  This paper revisits polar code design for a binary-input additive white
Gaussian noise (BI-AWGN) channel when successive cancellation (SC) decoding is
applied at the receiver. We focus on the reciprocal channel approximation
(RCA), which is often adopted in the design of low-density parity-check (LDPC)
codes. In order to apply RCA to polar code design for various codeword lengths,
we derive rigorous closed-form approximations that are valid over a wide range
of SNR over an AWGN channel, for both the mutual information of BPSK signaling
and the corresponding reciprocal channel mapping. As a result, the
computational complexity required for evaluating channel polarization is thus
equivalent to that based on the popular Gaussian approximation (GA) approach.
Simulation results show that the proposed polar code design based on RCA
outperforms those based on GA as well as the so-called improved GA (IGA)
approach, especially as the codeword length is increased. Furthermore, the
RCA-based design yields a better block error rate (BLER) estimate compared to
GA-based approaches.
