# Improvement of Fast Simplified Successive-Cancellation Decoder for Polar Codes
## 3 Proposed Method
### Performance Analysis  
The Fast-ISSS algorithm includes the parallel process for the constituent code and bit process for the leaf node, and the parallel process is similar to the multi-bit parallel process with the Symbol-Decision SC Algorithm [13]. During decoding, the constituent code length of the Fast-ISSS algorithm with parallel process is not fixed; however, the symbol length with the parallel process of the Symbol-Decision SC Algorithm is fixed. For the convenience of analysis, two assumptions are made: (1) A length-\(N\) polar code is unevenly divided into \(K\) segments, and the length \(M_{i}\) of each segment can be changed, where \(i=0,1,...,K-1\), and \(M_{i}\) is a power of 2 (i.e., \(M_{i}\) can be 1). The bit-decision SC and Fast-ISSS algorithms are used to decode each segment. Both decoding schedules are the same from a segment to the next segment. (2) During the local decoding, the process length \(M_{i}\) of the two algorithms is not fixed, and are changed simultaneously; further, the length \(M_{i}\) for both algorithms is same. In terms of the FER performance, we have the following. **Proposition 1**. : _If the data sequence is independent and equally likely, for an \(\left(N,K\right)\) polar code over the given channel, the FER of the Fast-ISSS algorithm \(\mathrm{P_{M+B}}\left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right)\) and the FER of the bit-decision SC algorithm \(\mathrm{P_{B}}\left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right)\) satisfy:_  
\[\mathrm{P_{M+B}}\left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right)=\mathrm{P_{B}} \left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right) \tag{7}\]  
Proof. : The FER of a polar code with the bit-decision SC algorithm is given in [14]. \[\mathrm{P_{B}}\left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right)=1-\prod_{i=0}^{ K-1}\left(1-p\left(C_{i}\right)\right) \tag{8}\]  
For the Fast-ISSS algorithm,  
\[\mathrm{P_{M+B}}\left(\hat{u}_{0}^{N-1}\
eq\ u_{0}^{N-1}\right)=1-\prod_{i=0} ^{K-1}\left(1-p\left(C_{i}^{\prime}\right)\right) \tag{9}\]  
Let \(p\left(C_{0}\right)=\mathrm{P_{SC}}\left(\hat{u}_{0}^{M_{0}-1}\
eq\ u_{0}^{M_{ 0}-1}\right)\) and \(p\left(C_{0}^{\prime}\right)=\mathrm{P_{Fast-ISS}}\left(\hat{u}_{0}^{M_{0}-1} \
eq\ u_{0}^{M_{0}-1}\right)\) represent the probability that the first segment is erroneously decoded by the SC and M-bit Fast-ISSS algorithms, respectively. For \(i=1,2,...,K-1\), let \(p\left(C_{i}\right)=\mathrm{P_{SC}}\left(\hat{u}_{M_{i}}^{M_{i}+M_{i+1}-1} \
eq\ u_{M_{i}}^{M_{i}+M_{i+1}-1}\middle|\hat{u}_{0}^{M_{i}-1}=u_{0}^{M_{i}-1}\right)\) and \(p\left(C_{i}^{\prime}\right)=\mathrm{P_{Fast-ISS}}\left(\hat{u}_{M_{i}}^{M_{i}+ M_{i+1}-1}\
eq\ u_{M_{i}}^{M_{i}+M_{i+1}-1}\middle|\hat{u}_{0}^{M_{i}-1}=u_{0}^{M_{i}-1}\right)\) denote the segment error probabilities of the \(i\)th segment by the SC and M-bit Fast-ISSS decoding algorithms, provided that all previous segments are correctly decoded. We need to analyze \(\mathrm{P_{Fast-ISS}}\left(\hat{u}_{0}^{M_{0}-1}\
eq\ u_{0}^{M_{0}-1}\right)\) of the Fast-ISSS algorithm compared with \(\mathrm{P_{SC}}\left(\hat{u}_{0}^{M_{0}-1}\
eq\ u_{0}^{M_{0}-1}\right)\) of the bit-decision SC decoding on the first segment in possible five cases: (a) For an R-0 node, it does not affect performance. (b) For an R-1 node, it shows that for any R-1 node \(v\), the calculated \(\beta_{v}\) of both methods has the same results, and that the decisions at leave nodes with index \(i\subset l_{v}\) made by both methods also agree. These results are proved in [4]. (c) For an REP or SPC node, because the special positions of the frozen bits, the constituent code performs low complexity ML decoding in parallel based on hard decision, that is, the performance is no worse than the original SC decoding [5]. (d) If an R-other node \(v\) satisfies the FC, the output of the corresponding node \(v\) can be computed by hard decision, i.e., \(\beta_{v}=\mathrm{h}(\alpha_{v})\), \(\hat{u}=\beta_{v}G_{m-d_{v}}\), which is consistent with the node in the same case as the SSC decoding.