SC Successive Cancellation

Successive cancellation (SC) is a decoding algorithm commonly used in the context of polar codes, a class of error-correcting codes introduced by Erdal Arikan in 2009. The SC decoding algorithm is designed to efficiently decode polar codes and achieve near-optimal error-correction performance.

To understand SC decoding, let's first briefly discuss polar codes. Polar codes are constructed by transforming a given binary input sequence into a longer code sequence through a process called channel polarization. This transformation converts a set of noisy channels into a set of "good" and "bad" channels, where the good channels can reliably transmit information while the bad channels are highly susceptible to errors. By exploiting this polarization effect, polar codes can achieve the capacity of the underlying channels, making them a powerful tool for reliable communication.

Now, let's dive into the details of the SC decoding algorithm. The goal of the decoder is to recover the original binary input sequence from the received noisy codeword. The SC algorithm achieves this by recursively decoding the bits of the codeword one by one, starting from the most reliable bit and proceeding towards the least reliable bit.

The decoding process begins by considering the entire codeword as a single block, assuming no errors have occurred. This block is then split into two smaller blocks of equal size. The decoder treats one of these blocks as an estimate of the input sequence and the other as the frozen bits whose values are known to be transmitted without errors. The frozen bits are typically assigned with reliable information based on the channel polarization transformation.

Next, the decoder repeats the same splitting process for each of the smaller blocks. It estimates the bits in one block while treating the other block as frozen. This recursive splitting and decoding process continues until individual bits are reached.

When decoding a specific bit, the SC algorithm employs a technique called successive cancellation. This cancellation process relies on the fact that the frozen bits have already been correctly decoded and can be used to estimate the channel outputs associated with the remaining unknown bits.

To perform successive cancellation, the decoder calculates the likelihood ratios of the received channel outputs for each unknown bit in the current block. These likelihood ratios represent the likelihood of a bit being 0 or 1 given the received channel output. The decoder then applies these likelihood ratios to cancel out the effect of the estimated bits on the channel outputs, revealing more reliable estimates for the remaining bits.

After performing successive cancellation, the decoder splits the current block into two smaller blocks and repeats the process for each sub-block. This continues until the decoding process reaches the individual bits of the codeword.

Once all the bits are decoded, the decoder outputs the estimated binary input sequence. The SC algorithm is known to achieve near-optimal decoding performance for polar codes, making it a widely used decoding technique in practical applications.

In summary, successive cancellation (SC) is a decoding algorithm used for polar codes. It recursively splits the codeword into smaller blocks, estimates the bits in each block while utilizing the known frozen bits, and applies successive cancellation to refine the estimates of the remaining bits. This iterative process allows SC decoding to efficiently recover the original binary input sequence from the received noisy codeword.