  Polar codes are newly discovered capacity-achieving codes, which have
attracted lots of research efforts. Polar codes can be efficiently decoded by
the low-complexity successive cancelation (SC) algorithm and the SC list (SCL)
decoding algorithm. The belief propagation (BP) decoding algorithm not only is
an alternative to the SC and SCL decoders, but also provides soft outputs that
are necessary for joint detection and decoding. Both the BP decoder and the
soft cancelation (SCAN) decoder were proposed for polar codes to output soft
information about the coded bits. In this paper, first a belief propagation
decoding algorithm, called reduced complexity soft cancelation (RCSC) decoding
algorithm, is proposed. Let $N$ denote the block length. Our RCSC decoding
algorithm needs to store only $5N-3$ log-likelihood ratios (LLRs),
significantly less than $4N-2+\frac{N\log_2N}{2}$ and $N(\log_2N+1)$ LLRs
needed by the BP and SCAN decoders, respectively, when $N\geqslant 64$.
Besides, compared to the SCAN decoding algorithm, our RCSC decoding algorithm
eliminates unnecessary additions over the real field. Then the simplified SC
(SSC) principle is applied to our RCSC decoding algorithm, and the resulting
SSC-aided RCSC (S-RCSC) decoding algorithm further reduces the computational
complexity. Finally, based on the S-RCSC decoding algorithm, we propose a
corresponding memory efficient decoder architecture, which has better error
performance than existing architectures. Besides, our decoder architecture
consumes less energy on updating LLRs.
