Analysis on crossover probability estimation using LDPC syndrome

. RESEARCH PAPERS .

SCIENCE CHINAInformation Sciences

September 2011 Vol. 54 No. 9: 1895–1904

doi: 10.1007/s11432-011-4311-y

c© Science China Press and Springer-Verlag Berlin Heidelberg 2011 info.scichina.com www.springerlink.com

Analysis on crossover probability estimation usingLDPC syndrome

FANG Yong

College of Information Engineering, Northwest A&F University, Yangling 712100, China

Received March 19, 2010; accepted April 12, 2010; published online June 24, 2011

Abstract Correlation estimation is a critical issue that impacts the application of Slepian-Wolf coding (SWC)

in practice. Dynamic online correlation estimation is a type of newly-appearing approaches, in which the

decoder estimates the virtual correlation channel between two correlated sources using both side information

and the compressed SWC bitstream of the source. Since the compressed SWC bitstream usually contains

partial information of the source, the emergence of dynamic online correlation estimation is helpful to solving

the problem of correlation estimation in the SWC and further makes the SWC realisable. Currently, the SWC is

usually implemented by LDPC codes. In this case, the SWC bitstream is just the LDPC syndrome of the source.

It has been revealed that there are residual redundancies in LDPC syndromes, which can be used to estimate

the crossover probability between two correlated binary sequences. However, this algorithm has not been well

justified yet. This paper makes use of the central limit theorem (CLT) to establish a mathematic model for

analyzing the performance of this algorithm. Especially, for irregular LDPC codes, the optimization of weight

vectors is discussed in detail. Representative experimental results are provided to validate the analysis.

Keywords distributed source coding, Slepian-Wolf coding, LDPC syndrome, correlation estimation

Citation Fang Y. Analysis on crossover probability estimation using LDPC syndrome. Sci China Inf Sci, 2011,

54: 1895–1904, doi: 10.1007/s11432-011-4311-y

1 Introduction

Nowadays, distributed source coding (DSC) is getting more and more applications in practice. In [1],Slepian and Wolf proved that correlated discrete sources can be compressed without loss efficiently,whether or not the encoders of separate sources communicate with each other. Wyner and Ziv soonthereafter considered the problem of lossy DSC with decoder side information (SI). Both Slepian-Wolftheorem and Wyner-Ziv theorem lay the theoretical foundation for DSC. In general, a Wyner-Ziv coding(WZC) encoder is implemented by serially concatenating a quantizer with a Slepian-Wolf coding (SWC)encoder [3].

This paper concerns the problem of SWC with decoder SI, i.e., the encoder compresses discrete sourceX with no access to correlated discrete SI Y . Slepian-Wolf theorem states that lossless compression ispossible at rates R � H(X |Y ) [1]. The SWC is traditionally implemented by channel codes, e.g., Turbocodes [4] and low-density parity-check (LDPC) codes [5]. Recently, some entropy-coding-based SWCtechniques emerge, e.g., distributed arithmetic coding [6].

(email: [email protected])

1896 Fang Y Sci China Inf Sci September 2011 Vol. 54 No. 9

As stated in [7], two critical issues remain in practical DSC systems: “estimating at encoder or decoderthe virtual correlation channel from unknown or only partially known data” and “finding the best SI atthe decoder for data not or only partially known”. It has been recognized that both problems finally boildown to the former: given multiple SIs at the decoder, once the virtual correlation channels between eachSI and the source are known, the best SI can be selected without any problem [8]. Many approaches havebeen devoted to the topic of correlation estimation, which can be roughly divided into two classes: offlineestimation and online estimation [9]. In offline correlation estimation, both source and SI (or at leastpartial SI) are assumed to be accessible at the encoder and used jointly by the encoder to estimate thevirtual corrlation channel between them. To compromise between coding efficiency and computationalcomplexity, [10] uses only partial SI samples during this process. In contrast, online correlation estimationoccurs at the decoder, which can be further divided into two sub-classes: static estimation and dynamicestimation. The static online approaches do correlation estimation using only SI, regardless of the receivedSWC bitstream (i.e., LDPC syndrome if the LDPC-based SWC is considered), e.g., [9, 11] exploit boththe previous SI frame and the next SI frame to do correlation estimation for the current frame in betweentwo SI frames. In some sense, the SWC bitstream can be deemed a coarse description of the source,so there are residual redundancies in LDPC syndromes [12, 13]. This property inspires dynamic onlinecorrelation estimation: The decoder exploits both SI and the received SWC bitstream to do correlationestimation in a dynamic way [8]. The work in [8] is then simplified and extended to irregular LDPC codes[12]. Dynamic online correlation estimation for DAC-based SWC is also found in [14].

The realization of dynamic online correlation estimation requires the “estimate-(request)-decode”scheme [8], which is an improvement of the “request-decode” scheme [15]. Initially, the encoder emitsa short bitstream and the decoder tries to estimate the virtual correlation channel from the receivedbitstream. Knowing the estimate of the virtual correlation channel, the decoder can decide whetherthe received bitstream is enough for a successful decoding. Only when the received bitstream seemsenough, will the decoder try decoding; otherwise it will bypass the decoding and feedback the estimateof the virtual correlation channel to the encoder, which replies with additional bits correspondingly.Such process is iterated until the decoding succeeds. If the estimate of the virtual correlation channelis accurate enough, feedback is needed only once and decoding is run only once. To implement the“estimate-(request)-decode” scheme, rate-adaptive SWC codes [16] are needed.

The process of correlation estimation consists of two stages: modeling the virtual correlation channeland estimating the model parameter. A widely used model for the virtual correlation channel is thebinary symmetric channel (BSC), which is characterized by crossover probability. Assume that binarysource X = {xt}n

t=1 and SI Y = {yt}nt=1 are correlated by X = Y ⊕ Z, where Z = {zt}n

t=1. Whenthe BSC correlation is considered, Z is a sequence of independent and identically-distributed (i.i.d.)random variables with bias probability p = Pr(zt = 1) < 0.5 (p is also called the crossover probabilitybetween X and Y ). For the LDPC-based SWC, the intrinsic log-likelihood ratio (LLR) of xt is givenby (1 − 2yt) log((1 − p)/p) to trigger the decoding. This paper will concentrate on crossover probabilityestimation using LDPC syndromes for BSC-correlated sources.

The mean-intrinsic-LLRs of LDPC syndromes are used in [12] to estimate crossover probability p.Though this algorithm works well in simulations, it is quite empirical and not well justified. For irregularLDPC codes, no rule is given to select weights so that it is not clear why the weights given by [12]are good. Therefore, this paper will establish a mathematic model to analyze the performance of thealgorithm in [12] and show why it is good.

This paper is arranged as follows. Section 2 first describes briefly the proposed framework of crossoverprobability estimator using LDPC syndrome [12] and then establishes a mathematic model to analyzethe performance of the proposed method. Section 3 discusses the optimization of weight vectors forirregular LDPC codes. Section 4 considers some implementation issues of this algorithm. Finally, section5 concludes this paper.

Fang Y Sci China Inf Sci September 2011 Vol. 54 No. 9 1897

Figure 1 Framework of crossover probability estimator using LDPC syndrome, including a local encoder at the receiver.

2 Performance analysis

2.1 Framework description

Figure 1 illustrates the framework of crossover probability estimator using LDPC syndrome [12]. Thisframework introduces a local encoder at the receiver. Let SX = HX be the syndrome of source X , whereH is the parity-check matrix. The receiver first calculates the syndrome of Y , SY = HY and then setsS = SX ⊕ SY = H(X ⊕ Y ) = HZ. Now the problem of estimating the crossover probability betweenX and Y boils down to estimating the bias probability of Z. Note that such “local-encoder-at-decoder”structure in fact mirrors the “local-decoder-at-encoder” structure in conventional source coding system.

2.2 Central limit theorem (CLT)

If {xt}nt=1 is a sequence of n i.i.d. random variables each having finite mean μ and variance σ2, then

the CLT states that as n increases the distribution of the average of {xt}nt=1 approaches the normal

distribution with mean μ and variance σ2/n, irrespectively of the distribution of xt. The applicationsof the CLT are found in many fields, e.g., the CLT is used to analyze the system performance of CodeDivision Multiple Access (CDMA) cellular mobile communications [17].

2.3 Mathematic model

Let us compress X with an irregular LDPC code defined by λ(x) =∑dv

i=1 λixi−1 and ρ(x) =

∑dc

i=1 ρixi−1,

where λi and ρi are the fractions of degree-i variable and check nodes, respectively (note that the meaningsof λi and ρi here are different from the conventions). This LDPC code gives rate

R =∑dv

i=1 iλi∑dc

i=1 iρi

. (1)

It has been shown in [12] that λi is not involved in the procedure of crossover probability estimation.Thus we need to consider only ρi.

Let Si be the set of degree-i syndrome nodes. Then S = {Si}dc

i=1. When Z is a sequence of i.i.d.random variables with bias probability p, Si should be a sequence of i.i.d. random variables with biasprobability Qi, where (see [18])

1 − 2Qi = (1 − 2p)i. (2)

This implies that there are residual redundancies in Si, which makes it possible to estimate p from Si.Firstly, Qi is estimated by E(Si), the mean of Si. Then, p can be found from (2).

According to the CLT, E(Si) is approximately normally distributed with mean Qi and variance Qi(1−Qi)/mi, for large mi and Qi not too close to 1 or 0, where mi is the cardinality of Si, i.e.,

E(Si) ∼ N(Qi, Qi(1 − Qi)/mi). (3)

Then1 − 2E(Si) ∼ N(μi, σ

2i ), (4)

where μi = 1 − 2Qi and σ2i = 4Qi(1 − Qi)/mi.


Let p̃i be the estimate of p given E(Si), i.e.,

1 − 2E(Si) = (1 − 2p̃i)i. (5)

The probability density function (PDF) of p̃i is given by

f(p̃i) ≈ |g′(p̃i)|√2πσi

exp(

− (g(p̃i) − μi)2

2σ2i

)

, (6)

where g(p̃i) = (1 − 2p̃i)i and |g′(p̃i)| = 2i(1 − 2p̃i)i−1.The reliability of p̃i is subject to two factors: mi and i. Since σ2

i ∝ 1/mi, p̃i will become arbitrarilyaccurate as mi increases. Hence, we can use long LDPC codes to improve estimation precision. To studythe impact of i on p̃i, some examples of f(p̃i) for various i are plotted in Figure 2(a), where mi and p arefixed at 768 and 0.02, respectively. It is shown that as i increases, f(p̃i) is spread out over a larger rangearound p, meaning that estimation precision is degraded.

2.4 Problem formulation

Since estimates given by multiple observations are usually more accurate than that given by a singleobservation, jointly using the set {E(Si)}dc

i=1 can result in more accurate estimate of p. However, beforedoing so, the impacts of mi and i on E(Si) must be considered. Naturally, to compensate for the impactof mi, ρi must be taken into account. To compensate for the impact of i, [12] introduces a weight. Thus,this problem is formatted as

dc∑

i=1

wiρi(1 − 2E(Si)) =dc∑

i=1

wiρi(1 − 2p̃)i, (7)

where p̃ is the estimate of p and wi is a weight reflecting the reliability of (1−2E(Si)). This is an equationregarding p̃. By solving this equation, p̃ can be found.

For simple analysis, we let ui = 1 − 2E(Si), v = 1 − 2p̃, and make the following definitions:

U = (u1, . . . , udc)T, V = (v, . . . , vdc)T,

W = (w1, . . . , wdc)T, P = (ρ1, . . . , ρdc)

T. (8)

This problem becomes(W · P )TU = (W · P )TV, (9)

where · is the Hadamard product. Let u = (W · P )TU and g(p̃) = (W · P )TV . Then

u = g(p̃). (10)

To calculate the PDF of p̃, one must find the PDF of u first. It is easy to obtain

wiρiui ∼ N(wiρiμi, w2i ρ2

i σ2i ). (11)

Let M = (μ1, . . . , μdc)T and

Σ =

⎛

⎜⎜⎝

σ21 · · · 0...

. . ....

0 · · · σ2dc

⎞

⎟⎟⎠ . (12)

Due to the i.i.d. property, ∀i ∈ Idc = (1, . . . , dc)T, E(Si) are independent of each other. Then

u ∼ N(μ, σ2), (13)


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

20

40

60

80

100

120

140

160

180(a)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

50

100

150

200

250(b)

f (p i

)

f (p)

pi p

i=48i=51i=99

{1,0,0}{1,1,1}{l48,l51,l99}

p=0.02, mi≡768 p=0.02, ρ48=ρ51=ρ99=1/3, mi≡768~ ~

~ ~

Figure 2 (a) Some examples of f(p̃i). It is shown that as i increases, f(p̃i) is spread out over a larger range around

p, i.e., p̃i becomes less reliable. (b) Some examples of f(p̃). Given an irregular LDPC code defined by ρ48 = ρ51 =

ρ99 = 1/3 and mi ≡ 768, three weight vectors are tested: {w48, w51, w99} = {1, 0, 0}, {1, 1, 1}, and {l48, l51, l99}, where

li = log((1−Qi)/Qi). It is shown that weight vector {l48, l51, l99} performs the best and {1, 0, 0} performs the worst, while

the performance of {1, 1, 1} is in between.

where μ = (W · P )TM and σ2 = (W · P )TΣ(W · P ). Therefore

f(p̃) ≈ |g′(p̃)|√2πσ

exp(

− (g(p̃) − μ)2

2σ2

)

, (14)

where |g′(p̃)| = 2(W · P )TV ′. It is easy to verify

V ′ = v−1(W · P )T(Idc · V ). (15)

To show how weight vector W impacts f(p̃), consider the example in Figure 2(b). For an irregu-lar LDPC code defined by ρ48 = ρ51 = ρ99 = 1/3 and mi ≡ 768, three weight vectors are tested:{w48, w51, w99} = {1, 0, 0}, {1, 1, 1}, and {l48, l51, l99}, where li = log((1−Qi)/Qi). Among them, weightvector {1, 0, 0} performs the worst and {l48, l51, l99} performs the best, while the performance of {1, 1, 1}is in between.

3 Optimization of weight vector

Figure 2(b) shows that weight vector W plays a critical role for f(p̃). Intuitively, wi should satisfy thefollowing three constraints: i) wi should be monotonically decreasing with respect to Qi. ii) If Qi → 0,then wi → ∞. iii) If Qi → 0.5, then wi → 0. In [12], wi is set at log((1 − Qi)/Qi), which suggests thatu is the mean-intrinsic-LLR of LDPC syndrome. This weight vector satisfies the above three constraintsand performs well in simulations. Nevertheless, it is not clear whether it is optimal or near-optimal. Tofind W ∗, the optimal W , we need a criterion. This section will first define a measurement and then useit to optimize W .

3.1 Pseudo signal-noise ratio (PSNR)

A good cost function is the mean squared error (MSE):

MSE(p̃; W ) =∫ 1

0

(p̃ − p)2f(p̃)dp̃. (16)

Because MSE(p̃; W ) is a function of W , the problem is formatted as deriving W ∗ that minimizesMSE(p̃; W ):

W ∗ = arg minW

MSE(p̃; W ). (17)


Due to the linearity, there are in fact infinite numbers of W ∗. One can impose a constraint on W ∗ tomake it unique. A possible constraint is

∑dc

i=1 wi = 1, while an alternative is to fix some weight, say wi′

(given ρi′ > 0), to 1.Usually, MSE(p̃; W ) is very small. Therefore we introduce the pseudo signal-noise ratio (PSNR) which

is defined as

PSNR(p̃; W ) =p2

MSE(p̃; W ). (18)

To make it more readable, as usual we express PSNR(p̃; W ) in terms of the logarithmic decibel scale:

PSNR(p̃; W ) = 10 log10

(p2

MSE(p̃; W )

)

(dB). (19)

As MSE(p̃; W ), PSNR(p̃; W ) is a function of W . The problem is formatted as deriving W ∗ that maximizesPSNR(p̃; W ):

W ∗ = argmaxW

PSNR(p̃; W ). (20)

3.2 Search for optimal weight vector

No closed form exisits for PSNR(p̃; W ), so we have to resort to numeric method. Thanks to VisualNumerics IMSL, it is possible to find W ∗ very fast on PCs. As follows are some examples. Six irregularLDPC codes are used (Table 1). Code B and code D have two kinds of right (syndrome) nodes, whilefour other codes have three kinds of right nodes. Codes A–C have uniform ρi, while codes D–F havenonuniform ρi. All these codes can be obtained from the LDPC code given in [16].

For each LDPC code, L and W ∗ are enlisted in Table 2 for various p, where L = {li}dc

i=1. For reference,the entropies for various p are enlisted in Table 3. As shown in [8, 12], it makes little sence if R � H(X |Y ),so only the cases R < H(X |Y ) are tested. In Table 2, W ∗ is obtained with L as the initial guess (otherinitial guess gives the same results). From Table 2, we find that L is very close to W ∗.

Correspondingly, PSNR(p̃; L) and PSNR(p̃; W ∗) are plotted in Figure 3, denoted by “L, theory” and“W ∗, theory” respectively. From Figure 3, we find that PSNR(p̃; L) is very close to PSNR(p̃; W ∗) andas p increases, PSNR(p̃; L) and PSNR(p̃; W ∗) are approaching to each other. Especially for code B andcode D, PSNR(p̃; L) almost coincides with PSNR(p̃; W ∗) at all p. From these observations, we concludethat L is near-optimal.

4 Implementation issues

The process of finding W ∗ has been described in detail using p. However our purpose is to find p so thatp is unavailable at the decoder in practice. Though L proves to be near-optimal, its caculation also needsp. Thus, some approximations to L should be used. There are two natural approximations to L. Thefirst is to use L̃ = {l̃i}dc

i=1, where

l̃i = log1 − E(Si)

E(Si). (21)

The second is to use L̂ = {l̂i}dc

i=1, where

l̂i = log1 + vi

1 − vi. (22)

Two approximations to L will result in four combinations:

(L̃ · P )TU = (L̃ · P )TV, (L̃ · P )TU = (L̂ · P )TV,

(L̂ · P )TU = (L̂ · P )TV, (L̂ · P )TU = (L̃ · P )TV.(23)


Table 1 Example of irregular LDPC codes

Code ID {ρi} {mi} R

Code A (ρ48, ρ51, ρ99) = (1/3, 1/3, 1/3) mi ≡ 768 1/22 ≈ 0.045

Code B (ρ48, ρ51) = (1/2, 1/2) mi ≡ 1536 2/33 ≈ 0.06

Code C (ρ24, ρ27, ρ48) = (1/3, 1/3, 1/3) mi ≡ 1536 1/11 ≈ 0.09

Code D (ρ24, ρ27) = (3/4, 1/4) m24 = 4608, m27 = 1536 4/33 ≈ 0.12

Code E (ρ12, ρ15, ρ24) = (1/5, 1/5, 3/5) m12 = m15 = 1536, m24 = 4608 5/33 ≈ 0.15

Code F (ρ12, ρ15, ρ24) = (1/2, 1/6, 1/3) m12 = 4608, m15 = 1536, m24 = 3072 2/11 ≈ 0.18

Table 2 Search results of W ∗

Code ID p L W ∗

0.01 (0.798216, 0.746626, 0.272323) (0.744154, 0.729744, 0.473165)

0.015 (0.472103, 0.429531, 0.0981251) (0.450903, 0.432206, 0.180892)

Code A 0.02 (0.283759, 0.250685, 0.0351501) (0.275125, 0.255699, 0.0630373)

0.025 (0.17093, 0.146457, 0.0124644) (0.167635, 0.149749, 0.0191344)

0.03 (0.102695, 0.0852734, 0.00437208) (0.101035, 0.0914886, 0.00842548)

0.01 (0.798216, 0.746626) (0.78053, 0.765523)

0.015 (0.472103, 0.429531) (0.460698, 0.442034)

Code B 0.02 (0.283759, 0.250685) (0.276924, 0.25834)

0.025 (0.17093, 0.146457) (0.167527, 0.150679)

0.03 (0.102695, 0.0852734) (0.101, 0.0879135)

0.02 (0.789419, 0.690467, 0.283759) (0.720011, 0.691785, 0.471882)

0.03 (0.460995, 0.38079, 0.102695) (0.430633, 0.394885, 0.182669)

Code C 0.04 (0.272022, 0.211307, 0.0365506) (0.258227, 0.222887, 0.0651701)

0.05 (0.159873, 0.116431, 0.0127255) (0.154574, 0.12289, 0.0203364)

0.06 (0.0930953, 0.0634173, 0.00432712) (0.0943809, 0.0720921, 0.00734431)

0.02 (0.789419, 0.690467) (0.757197, 0.727293)

0.03 (0.460995, 0.38079) (0.441117, 0.404815)

Code D 0.04 (0.272022, 0.211307) (0.260631, 0.225855)

0.05 (0.159873, 0.116431) (0.153847, 0.124496)

0.06 (0.0930953, 0.0634173) (0.0905419, 0.0670036)

0.03 (1.03539, 0.836113, 0.460995) (0.904571, 0.864024, 0.702883)

0.05 (0.58064, 0.417754, 0.159873) (0.521431, 0.455555, 0.270063)

Code E 0.07 (0.33032, 0.20897, 0.0535916) (0.304913, 0.235985, 0.0935946)

0.09 (0.185369, 0.102003, 0.0170833) (0.177093, 0.115907, 0.0277809)

0.11 (0.101517, 0.048143, 0.00514401) (0.0982611, 0.0572934, 0.00847355)

0.04 (0.771444, 0.589057, 0.272022) (0.682913, 0.626334, 0.441509)

0.06 (0.438223, 0.296092, 0.0930953) (0.399384, 0.330896, 0.163132)

Code F 0.08 (0.248085, 0.146553, 0.0304626) (0.23151, 0.16847, 0.0547363)

0.1 (0.137656, 0.0703978, 0.0094448) (0.131819, 0.0815278, 0.0162636)

0.12 (0.0743007, 0.0326041, 0.00275776) (0.0725875, 0.037589, 0.00426584)

These combinations are denoted by (L̃, L̃), (L̃, L̂), (L̂, L̂), and (L̂, L̃) in turn. All these combinationsformat the problem as an equation containing one unknown variable p̃, given {E(Si)}dc

i=1 as inputs. Itdeserves to point out that the results given in [12] are obtained using the combination (L̂, L̂).

To investigate how well these combinations work in practice, plenty of simulations are run and theresults are included in Figure 3, where simulation settings are the same as those in [12]. The results areaveraged over 10000 trials. Note that this paper considers only the case p < 0.5, so E(Si) is clipped to[0, 0.5] in these experiments.

In Figure 3, for each LDPC code, five algorithms are compared. The first algorithm sets W in bothsides of (9) to W ∗, giving the name “W ∗, simulation” in Figure 3. This algorithm is impracticable infact because the calculation of W ∗ needs p. The rest four algorithms are defined in (23). Then for eachalgorithm, several experiments are run for various p. Note that when v = 0 (i.e., p = 0.5),vi ≡ log(1+vi)/


Table 3 Entropies for various p

p 0.0100 0.0200 0.0300 0.0400 0.0500 0.0600

H(X|Y ) 0.0808 0.1414 0.1944 0.2423 0.2864 0.3274

p 0.0700 0.0800 0.0900 0.1000 0.1100 0.1200

H(X|Y ) 0.3659 0.4022 0.4365 0.4690 0.4999 0.5294

0.01 0.015 0.02 0.025 0.03−5

0

5

10

15

20

25

p

PS

NR

(dB

)

Code A

(a)

0.01 0.015 0.02 0.025 0.03

0

5

10

15

20

25

30

p

PS

NR

(dB

)

Code B

(b)

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.065

10

15

20

25

30

p

PS

NR

(dB

)

Code C

(c)

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.0618

20

22

24

26

28

30

p

PS

NR

(dB

)

Code D

(d)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.1112

14

16

18

20

22

24

26

28

30

32

p

PS

NR

(dB

)

Code E

(e)

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.1214

16

18

20

22

24

26

28

30

32

p

PS

NR

(dB

)

Code F

(f)

L, theoryW*, theoryW*, simulation(L, L)(L, L)(L, L)(L, L)

~ ~~

~

^^ ^^


~ ~~

~

^^ ^^


~ ~~

~

^^ ^^


~ ~~

~

^^ ^^

L, theoryW*, theoryW*, simulation

(L, L)(L, L)(L, L)

(L, L)

~ ~~

~

^^ ^^


~ ~~

~

^^ ^^

Figure 3 Theoretical PSNR results and average PSNR results over 10000 trials.

(1− vi) ≡ 0, so for (L̂, L̂) and (L̂, L̃), we should be carefully in programing to prevent p̃ from convergingto 0.5.

4.1 Model verification

To verify the proposed mathematic model, we compare the curve “W ∗, simulation” with the curve “W ∗,theory”. At small p, the curve “W ∗, simulation” is around 2 dB lower than the curve “W ∗, theory”. As


p increases, both curves go closer and closer to each other. These observations justify the validity of theproposed model. However, once p is greater than a certain value, the curve “W ∗, simulation” descendssharply. For example, turning points are observed for codes A–C at p = 0.0225, p = 0.025, and p = 0.05,respectively. This phenomenon is somewhat like the error floor in the decoding of LDPC codes.

4.2 Robustness

In general, none of these four combinations shows significant superiority over others at small p, while atlarge p, (L̂, L̃) defeats the other three combinations due to its robustness. The curves of (L̃, L̃), (L̃, L̂),and (L̂, L̂) seem very similar at all cases. They almost coincide with each other at small p and exhibitsimilar “error floor” phenomenon at large p. In contrast, the curve of (L̂, L̃) is different. Though (L̂, L̂)shows no significant advantage at small p, it performs very well at large p. For code A and code B,even until p = 0.03, no sharp descent is found for the curve of (L̂, L̃). In other words, no “error floor”phenomenon occurs. For code C, until p = 0.06, no “error floor” phenomenon is observed for (L̂, L̃).For other codes, (L̂, L̃) also shows the best PSNR at large p, meaning that, (L̂, L̃) performs much morerobustly at large p than the other three combinations.

5 Conclusions

This paper establishes a mathematic model for analyzing the algorithm of crossover probability estimationusing LDPC syndrome [12]. To evaluate the performance of this algorithm, the pseudo signal-noise ratiois defined as the measurement. The optimization process is formatted into maximizing the PSNR forgiven p and an LDPC code. This mathematic model is well verified in simulations and the algorithmin [12] also proves to be near-optimal. Considering the implementation, four practical algorithms arecompared in the experiments and (L̂, L̃) wins due to its robustness at large p.

Acknowledgements

I thank Dr. Wu JiaJi, an associate professor from Xidian University, Xi’an, China, for his kind assistance in

the programming. This work was supported by the National Natural Science Foundation of China (Grant Nos.

61001100, 61077009, 60975007) and the Provincial Science Foundation of Shaanxi, China (Grant Nos. 2010K06-15,

2010JQ8019).

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Analysis on crossover probability estimation using LDPC syndrome