Prerequisites

Before You Begin

This chapter develops the theory of reconfigurable intelligent surfaces (RIS) for wireless communications. The material builds on linear algebra (Chapter 1) for matrix operations and eigenvalue analysis, optimisation theory (Chapter 3) for alternating optimisation and semidefinite relaxation, antenna and array fundamentals (Chapter 7) for array response vectors and beamforming, and MIMO capacity and transceiver design (Chapters 15--16) for precoding and spatial multiplexing. The chapter is at the level of current research papers in IEEE Transactions on Wireless Communications and requires comfort with complex-valued optimisation under non-convex constraints.

Linear algebra: complex matrices, Hermitian forms, rank-one decompositions(Review ch01)
Self-check: Can you express a rank-one positive semidefinite matrix as $\mathbf{V} = \mathbf{v}\mathbf{v}^H$ and apply the trace identity $\mathbf{a}^H \mathbf{V} \mathbf{a} = \mathrm{tr}(\mathbf{V}\mathbf{a}\mathbf{a}^H)$ to reformulate quadratic forms?
Optimisation: convex relaxation, semidefinite programming, KKT conditions(Review ch03)
Self-check: Can you formulate a semidefinite relaxation (SDR) by lifting a vector variable $\mathbf{x}$ to a matrix $\mathbf{X} = \mathbf{x}\mathbf{x}^H$ , relaxing $\mathrm{rank}(\mathbf{X}) = 1$ to $\mathbf{X} \succeq \mathbf{0}$ , and solve the resulting SDP? Can you state the conditions under which the SDR is tight?
Antennas and arrays: array response vectors, element patterns, beamforming(Review ch07)
Self-check: Can you write the array response vector for a uniform linear array (ULA) as $\mathbf{a}(\theta) = [1, e^{j2\pi d\sin\theta/\lambda}, \ldots, e^{j2\pi(N-1)d\sin\theta/\lambda}]^T$ and explain how element-wise phase shifts steer the beam?
MIMO capacity: channel capacity with spatial multiplexing(Review ch15)
Self-check: Can you derive the MIMO capacity $C = \log_2\det(\mathbf{I} + \frac{P}{M}\mathbf{H}\mathbf{H}^{H}/\sigma^2)$ and explain the role of channel rank, condition number, and spatial degrees of freedom?
MIMO transceivers: precoding, beamforming, and receiver design(Review ch16)
Self-check: Can you design the optimal MIMO beamformer as the dominant right singular vector of the channel and explain the relationship between maximum ratio transmission (MRT) and the matched filter?

Chapter 28 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$N$	Number of RIS reflecting elements	s01
$M$	Number of BS transmit antennas	s01
$\boldsymbol{\Theta}$	RIS diagonal phase-shift matrix, $\mathrm{diag}(\beta_1 e^{j\theta_1}, \ldots, \beta_N e^{j\theta_N})$	s01
$\mathbf{G}$	BS-to-RIS channel matrix ( $N \times M$ )	s01
$\mathbf{h}_r$	RIS-to-user channel vector ( $N \times 1$ )	s01
$\mathbf{h}_d$	Direct BS-to-user channel vector ( $M \times 1$ )	s01
$\mathbf{w}$	BS active beamforming vector ( $M \times 1$ )	s01
$\boldsymbol{\phi}$	RIS phase-shift vector, $[\beta_1 e^{j\theta_1}, \ldots, \beta_N e^{j\theta_N}]^T$	s01
$\mathbf{H}_{\mathrm{eff}}$	Effective cascaded channel including direct and reflected paths	s01
$b$	Number of phase quantisation bits per RIS element	s05
$\mathcal{F}$	Discrete phase-shift codebook, $\{0, 2\pi/2^b, \ldots, 2\pi(2^b-1)/2^b\}$	s05

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