Prerequisites & Notation

Before You Begin

This chapter assumes you have already studied MIMO fundamentals from the Telecom book (Chapters 15–18) and multi-user capacity from the ITA book (Chapters 13–16). The following specific tools are used without re-derivation.

MIMO channel model: $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ , singular value decomposition, water-filling capacity(Review ch15)
Self-check: Can you write the ergodic capacity of an $n_r \times n_t$ MIMO channel in terms of singular values and explain the water-filling solution?
Multi-user MIMO uplink (MAC) and downlink (BC) capacity regions; dirty-paper coding is optimal but complex(Review ch16)
Self-check: Can you sketch the sum-rate capacity of a $K$ -user AWGN MAC and explain why treating interference as noise is suboptimal in general?
Law of large numbers and concentration inequalities: $\frac{1}{n}\sum_i X_i \to \mathbb{E}[X]$ a.s. as $n \to \infty$
Self-check: Can you state the strong law of large numbers and give a rough argument for why sample means concentrate around their expectations?
Linear algebra: trace, determinant, Frobenius norm, $\text{tr}(\mathbf{AB}) = \text{tr}(\mathbf{BA})$ , Woodbury identity(Review ch01)
Self-check: Can you compute $\text{tr}(\mathbf{A}^H\mathbf{A})$ and explain why it equals $\|\mathbf{A}\|_F^2$ ?
Channel estimation via least squares and MMSE; pilot-based training sequences(Review ch07)
Self-check: Given $\mathbf{y} = \mathbf{h}s + \mathbf{w}$ , can you derive the MMSE estimate of $\mathbf{h}$ from a training signal $s$ ?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter. All customizable symbols use $\ntn{}$ tokens; the values shown are the defaults from the notation registry.

Symbol	Meaning	Introduced
$N_t$	Number of base station antennas (also written $M$ in some literature)	s01
$K$	Number of single-antenna users	s01
$\mathbf{H}$	Uplink channel matrix, $N_t \times K$ (columns = per-user channel vectors $\mathbf{h}_k$ )	s01
$\mathbf{h}_k$	$k$ -th column of $\mathbf{H}$ ; uplink channel vector from user $k$ to the BS, $N_t \times 1$	s01
$\mathbf{w}$	Additive white Gaussian noise vector, $\sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$	s01
$\sigma^2$	Noise variance per antenna	s01
$\text{SNR}$	Per-user transmit SNR $= P_k / \sigma^2$	s01
$\mathbf{W}$	Precoding / receive combining matrix, $N_t \times K$	s04
$\mathbf{v}$	Beamforming / combining vector for a single user, $N_t \times 1$	s04
$\beta$	Large-scale fading (path-loss + shadowing) coefficient for user $k$	s02
$\mathbf{R}_k$	Spatial covariance matrix of user $k$ , $\mathbb{E}[\mathbf{h}_k \mathbf{h}_k^H]$ , size $N_t \times N_t$	s03
$\tau_p$	Pilot sequence length (symbols per coherence block allocated to uplink training)	s05
$\tau_c$	Coherence block length (symbols): $\tau_c = B_c \cdot T_c$ in discrete symbols	s05

Back to chapter overview Capacity Scaling: Why More Antennas?