# All-Integer Quantization for Low-Complexity Min-Sum Successive Cancellation Polar Decoder
## 5 Experimental Results
### Decoding Performance Under Integer Quantization  
This section examines the impact of integer calculations on QMS-SC decoding (7) and quantized SC (QSC) decoding (3) in relation to BER and FER performance, by varying the quantization level from 3-bit to 5-bit. The BER performance is presented because the non-systematic SC decoding algorithm is sensitive to error propagation, as shown in [27]. Consequently, the numerical results, which are quantized to integer values, within the SC decoding calculation may lead to error propagation. The results are presented in Figure 3 for polar codes with \(N=256\) and \(R=0.5\), and in Figure 4 for \(N=4096\) and \(R=0.83\). For short-length polar codes, the performance improves with increasing quantization levels in both BER and FER metrics. Notably, QMS-SC decoding outperforms QSC decoding, demonstrating performance gains of 0.84 dB, 1.68 dB, and 2.45 dB for 3-bit, 4-bit, and 5-bit quantization levels, respectively. For polar codes with a higher length of 4096 and a rate of 0.83, the min-sum SC decoding also outperforms the conventional SC decoding in the integer-based quantization domain. To deeply evaluate integer-based decoding algorithms, it is necessary to compare the LLR outputs of QMS-SC and QSC decoding with those of floating-point SC decoding under identical conditions, including the same transmitted information and noise levels. Unfortunately, the original SC decoding output cannot be directly used as the exact value because (4) may occasionally produce excessively large outputs. In the sequential calculations of the SC decoding algorithm, such large values can serve as inputs to the transcendental Equation (3), potentially resulting in infinite outputs. Ensuring an accurate comparison, the output of (4) is clamped at \(\pm 2^{q-1}-1\). To quantify the difference in LLRs, let \(\epsilon\) represent the difference between the approximate LLR outputs and the exact LLR outputs (infinite-precision values), as defined by  
\[\epsilon=\frac{1}{K}\sum_{i\in\mathcal{A}}\left(\hat{L}_{i}-L_{i}\right)^{2}, \tag{9}\]  
where quantized LLR, denoted as \(\hat{L}_{i}\), at the decoder output for the index set \(\mathcal{A}\) are compared with the exact LLR, denoted by \(L_{i}\). To illustrate the error in the LLR outputs of the decoding methods with full-integer quantization compared to infinite-precision values, we consider \(E_{\text{b}}/N_{0}\) that achieves a BER of \(10^{-5}\) for block lengths of 256 bits and 4096 bits. The quantization level varies from 3-bit to 5-bit. The results are obtained by (9) and presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Decoding** & **Rate** & **Length** & \multicolumn{3}{c}{**Average \(\epsilon\)**} \\  & & & **3-Bit** & **4-Bit** & **5-Bit** \\ \hline \multirow{2}{*}{Quantized MS-SC} & 0.5 & 256 & \(\approx\) 0.84 & \(\approx\) 0.75 & \(\approx\) 0.72 \\ \cline{2-5}  & 0.83 & 4096 & \(\approx\) 2.02 & \(\approx\) 1.29 & \(\approx\) 1.28 \\ \multirow{2}{*}{Quantized SC} & 0.5 & 256 & \(\approx\) 1.26 & \(\approx\) 3.77 & \(\approx\) 15.7 \\ \multirow{2}{*}{Quantized SC} & 0.83 & 4096 & \(\approx\) 2.06 & \(\approx\) 1.54 & \(\approx\) 3.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error measurement between integer LLR outputs relative to exact LLRs of SC decoding. Figure 4: Performance of SC decoder (polar codes, \(N=4096\), rate = 0.33) with integer calculation on AWGN channels: (**a**) BER, and (**b**) FER. Figure 3: Performance of SC decoder (polar codes, \(N=256\), rate = 0.5) with integer calculation on AWGN channels: (**a**) BER, and (**b**) FER. Considering the performance metrics, the conventional SC decoding algorithm is highly sensitive to integer computation, resulting in a poor decoding performance for polar codes. Moreover, the significant discrepancy in LLRs highlights the inaccuracies in the decoding calculations, which result in an inferior performance. In contrast, the min-sum SC decoding algorithm is well suited to integer calculations.