  Polar codes are the first class of constructive channel codes achieving the
symmetric capacity of the binary-input discrete memoryless channels. But the
corresponding code length is limited to the power of two. In this paper, we
establish a systematic framework to design the rate-compatible punctured polar
(RCPP) codes with arbitrary code length. A new theoretic tool, called polar
spectra, is proposed to count the number of paths on the code tree with the
same number of zeros or ones respectively. Furthermore, a spectrum distance SD0
(SD1) and a joint spectrum distance (JSD) are presented as performance criteria
to optimize the puncturing tables. For the capacity-zero puncturing mode
(punctured bits are unknown to the decoder), we propose a quasi-uniform
puncturing algorithm, analyze the number of equivalent puncturings and prove
that this scheme can maximize SD1 and JSD. Similarly, for the capacity-one mode
(punctured bits are known to the decoder), we also devise a reversal
quasi-uniform puncturing scheme and prove that it has the maximum SD0 and JSD.
Both schemes have a universal puncturing table without any exhausted search.
These optimal RCPP codes outperform the performance of turbo codes in LTE
wireless communication systems.
