Hyperbolic Tangent Function Avoided for Encoded Pilot Low Density Parity Check Decoding Ashutosh Goyal and Hyuck M. Kwon, Wichita State University, Electrical and Computer Engineering, Wichita, KS 67260, USA Email: {axgoyal, hyuck.kwon@wichita.edu} Abstract — Very recently the same authors of this paper proposed an encoded pilot system where pilot bits are encoded with the message bits using a systematic code, e.g., a systematic low density parity check (LDPC) code. The authors claimed that the encoded pilot system can have significantly lower complexity and better energy efficiency at the same bandwidth efficiency or even higher bandwidth efficiency over the no pilot systems or the conventional time multiplexed pilot systems. The pilot bits have not been encoded with message bits but multiplexed after the encoder and de-multiplexed before decoding at the conventional receiver. The last case is called a deleted case because its corresponding parity check matrix is obtained by deleting the corresponding columns of the original parity check matrix. This paper finds a difficulty to use the well known sum-product LDPC decoding algorithm for the deleted case because hyperbolic tangent function causes overflow and underflow. In this paper the sum-product algorithm is modified. And this paper takes a well known Hamming code of (3,7) parity check matrix H under additive white Gaussian noise environment (AWGN) and demonstrates that the encoded pilot system is superior to the other systems, using the modified algorithm. I. INTRODUCTION A typical low density parity check code (LDPC) decoding is based on the log-likelihood ratio calculations [1] which are typically written in a hyperbolic tangent (tanh) function. But, the tanh(x) is saturated at ±1 even if |x| is slightly larger than 3. So, the accuracy of numerical calculation for tanh(x) function is low and causing a numerical problem. For example, the values of tanh(x) at the pilot bit nodes (PBN) is set to ±1, regardless of the received values at the PBNs to provide the highest reliability. But the value of tanh(x) at a non-pilot BN could also be ±1 if the absolute value of x is larger than 3. This causes a difficulty to distinguish the highest reliability at the PBNs from the saturated reliability of tanh(x) values at BNs. In this paper the sum-product algorithm is modified to avoid such overflow and underflow problem for the computation of tanh function. Section II describes the system model. Section III presents three LDPC systems, Information Case, Encoded pilot case (proposed scheme) and Deleted Case. Section IV shows simulation results. And Section V concludes this paper. II. SYSTEM MODEL This system model describes only a single user single transmit and single receive antenna LDPC encoded system though it can be extended to a multiple input and multiple output antenna element system. We consider a discrete-time equivalent baseband model of an LDPC coded system under additive white Gaussian noise (AWGN) channel. The modulation is binary phase shift keying (BPSK). Fig. 1 represents the scheme given by Valenti et al [2], in this scheme the pilots are added after turbo encoding and interleaving and removed at the receiver before decoding. Fig. 2 illustrates a proposed transmission scheme, message bits m are encoded using LDPC encoder and encoded data X t is modulated using BPSK modulation and sent to the transmitter. In the proposed encoded pilot scheme the message part m contains information data bits as well as pilot bits, i.e. m=m b +m p where m b denotes useful message bits, m p pilot bits, and + the multiplexing. In this study to analyze the effect of pilots on decoder performance, it is assumed that the channel is perfectly known, i.e. no channel estimation is required for coherent demodulation and detection. For fading channel, the same authors presented corresponding results in [3]. So, the channel is assumed as AWGN, therefore the received vector Y t can be given as: . ) 1 2 ( t t t n X Y + − = (1) where, t=1,……n, X t =0 or 1 the code bit and n is the number of columns of generator matrix G sys , i.e., the codeword length. The code structure of LDPC is (n,j,k), where n is the number of columns of parity check matrix H and j,k are the number of 1’s in column and row of H-matrix, respectively. The Parity check H-matrix, corresponding Generator G-matrix and Tanner Graph [3] for (3,7) Hamming matrix are given in Fig. 3(a), (b) and (c), respectively. MODIFIED LDPC DECODING ALGORITHM: The notations used in this paper are similar to the Gallager’s monograph [1]. It is well known by Bayes’ rule that when the transmitted bit information X t =0 or 1 are equiprobable then Maximum a posteriori (MAP) rule becomes the Maximum likelihood (ML) rule. For given channel conditions and AWGN environment ML rule and MAP rule can be given as: ( ) ( ) ( ) ( ) 0 1 0 1 = = = ≡ = = = t t t t t t t t X Y P X Y P ML Y X P Y X P MAP . (2) According to [1] the likelihood ratio under AWGN can be further given as: ( ) ( ) ( ) ( ) 2 2 2 2 2 ) 1 ( 2 2 ) 1 ( 2 2 1 2 1 1 1 0 1 n t n t y n y n t t t t t t t t e e n Y P n Y P X Y P X Y P σ σ πσ πσ + − − − = + − = + = = = = (3)