Prerequisites & Notation

Prerequisites for Chapter 4

This chapter builds directly on the channel estimation framework of Chapter 3 and the system model from Chapter 1. We assume the reader is comfortable with MMSE channel estimation, pilot contamination, and the basic massive MIMO uplink model.

Massive MIMO system model and favorable propagation(Review mimo-ch01)
Self-check: Can you write the uplink received signal $\mathbf{y} = \sum_{k=1}^{K} \sqrt{{P_t}_{k}} \, \mathbf{H}_{k} x_k + \mathbf{w}$ ?
Channel estimation: LS and MMSE estimators(Review mimo-ch03)
Self-check: Can you state the MMSE estimate $\hat{\mathbf{H}}_k$ and the estimation error covariance?
Pilot contamination and its effect on estimation quality(Review mimo-ch03)
Self-check: Can you explain why pilot contamination persists as $N_t \to \infty$ ?
Linear combining: MRC, ZF, MMSE(Review mimo-ch01)
Self-check: Do you know the combining vectors $\mathbf{v}_{k}^{\text{MRC}}$ , $\mathbf{v}_{k}^{\text{ZF}}$ , $\mathbf{v}_{k}^{\text{MMSE}}$ ?
Jensen's inequality and concavity of the logarithm
Self-check: Can you state Jensen's inequality for concave functions?

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MMSE channel estimation framework and pilot contamination model

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Massive MIMO system model, channel hardening, favorable propagation

Chapter 4 Notation

Key symbols used throughout this chapter. We follow the conventions established in Chapters 1-3. New symbols specific to rate analysis are introduced here.

Symbol	Meaning	Introduced
$\mathbf{H}_{k} \in \mathbb{C}^{N_t}$	Channel vector from user $k$ to the base station
$\hat{\mathbf{H}}_k$	MMSE estimate of $\mathbf{H}_{k}$
$\tilde{\mathbf{H}}_k = \mathbf{H}_{k} - \hat{\mathbf{H}}_k$	Channel estimation error
$\beta_{k}$	Large-scale fading coefficient (path loss) of user $k$
$\gamma_k$	Estimation quality: $\gamma_k = \mathbb{E}[\\|\hat{\mathbf{H}}_k\\|^2] / N_t$
$\mathbf{v}_{k} \in \mathbb{C}^{N_t}$	Combining vector for user $k$
$\text{SINR}_k$	Effective signal-to-interference-plus-noise ratio for user $k$
$R_k$	Achievable ergodic rate for user $k$ (bits/s/Hz)
$K$	Number of active single-antenna users
$N_t$	Number of base station antennas
$\sigma^2$	Noise variance
$P_t$	Uplink transmit power
$E_t$	Transmit energy (used in power scaling: $P_t = E_t / N_t^\alpha$ )

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