Prerequisites & Notation

Prerequisites for Chapter 12

MIMO detection sits at the intersection of linear algebra (Chapter 1), hypothesis testing and ML estimation (Chapters 3-5), and lattice geometry. The reader should be fluent with the first; comfortable with the second; and willing to pick up lattice theory on the fly.

Gram matrices, QR decomposition, and least squares projection(Review ch01)
Self-check: Given $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ with full column rank, can you write the projector onto the column space of $\mathbf{H}$ and compute its complement?
Complex Gaussian random vectors and their covariance structure(Review ch02)
Self-check: If $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$ , what is the distribution of $\mathbf{A}\mathbf{w}$ ?
Maximum likelihood detection for discrete hypotheses(Review ch05)
Self-check: For $\mathbf{y} = \mathbf{s}_i + \mathbf{w}$ with $\mathbf{w}$ white Gaussian, why does ML reduce to minimum distance?
Linear MMSE estimation under a linear Gaussian model(Review ch06)
Self-check: Can you write the LMMSE estimator for $\mathbf{x}$ from $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ ?
MIMO capacity and channel chain rule (background)
Self-check: Can you state the mutual information $I(\mathbf{x}; \mathbf{y} | \mathbf{H})$ for a Gaussian MIMO channel with $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{Q})$ ?

Notation for This Chapter

We use $\mathbf{H}$ for the MIMO channel matrix throughout. The transmitted vector $\mathbf{x}$ is drawn from a discrete constellation alphabet $\mathcal{A}$ (QPSK, 16-QAM, etc.); the noise $\mathbf{w}$ is circularly symmetric complex Gaussian.

Symbol	Meaning	Introduced
$\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$	MIMO channel matrix; $n_r$ receive antennas, $n_t$ transmit antennas	s01
$\mathcal{A}$	Constellation alphabet (e.g., QPSK: $\mathcal{A} = \{\pm 1 \pm j\}/\sqrt{2}$ )	s01
$\mathbf{x} \in \mathcal{A}^{n_t}$	Transmitted symbol vector	s01
$\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$	Circularly symmetric complex Gaussian noise	s01
$\hat{\mathbf{x}}_{\text{ML}}$	Maximum-likelihood estimate of $\mathbf{x}$	s01
$\mathbf{G}_{\text{ZF}}, \mathbf{G}_{\text{MMSE}}$	Linear detector filters (zero-forcing, MMSE)	s02
$\gamma_k$	Post-detection SINR on the $k$ -th stream	s02
$\Lambda(\mathbf{B})$	Lattice generated by basis $\mathbf{B}$	s04
$r$	Search radius in sphere decoding	s04
$\delta$	LLL reduction parameter, $\delta \in (1/4, 1)$	s05

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