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Programming the Propagation Environment

Classical wireless engineering accepts the propagation channel as a given and designs the transmitter and receiver around it. The environment — walls, streets, furniture — is a fixed nuisance. A reconfigurable intelligent surface (RIS) inverts this picture: the environment becomes one more variable that the system can optimize. By coating a reflecting surface with a dense grid of sub-wavelength elements whose reflection phase can be reprogrammed in real time, we turn the "channel" into a controlled object.

This single idea is the golden thread that runs through the 18 chapters of this book. Every algorithm, every architecture, and every signal model we develop will be in service of one question: given that we can now choose the phase shifts of $N$ passive reflecting elements, how much do we gain, and how do we compute the optimal choice?

Definition:
Reconfigurable Intelligent Surface (RIS)

A reconfigurable intelligent surface is a planar (or conformal) array of $N$ sub-wavelength metasurface elements (also called unit cells or reflecting elements) whose scattering response can be electronically reconfigured. Each element $n \in \{1, \ldots, N\}$ applies a programmable reflection coefficient

$\phi_n = a_n\, e^{j\theta_n},$

where $a_n \in [0, 1]$ is the reflection amplitude (ideally $a_n = 1$ , corresponding to a lossless element) and $\theta_n \in [0, 2\pi)$ is the phase shift. In the ideal diagonal model, the RIS acts as a reconfigurable diagonal multiplier between the incoming and outgoing fields: incoming wave at element $n$ is multiplied by $\phi_n$ and re-radiated.

Assembling the $N$ coefficients into a vector $\boldsymbol{\phi} = [\phi_1, \ldots, \phi_N]^T$ , the RIS applies the diagonal phase-shift matrix

$\boldsymbol{\Phi} = \text{diag}(\phi_1, \ldots, \phi_N) = \text{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})$

under the unit-modulus assumption $a_n = 1$ for all $n$ .

The unit-modulus constraint $|\phi_n| = 1$ is the defining hardware feature of a passive RIS — each element is lossless and can only rotate the phase of the reflected wave, not amplify it. This constraint is what makes RIS optimization non-convex; we will return to this in Chapter 6.

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Reconfigurable Intelligent Surface

A nearly-passive planar metasurface composed of $N$ sub-wavelength elements whose reflection phases are electronically programmable. The RIS does not generate RF signals; it reflects incident waves with controlled phase shifts, shaping the propagation environment.

Unit-Modulus Constraint

The constraint $|\phi_n| = 1$ on each RIS element's reflection coefficient, corresponding to lossless reflection. The set $\{\boldsymbol{\phi} \in \mathbb{C}^N : |\phi_n| = 1\ \forall n\}$ is the $N$ -dimensional complex torus — a non-convex manifold. It is the defining geometric object of RIS optimization.

Definition:
Passive Beamforming

Passive beamforming is the practice of choosing the RIS phase vector $\boldsymbol{\theta} \in [0, 2\pi)^N$ to shape the reflected electromagnetic field. Unlike conventional (active) beamforming, which weights the signal by a vector $\mathbf{v}$ at the transmitter's power amplifiers, passive beamforming requires no transmit power at the RIS — it only reprograms the reflection phases. The energy radiated by the RIS comes entirely from the incident wave.

RIS-Aided Wireless System — Canonical single-RIS deployment. A base station (BS) communicates with a user equipment (UE) via a direct path $\mathbf{h}_d$ (often blocked or weak) and an indirect path through the RIS. The RIS receives the incident wave through $\mathbf{H}_1$ , applies the diagonal phase shifts $\boldsymbol{\Phi}$ , and re-radiates via $\mathbf{H}_2$ . A low-rate control link from the BS programs $\boldsymbol{\theta}$ .

Historical Note: From Reflectarrays to Programmable Metasurfaces

1963–2019

The idea of a passive surface that reflects waves with controlled phase is not new. Reflectarrays — large planar antennas that mimic a parabolic dish by engineered phase delays — were proposed by Berry, Malech, and Kennedy in 1963. What is new is the reconfigurability. The ability to electronically reprogram each element's phase in microseconds, using varactor diodes or PIN diodes, emerged from the metamaterials community in the 2010s. The term reconfigurable intelligent surface was popularized by Basar et al. (2019), who argued that 6G wireless networks should treat the environment itself as a controllable design variable — the phrase "programmable wireless environment" captured the imagination of the field and launched a publication avalanche.

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When Does the Diagonal Model Hold?

The diagonal phase-shift model $\boldsymbol{\Phi} = \text{diag}(\phi_n)$ assumes that (i) each element reflects only the wave incident on it, and (ii) the reflection coefficient $\phi_n$ depends neither on the angle of incidence nor on the state of neighbouring elements. Both assumptions are idealizations. Mutual coupling between adjacent elements introduces off-diagonal terms; strong angular selectivity makes $\phi_n$ depend on the incidence angle. Chapter 2 develops the full electromagnetic model and quantifies when the diagonal approximation is accurate. For most of this book, we work in the diagonal regime — it is where the optimization theory is cleanest and where most papers live.

Why This Matters: The RIS as a Reconfigurable Lens

An intuitive way to picture a passive RIS is as a digital version of a curved mirror or a Fresnel lens. A parabolic mirror focuses parallel rays to a fixed focal point — its geometry is locked in at manufacture. An RIS does the same thing, but the "focal point" is computed in real time: by choosing the phase profile $\{\theta_n\}$ to compensate for the path-length difference from each element to a desired focal point, the RIS focuses the reflected wave wherever the controller dictates. When the user moves, the controller updates $\boldsymbol{\theta}$ and the focus follows.

See full treatment in Semidefinite Relaxation and Gaussian Randomization

What is a Reconfigurable Intelligent Surface?