A New Perspective on Polar Codes: Analysis of Bit Error Probability

This paper conducts the first analysis of the bit error probability
$P_{b}$
of polar codes under successive cancellation (SC) decoding, with a focus on information bits. We introduce the concept of component bit error probability
$P_{b}(i)$
, which corresponds to the
$i$
-th stage of SC decoding, and
$P_{b}$
is computed as the sum of all
$P_{b}(i)$
over the information set. To facilitate the evaluation of
$P_{b}(i)$
and
$P_{b}$
, we propose an analytical approach that leverages the algebraic structure of a particular type of polar codes termed polar subcodes. This approach allows us to derive closed-form approximations for the upper bounds on both
$P_{b}(i)$
and
$P_{b}$
. Our analysis applies to both systematic and non-systematic polar codes, where we further investigate the homogeneity and introduce the input-output weight distribution of polar subcodes to obtain more concise expressions for the proposed approximate upper bounds. Experimental results are also presented to validate the effectiveness of these approximations.