Prerequisites & Notation

Before You Begin

Given the channel estimate from Chapter 7 and the DD input-output relation of Chapter 4, this chapter's problem is: recover the data symbols. We assume the full machinery of Chapters 4, 6, 7.

DD channel matrix $\mathbf{H}_{DD}$ : sparse Kronecker structure(Review OTFS Ch. 4)
Self-check: Can you write $\mathbf{H}_{DD} = \sum_i h_i (\boldsymbol{\Pi}_N^{k_i}\boldsymbol{\Delta}_N^{\ell_i}) \otimes \boldsymbol{\Pi}_M^{\ell_i}$ ?
Channel estimates from Chapter 7(Review OTFS Ch. 7)
Self-check: Do you understand that channel estimation and detection are decoupled — we treat the channel as known in this chapter?
LMMSE estimation(Review FSI Ch. 7)
Self-check: Can you derive the LMMSE estimator $\hat{\mathbf{x}} = (\mathbf{H}_{DD}^{H}\mathbf{H}_{DD} + \sigma^2\mathbf{I})^{-1}\mathbf{H}_{DD}^{H}\mathbf{y}$ ?
Belief propagation on factor graphs(Review FSI Ch. 17)
Self-check: Do you recall the sum-product message updates on a factor graph?
MIMO detection algorithms(Review Telecom Ch. 16)
Self-check: Are V-BLAST, sphere decoding, and MMSE detection familiar?

Notation for This Chapter

Symbols introduced in this chapter.

Symbol	Meaning	Introduced
$\hat{X}_{DD}, \hat{\mathbf{x}}$	Detected DD grid (matrix) or vectorized form	s01
$\mathcal{X}$	QAM constellation (data-symbol alphabet)	s01
$\mathcal{N}_i$	Set of data symbols that interfere with symbol $i$	s03
$\mu_{a \to b}(x)$	Message from factor $a$ to variable $b$ about the value of $x$	s03
$\lambda_i$	Log-likelihood ratio (LLR) for data symbol $i$	s05
$T_{\text{iter}}$	Number of detection iterations (MP or iterative decoding)	s03

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