  Systematic polar codes are shown to outperform non-systematic polar codes in
terms of the bit-error-rate (BER) performance. However theoretically the
mechanism behind the better performance of systematic polar codes is not yet
clear. In this paper, we set the theoretical framework to analyze the
performance of systematic polar codes. The exact evaluation of the BER of
systematic polar codes conditioned on the BER of non-systematic polar codes
involves in $2^{NR}$ terms where $N$ is the code block length and $R$ is the
code rate, resulting in a prohibitive number of computations for large block
lengths. By analyzing the polar code construction and the
successive-cancellation (SC) decoding process, we use a statistical model to
quantify the advantage of systematic polar codes over non-systematic polar
codes, so called the systematic gain in this paper. A composite model is
proposed to approximate the dominant error cases in the SC decoding process.
This composite model divides the errors into independent regions and coupled
regions, controlled by a coupling coefficient. Based on this model, the
systematic gain can be conveniently calculated. Numerical simulations are
provided in the paper showing very close approximations of the proposed model
in quantifying the systematic gain.
