Prerequisites & Notation

Chapter 3 builds directly on the system model from Chapter 1 and the channel models from Chapter 2. Make sure you are comfortable with the following.

Massive MIMO system model: uplink/downlink, TDD, channel matrix(Review mimo/ch01)
Self-check: Can you write the received signal model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ and state what each term is?
Spatially correlated channels: covariance matrix $\mathbf{R}_k$ , one-ring model(Review mimo/ch02)
Self-check: Can you define $\mathbf{R}_k = \mathbb{E}[\mathbf{h}_k \mathbf{h}_k^H]$ and explain what its eigenvalue spread tells you about the channel?
MMSE estimation: posterior mean, MSE expression, LMMSE as linear special case(Review fsi/ch07)
Self-check: Given a joint Gaussian model, can you write the MMSE estimator and its error covariance without looking it up?
Projection matrices: $\mathbf{P} = \mathbf{A}(\mathbf{A}^H\mathbf{A})^{-1}\mathbf{A}^H$ , orthogonal complement
Self-check: Can you show that $\mathbf{P}^2 = \mathbf{P}$ and that $(\mathbf{I} - \mathbf{P})$ projects onto the null space of $\mathbf{A}^H$ ?
Matrix inversion lemma: $(A + UCV)^{-1}$ formula
Self-check: Can you state the Woodbury identity and apply it to invert $(\sigma^2 \mathbf{I} + \mathbf{R}_k \mathbf{S}_p \mathbf{S}_p^H \mathbf{R}_k^H)$ ?

Symbols introduced or specialized in Chapter 3. For global conventions, see the master notation table.

Symbol	Meaning	Introduced
$\mathbf{H}_{k} \in \mathbb{C}^{N_t}$	Channel vector from BS to user $k$ ( $N_t$ base-station antennas)	s01
$\mathbf{R}_k = \mathbb{E}[\mathbf{H}_{k} \mathbf{H}_{k}^{H}] \in \mathbb{C}^{N_t \times N_t}$	Spatial covariance matrix of user $k$	s01
$\boldsymbol{\phi}_k \in \mathbb{C}^{\tau_p}$	Pilot sequence of user $k$ , length $\tau_p$ , $\\|\boldsymbol{\phi}_k\\|^2 = \tau_p$	s01
$\tau_c$	Coherence interval length (samples)	s01
$\tau_p$	Pilot length (samples), $\tau_p \leq \tau_c$	s01
$\mathbf{Y}_p \in \mathbb{C}^{N_t \times \tau_p}$	Received pilot signal matrix at base station	s01
$\hat{\mathbf{h}}_k$	Channel estimate of user $k$	s02
$\tilde{\mathbf{h}}_k = \mathbf{h}_k - \hat{\mathbf{h}}_k$	Channel estimation error	s02
$\mathbf{C}_k$	Error covariance matrix of MMSE estimate, $\mathbf{C}_k = \mathbb{E}[\tilde{\mathbf{h}}_k \tilde{\mathbf{h}}_k^H]$	s02
$\text{MSE}_k$	Mean squared error of channel estimate, $\text{MSE}_k = \text{tr}(\mathbf{C}_k)$	s02
$\mathcal{P}_k \subseteq \{1,\ldots,K_{\text{tot}}\}$	Set of users assigned to the same pilot as user $k$	s03
$L$	Number of cells in multi-cell system	s03
$p_u$	Uplink transmit power per user	s02