On the encoding complexity of systematic polar codes

Systematic polar codes are proposed by Arikan and are shown to have better BER performance than non-systematic polar codes. From a recursive decomposition of the generator matrix of polar codes, Arikan showed that the encoding complexity of systematic polar codes is also O(N log N) where N is the code block length. But the recursive process involves some additional calculations in transforming the problem instances back and forth. In this paper, by using the sparsity property of the generator matrix, we propose an encoding process which has the same complexity as non-systematic polar codes in the presence of an additional memory array. Without the additional memory elements, the number of additions of the proposed encoding process increases compared with non-systematic polar codes. We also provide an analysis to quantify this additional increase of the complexity.