LDPC codes for the Cascaded BSC-BAWGN channel

Transcription

1 LDPC codes for the Cascaded BSC-BAWGN channel Aravind R. Iyengar, Paul H. Siegel, and Jack K. Wolf University of California, San Diego 9500 Gilman Dr. La Jolla CA 9093 Abstract We study the erformance of LDPC codes over the cascaded BSC-BAWGN channel. This channel belongs to a family of binary-inut, memoryless, symmetric-outut channels, one that we call the CBMSC(,) family. We analyze the belief roagation (BP) decoder over this channel by characterizing the decodable region of an ensemble of LDPC codes. We then give inner and outer bounds for this decodable region based on existing universal bounds on the erformance of a BP decoder. We numerically evaluate the decodable region using density evolution. We also roose other message-assing schemes of interest and give their decodable regions. The erformance of each roosed decoder over the CBMS channel family is evaluated through simulations. Finally, we exlore caacity-aroaching LDPC code ensembles for the CBMSC(, ) family. I. INTRODUCTION Low-density arity-check (LDPC) codes [1], [] have been shown to achieve erformance close to caacity over binaryinut, memoryless, symmetric-outut (BMS [3]) channels like the binary erasure channel (BEC) [4], the binary symmetric channel (BSC) [5] and the binary-inut additive white Gaussian noise channel (BAWGNC) [5], [6], [7]. We consider a class of channels which belong to the same BMS channel family as the BEC, BSC and BAWGNC. Owing to this similarity, much of the terminology and notation used in this aer are reminiscent of the analysis of LDPC ensemble erformance in [3]. However, the channels under consideration exhibit certain differences from the BMS channels in [3] because of their multidimensional channel sace. This aer is organized as follows. In Section II, we introduce the CBMS(, ) channel family and discuss some characteristics of this family of channels. We then consider coding over this channel using LDPC codes and analyze the belief roagation (BP) decoder in Section III. On similar lines, we roose and analyze two other message-assing schemes in Section IV. We give exerimental results for the erformance of the three message-assing decoders in Section V. In Section VI, we exlore good LDPC codes for the CBMS(, ) channel. We conclude by summarizing our findings in Section VII. A. Cascaded channels II. THE CHANNEL MODEL A cascaded channel is one where the outut of one channel is fed as inut to another. The constituent channels of a cascaded channel are called subchannels. The inut of the cascaded channel is the inut to its first subchannel, and the outut is the outut of the last subchannel in the cascade. The oututs (res. inuts) of all subchannels other than the last (res. first) are inaccessible. Note that from the definition, it follows that two subchannels can be cascaded only if the outut alhabet of the first is the same as the inut alhabet of the next. We will assume that the cascaded channel consists of only two subchannels. A generalization to an arbitrary number of subchannels is straightforward. Further, we will call a cascaded channel a cascaded BMS (CBMS) channel if it is a cascade of a BMS channel with a memoryless, symmetric channel. It is easy to see that any CBMS channel is symmetric and memoryless. Since the inut to a CBMS channel is binary, CBMS channels belong to the family of BMS channels. However, unlike simle BMS channels 1, CBMS channels are arameterized by the fidelities of both the constituent subchannels. We note here that a CBMS channel could be a degraded BMS channel belonging to the same channel family as the first subchannel [8], e.g. a BAWGN channel followed by another AWGN channel will give a CBMS channel which is also a BAWGN channel with a larger noise variance. In this case the CBMS channel is in the same family as the first subchannel. However, a BAWGN channel followed by a Lalace channel is also a CBMS channel, one not belonging to the family of BAWGN channels. B. Cascaded BSC-BAWGN channel The cascaded BSC-BAWGN channel, as the name suggests, is a CBMS channel where the first subchannel is a BSC with crossover robability and the second subchannel is a BAWGNC with noise variance. It is denoted as CBMSC(, ) and is deicted in Figure 1. Note that the Fig. 1. The CBMS(, ) channel. channel sace for this channel is given as S = [0, 1/] R +. The family of channels over S is denoted CBMSC(, ). The caacity of cascaded channels has been considered under certain cases of interest in [9], [10], [11], [1]. Since 1 We will refer to BMS channels arameterized by a single variable as simle BMS channels.

2 the CBMS(, ) channels are BMS channels, the caacity of the family of these channels can be numerically evaluated [3]. Figure shows the contours of equal caacity in a subset of S. Though we write the arameters of the channel as the ordered air (, ) in accordance with the order of the subchannels, all figures in this aer are deicted with as the ordinate for convenience. Note that the extremal The CBMS(, ) channel can also model other scenarios like an earth-satellite-earth link with a simlistic hard-decision decoder on the satellite and a more owerful soft-decision decoder at the receiver on the earth. In general, all memoryless (multiho) relay channels with thresholding at intermediate nodes can be modeled as CBMS(, ) channels. D. Coding for the CBMS(, ) channel Owing to the asymtotically close-to-otimal erformance of LDPC codes over the subchannels of the CBMS(, ) channel, we will consider coding for the CBMS(, ) channel using LDPC codes and the erformance of these codes with iterative message-assing decoders with emhasis on the asymtotically bit-otimal BP decoder. Note that since message-assing schemes are in general hyothesized on some channel model, estimation of channel arameters lays a key role in decoding. In this case, this translates to the availability of good estimates for and the SNR (defined in db as SNR := 0 log) of the constituent subchannels of the CBMS(, ) channel, which will be assumed. Fig.. Contours of Caacity C(, ) of the family CBMSC(, ) in bits er channel use. channels in this family corresond to the families BSC() and BAWGNC() when = 0 and = 0 resectively. C. Motivation The motivation for considering the CBMS(, ) channel family comes from a new-generation magnetic recording media called bit atterned media (BPM). BPM is claimed to surass conventional magnetic recording media in storage caacities owing to its structure. The sueraramagnetic effect that makes it difficult to reduce the size of bit cells in conventional media is avoided by storing bits on magnetic islands isolated by non-magnetic material. However, this structure brings to fore the challenge of good timing and ositional synchronization of the write head during the write rocess. There is therefore a ossibility of writing data erroneously on the disk [13]. We model this write channel as a BSC, which is followed by a memoryless read channel modelled as a BAWGNC. The CBMS(, ) channel therefore serves as a first-order aroximation of the BPM channel the (coded) information X n (See Figure 1) gets written onto the disk as W n, which is then read as Y n. A more realistic model would be a similar write and read channel cascade where the subchannels have memory, which is a subject for future work. Throughout the aer, we will assume that information theoretic quantities like entroy and caacity are measured in terms of bits. III. THE BELIEF PROPAGATION DECODER Since the CBMS(, ) channel is a BMS channel, the Log- Likelihood Ratio (LLR) of the channel outut is a sufficient statistic for decoding [3]. The BP decoding scheme thus remains the same as in the case of simle BMS channels, excet for the initial channel LLR density. Hence, the analysis of the BP decoder over the CBMS(, ) channel can be erformed using the standard density evolution [3], [6], [8] of the LLR of the channel outut, i.e. the convergence to the tree channel, the restriction to the all-zero codeword and the concentration around the ensemble average arguments [3] can be carried forward to this case. In this case, we have the LLR l to be [ ] (1 )e l + l = ln (1) (1 ) + e l where l = y/, a function of the channel outut y. Note that in case of a BAWGNC, l gives the actual LLR (which can be readily seen by setting = 0). We fix a code ensemble [3] (λ, ρ), and define the decodable region of the ensemble as R (λ,ρ) := (, ) : P (,) (λ,ρ) = 0 () where P (,) (λ,ρ) denotes the robability that the message on a random edge of the Tanner grah of a code in the ensemble (λ, ρ) is in error, averaged over the ensemble, in the limit of infinite blocklength and infinite rounds of message-assing. The boundary of the decodable region is analogous to the threshold value of simle BMS channels [3], [8]. Whereas in the case of simle BMS channels the channel families are totally ordered by degradation [8], the CBMSC(, ) channel family is only artially ordered by degradation. We therefore devote attention to characterizing and bounding the decodable region of an LDPC code ensemble

3 for the family of CBMS(, ) channels. For comleteness, we revisit the definition of degradation. Definition 1 (Degraded channels): A channel Z X is outut degraded with resect to Y X if X Y Z, i.e. X, Y and Z form a Markov chain. A. Characterization of R (λ,ρ) The following can be deduced directly from the channel degradation argument of [8]. Lemma 1 (Monotonicity in ): If (, ) R (λ,ρ), then (, ) R (λ,ρ). In order to rove a similar monotonicity in the arameter, we construct a channel that is equivalent to the CBMS(, ) channel, but is degraded with resect to BAWGNC(). Note that it is not immediately obvious that the CBMS(, ) channel, as in Figure 1, is outut degraded with resect to BAWGNC(). Definition (Inut-distorted channels): Let X Y be a BSC with crossover robability, and let Y Z be any BMS channel with transition robability f Z Y. Then the channel Z X is said to be inut-distorted with resect to the channel Z Y. Note that any CBMS channel with its second subchannel being a BMS channel itself is an inut-distorted channel by definition. Theorem 1 (Equivalence of inut-distortion & degradation): Any inut-distorted channel X Y Z is degraded with resect to the channel Y Z. Proof: By definition, we have X X = ±1, Y Y = X, f Y X (y x) = (1 )δ(y x) + δ(y + x) for some [0, 1/], where δ(x) = f Z Y (z y). Then, f Z X (z x) = y Y f Y,Z X (y, z x) = y Y f Y X (y x)f Z Y (z y) 1 x = 0, and f Z Y ( z y) = 0 else = y Y(1 )δ(y x) + δ(y + x)f Z Y (z y) = (1 )f Z Y (z x) + f Z Y (z x) = (1 )f Z Y (z x) + f Z Y ( z x) (3) Consider the channel system Y Z W where the channel Y Z is given as before, and the channel Z W satisfies W W = Z and f W Z (w z) = (1 )δ(w z)+δ(w+z). Then f W Y (w y) = f Z,W Y (z, w y)dz z Z = f Z Y (z y)f W Z (w z)dz Z = (1 )f Z Y (w y) + f Z Y ( w y) (4) From (3) and (4), we have f Z X = f W Y. Since the channel Y Z degrades to Y W, we conclude that Y Z degrades to X Z. From Theorem 1 and the channel degradation argument of [8], we have the following lemma Lemma (Monotonicity in ): If (, ) R (λ,ρ), then (, ) R (λ,ρ). A consequence of Lemmas 1 and is that if a oint (, ) is found to be in R (λ,ρ), then we can conclude that all channel oints with a smaller and a smaller (oints to the left and below (, ) in the channel sace S ) are also guaranteed to belong to R (λ,ρ). B. Bounds for R (λ,ρ) We now give inner and outer bounds for the decodable region of a code ensemble based on existing universal bounds on the erformance of the BP decoder. Proosition 1 (Bhattacharyya outer bound): Let B (λ,ρ) = (, ) S : B(, ) < B (λ,ρ), where B(, ) is the Bhattacharyya constant [3] associated with the channel CBMSC(, ) and B (λ,ρ) := inf b(x) : b(x) x, x (0, 1], with x b(x) = ). λ( k ρ k 1 (1 x ) k 1 Then, R (λ,ρ) B (λ,ρ). Discussion This follows from the universal uer bound on the Bhattacharyya constant given in [14], [15]. First, any BMS channel is exressed as a convex combination of BSCs of crossover robabilities ranging from 0 to 1/. Noting that the check node rocessing for a density corresonding to a BSC gives the smallest Bhattacharyya constant for the outut density, the above uer bound on the Bhattacharyya arameter can be obtained. Proosition (Entroy outer bound): Let H(λ,ρ) = (, ) S : H(, ) < H (λ,ρ), where H(, ) is the entroy (channel equivocation) [3], [8], [16] associated with the channel CBMSC(, ) and H (λ,ρ) := inf h(x) : h(x) x, x (0, 1/], with h (x) h(x) = ( ( )), λ k ρ 1+(1 x) k 1 kh where h ( ) is the binary entroy function. Then, R (λ,ρ) H (λ,ρ). Discussion This follows from the best case of the extremes of information combining [16]. As in Proosition 1, the smallest entroy of the outut density is obtained for inut densities corresonding to BSC at the check node and BEC at the variable node. Using these best case densities, we obtain the uer bound for the entroy. Proosition 3 (Caacity outer bound): Let C(λ,ρ) := (, ) S : C(, ) > r(λ, ρ) wherec(, ) is the caacity of the channel CBMSC(, ) and r(λ, ρ) = 1 R 1 R0 ρ(x)dx 1 is λ(x)dx 0 the design rate of the code. Then R (λ,ρ) C (λ,ρ). Proof: Since the asymtotic (in blocklength) actual rate r of a code is equal to its design rate r [3], the result follows from the channel coding theorem.

4 Proosition 4 (Degradation outer bound): Let D(λ,ρ) = (, ) S : < (λ,ρ), < (λ,ρ), where (λ,ρ) is the threshold for the (λ, ρ) code ensemble over the family BAWGNC(), and (λ,ρ) is that over the family BSC(). Then, R (λ,ρ) D (λ,ρ). Proof: Suose to the contrary that P (,) (λ,ρ) = 0 for some > (λ,ρ). Then, since BAWGNC() degrades to CBMSC(, ), we have P (,0) (λ,ρ) = 0 (, 0) R (λ,ρ) which contradicts the assumtion that (λ,ρ) is the threshold for the BAWGN channel family. A similar argument holds for (λ,ρ). This bound is more easily stated as the contraositive of the claims made in Lemmas 1 and. Figure 3 shows the outer bounds discussed here for two code ensembles. The set B (λ,ρ) H (λ,ρ) C (λ,ρ) D (λ,ρ) is therefore an outer bound for the decodable region. We have observed that the Bhattacharyya bound and the entroy bound have been strictly looser than the caacity bound in all cases we considered, with the Bhattacharyya bound being the loosest. Proosition 5 (Bhattacharyya inner bound): Let B (λ,ρ) = (, ) S : B(, ) < B (λ,ρ), where B (λ,ρ) := infb(x) : b(x) x, x (0, 1], with b(x) = Then, B (λ,ρ) R (λ,ρ). x λ(1 ρ(1 x)). Discussion This follows from the universal lower bound on the Bhattacharyya constant of a BMS channel [14], [15]. In this case, the lower bound is obtained as in Proosition 1, by noting that the check node rocessing for a density corresonding to a BEC gives the largest Bhattacharyya constant for the outut density. Note that the above bound is tight for the BEC, in which case the Bhattacharyya constant is given by the channel erasure robability. The above definition of B (λ,ρ) therefore coincides with the definition of the threshold for the BEC in [3], [8]. Proosition 6 (Entroy inner bound): Let H (λ,ρ) = (, ) S : H(, ) < H (λ,ρ), where H (λ,ρ) := infh (), (0, 1/) : x (0, 1/) : h(x, ) 0, with ) h(x, ) = 1 h (x) ρ (1 ĥ(λ, x, ), ĥ(λ, x, ) = j ( ) j 1 x l (1 x) j 1 l h(, x, j, l), l λ j j 1 l=0 and h(, x, j, l) as in (5). Then, H (λ,ρ) R (λ,ρ). Discussion This follows from the worst case of the extremes of information combining [16]. As in Proosition, the lower bound is obtained by noting that the largest entroy of the outut density is obtained for inut densities corresonding to BEC at the check node and BSC at the variable node. Proosition 7 (Degradation inner bound): Let D (λ,ρ) = (, ) S : Q ( ) 1 < (λ,ρ) where (λ,ρ) is as in Proosition 4, a b = a(1 b) + b(1 a), a, b [0, 1/] D (λ,ρ) C (λ,ρ) H (λ,ρ) B (λ,ρ) (a) (3, 6) regular code ensemble λ(x) = x, ρ(x) = x x 10 3 D (λ,ρ) C (λ,ρ) H (λ,ρ) B (λ,ρ) (b) Rate-0.9 code ensemble otimized[5] for the BAWGNC λ(x) = x x x x x, ρ(x) = 0.5x x 45, henceforth referred to as the Otimized rate-0.9 code ensemble. Fig. 3. Outer bounds for the decodable region R (λ,ρ). The sets corresond to the region under the curves marked in the figure. and Q( ) is the Q-function of the standard normal distribution. Then, D (λ,ρ) R (λ,ρ). Proof: Let X Y Z be the CBMS(, ) channel. Consider the channel X Y Z W, where W = +1, Z 0 sgn(z) =. We have W W = ±1 = X, 1, Z < 0 and W X (w x) = (1 )δ(w x) + δ(w + x), with = Q ( 1 ). Thus, X W is a BSC with crossover robability. Thus, if < (λ,ρ), the threshold for the family BSC(),

5 " h(, x, j,l) = log «# " x j 1 l + (1 ) log 1 x 1 + «# x j 1 l. (5) 1 1 x we have P (λ,ρ) = 0. Since the channel X Z degrades to the channel X W, P (λ,ρ) = 0 P(,) (λ,ρ) = 0. Hence, if (, ) D (λ,ρ), then (, ) R (λ,ρ). Proosition 8 (Soft bit inner bound): Let S (λ,ρ) = (, ) S : S(, ) < 4 (λ,ρ) (1 (λ,ρ) ) where S(, ) is the Soft bit value [15] of the CBMS(, ) channel and (λ,ρ) is as in Proosition 4. Then, S (λ,ρ) R (λ,ρ). Discussion This follows from the one-dimensional noniterative bound based on the Soft bit value given in [15]. Figure 4 shows the inner bounds for the two code ensembles considered before. We note here that the degradation inner bound and the soft bit inner bound are tight on the -axis, i.e. on the -axis, the D and S bounds meet the D bound. The set B (λ,ρ) H (λ,ρ) D (λ,ρ) S (λ,ρ) is therefore an inner bound to the decodable region R (λ,ρ). Note that in the case of the (3, 6) regular code ensemble, the soft bit inner bound is tighter than all other inner bounds. This, however, is not true in the case of the otimized rate-0.9 code ensemble (See Figure 4). C. Decodable region The actual decodable region was found using density evolution. By fixing, we are guaranteed by Lemma the existence of a threshold the largest for which the decoder gives a zero error. The boundary of R (λ,ρ) is therefore estimated by finding the for a fixed set of values within the range [0, (λ,ρ) ], where (λ,ρ) is as in Proosition 4. This is shown in Figure 5 for our examle code ensembles. We conjecture that the decodable region of a code ensemble, like most of the bounds we have obtained, is a convex subset of the channel sace S x D (λ,ρ) S (λ,ρ) H (λ,ρ) B (λ,ρ) (a) (3, 6) regular code ensemble. IV. MESSAGE-PASSING In this section, we roose two alternative message-assing schemes to the BP decoder. The schemes and the motivation for considering them are discussed below. A. Decoding schemes Hard Decision decoder (HD): Here we threshold the outut of the CBMS(, ) channel and consider the channel to now function as a BSC with crossover robability = Q ( ) 1 where and Q( ) are as defined in Proosition 7. We then erform BP for this modified BSC. The rationale is that a good code for the BSC should be able to tackle the written-in errors well. Gaussian Noise decoder (GN): In this case, we have a decoder that ignores the first subchannel and erforms BP under the assumtion that the channel is a simle BAWGN channel. Analysing the erformance of this decoder is useful, e.g. in the case of magnetic recording, to understand the enalty aid in naively assuming that the information written on to a disk is error-free. Note that since we ignore the first 4 B (λ,ρ) H (λ,ρ) D (λ,ρ) S (λ,ρ) (b) Otimized rate 0.9 code ensemble. Fig. 4. Inner bounds for the decodable region R (λ,ρ). subchannel here, the estimation of is not necessary, and therefore the decoder is slightly less comlex than the other two decoders in this resect. B. Decodable regions The degradation inner bound D of the BP decoder gives the decodable region for the HD decoder. This is true because the construction in the roof of Proosition 7 is the same as the oeration erformed in the HD decoder.

6 (a) (3, 6) regular code ensemble. Fig. 6. Decodable regions for the BP, HD, and GN decoders for (3, 6) regular code ensemble. (b) Otimized rate 0.9 code ensemble. Fig. 5. BP Decodable region R (λ,ρ). For the GN decoder, since the decoder assumes a wrong channel model, we use density evolution with the LLR calculated as l = y/, where y is the channel outut conditioned on the transmission of a 1 f Y X (y 1) = 1 π 1 π (y 1) (1 )ex ( ) (y + 1) ex (. ) + In this case, we can make no claim of the monotonicity in of the decoder erformance. The decodable region for the GN decoder is found as in the case of the BP decoder, however by using the modified LLR density. In Figure 6, we give the decodable regions for the HD, GN and BP decoders for the (3, 6) regular code ensemble. We make two observations from this figure first, the nonmonotonicity of the erformance of the GN decoder aarent from the shae of its decodable region, and second, the similarity in the erformance of the HD and BP decoders for small (large SNR) and large. Also note that the decodable region of the GN decoder is another inner bound to the decodable region of the BP decoder. The non-monotonicity of the GN decoder in, for large enough is because the decoder in this case is being fooled into believing that the channel is reliable when it is not. For large and small, the erformance of the HD decoder and the BP decoder coincide because the CBMS channel in this case is very similar to the BSC, i.e. the behaviour of the cascaded channel is dominated by the behavior of the first subchannel. Also, as exected, the HD and the GN decoders are as good as the BP decoder when the channel is dominated by the BSC and the BAWGNC resectively. V. EXPERIMENTAL RESULTS In this section, we shall exlore the erformance of binary LDPC codes on the CBMS(, ) channel with the three different message-assing schemes roosed in earlier sections. A rate-1/ binary LDPC code of blocklength 4096 bits that was randomly samled (with multile edges between nodes avoided) from the Gallager (3, 6) regular ensemble was used to estimate the decoder erformance. The values used for ranged from 1% through 5%. Since the CBMS(, ) channel is memoryless, all the decoding schemes have the same comlexity and this comlexity scales as O(Ln(d v q + rd c q log q)) where L is the maximum number of iterations erformed, n is the blocklength of the

7 BP 1% % 3% 4% 5% HD 1% % 3% 4% 5% GN 1% % 3% 4% 5% Fig. 7. Performance of the decoders for increasing. = 1% BP HD GN = 3% BP HD GN = 5% BP HD GN Fig. 8. Performance comarison of the BP, HD, and GN decoders for different values. code, d v and d c are the maximum variable and check node degrees resectively in the Tanner grah of the code, r is the normalized redundancy of the code, given as r = 1 r where r is the design rate. q is the size of the Galois field over which the code is defined (q = ). Figure 7 shows the erformance of each decoder over the range of values. In Figure 8, we comare the three decoders for different values of. We see from Figure 8 that the BP decoder outerforms both the HD and the GN decoders, as exected. However, with increasing, the HD decoder erforms better than the GN decoder at high SNR values, i.e. as the channel becomes more like the BSC (small, large ), the HD decoder handles the channel better than the GN decoder. Note that beyond a certain, the erformance of the GN decoder is non-monotonic in, e.g. when = 5%, the first reduces and then increases with increasing SNR (See Figure 7). Also notice that the ga between the BP and the HD decoder (Figure 8) at a high SNR reduces as increases, going from 1.5 db when = 1% to 0.8 db when = 5%. These agree with the observations made in Section IV-B from the decodable regions of these decoders. VI. GOOD CODES FOR THE CBMS(, ) CHANNEL From Figure 3(b), we see that the otimized rate-0.9 code comes very close to achieving caacity on the BAWGNC

8 with a threshold of while caacity is C However, it falls well short of the caacity on the BSC. The threshold for this code on the BSC was observed to be while a caacity of 0.9 bits er channel use was achieved by a channel with C This suggests that a code otimized for the BAWGNC might be far from otimal on the BSC. In Figure 9, we show the decodable regions of the BP decoder for two code ensembles of rate 0.75, one otimized for the BAWGNC C g = (0.4035x x x x x x x, 0.5x x 16 ) with the decodable region R g, and another otimized for the BSC C s = ( x x x x x x, x 3 ) with decodable region R s. These otimized codes were obtained from [17] C (λ,ρ) R s R g Fig. 9. Decodable region estimates for the two otimized code ensembles C g and C s of rate Also shown in the figure is the caacity outer bound for the decodable region of a code of rate The ideal rate-0.75 code for the CBMSC(, ) family has C (λ,ρ) as its decodable region. We see from Figure 9 that the code otimized for the BSC erforms better (is closer to the caacity bound) than the code otimized for the BAWGNc excet near the - axis, i.e. the BAWGNC otimized code is suerior only when the channel is dominated by the BAWGN subchannel. This suggests that a good strategy to obtain good codes for the CBMS(, ) channel is to otimize a code for the BSC, rather than otimizing for the BAWGNC. A better scheme would be to otimize the degree distributions for a fixed rate code over both the BAWGNC and BSC simultaneously, which is clearly a more comlex otimization roblem. VII. CONCLUSION We have introduced a new class of BMS channels that model many scenarios including the BPM recording channel. We analyzed the theoretical decodable regions of LDPC codes under BP decoding over the family of CBMS(, ) channels by giving inner and outer bounds to the decodable region based on channel arameters like the entroy, the Bhattacharyya constant, the Soft bit value and the caacity, and also based on the channel ordering introduced by degradation. We also numerically estimated the decodable regions for three roosed message-assing decoders, including the BP decoder using density evolution. We conducted erformance estimation for a rate-1/ (3, 6)- regular LDPC code of blocklength 4096 bits with the three roosed message-assing schemes and showed the decoder characteristics suggested by theory. In articular, we showed that ignoring the BSC (as in the GN decoder) can result in a large enalty in erformance, more so when the channel SNR is high. By looking at codes otmized for the BSC and BAWGNC, we noted that otimizing a code over the BSC works better than otimizing for the BAWGNC. 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