In this paper, we propose a new family of rate-compatible regular low-density parity-check (LDPC) convolutional codes. The construction is based on graph extension, i.e., the codes of lower rates are generated by successively extending the graph of the base code with the highest rate. Theoretically, the proposed rate-compatible family can cover all the rational rates from 0 to 1. In addition, the regularity of degree distributions simplifies the code optimization. We prove analytically that all the LDPC convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel (BEC). The analysis is extended to the general binary memoryless symmetric channel, for which a capacity-approaching performance can be achieved. Analytical thresholds and simulation results for finite check and variable node degrees are provided for both BECs and binary-input additive white Gaussian noise channels. The results confirm that the decoding thresholds of the rate-compatible codes approach the corresponding Shannon limits over both channels.
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