Prerequisites & Notation

Before You Begin

This chapter develops linear and nonlinear detection strategies for the massive MIMO uplink. The reader should be comfortable with the following before proceeding.

Massive MIMO system model and channel hardening(Review mimo/ch01)
Self-check: Can you write down the uplink received signal model for $N_t$ BS antennas and $K$ single-antenna users?
Channel estimation: LS and MMSE estimators, pilot contamination(Review mimo/ch03)
Self-check: Can you state the MMSE channel estimate and its error covariance?
Achievable rate expressions with imperfect CSI (UatF bound)(Review mimo/ch04)
Self-check: Can you write the UatF rate expression for MRC processing?
MAC capacity region and successive decoding(Review ita/ch15)
Self-check: Can you sketch the MAC rate region for two users and identify the corner points?
MMSE and LMMSE estimation theory(Review fsi/ch12)
Self-check: Can you derive the LMMSE estimator $\hat{\mathbf{x}} = \mathbf{R}_{xy}\mathbf{R}_y^{-1}\mathbf{y}$ ?
Matrix inversion lemma (Woodbury identity)
Self-check: Can you state and apply the identity $(\mathbf{A} + \mathbf{U}\mathbf{C}\mathbf{V})^{-1} = \mathbf{A}^{-1} - \mathbf{A}^{-1}\mathbf{U}(\mathbf{C}^{-1} + \mathbf{V}\mathbf{A}^{-1}\mathbf{U})^{-1}\mathbf{V}\mathbf{A}^{-1}$ ?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{y}$	Received signal vector at the BS, $\mathbf{y} \in \mathbb{C}^{N_t}$	s01
$\mathbf{H}$	Channel matrix $\mathbf{H} \in \mathbb{C}^{N_t \times K}$ , columns $\mathbf{h}_k$	s01
$\mathbf{h}_k$	Channel vector from user $k$ to the BS, $\mathbf{h}_k \in \mathbb{C}^{N_t}$	s01
$x_k$	Transmitted symbol of user $k$ , $\mathbb{E}[\|x_k\|^2] = P_k$	s01
$\mathbf{w}$	AWGN noise vector, $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$	s01
$\mathbf{G}$	Linear combining/detection matrix, $\hat{\mathbf{x}} = \mathbf{G}^H \mathbf{y}$	s01
$\text{SINR}_k$	Post-detection signal-to-interference-plus-noise ratio for user $k$	s02
$\mathbf{R}_k$	Spatial covariance matrix of user $k$ : $\mathbf{R}_k = \mathbb{E}[\mathbf{h}_k \mathbf{h}_k^H]$	s01
$\kappa(\mathbf{A})$	Condition number of matrix $\mathbf{A}$ : $\kappa(\mathbf{A}) = \sigma_{\max}/\sigma_{\min}$	s03
$\mathbf{Q}$	Bussgang equivalent channel matrix (quantized systems)	s07
$\mathbf{R}_{\eta}$	Quantization distortion covariance matrix	s07

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