Convergence of the Min-Sum Decoding Scheme for LDPC Codes from ...

2014 Fourth International Conference of Emerging Applications of Information Technology

Convergence of the Min-Sum Decoding Scheme for LDPC Codes from a Dynamical Systems Perspective Monosij Maitra and Abhik Mukherjee Department of Computer Science and Technology Indian Institute of Engineering Science and Technology, Shibpur (Erstwhile Bengal Engineering and Science University, Shibpur) Howrah, West Bengal, India Email: [email protected], [email protected]

highly suitable for a hardware implementation involving low power systems, it suffers from a poor performance when the block length of codewords get large. In that case the general Soft decision or the Sum-Product decoder (SPD) gets preference for a guaranteed optimal performance, though from a point of view of implementation and application, they are computationally expensive. The Min-Sum decoder [6] (MSD) is an approximation of the general the SPD in the sense that it removes the most computationally expensive operations from the SPD as a result of which the decoder’s optimality in reaching the Shannon limit for the corresponding channel gets hampered by a small margin [7]. Nevertheless, due to the removal of the computational overheads, MSD can be a good choice for hardware implementation optimizing both the decoder performance and computational power of the system in which it is implemented [8], [9]. Some major past studies on the iterative Turbo decoders, from the aspect of a nonlinear dynamical system, have revealed some important characteristics about their nature [10], [11], [12]. The dynamics of the SPD in high noise (low SNR) and moderate noise (waterfall SNR) regions has already been studied in the past in an Additive White Gaussian Noise (AWGN) channel [13] and is shown to exhibit multiple types of bifurcations and chaotic trajectories thus leading to a decoding failure in these SNR regions for two specic LDPC codes. When the SNR stays above the waterfall region, the decoder converges smoothly into a valid codeword. In this paper we have studied the dynamics of the MSD for decoding a particular Gallager code in its waterfall SNR region. It also describes the methodology of nding the SNR zone where the MSD converges to correct decoding for any particular code. In Section II we dene LDPC codes due to Gallager [1] and its Tanner graph representation [14]. We also describe the SPD in its Log-Likelihood Ratio (LLR) domain and the MSD as an approximation of the SPD. In Section III we briey present how the LDPC decoders and specially the MSD can be viewed as a discrete dynamical system and our experimental setup to study the dynamics. In Section IV we provide some major results obtained from our simulations leading to a generic discussion on them. The limitations of

Abstract—LDPC codes represent a class of codes for a wide variety of modern day coding applications including wireless communications and also some aspects of Cryptography. The general decoder for these codes, the Sum-Product algorithm can be viewed as a nonlinear dynamical system and has been shown to exhibit bifurcations and chaotic phenomena in the low and waterfall SNR zone in an AWGN channel. It has been attempted to investigate whether the Min-Sum decoder, a major approximation of the Sum-Product decoder, exhibits similar phenomena in its corresponding waterfall SNR zone for a particular Gallager code in the AWGN Channel. The results obtained indicate that the decoder does not show bifurcations and chaos in the waterfall SNR zone. Nevertheless, the decoder converges smoothly when the SNR stays above the waterfall region. This work guides how to nd the convergent SNR zone for decoding any particular LDPC code with the Min-Sum decoder. Keywords-Gallager codes; Iterative Deocding; Nonlinear Dynamics; Decoder Dynamics; Gaussian Channel

I. I NTRODUCTION Low Density Parity Check (LDPC) codes were discovered by Robert Gallager [1] and re-discovered by David MacKay [2] and others in 1962-63 and 1996-99, respectively. This class of code is special in the sense that it approaches the Shannon’s denition of channel capacity [3] i.e. the maximum capacity for the information transmission rate for any specic communication channel. Other than the Turbo codes [4], till now, LDPC codes are known to be the best capacity achieving codes and represent the state-of-the-art coding for almost every possible practical application. The application areas of these codes include types of communication systems like Mobile Systems and also some aspects of Cryptography. They have gained a lot of attention during the last decade due to their excellent error-correcting capability and have been chosen as an alternative coding scheme for many practical standards in wireless communications [5]. One of the major advantages of using these codes in practical data transmission is that their decoding algorithms namely the Sum-Product algorithm and the Min-Sum algorithm have an asymptotic time complexity linear in the size of a codeword. LDPC decoders are message passing iterative systems in general. While the Hard decision decoder for LDPC codes is 978-1-4799-4272-5/14 $31.00 © 2014 IEEE DOI 10.1109/EAIT.2014.40

107

D(fj ) = wr , ∀j = 1, 2, . . . , M while, for an irregular code, D(ci ) and D(fj ) are characterized by the degree distribution polynomials of the Tanner graph.

our present study and a wide area of future work in this direction is discussed in Section V. II. BACKGROUND

ON

LDPC C ODES

C. Decoding Algorithms for LDPC Codes

LDPC codes are dened in terms of their parity check matrices. We are interested in binary parity check matrices with elements from F2 = {0, 1}. The term low density corresponds to a binary, sparse parity check matrix, H of order (M × N ). We present Gallager’s description of LDPC codes [1] and refer to these codes as simply Gallager codes.

Let L(Pi ) denote the LLR for the initially computed bitprobability for ci after transmission through an AWGN channel with variance σ 2 and received vector ri . Let Lk (vji ) and Lk (uij ) represent the LLR messages from the check node, fj to the message node, ci and vice-versa, respectively, computed in the k th iteration and let Lk (Qi ) represent the LLR based on the total a-posteriori probabilities of bit ci computed in the k th iteration. Fj and Ci are dened with respect to the Tanner graph of the associated code as Fj = {i|(ci , fj ) ∈ E} and Ci = {j|(fj , ci ) ∈ E}. The LLR domain SPD used to decode LDPC codes is shown as follows.

A. Gallager Codes Denition 2.1 (Gallager Codes): A (N, wc , wr ) LDPC code is a Gallager code (also called a (wc , wr ) regular LDPC code) with block length N and a parity check matrix H with M rows and N columns where the Hamming weight of each row and each column are two xed integers, wr and wc , respectively, where wr > wc ≥ 2 and wr  N , wc  M . Therefore the total number of 1s in H = M.wr (= N.wc ). c Thus, N.w ∈ N. Gallager also showed a pseudorandom wr construction for such an ensemble of LDPC codes as follows. We divide the matrix H into wc sub-matrices each of size M (w × N ) and let the rst one of these sub-matrices be c denoted by H0 . Each row of H0 has wr 1s in such a way that the ith row of H0 contains all its wr 1’s in the columns M [(i−1).wr +1] to (i.wr ), ∀i = 1, 2, . . . , w . The sub-matrices c other than H0 are formed by (wc − 1) distinct, random permutations, πk (H0 ), ∀k = 2, . . . , wc of the columns of H0 as shown below. Note that π1 (H0 ) = H0 . ⎡ ⎢ ⎢ ⎢ H=⎢ ⎢ ⎢ ⎣

π1 (H0 ) π2 (H0 ) π3 (H0 ) . . . πwc (H0 )

1) Initialization : Set k = 0 and L(Pi ) = −

2.ri σ2

(1)

2) Iterative Processing : (i) Check Node Processing (Horizontal step): ⎡ ⎤

(k−1)  j ) L (u i ⎦ L(k) (vji ) = 2 tanh−1 ⎣ tanh 2  i ∈Fj =i

(2) (ii)Message Node Processing (Vertical step): L(k) (vj  i ) L(k) (uij ) = L(Pi ) +

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3)

j  ∈Ci =j

3) Hard Decision :

(k) Lk (Qi ) = L(Pi ) + L (vji ) and  cˆi =

When wr and wc does not remain xed for the rows and columns of a parity check matrix H, then the code is called an irregular LDPC code.

j∈Ci

0 1

if Lk (Qi ) > 0; if Lk (Qi ) ≤ 0

If cˆ.H T = 0, declare cˆ as the transmitted codeword; else go to 2 until k exceeds Maximum number of iterations. If k = Maximum number of iterations and still cˆ.H T = 0, declare a decoding failure.

B. Tanner Graph Tanner studied LDPC codes in 1981 and introduced a new graph theoretic way of representing them in [14]. He showed that there exists a bijection between the parity check matrix, [H] M×N of any block code and a bipartite graph G = ((V1 V2 ) , E), where the two disjoint sets of vertices V1 and V2 represents the N code bits or message nodes and the M check sums or check nodes, respectively. An edge e = (fj , ci ) ∈ E iff message node ci participates in the check sum fj i.e. whenever H(j, i) = 1, i = 1, 2, . . . , N and j = 1, 2, . . . , M . Thus, in the case of a regular code, the associated Tanner graph degree distribution is as follows: D(ci ) = wc , ∀i = 1, 2, . . . , N and

The Min-Sum Approximation : The tanh(.) and tanh−1 (.) operations are pretty complex for hardware implementation. To overcome this difculty an ingenious approximation was discovered by M. Fossorier [6] as follows: let L(ui j ) = φi j . βi j , where φi j = sign(L(ui j )) and βi j = |L(ui j )|. Then Eqn.(2) can be written as ⎛ ⎞ φi j ⎠ .  min (βi j ) (4) L(k) (vji ) = ⎝ i ∈Fj =i

108

i ∈Fj =i

provided all the βi j ’s are statistically independent. It has been proved in [7] that MSD is an overestimation of the SPD in general. III. LDPC D ECODERS

AS

expressed in terms of the LLR based on the total a-posteriori probabilities of bit ci i.e. Lk (Qi ) as follows:  (k)   L (Qi ) 1 + tanh 2 (k) (10) Pi (0) = 2 The following standard identity is used to prove the above result. x ex − 1 tanh = x (11) 2 e +1 Since L(Qi ) is the Log-Likelihood ratio of the probability that the code bit ci is a 0 to the probability that the code bit ci is a 1, therefore in the k th iteration of the algorithm L(k) (Qi ) has the following expression.   (k) Pi (0) (k) (12) L (Qi ) = ln (k) Pi (1)

DYNAMICAL S YSTEMS

The message passing strategy for decoding LDPC codes is, in general, an iterative process. The probability domain SPD has already been shown as a discrete, nonlinear dynamical system in [13]. We show how the MSD can also be treated in a similar way. We represent the MSD as a pair of iterative functions f1 (.) and f2 (.) populating the vectors V(k) (c, σ) and U(k) (c, σ) parameterized by the transmitted codeword, c and σ, representing the SNR. V(k) (c, σ) and U(k) (c, σ) represent two vectors, each of length N.wc (= M.wr ), containing all the extrinsic LLR messages sent from the check nodes to the message nodes and vice-versa, respectively, computed in the k th iterate of the algorithm. Then, the iterative process can be written mathematically as   V(k) (c, σ) = f1 U(k−1) (c, σ) (5)   U(k) (c, σ) = f2 V(k) (c, σ) (6) where V

(k)

U

(7)

  (c, σ) = L(k) (uij )

(8)

(1×N.wc )

(1×M.wr )

(k)

Pi (0)

(k) Pi (1)

⇒

  (c, σ) = L(k) (vji )

(k)

Thus we can write

(k)

(k)

(k)

(k)

(k)

Pi (0) − Pi (1) Pi (0) + Pi (1)

= =

(k)

eL

(Qi )

(k)

(Qi ) − 1

(k)

(Qi ) + 1

eL

eL

(13)

(14)

(k)

Since Pi (0) + Pi (1) = 1, thus application of Eqn.(11) reduces Eqn.(14) to  (k)    L (Qi ) (k) (k) Pi (0) − 1 − Pi (0) = tanh 2    (k) L (Qi ) 1 + tanh 2 (k) ⇒ Pi (0) = 2 To understand the dynamics of the decoder we have used histograms of the a-posteriori probabilities in the waterfall and high SNR regions for both the decoders along with plots of E(l) vs. l (time-series diagrams), E(l + 1) vs. E(l) (phase portraits) and E(SNR) vs. SNR i.e. the average values of E(l) computed as a function of SNR (bifurcation diagram). The Lyapunov exponent for the SPD has also been calculated as a function of SNR for verifying the existence of chaos.

In general, there are two types of xed points for iterative decoders occurring at low and high values of SNR, respectively [15], [16]. The rst type of xed point corresponds to a decoding failure. In this case, the probabilities of most of the code bits heavily cluster around 0.5 leading to the difculty of deciding a valid codeword. This type of xed point is called an indecisive xed point. Interestingly, the second type of xed point correspond to the convergence of probabilities to either 0 or 1 leading to a valid codeword. This type of xed point is called an unequivocal xed point. A. Experimental Setup For our simulations to study the dynamics of the MSD we have used the all zero codeword, 0 and the measure following [13]. N 1 (l) E(l) = [P (0)]2 (9) N i=1 i

IV. I MPLEMENTATION R ESULTS Extensive simulations have been performed on a (1044, 3, 6) Gallager code where we have rst, identied the waterfall SNR region of this code with respect to the SPD and the MSD by comparing their Bit-Error-Rate(BER) vs. SNR data and plots. We have simulated both the decoders and computed the BER at each SNR interval for 500 received vector samples of the all zero codeword. Averaging the BER over those samples at each SNR, the BER vs. SNR plot is obtained. Figure 1 shows the BER shift of the waterfall SNR region for the MSD from SPD. The waterfall SNR regions for SPD and MSD were found out to be [0.5 dB, 1.5 dB] and [1.0 dB, 2.0 dB], respectively.

E(l) represents the mean-square value of the a-posteriori probabilities of the code bits being 0 at the lth iteration. A (l) correct decoding of the received vector makes Pi (0) = 1, ∀i = 1, 2, . . . , N which forces E(l) = 1 in the ideal case. MSD is intrinsically based on the algebra of LLR. Thus, (l) there is no direct way of nding Pi (0) which is needed to compute E(l). Therefore, the relation between L(l) (Qi ) (l) and Pi (0) has been established as follows. The probability (k) that the bit ci is a 0 after the k th iteration i.e. Pi (0) can be

109

10

500 Frequency

Frequency

200 150 100 50 0 0.3

400 300 200 100

0.4

0.5 0.6 P(0)

0 0

0.7

0.5 P(0)

1

(a) Sum-Product decoder: Left plot shows the existence of an Indecisive xed point at SNR = 0.8 dB; Right plot shows the existence of an Unequivocal xed point at SNR = 1.5 dB

SP decoding MS decoding

1

600

250

This SNR region has been studied carefully by subdividing the region using resolution of 10−3 . Next we have performed our intended study for the decoder dynamics over the waterfall SNR region for our chosen LDPC code. Figure 2 shows the histograms of the a-posteriori probabilities for SPD and MSD in both of their waterfall and high SNR regions for the chosen Gallager code. For both the decoders the time series plots and the phase trajectory plots have been computed with random noise samples at each SNR.

25 1000

10

20

3

10

0

15 10 5

4

10

Frequency

800 Frequency

BER

2

0.5

1

1.5 SNR (in dB)

2

2.5

0 0

Figure 1: Comparison of the SPD and the MSD for the (1044,3,6) Gallager code used in the study

600 400 200

0.5 P(0)

1

0

0.992 0.994 0.996 0.998 P(0)

1

(b) Min-Sum decoder: Left plot shows the absence of any Indecisive xed points at SNR = 1.05 dB; Right plot shows the existence of the Unequivocal xed point at SNR = 1.9 dB

We have veried that the SPD shows indecisive xed points at different values of SNR over its corresponding waterfall region. But we found that the MSD could not show any indecisive xed point its waterfall region for our chosen code. The SPD has also been veried to show bifurcations and a region of chaotic trajectories. Figure 3(a) and Figure 3(b) shows the behaviour of the SPD and the MSD, respectively at two different values of SNR. For the MSD it has been observed that till SNR = 1.78 dB, for different noise values the average value of E(l) increases slowly. At each SNR the decoder toggles between a nite set of vectors of variable cardinality no one of which is a valid codeword. Thus no such periodic nature is also found. The rate of change of E(l) increases sharply when SNR crosses 1.78 dB. For SNR > 1.82 dB the MSD bifurcates from this random behaviour and converges to the unequivocal xed point with a correct decision of the all zero codeword. Figure 4 shows the convergence of the MSD for two different values of SNR. Figure 5 shows the E(SNR) vs. SNR plots of both the SPD and the MSD for the (1044, 3, 6) Gallager code. This diagram is obtained by increasing the SNR with a gap of 0.001 dB and computing the average E(l) over 1000 received vector samples for the all zero codeword at each SNR. The plots shown for the MSD suggests that this decoder uctuates from converging to an indecisive xed point at low and waterfall SNR regions.

Figure 2: Aposteriori probability distribution of the decoders plots where a rate of change occurs for E(SNR). Table I shows the SNR zones where the decoding failures with presence and absence of indecisive xed points in the SPD and the MSD respectively occurs. The table also shows the SNR range where the decoders converge to unequivocal xed points giving the correct codeword as output. This clearly indicates that in the MSD there is a sharp increase in E(SNR) as the SNR crosses 1.78 dB but there are no indecisive xed points associated with this zone. In contrast, in the SPD, when the SNR lies between 1.2 dB and 1.4 dB, we have found that the decoder shows chaotic phenomena in its corresponding waterfall region for the chosen Gallager code. We explain this degradation occurring in the MSD as associated with the approximation done in the check node update step of the algorithm. Eqn.(4) implies that the minimum of all the |L(ui j )| is taken to compute L(vji ) instead of doing some complex operations as shown in Eqn.(2). in  This means  L(ui j ) the corresponding SPD each of the tanh values 2 other than one which corresponds to the minimum value of |L(ui j )|, are replaced by  Eqn.(12) implies that  a 1. Thus, L(ui j ) forcing a particular tanh = 1 ⇒ L(ui j ) → ∞ 2 (k) (k) i.e. Pi (1) → 0 and Pi (0) → 1. Therefore, this becomes a type of hard decision imposed on the check node update step in every iteration. But it can be said  with a high  degree L(ui j ) of certainty that all of these other tanh values 2 cannot be 1 simultaneously in the low SNR region. This fact

A. Discussions Figure 5(a) and Figure 5(b) suggests that there are some critical points i.e. (SNR, E(SNR)) pairs in the respective

110

0.9

0.9

0.88

0.88

0.86

0.86

0.84

0.84

1 Unequivocal Fixed Point

0.82

0.82

0.8

0.8

0.78

0.78

Chaos

E(SNR)

E(l+1)

E(l)

0.9

Indecisive Fixed Points

0.8 Bifurcations

0.76

0

500 l

0.76 0.7

1000

0.75

0.8 E(l)

0.85

0.9

0.7 0

0.74

0.72

0.72

0.7

0.7

0.8

1.2

1.6

1

Unequivocal Fixed Point

E(SNR)

E(l)

E(l+1)

Decoding Failure

0.68

0.66

0.66

0.64 0

500 l

0.64 0.59

1000

2.4

(a) Bifurcation diagram of SPD for the (1044, 3, 6) Gallager code

0.9

0.68

2

SNR

(a) Time series plot and Phase portrait at SNR = 0.915 dB for SPD 0.74

0.4

0.8

0.7

0.63

0.67 E(l)

0.71 0.74

0.6 0

0.4

0.8

1.2

1.6

2

2.4

SNR

(b) Time series plot and Phase portrait at SNR = 1.112 dB for MSD

(b) E(SNR) vs. SNR plot of MSD for the (1044, 3, 6) Gallager code

Figure 3: Dynamics of SPD and MSD at two relevant values of SNR in the waterfall zone for the chosen Gallager code

Figure 5: Route to convergence as a function of SNR for the all zero codeword in case of both the iterative decoders

120 Number of Bits in Error

1 0.95

E(l)

0.9 0.85 0.8

0

100

l

200

Nevertheless, due to the approximation done in the MSD, there is a shift of the waterfall SNR region for the same code from that of the SPD. Due to this, there is an expansion in the SNR zone corresponding to the decoding failure in case of the MSD. But in both decoding scenarios, when the SNR stays above the waterfall region for that corresponding decoder, the decoders converge to the unequivocal xed point which also turns out to be the correct codeword that was transmitted. This happens primarily due to the high SNR zone in which the decoders operate now and the decoder’s own merit in computing the codeword with respect to an LDPC code.

100 80 60 40 20 0 0

300

100

l

200

300

(a) Dynamics of the MSD at SNR = 1.823 dB 140 Number of Bits in Error

1 0.95

E(l)

0.9 0.85 0.8 0.75 0

5

10 l

15

20

120

B. Summary

100

Now we summarize the steps involved in studying the MSD dynamics to nd an operative SNR region for a particular LDPC code. These generic sequential steps should be undertaken for any iterative decoding scheme for a particular code before putting it in practice. The MSD has to be simulated over a number of iterations depending on the convergence criterion and the maximum number of iterations has to be xed beforehand.

80 60 40 20 0 0

5

10 l

15

20

(b) Dynamics of the MSD at SNR = 1.934 dB

Figure 4: Convergence of the Min-Sum decoder for the chosen Gallager code near the end of its waterfall region

Step 1. The average BER values obtained by simulating MSD at different SNR values should be plotted for ascertaining the waterfall region of this code alike Figure 1. The SNR values at the two ends of the waterfall region may be denoted as Swl dB and Swu dB with Swl < Swu .

introduces a heavy error computationally which restricts the decoding process to give a correct estimation of the codeword in every iteration which serves as the major source for a random oscillation of the measure E(l) which explains the absence of typical bifurcations.

111

fruitful to study the dynamics of this decoder with respect to communication channels in presence of high noise prior to its usage in practical scenarios. Such studies can lead to a methodology of predicting the SNR region of decoding failure with more certainty for the chosen code.

Step 2. The region [Swl dB,Swu dB] must be divided into intervals of r dB. The MSD has to be simulated at each SNR pair [(Swl + i.r) dB, (Swu − i.r) dB] and the histogram of the probabilities, P (0) has to be generated (alike Figure 2(b)). The histogram properties are indicative of the decoder convergence and i for which the transition occurs from the decoding failure zone to an unequivocal xed point is denoted as Swt .

R EFERENCES [1] Gallager R. G.; Low Density Parity Check Codes; MIT Press, Cambridge, 1963. [2] MacKay D. J. C.; Good Error-Correcting Codes Based on Very Sparse Matrices; IEEE Trans. on Info. Th.; vol. 45, no. 2, pp. 399-431, March 1999.

Step 3. The measure E(l), as in Eqn.(9), has to be computed at each SNR to analyze how P (0) progresses. The E(l) values must be averaged at each SNR over large number of received vector samples of the all zero codeword. Thus a plot alike Figure 5 can be generated to see how the decoder dynamics depends on change of SNR and a table alike Table I can consolidate the ndings about the operative region for any code with an iterative decoding scheme.

[3] Shannon C. E.; A Mathematical Theory of Communication; Bell System Technical Journal 27 (3); pp. 379-423, (Jul./Oct. 1948). [4] Berrou G., Glavieux A., Thitimajshima P.; Near Shannon-Limit Error Correcting Coding and Decoding: Turbo Codes; Proc. Int. Conf. Comm.; pp. 1064-1070, May, 1993. [5] Ismail M., Ahmed I., Coon J., Armour S., Kocak T., McGeehan J.; Low Latency Low Power Bit Flipping Algorithms For LDPC Decoding; IEEE 21st Int. Symp. on Personal Indoor and Mobile Radio Communications, pp. 278-282, Sept. 2010.

Table I: Forbidden and Convergent SNR regions for the (1044,3,6) Gallager code with the SPD and the MSD Different SNR zones of importance Zone of Indecisive xed points Chaotic Zone Lack of signicant xed points SNR Zone for Correct Decoding (Zone of Unequivocal xed point)

SNR Zone for Decoding Failure

[6] Fossorier M. P. C., Mihaljevic M., Imai H.; Reduced Complexity Iterative Decoding of Low-Density Parity Check Codes Based on Belief Propagation; IEEE Trans. on Comm.; vol. 47, pp. 673680, 1999.

Decoding Algorithm SPD MSD 1.82 dB

[7] Chen J., Fossorier M. P. C.; Near Optimum Universal Belief Propagation Based Decoding of Low Density Parity Check Codes; IEEE Trans. on Comm.; vol. 50, pp. 406-414, 2002. [8] Islam M. R., Shaullah D. S., Faisal M. M. A., Rahman I.; Optimized Min-Sum Decoding Algorithm for Low Density Parity Check Codes; 14th Int. Conf. on Adv. Comm. Tech., pp. 475480, Feb. 2012. [9] Wang C., Chen X., Li Z., Yang S.; A Simplied Min-Sum Decoding Algorithm for Non-Binary LDPC Codes; IEEE Trans. on Comm., vol. 61, pp. 24-32, Jan. 2013.

V. C ONCLUSION In this work, it has been attempted to see whether the bifurcation and chaotic phenomena still persist in an approximated iterative decoding system for a large Gallager code. We could not nd any such behaviour in the system and have suggested briey the cause for this degradation with respect to the Tanner graph of the underlying code, that the system fails to converge to indecisive xed points in its waterfall region. At this point we would like to investigate empirically the existence of other codes where this departure in performance is occurring in the case of MSD. A wide area for further work in this direction still prevails. Since we performed all our simulations on a regular LDPC code, thus the MSD dynamics for irregular codes also needs to be worked out. There exist practical scenarios when highly noisy transmissions occur through communication channels and in the last decade several modications of the SPD have been proposed which have provided major contributions in hardware implementations, mobile communications and deep space missions. Since the MSD is advantageous for hardware implementation over SPD in many applications, it can be

[10] Richardson T.; The geometry of Turbo-Decoding Dynamics; IEEE Trans. on Info. Th.; vol. 46 pp. 9 - 23, Jan., 2000. [11] Agrawal D., Vardy A.; The Turbo Decoding Algorithm and Its Phase Trajectories; IEEE Trans. on Info. Th.; vol. 47, pp. 699722, 2001. [12] Kocarev L., Tasev Z., Vardy A.; Improving Turbo Codes by Control of Transient Chaos in Turbo-Decoding Algorithms; Electronics Letters; vol. 38, pp. 1184-1186, Dec, 2002. [13] Zheng X., Lau F. C. M., Tse C. K., Wong S. C.; Study of Bifurcation Behavior of LDPC Decoders; Int. J. of Bifurcation and Chaos; vol. 16, no. 11, pp. 3435-3449, Nov. 2005. [14] Tanner R. M.; A Recursive Approach to Low Complexity Codes; IEEE Trans. on Info. Th.; IT-27: 533-547, Sept 1981. [15] Alligood K. T., Sauer T.D., Yorke J.A.; Chaos: An Introduction to Dynamical Systems; Springer Int. Ed., 2000. [16] Hilborn R. C.; Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers; Oxford Univ. Press, 2001.

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