Prerequisites & Notation

Before You Begin

This chapter assumes fluency with MIMO precoding, mmWave propagation, and the concept of antenna arrays with phase shifters. The algorithmic content of Section 4 builds on compressed sensing (orthogonal matching pursuit) from FSI Chapter 12.

MIMO channel model and capacity: $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ , water-filling, SVD-based precoding(Review ch15)
Self-check: Can you write the optimal fully-digital precoder for a rank- $L$ channel and explain why it has unconstrained complex entries?
Uniform linear and planar array steering vectors, array factor, beamwidth(Review ch07)
Self-check: Can you write the $N_t$ -element ULA steering vector $\mathbf{a}(\theta)$ and sketch its DFT beampattern?
mmWave propagation: narrow angular spread, sparse multipath, large path loss(Review ch27)
Self-check: Can you explain why a mmWave channel typically has $L \leq 5$ dominant paths and why the channel matrix has low rank?
Orthogonal matching pursuit (OMP) for sparse signal recovery(Review ch12)
Self-check: Can you write OMP as an iterative greedy residual-matching procedure and state when it recovers the true support?
DFT matrix, Butler matrix, and analog Fourier operations(Review ch14)
Self-check: Can you write the DFT matrix and explain how a hybrid coupler implements a $2 \times 2$ unitary?

Notation for This Chapter

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

Symbol	Meaning	Introduced
$N_t$	Number of transmit antennas at the base station	s01
$N_r$	Number of receive antennas at the user	s01
$N_{\text{RF}}$	Number of RF chains on one side of the link; $K \leq N_{\text{RF}} \leq N_t$	s01
$K$	Number of users / data streams	s01
$\mathbf{F}_{\text{RF}}$	Analog / RF precoder, $N_t \times N_{\text{RF}}$ , entries of constant modulus $\|[\mathbf{F}_{\text{RF}}]_{m,n}\| = 1/\sqrt{N_t}$	s01
$\mathbf{F}_{\text{BB}}$	Digital / baseband precoder, $N_{\text{RF}} \times K$ , arbitrary complex entries	s01
$\mathbf{W}$	Effective (fully-digital) precoder $= \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}$ in the hybrid architecture	s01
$\mathbf{H}$	mmWave channel matrix, $N_r \times N_t$ , typically low-rank with $L$ dominant paths	s01
$\mathbf{a}(\phi)$	Tx array response (steering) vector at azimuth $\phi$ , $N_t \times 1$	s01
$\hat{\mathbf{a}}(\phi)$	Rx array response (steering) vector at azimuth $\phi$ , $N_r \times 1$	s01
$\mathbf{v}$	Analog beamforming vector (one RF chain), $N_t \times 1$ , $\|[\mathbf{v}]_m\| = 1/\sqrt{N_t}$	s01
$\lambda$	Carrier wavelength $= c/f_0$ ; at 28 GHz, $\lambda \approx 1.07$ cm	s01
$\text{SNR}$	Signal-to-noise ratio in linear scale	s01
$\sigma^2$	AWGN noise variance per receive antenna	s01
$b$	Number of phase-shifter resolution bits; $2^b$ discrete phase levels	s05
$\mathcal{F}$	Beam codebook, a finite set of analog precoding vectors	s03
$P_{\text{cir}}$	Per-RF-chain circuit power (ADC + DAC + mixer + LNA); scales linearly with $N_{\text{RF}}$	s01

← Ch 19 Analog, Digital, and Hybrid Architectures