Prerequisites & Notation

Before You Begin

This chapter builds directly on belief propagation from Chapter 18 and on classical linear MMSE estimation from earlier FSI chapters. The turbo principle is the practical incarnation of message passing on loopy factor graphs: iterative exchange of extrinsic information between soft-input soft-output (SISO) blocks. Readers should be comfortable with the following.

Factor graphs and the sum-product algorithm(Review ch18)
Self-check: Can you write the variable-to-factor and factor-to-variable update rules and describe why BP is exact on trees?
Log-likelihood ratios and soft-input soft-output decoding(Review ch18)
Self-check: Can you express a posterior probability as a sigmoid of an LLR and explain the distinction between a-priori, a-posteriori, and extrinsic information?
Linear MMSE estimation
Self-check: Given $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ with known second-order statistics, can you write the LMMSE estimator and its MSE matrix?
Convolutional codes and the BCJR algorithm
Self-check: Can you describe the forward-backward recursion on a trellis and obtain bitwise a-posteriori LLRs?
ISI channel model and MMSE equalization
Self-check: Can you derive the block-form LMMSE equalizer for a finite-length ISI channel?
Mutual information between a binary input and an LLR
Self-check: Given a symmetric Gaussian LLR model, can you compute $I(X; L) = J(\sigma)$ ?

Notation for This Chapter

Symbols introduced and used in Chapter 19. The mutual-information axis variables $I_A$ and $I_E$ are the workhorses of EXIT analysis.

Symbol	Meaning	Introduced
$L_A$	A-priori log-likelihood ratio input to a SISO block	s01
$L_E$	Extrinsic log-likelihood ratio output of a SISO block	s01
$L_D$	A-posteriori (decision) log-likelihood ratio, $L_D = L_A + L_E + L_{ch}$	s01
$I_A, I_E$	Mutual information between the coded bit and the a-priori / extrinsic LLR	s01
$T_{\text{dec}}(I_A)$	EXIT transfer function of a soft-input soft-output decoder	s01
$T_{\text{eq}}(I_A)$	EXIT transfer function of a soft-input soft-output equalizer/detector	s02
$J(\sigma)$	Mutual information between an equiprobable binary symbol and its LLR observed through a symmetric Gaussian channel of LLR std. deviation $\sigma$	s01
$\bar{x}_k$	Soft (posterior mean) symbol estimate of data symbol $x_k$	s02
$v_k$	Residual variance of the soft symbol estimate, $v_k = \mathbb{E}[\|x_k - \bar{x}_k\|^2]$	s02
$\boldsymbol{\Pi}$	Interleaver permutation matrix	s01
$q^\star(x)$	Projected (Gaussian) message produced by the EP moment-matching step	s04
$\text{Proj}_{\mathcal{F}}[\cdot]$	Projection onto exponential family $\mathcal{F}$ via KL minimization	s04

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