 3 types of generalized erasure channels ǫ[ ], ǫ[ ], ǫ[ ] and PR3ǫ[ ] Example 1 Binary Erasure Channel For the binary erasure channel ǫ with erasure probability ǫ, φx;ǫ,φx;ǫ,iǫ and ǫ[ ] are respectively given as φ x;ǫ = ǫ, Φ x;ǫ = ǫx, I ǫ = 1 ǫ, ǫ[ ] = ǫ [ ], given as 4ǫ 2 φ x;ǫ = 2 x1 ǫ 2 4ǫ 2 Φ x;ǫ = 1 ǫ2 1 ǫx, I ǫ = 1 2ǫ2 1+ǫ, ǫ[ ] = ǫ [ ], where ǫ [ ] is the unique solution ǫ of the following equation x 2 [ ] =φ 1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ [ ] 1 1 dr 1 x 2 [ ] dg 1 } dg 1 ǫ [ ] can be written in an explicit form III-B The potential function is given as U,x 2 [ ];ǫ[ ] 17 := 1 ǫ [ ] 1 1 dr 1 x 2 [ ] dg 1 } dg 1 1 dr 1 1 x 2 [ ] dg } 1 dr 1 x 2 [ ] dg 1 1 x 2 [ ] Example 2 Dicode Erasure Channel For the dicode erasure channel ǫ with erasure probability ǫ, the output y,1, 1,?} n for given x,1} n is defined as follows y j = x j x j 1 wp 1 ǫ,? wp ǫ, where x := φx;ǫ,φx;ǫ,iǫ and ǫ[ ] are respectively where ǫ [ ] is the unique solution ǫ of the following equation x 2 [ ] =φ 1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ 1 1 dr 1 x 2 [ ] dg 1 } dg 1 In the case of ǫ, ǫ [ ] can be written in an explicit form 1/2 x 2 ψ[ ] 2[] ψ[x ǫ [ ] := 1] dg 1/dg 2 ψ[ ] 1/2 x 2[] ψ[] dg 1/dg In general, we do not need the explicit form of ǫ[ ] for the purpose of this paper The potential function is given as follows U,x 2 [ ];ǫ[ ] := 1 Φ 1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ [ ] 1 1 dr 1 1 x 2 [ ] dg } 1 dr 1 x 2 [ ] dg 1 1 x 2 [ ] Example 3 Partial Response 2 Channel For the channel ǫ with erasure probability ǫ, the output y

7 ,1,2,3,4,?} n for given x,1} n is defined as follows x j +2x j 1 +x j 2 wp 1 ǫ, y j =? wp ǫ, where x := φx;ǫ,φx;ǫ,iǫ and ǫ[ ] are respectively given as φ x;ǫ = 4ǫ ǫx+1 ǫx ǫ 2 x 1 ǫǫ 2 x Φ x;ǫ = 4ǫ 2 2 x 4 21 ǫ 2 x 1 ǫǫ 2 x 2, I ǫ = 1 2ǫ3 ǫ+1 ǫ 3 +ǫ 2 +2, ǫ[ ] = ǫ [ ], where ǫ [ ] is the unique solution ǫ of the following equation x 2 [ ] =φ 1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ 1 1 dr 1 x 2 [ ] dg 1 } dg 1 The potential function is given as follows U,x 2 [ ];ǫ[ ] := 1 Φ 1 1 dr 1 x 2 [ ] dg 1 } dg ;ǫ [ ] 1 1 dr 1 1 x 2 [ ] dg } 1 dr 1 x 2 [ ] dg 1 1 x 2 [ ]