Summary

Chapter 12 Summary: Channel Coding

Key Points

• 1.

  Block codes map $k$ information bits to $n$ coded bits using a generator matrix $\mathbf{G}$. The minimum Hamming distance $d_{\min}$ determines the error-correcting capability $t = \lfloor (d_{\min}-1)/2 \rfloor$. Soft-decision decoding provides 2-3 dB gain over hard-decision decoding.

• 2.

  Convolutional codes encode a continuous bit stream using shift registers with constraint length $K$. The Viterbi algorithm provides optimal ML sequence decoding with complexity $O(2^{K-1})$ per bit, while the BCJR algorithm provides soft-output MAP decoding essential for iterative receivers.

• 3.

  Turbo codes (parallel concatenated convolutional codes with iterative decoding) and LDPC codes (sparse parity-check matrices with belief-propagation decoding) both achieve performance within 1 dB of the Shannon limit. EXIT charts predict the convergence threshold of iterative decoders. 5G NR uses LDPC for data and polar codes for control.

• 4.

  Polar codes exploit channel polarization to achieve capacity with explicit construction. CRC-aided successive cancellation list (CA-SCL) decoding overcomes the poor finite-length performance of basic SC decoding and is used in 5G NR control channels.

• 5.

  BICM separates binary coding from higher-order modulation via a bit interleaver, achieving near-optimal performance with Gray mapping (capacity gap $< 0.1$ dB). This pragmatic approach is universal in modern wireless standards.

• 6.

  HARQ with incremental redundancy adapts the effective code rate to channel conditions through retransmissions. On fading channels, interleaving converts burst errors to random errors, enabling channel codes to provide both coding gain and diversity gain (diversity order equal to $d_{\min}$).