# Area-Efficient Early-Termination Technique for Belief-Propagation Polar Decoders
## 2 Belief-Propagation (BP) Decoder and Previous Early-Termination Techniques
In addition, References [17; 16] propose area-efficient windowing methods that check the changes on the bit channels within the consecutive iteration numbers, known as the window size. Algorithm 2  
Figure 3: Hardware architecture of threshold-based early termination method [15] of (8, 4) polar. Figure 2: Basic processing element of BP decoder. and Algorithm 3 depict information-BER-based [16] and frozen-BER-based [17] early-termination processes, respectively. According to Algorithm 2 and Algorithm 3, information-BER-based [16] and frozen-BER-based [17] early-termination processes calculate the changes within the information set \(A\) and the frozen set \(A^{c}\) during the window size \(W\). Although information-BER-based [16] and frozen-BER-based [17] early-terminations can save hardware resources significantly, they deteriorate the decoding throughput compared to the threshold-based method [15]. Figures 4 and 5 show the hardware architecture of information-BER-based [16] and frozen-BER-based [17] for (8, 4) polar codes, respectively. Since we assumed that the information set \(A\) is [4; 6; 7; 8] and the frozen set \(A^{c}\) is [1, 2, 3, 5], information-BER-based [16] and frozen-BER-based [17] require only LLR values of information set \(A\) and frozen set \(A^{c}\), respectively. In this paper, we propose a new early-termination method that introduces no throughput or error-correcting degradations with the smallest hardware resources. Figure 4: Hardware architecture of information-BER-based early-termination method [16] of (8, 4) polar codes. Figure 5: Hardware architecture of frozen-BER-based early termination method [17] of (8, 4) polar codes. ```
Initialization: If (\(s==n+1\)) then \(L^{t}_{i,n+1}=\) LLR (\(y_{i}\)) Else if (\(s==1\)) & (\(i\in A^{\prime}\)) then \(R^{t}_{i,1}=\infty\) Else \(L^{t}_{i,s}=R^{t}_{i,s}=0\) Decoding iteration: While\(t\leq T\) Update \(L^{t}_{i,s}\) and \(R^{t}_{i,s}\) based on (3) and (4) If \(\sum\limits_{i\in A^{\prime}=-W+1}^{t}\left(a^{2}_{i}\oplus a^{3-1}_{i}\right)==0\)then Stop decoding iteration \(t=t+1\)
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**Algorithm 2**Information-BER-based [16] early-termination technique  
```
Initialization: If (\(s==n+1\)) then \(L^{t}_{i,n+1}=\) LLR (\(y_{i}\)) Else if (\(s==1\)) & (\(i\in A^{\prime}\)) then \(R^{t}_{i,1}=\infty\) Else \(L^{t}_{i,s}=R^{t}_{i,s}=0\) Decoding iteration: While\(t\leq T\) Update \(L^{t}_{i,s}\) and \(R^{t}_{i,s}\) based on (3) and (4) \(F^{t}_{\text{hor}}=\frac{1}{|A|}\sum\limits_{i\in A^{\prime}}\left(a_{i}\oplus 0\right)\) If \(\sum\limits_{v=t-W+1}^{t}\left|F^{v}_{\text{hor}}-F^{v-1}_{\text{hor}}\right|==0\)then Stop decoding iteration \(t=t+1\)
```  
**Algorithm 3**Frozen-BER-based [17] early-termination technique  
```
Initialization: If (\(s==n+1\)) then \(L^{t}_{i,n+1}=\) LLR (\(y_{i}\)) Else if (\(s==1\)) & (\(i\in A^{\prime}\)) then \(R^{t}_{i,1}=\infty\) Else \(L^{t}_{i,s}=R^{t}_{i,s}=0\) Decoding iteration: While\(t\leq T\) Update \(L^{t}_{i,s}\) and \(R^{t}_{i,s}\) based on (3) and (4) If \(i\in A\otimes\left|\frac{L^{t}_{i,1}}{L^{t}_{i,1}}\right|\geq\beta\) (\(\beta\): threshold value) then Stop decoding iteration \(t=t+1\)
```  
**Algorithm 4**Proposed early-termination technique