Hyperbolic tangent function avoided for encoded pilot low density parity check decoding

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