Simplified Soft Decision Decoding of LDPC codes

Simplified Soft Decision Decoding of LDPC codesMARUTHI L NK S GURUMURTHYAbstractLDPC codes gained importance since its re-discovery by Mackay and Neil based on sixpence Graph. This paper presents the implementation of out front error subject soft-input, soft-output decoding process that expeditiously decrypts the sure set of selective in engineeration under low-pitched signal-to-noise ratios collectable to which the errors are reduced and hence transmission time is greatly reduced. The implemented algorithmic programic program is less complex and does not require knowledge of signal-to-noise-ratio of the received data-path. entreError correction coding techniques came into existence after existence of channel capacity theorem by Shannon, the father of information theory in 1948 on authoritative communication over noisy transmission channels. Thereafter, many coding techniques were authentic for efficient coding techniques, like Hamming, Golay and many other techniques were v eritable. Though Gallagher in 1962 developed LDPC codes, the method employed was not optimal. Hence it was not reviewed for last 3 decades, until 1992, when a paper Near Shannon limit error correcting coding and decoding turbo codes presented by Berrou, Glavieux and Thitimajshima changed the trend followed by researchers for the past 5 decades. And now, we are aiming to develop such codes through another strategy. The coding gain provided by this method is much high(prenominal) when compared to other coding systems. The LDPC codes are becoming more popular because of their reduced power for transmission and less complex logic to achieve low BERs which is very essential for reliable transmission over noisy channels.LDPC CODESLow Density space-reflection symmetry Check (LDPC) codes are a powerful class of forward error correction codes developed by Gallagher in 1962, practically implemented by MacKay in 1995. Low Density refers to less number of 1s in the Parity Check Matrix (H) of Block Codes.Block codes are a type of Error Correction Codes which uses a Generator Matrix(G) to produces a Code-word of continuance n for message(D) of length k, where n-k is the redundancy bits added to make a Code-word of length n, higher the redundancy higher the error correction capability and higher the Bandwidth. Hence, there is a trade-off between error correction capability and bandwidth.C=D.GThe size of the Matrix G is (n, n-k) and the size of the Matrix H is (n-k, n).Usually redundancy is defined by the term code rate which is given byR=k/nThis varies from to 1/6.The Encoder produces a Code-word which satisfy the conditionC.HT=0If R is the received data from the Transmitter then the acknowledgement of the correct data received is given by the Syndrome(S).S=R.HTThe value of S obtained by the supra calculation determines whether the received data is free from error or not, i.e. if S=0 then the received data is free from error, otherwise the received data has an error.Sin ce this property is very useful to reduce the calculation time if we receive the data correctly as soon as we receive data. The probability that the received data needs to be retransmitted which in the range of 10-6 to 10-8.Encoding the data as required is an easy process, but decoding the data effectively and efficiently is a very hard process.The proposed algorithm uses the basic concept of block codes, i.e.R=C+EWhere R=noise corrupted vectorC=Code vector transmitted over noisy channelE=Error vector.The receiver does not know C and E its function is to decode C from R, and the message D from C.PROPOSED ALGORITHMThe base for LDPC decoding algorithm proposed is as described by MacKay and Neil in 1997.Following are the steps that curtly describe the algorithmSTEP 1 INITIALIZATIONLet rj be the received vector, i.e. the data received from the Gaussian channelWe calculate the components of the vectors d02( j), and d12( j) asd02( j)= (rj+1)2d12( j)= (rj-1)2 j = 1, 2, . . . , n (1)These first soft estimates of the code symbols are used to initialise the algorithm by setting the following coefficients q0ij and q1ij at each symbol lymph glandq0ij= d02( j)q1ij= d12( j) j = 1, 2, . . . , n,i = 1, 2. . . , . n-k, (2)To eliminate the effect of zero in the calculation the expressions for the calculations can be reduced as shown belowln(em+en) = max(m,n) + ln(1+ e-m-n)log2(2m+2n) = max(m,n) + ln(1+ 2-m-n)(3)The tables required for calculations are computed employ the above formula. The complexity is greatly reduced and the reliability of the transmission of signal is increased.STEP 2 HORIZONTAL STEPaij= f+(q0ij , q1 ij )bij= f(q0ij , q1ij)if (q 0ij ) (q1ij )sij= 0 elsesij= 1cij= ikikif ik is evenr0n,ij= f+(0, cij )r1n,ij= f(0, cij )if ik is oddr0n,ij= f(0, cij )r1n,ij= f+(0, cij )STEP 3 VERTICAL STEPq0ij= d02 (j) + 0n,ijq1ij= d12 (j) + 1n,ijSTEP 4 DECISIONrm0ij=r0n,ij+ q0ijrm1ij=r1n,ij+ q1ijif rm02(j)12(j) thenc(j) = 1elsec(j) = 0The LUT entries used in the proposed algo rithm for equation (3) and the exact value obtained will vary as shown in the figure. equation of BER performance of (8 * 12) LDPC SSD decoder (10 iterations)Comparison of BER performance of (8 * 12) LDPC SSD decoder (50 iterations)As explained above, for higher precision the number of iterations is increased to obtain the same performance of the exact equation.The obtained results approach towards the complex algorithm developed by Gallagher with simple iterative approach and provides a high coding gain compared to uncoded signal. And it provides higher performance at larger iterations. This algorithm provides even higher performance for large length codes.CONCLUSIONIn this paper we have described Low Density Parity Check (LDPC) codes and decoding of these codes using low complexity algorithms. LDPC codes are used now-a-days in communication systems that take advantage of parallelism, good error correction and high through put. This led to the new algorithm which could decode the e rrors and yet give similar BER performance as the complex algorithms without the knowledge of the channel noise parameters like variance. This new algorithm is based on repeated use of an antilog-sum operation, and has been simulated on the Tanner graph representation of several LDPC codes and this algorithm can be regarded as a generalized form of belief propagation, where the belief propagated is euclidian distance estimate rather than a probability estimate. The advantages of the new algorithm are that the performance is as good as the complex algorithm provided the value of base of the logarithm is used properly that knowledge of noise is not required and that in the simplified form the algorithm needs only additions/subtractions, comparisons and two look-up tables avoiding the use of quotients and products operations that are of high complexity in practical implementations especially using FPGA technology.REFERENCES1 R.G. Gallager, Low Density Parity Check Codes, IRE Trans. Inf ormation Theory, IT-8, 21-28 (1962).2 D.J.C. Mackay and R.M. Neal, Near Shannon limit performance of low density parity check codes, Electronics Letters, vol. 33, pp 457-458 (1997).3 L. Arnone, C. Gayaso, C. Gonzalez and J. Castineira, Sum-Subtract Fixed Point LDPC Decoder, Latin American apply Research, vol. 37, pp 17-20 (2007).4 Castineira Moreira, Farrell P.G. Essentials of error control coding, Wiley (2006).5 Castineira Moreira J., Farrell, P.G. 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