Polar Codes with Successive Cancellation Decoding in AWGN Channels

Polar codes are a coding scheme that employs successive cancellation decoding procedures in AWGN (Additive White Gaussian Noise) channels. This coding methodology is widely implemented in communication systems to enhance channel transmission reliability and efficiency. The core algorithm utilizes channel polarization transformation through recursive matrix operations (typically using Kronecker products with the base matrix F = [1,0;1,1]), where information bits are strategically allocated to the most reliable virtual channels. The successive cancellation decoder operates through recursive likelihood ratio calculations, sequentially estimating each bit from index 1 to N using a tree-structured computation that reduces complexity to O(N log N). This coding approach effectively corrects channel errors and achieves high-quality data transmission through its inherent channel capacity-achieving properties. The implementation involves key functions such as code construction (using Gaussian approximation or Bhattacharyya parameter methods), bit-reversal permutation, and recursive log-likelihood ratio propagation. With straightforward design principles and relatively low implementation complexity, polar codes find extensive applications in wireless communications, satellite systems, mobile networks, and 5G NR control channels. Their mathematical elegance and proven optimality make them particularly favored by researchers and communication engineers for modern error correction implementations.