## 3 Polar Coded SDF Cooperation  
### Polar Encoding and Decoding  
A polar code is specified completely by a parameter pair \((N,K,f)\), where \(N\) is the code length, \(K\) is the number of information bits per code word, and \(f\) is a set of \(N-K\) integer indices, which are called frozen bit locations, from \(1,2,\ldots,N\). Let \(W\) be a binary discrete memoryless channel (DMC) with an input alphabet \(x=0,1\), output alphabet \(y\), and transition probability \(W(y|x)\). The channel mutual information with equiprobable inputs, or symmetric capacity, is then defined by  
\[I(W)=\sum_{y\in Y}\sum_{x\in X}\frac{1}{2}W(y|x)\log\frac{W(y|x)}{\frac{1}{2} W(y|0)+\frac{1}{2}W(y|1)}, \tag{5}\]  
with Barth's parameters:  
\[Z(W)=\sum_{y\in Y}\sqrt{W(y|0)W(y|1)}. \tag{6}\]  
With a code block of length \(N\), the channel input and output sequences are denoted by \(x_{1}^{N}\) and \(y_{1}^{N}\), respectively, with the corresponding vector channel \(W^{N}(y_{1}^{N}|x_{1}^{N})\). Then, the procedure of polar coding is summarized as follows. * **Channel polarization** Considering the matrix \(\mathbf{G_{2}}=\left[ \begin{array}{cc}1&0\\ 1&1\end{array} \right]\), let \(\mathbf{G_{N}}=\mathbf{G}_{2}^{\otimes n}\) be the \(n\)-th Kronecker power of \(\mathbf{G_{2}}\), where \(n=log_{2}N\). Input bits are denoted by \(u_{1}^{N}\in\{0,1\}^{N}\). The vector channel is \(W^{N}(y_{1}^{N}|u_{1}^{N})=W^{N}(y_{1}^{N}|u_{1}^{N}\mathbf{G_{N}})\). From \(W^{N}(y_{1}^{N}|u_{1}^{N})\), a successive cancellation (SC) decoder implicitly defines the bit channel (with \(i\in[N]\)) as follows: \[W_{N}^{(i)}(y_{1}^{N},u_{1}^{i-1}|u_{i})=\sum_{u_{1+1}^{N}}W_{N}(y_{1}^{N}|u_{ 1}^{N}).\] (7)  
The channel polarization theorem [10] states that \(I(W_{N}^{(i)})\) converges to either \"0\" or \"1\" as \(N\) tends to infinity and the fraction of the noiseless channel tends to \(I(W_{N}^{(i)})\). Then, polar codes with rate \(R=K/N\) are constructed by selecting the \(N-K\) indexes with the smallest\(I(W_{N}^{(i)})\) or the largest \(Z(W_{N}^{(i)})\) for \(i\in[N]\). These are called the frozen set, denoted as \(f\), and the corresponding input bits are set to zero. The complementary unfrozen set of \(K\) indexes, e.g., \(f^{c}\), correspond to information bits. * **Polar encoding** For a polar code \((N,K,f)\), a code word is generated as follows: \[x_{1}^{N}=u_{f^{c}}\mathbf{G}_{N}(f^{c})\oplus u_{f}\mathbf{G}_{N}(f),\] (8) where \(u_{f^{c}}\) is a \(K\)-length information bit vector, \(u_{f}\) is the frozen information of zero, and \(\mathbf{G}_{N}\) is the generator matrix of the polar code. An efficient implementation with typical computational complexity \(O(NlogN)\) as encoded by (7) is illustrated in Figure 2.
* **Polar successive cancellation (SC) decoding** In SC decoding, the information bits are estimated as \[\hat{u}_{i}=\arg\max_{u_{i}\in\{0,1\}}W_{N}^{(i)}(y_{1}^{N},u_{1}^{i-1}|u_{i}),i\in f^{c}.\] (9) SC decoding follows essentially the same diagram as in Figure 2 using a decoding operation that resembles one iteration of the existing belief propagation decoding. With a code length \(N\), the complexity of the SC decoding is \(O(NlogN)\)[3]. * **Construction of polar codes in fading channels** The polarization phenomenon appears in arbitrary discrete memoryless channels as well [31]. The transmission in a general fading channel is expressed as \[y=h\cdot x+n,\] (10) where \(h\) is the channel coefficient of a fading channel, and fits a normal distribution \(N(0,1)\). Here, \(n\) is the AWGN with zero mean and variance \(\sigma^{2}\). Due to limitations of the channel estimation, there is a limit of the channel values below which the estimation is unreliable; this is represented as a variable \(\alpha\). For a given value of \(\alpha\), a percentage is calculated as \[p=\Pr\{|h|\leq\alpha\},\] (11) where \(N\times p\) unreliable observations are performed at the receiver. These observations in return cause poor BER performance.