Chapter 8: BICM with Iterative Decoding (BICM-ID)

Learning Objectives

• Explain the BICM-ID architecture: feed the decoder's extrinsic soft output back as a priori information to the demapper, iterate, and close the decoding gap left open by one-shot BICM
• State and apply the Gaussian-LLR assumption that turns the demapper and decoder into scalar maps between the a-priori mutual information $I_A$ and the extrinsic mutual information $I_E$, and recognise both its power and its limits
• Construct the EXIT chart (ten Brink, 2001): draw the demapper curve $I_E = T_{\mathrm{dem}}(I_A, \mathrm{SNR})$ and the inverted decoder curve, read off the iterative trajectory as a staircase, and identify the convergence tunnel
• Prove the convergence theorem: BICM-ID reaches $(I_A, I_E) = (1, 1)$ iff the demapper curve lies strictly above the inverted decoder curve on $[0, 1)$; locate the convergence threshold SNR as the SNR at which the tunnel just closes
• Contrast Gray, set-partition (SP) and anti-Gray labelings under iteration: SP has a steep demapper EXIT curve and a better $I_E(I_A = 1)$ endpoint, so SP-BICM-ID beats Gray-BICM-ID in the low-BER floor at fixed code rate
• Design an EXIT-matched LDPC degree profile: given the demapper curve at the target SNR, optimise the variable-node degree distribution so that the inverted decoder curve lies JUST below the demapper curve, which maximises the achievable code rate
• Locate BICM-ID in practice: DVB-S2X iterative-BICM modes, 5G LDPC with iterative demapper/decoder exchange, and the gap-to-CM-capacity that remains in realistic receivers