Advance in Electronic and Electric Engineering. ISSN 2231-1297, Volume 4, Number 5 (2014), pp. 463-468 Research India Publications http://www.ripublication.com/aeee.htm Power Efficiency of LDPC Codes under Hard and Soft Decision QAM Modulated OFDM Kumar Raushan Ratnesh, Jai Sachith Paul, Sanket N Shettar Electronics and Communication, Christ University faculty of Engineering, Bangalore, Karnataka, INDIA. Abstract Quadrature Amplitude Modulation have many advantage when used in OFDM as they do not require sharp cut off band filters. So they are more used for cable communication of Direct TV and other cable modem systems. So with the increase in the order of modulation in OFDM, the bit error rate (BER) tends to increase. So at this instant LDPC codes which belong to Forward Error correction codes takes an edge which improves the BER system performance at a larger pace compared to other codes. The LDPC codes pattern closely matches with capacity of existing channels and efficient decoding linear time complex signals. Minimum distance of LDPC code is high and the power efficiency is directly proportional to the code length. Here it has been shown that LDPC coded OFDM over the AWGN channel under the soft decision decoding scheme ensures minimum bit error rate. It directly enhances the transmitter power gain. So in this paper different M-ary QAM OFDM schemes are compared and analyzed under hard coding and soft coding schemes to display the power efficiency. Keywords: Components: Orthogonal frequency divison multiplexing (OFDM), Lower density parity check code )(LDPC ), Binary Input Additive White Gaussian Noise (BIAWGN) channel, Aposteriori probability (APP), block codes, hard decision, soft decision, Inter symbol interference (ISI) 1. Introduction Orthogonal frequency division multiplexing (OFDM) transmission scheme is a type of a multichannel system transmission scheme which employs multiple subcarriers. It
464 Kumar Raushan Ratnesh et al gives an effective and low complexity mode of eliminating inter symbol interference for transmission over the frequency selective fading channels. The subcarrier frequencies in OFDM are selected in such a manner that the signals are orthogonal over one OFDM symbol period [2].Recently channel coding has been employed extensively in most digital transmission schemes to maneuver the bit error rate during communication. The channel coding has found its place in many optical communication links as it has edge over line impairment which includes amplified spontaneous emission, fiber aging losses, pulse distortion and cross talks. Channel coding is not only required for systems which require error detection but also in systems which require high code gain. 2. Overview of LDPC Code Low Density parity check code (LDPC) is an error correcting code employed to reduce the error in the noisy communication channel and probability of losing the information. This probability can be reduced to a desired value so that the data rate transmission can be as close to Shannon s limit [1]. A linear code has a parity check matrix representation and a bipartite graph but not all linear codes have a sparse representation. A n m is considered as sparse if the number of 1 s in any of the row which is the row weight W and the number of 1 s in any of the column which is column weight W is much less than the matrix dimension (W m, W n).the code represented by a sparse parity-check matrix is known as Low density parity-check code (LDPC). The sparse property of LDPC are the reason of its algorithm advantages. A LDPC is regular if W has a constant value for all column and W is constant for all the rows and W = W. A LDPC code which is not regular is irregular. 2.2 Irregular LDPC codes For an irregular low density parity check code the degree of all set of nodes are considered in accordance to some distribution. The first step involved is describing the desired numbers of column for each weight and desired numbers of rows for each weight. Next step involves a Construction method which includes an algorithm for determining edges between the vertices in such a way that the constraints are satisfied. The edges are completely random in nature. The number of 1 s in an irregular LDPC code matrix is determined by the equation M ( w i) = N( u i) (1) Where M denotes the number of parity check constraints, N is the length of code, w and u denotes the row and the column degree distribution respectively. Code word constraints is given by [HC ] = [0] (2) Where each row of H corresponds to parity check equation and every column of H corresponds to a bit in a code word. The (j,i) th value of H is equal to 1, if the i code word bit is substituted in the j parity check equation. There can be more than one
Power Efficiency of LDPC Codes under Hard and Soft Decision QAM Modulated 465 parity check matrices for a particular code since there can be more than one parity check matrices suitable and satisfying the given above constraint. But the number of rows of the two parity check matrices of the same code may not be the same. If for a H matrix there is no mathematical relationship among the row, it is denoted as linearly independent rows else they have dependent rows. So the number of parity check constraints is inversely proportional to the number of satisfied code words, provided there is a linear independent relationship between the parity check equations. The LDPC codes can be differentiated from block codes on the basis of sparseness of the parity check matrix. H is an essential parameter for an iterative decoding complexity which increases only linearly with the length of the code. But to get a sparse H matrix for an existing code is not possible. Whereas in LDPC codes, first parity check matrix is designed and then encoded with the suitable existing encoding algorithm which is then decoded with iteration using the graphical representation of the corresponding parity check matrix with the help of any iterative algorithm. On the other hand classical block codes are decoded with the help of Maximum Likelihood decoding algorithm. 3. Types of LDPC Algorithm 3.1 Hard-decision Algorithm It is also known as Bit-flipping algorithm. For C which represents each bit, the checks which are influenced by that particular bit are calculated first. If the number of nonzero checks is more than some threshold value, then that bit is decided to be incorrect and the bit is corrected by flipping its value [1]. With the help of this scheme, more than one error can be corrected. 3.1 Soft-decision Algorithm If a code word with N number of bits is transmitted then the aposteriori probability (APP) is used as the probability for the given bit in the transmitted code word which may be either equal to 0 or 1 which becomes the channel output [4] for the given bit. The APP ratio function or the likelihood ratio (LR) is given by lc = / / The log-app ration or the log likelihood ratio (LLR) is given by LC = log / / (3) (4) The output from band pass de-mapper is Lc where the value of C is the j bit of the transmitted code word C. The three prominent parameters in this algorithm are Lr, Lq andlq. The initial value of Lq is equal to Lc Lr = 2 a tanh ( tanh (.5L(q ))) (5) Lq = Lc + LQ = Lc + Lr Lr (6) (7)
466 Kumar Raushan Ratnesh et al 4. Analysis and Comparison of LDPC Coded OFDM The system is framed by using a Low density parity check encoder with a code rate of 1 2. The LDPC code used here is an irregular LDPC code and the parity check matrix is (32400, 64800). The system is simulated for 16, 32 and 64 QAM with OFDM scheme. The LDPC decoder used here is of both hard decision and soft decision. The channel is referred as BIAWGN channel since the information is binary in nature. The (figure 2) portrays the BER performance of hard decision mode of 16, 32 and 64 QAM modulated irregular LDPC coded OFDM [5]. Number of iteration used in the decoder is 50 and the convergence of value is more if the number of iteration is increased. It can be easily inferred from figure.2 and figure.3 that the LDPC coded 16 QAM and modulated with OFDM gives an error in the range of 10 with a 50 iteration decoder in hard decoding. While on the other hand in figure.3 with soft decision schema the error reduces to the range of 10 in just 15 iteration and around 10 in 20 iteration of decoder. So soft decision decoding reduces the time of actual computation giving fast result in real time applications and also improves the error performance and thus showing the power efficiency of the LDPC codes in soft decision compared to hard decision. Figure 1: Figure shows the bit error plot of the 64 QAM, 32 QAM and 16 QAM 0modulation technique with Orthogonal frequency division multiplexing under normal convolutional coding where the minimum bit error is less than 10 and no major change in the performance of BER takes place even after using higher order modulation. This figure act as a bench mark for the rest of the simulations where it is tried to reduce the bit error rate within a given range of SNR.
Power Efficiency of LDPC Codes under Hard and Soft Decision QAM Modulated 467 Figure 2: Figure shows the bit error plot of the 64 QAM, 32 QAM and 16 QAM modulation technique with Orthogonal frequency division multiplexing which is coded using Irregular Low density parity code under HARD decision schema. The minimum error achieved up to SNR range of 16 db is 10 with the hard decision coding and iteration value of 50. Figure 3: Figure shows the bit error plot of the 16 QAM, 32 QAM and 16 QAM modulation technique with orthogonal frequency division multiplexing which is coded using Low density parity code under Soft decision schema. The minimum error achieved is around 10 up to 16 db SNR with the SOFT decision coding and iteration value of only 20.
468 Kumar Raushan Ratnesh et al 5. Conclusion Soft-decision decoding with long Irregular LDPC code of the higher order QAM modulation technique gives an enhanced BER performance of the system when compared to the hard-decision decoded higher order QAM modulation. The error in case of hard decision decoding go as high as to 10 while for the soft decision it can be reduced to a range of 10. Soft decision decoding also proves useful for real time application where it takes only 20 iteration to reduce the error to a range of 10 and gives a better coding gain. So it finds a good application in higher information carrying capacity optical carriers. With the advantage of OFDM to reduce Inter symbol interference this combination provides a huge advantage in case of chromatic dispersion and polarization dispersion in optical communication. References [1] R.Gallager, (January 1962), Low density parity-check codes, IRE Trans, Information Theory, pp. 21-28. [2] D.Divsalar, H. Jin and R. McEliece September (1998), Coding theorems for turbo-like codes, Proc. 36th Annual Allerton Conf. on Comm., Control and Conputingce, pp. 201-210 [3] Marjan Karkooti, Predrag Radosavljevic and Joseph R. Cavallaro (September 01, 2006), Configurable, High Throughput, Irregular LDPC Decoder Irregular LDPC Decoder Architecture: Tradeoff Analysis and Implementation, Rice Digital Scholarship Archive. [4] Zongjie Tu, Shiyong Zhang (2007), Overview of LDPC Codes, Computer and Information Technology, 2007. CIT 2007. 7th IEEE International Conference, pp. 469 474 [5] Eldomabiala, Mathias,coinchon, KarimMaouche,, (December 1999 ), Study of OFDM Modulation, Ierucom Institute.