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Reconfigurable Intelligent Surfaces: A Signal Processing Perspective With Wireless Applications

Emil Björnson, Henk Wymeersch, Bho Matthiesen, Petar Popovski, Luca Sanguinetti, Elisabeth de Carvalho

arXiv:2102.00742v2eess.SPcs.IT

TL;DR

The article develops signal-processing approaches for RIS-aided systems, including channel estimation, reconfiguration, localization, and sensing. It shows that RIS placement and configuration affect localization coverage and that communications and localization benefit from different RIS and bandwidth requirements.

  • Problem

    The article addresses signal-processing challenges in analyzing and designing RIS-aided wireless systems.

  • Method

    The article uses signal-processing formulas, derivations, channel-estimation procedures, and RIS configuration strategies to analyze these systems.

  • Results

    RIS placement can significantly improve location coverage, while RIS configuration can be tailored to user location and differs from the optimal communication configuration.

  • Takeaways & Limitations

    Communications generally require many RIS elements and benefit from less bandwidth, whereas localization requires large bandwidths and can use a smaller RIS to add information from both paths.

Abstract

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Antenna array technology enables directional transmission and reception of wireless signals, for communications, localization, and sensing purposes. The signal processing algorithms that underpin this technology began to be developed several decades ago [1], but it is first with the ongoing deployment of the fifth-generation (5G) wireless mobile networks that it becomes a mainstream technology [2]. The number of antenna elements in the arrays of the 5G base stations and user devices can be measured at the order of 100 and 10, respectively. As the networks shift towards using higher frequency bands, more antennas fit into a given aperture. The 5G developments enhance the transmitter and receiver functionalities, but the wireless channel propagation remains an uncontrollable system. This is illustrated in Fig. 1(a) and its mathematical notation will be introduced later. Transmitted signals with three different frequencies are shown to illustrate the fact that attenuation can vary greatly across frequencies. Looking beyond 5G, the advent of electromagnetic components that can shape how they interact with wireless signals enables partial control of the propagation. A reconfigurable intelligent surface (RIS) is a two-dimensional surface of engineered material whose properties are reconfigurable rather than static [4]. This article provides a tutorial on the fundamental properties of the RIS technology from a signal processing perspective. It is meant as a complement to recent surveys of electromagnetic and hardware aspects [4], [7], [11], acoustics [12], communication theory [13], and localization [8]. We will provide the formulas and derivations that are required to understand and analyze RIS-aided systems using signal processing, and exemplify how they can be utilized for improved communication, localization, and sensing.

I. HISTORY AND FUNDAMENTALS

RIS technology introduces partial control over wireless propagation through reconfigurable surfaces, motivating new signal-processing models and algorithms. This tutorial develops a signal-processing foundation for analyzing RIS-aided communication, localization, and sensing systems.

  • RIS fundamentals: Adding an RIS with N elements creates N controllable propagation paths whose element amplitudes and phases can be tuned at the receiver.Each path includes propagation to an RIS element, passive filtering, and propagation from the element to the receiver.
  • RIS fundamentals: RIS elements passively filter incoming signals by changing amplitude, delay, or polarization without increasing signal power.A programmable controller changes each element’s impedance and therefore its reflection coefficient.
  • RIS fundamentals: A reconfigurable intelligent surface is a two-dimensional engineered surface whose element properties can be changed over time.Its elements are typically discrete, sub-wavelength scatterers with externally reconfigurable impedance.
  • RIS fundamentals: An RIS can shape reflected beams by configuring reflection coefficients across its elements, while receiving signals from directions in its facing half-space.The resulting signal-processing algorithms determine the desired reflection direction and beam shape.
  • Motivation: Unlike transmitter- or receiver-integrated surfaces, RISs are deployed between them, creating new use cases and signal-processing challenges.Their electromagnetic behavior requires revisiting established communication and localization models from first principles.
  • Motivation: The tutorial provides formulas and derivations for analyzing RIS-aided systems using signal processing.The stated objective is to provide a foundation for the technology’s communication, localization, and sensing applications.

A. Continuous-time system model with RIS elements as reconfigurable filters

The continuous-time RIS model represents each element as a reconfigurable passive filter within a cascade of propagation channels. The model is converted into a sampled finite-memory representation for digital signal processing.

  • Continuous-time model: Each RIS element filters and reradiates the incoming signal through a reconfigurable LTI impulse response controlled by an external variable θ_n.The control may be discrete or continuous, but one configuration is held during a transmission interval under the LTI assumption.
  • Element response: RIS element responses may vary substantially with frequency in both phase and amplitude because of delay and frequency-dependent impedance.In the example, four capacitance settings produce phase shifts π/2, 0, −π/2, and π at the carrier frequency.
  • Bandwidth assumptions: For signals with bandwidth limited to a few tens of MHz, frequency-dependent RIS nonlinearity can be neglected, while wireless channels usually determine whether the end-to-end channel is narrowband or wideband.The model assumes the RIS is narrowband even when the wireless channels may be wideband.
  • Element response: The RIS element’s few dB of signal loss is minor relative to wireless propagation losses that can be on the order of 100 dB.The largest element losses occur when tuning for zero phase response because of circuit resonance.
  • Continuous-time model: The end-to-end RIS channel is the sum over elements of cascaded transmitter-to-element, element-filter, and element-to-receiver impulse responses.For element n, the corresponding pseudo-baseband chain is (b_n ∗ ϑ_n;θ_n ∗ a_n)(t).
  • Discrete-time conversion: Sampling after receiver noise addition and lowpass filtering produces a causal finite impulse response with M ≥1 non-zero taps.The resulting discrete-time noise is circularly symmetric complex Gaussian and independent across samples.

C. Canonical multicarrier system model

The canonical multicarrier model uses OFDM to transform the RIS-aided dispersive channel into orthogonal frequency subcarriers. It expresses each subcarrier through the propagation channels and the common RIS configuration.

  • OFDM formulation: A dispersive discrete-time channel with memory M−1 causes intersymbol interference from previous transmitted symbols.OFDM is introduced as a way to separate the channel into frequency subcarriers.
  • OFDM applications: The OFDM model is used for both communication and localization.The receiver obtains frequency-domain observations by applying a Fourier transform to the received time-domain signal.
  • OFDM formulation: Appending a cyclic prefix converts the block transmission into cyclic convolution when K > M.The K-point DFT then diagonalizes the convolution into separate subcarriers.
  • OFDM formulation: The transformed system is a collection of K orthogonal subcarriers, each modeled as a transmitted signal multiplied by the channel frequency response plus noise.This structure removes intersymbol interference and yields a discrete memoryless AWGN channel per subcarrier.
  • Multipath RIS channel: For an RIS-assisted multipath channel, each transmitter–RIS-element–receiver combination contributes a path whose gain includes the product of propagation and element losses.With N elements, L_a paths to each element, and L_b paths from each element, the model contains N L_a L_b paths.
  • Multipath RIS channel: Propagation determines the frequency-dependent channel vectors, whereas the RIS determines a common controllable vector across subcarriers.The frequency response is consequently expressed using the propagation matrix, the RIS configuration, and a DFT matrix.

E. Simplified narrowband system model

The narrowband RIS model represents the end-to-end link as a memoryless channel whose coefficient depends on the RIS configuration, under explicit AWGN assumptions. It also distinguishes out-of-band and in-band RIS control, with in-band control changing the channel model and consuming useful bandwidth.

  • Narrowband channel model: With one strong path to and from the RIS, the channel can be modeled using the RIS steering vector and a common phase-shift pattern.The narrowband approximation also applies when multipath delay spread is much smaller than the sampling period 1/B.
  • Narrowband channel model: The narrowband input-output relation is z[k] = hθ x[k] + w[k], with hθ fixed over a codeword and known to transmitter and receiver.The noise is modeled as independent complex Gaussian noise, w[k] ∼ NC(0, N0).
  • Capacity objective: RIS configuration aims to maximize hθ capacity in narrowband systems or the sum capacity of K OFDM subcarriers.The same RIS configuration determines all subcarrier channel values, so they may not be independently optimized.
  • RIS control: Out-of-band control preserves useful bandwidth, whereas in-band control consumes bandwidth and can make the end-to-end channel non-AWGN.With simultaneous payload and control, the RIS decodes control information and causally changes hθ for later symbols, corresponding information-theoretically to a relay channel.

A. RIS design for narrowband capacity maximization

Narrowband RIS capacity maximization aligns the reradiated signals in phase and, when needed, uses delays to equalize propagation times. The resulting coherent combining yields quadratic SNR growth with RIS size, while discrete phase control incurs only a small loss in the analyzed case.

  • Continuous RIS design: The optimal RIS configuration makes all reradiated signals reach the receiver synchronously in phase.This follows from the Cauchy-Schwarz bound when the RIS elements can select continuous phase shifts.
  • Scaling with RIS size: Under equal propagation losses and far-field conditions, the SNR grows quadratically with the number of RIS elements.The surface intercepts energy proportional to N and focuses the reradiated signals through constructive interference, producing received energy proportional to N2.
  • Continuous RIS design: The RIS can synthesize a parabolic-reflector-like surface by adding delays that equalize the propagation time of reflected paths.Unlike a homogeneous flat surface, the RIS can electronically change the direction and shape of the reflected waveform.
  • Discrete RIS design: Four available phase configurations produce an SNR loss of only around 8/π2 = −0.9 dB.The result implies that a small number of configurations per element can be sufficient for RIS implementation.
  • Numerical behavior: When the uncontrollable channel is weak, the RIS-controlled path surpasses it at N = 32 elements and the SNR then grows as N2.With N = 1, the RIS-controlled path is 30 dB weaker than the uncontrollable path.
  • Numerical behavior: When the uncontrollable channel is strong, the RIS has limited capacity impact because N = 1000 elements are required for equal path strength.The ideal RIS and four-phase-shift cases show a small performance difference in the reported setup.

C. Reconfiguration under mobility

Mobility makes wireless channels time-variant and introduces Doppler effects, but RIS reconfiguration can model and mitigate these effects. The proposed LTV framework supports configurations that preserve SNR while controlling delay and Doppler behavior.

  • Mobility modeling: User mobility turns the RIS-assisted propagation paths into linear time-variant systems requiring LTV system theory.The RIS elements become time-varying filters when their delays and gains vary with time.
  • Mobility modeling: The LTV input-output model represents the end-to-end channel as the sum of the uncontrollable path and RIS-assisted time-varying paths.Each RIS path is modeled through cascaded transmitter-to-RIS, RIS-element, and RIS-to-receiver systems.
  • Doppler mitigation: RIS delays can be tuned to counteract the rate of change of the receiver-side path delay, making the Doppler effect unobservable in the received signal.This Doppler-cloaking condition requires dτθn(t)/dt = −dτn,b(t)/dt.
  • Doppler mitigation: When an uncontrollable direct path is present, simultaneously maximizing SNR and compensating the Doppler shifts introduced by that path is impossible.The RIS-assisted path can be configured to match the direct-path Doppler shift, yielding zero Doppler spread, but other configurations may create spread.
  • Doppler mitigation: A mobility-aware SNR-optimal configuration with minimum delay remains optimal and does not introduce additional Doppler spread.The configuration also minimizes delay spread, while avoiding phase discontinuities requires restricting delay-induced phase shifts to integer multiples of 2π.

D. RIS design for wideband capacity maximization

Wideband RIS design must select one configuration shared across all simultaneously transmitted subcarriers, creating a nontrivial optimization tradeoff. The strongest-tap maximization heuristic performs especially well with strong LOS paths but loses effectiveness for NLOS deployment.

  • Wideband optimization: Wideband RIS capacity maximization jointly selects power allocation and one RIS configuration across K parallel subcarriers.Waterfilling solves the power-allocation part, while the shared RIS vector creates the cross-subcarrier tradeoff.
  • Wideband optimization: Strongest-tap maximization selects the RIS configuration producing the largest time-domain channel tap, motivated by M ≪ K.This favors concentration of received signal power in a strong time-domain tap rather than optimizing an individual subcarrier.
  • LOS deployment: 96-98% of the upper-bound rate is achieved by the STM configuration for an LOS transmitter-to-RIS channel.The gap grows with bandwidth, while refined algorithms improve the rate by only a few percent.
  • LOS deployment: 2.7-2.9 times higher rate is achieved with an RIS than without one in the evaluated LOS case.A passive metal sheet can approach RIS performance in ideal orientations, but the RIS provides substantial gains across several MHz of bandwidth.
  • NLOS deployment: For an NLOS transmitter-to-RIS channel, the RIS improves the narrowband rate by 4%, but the gain vanishes as bandwidth increases.The weaker RIS path narrows the gap to the no-RIS case, while the lack of a dominant path prevents one configuration from fitting the entire band.
  • Deployment implications: Effective wideband deployment requires LOS from the access point to the RIS and LOS from the RIS to the served users.Under these conditions, the RIS can greatly improve the rate slope and approach the upper bound.

E. Protocol for channel estimation and reconfiguration

Practical RIS optimization requires estimating the cascaded channel before configuring the surface. Multiple OFDM blocks with different RIS settings provide the independent observations needed for linear estimation, while periodic pilots support mobility.

  • Channel estimation: The receiver must estimate the cascaded transmitter-RIS and RIS-receiver channel before computing a suitable RIS configuration.Estimating the cascaded matrix V is sufficient to predict the frequency response for any RIS vector.
  • Channel estimation: A single OFDM block provides M observations for MN unknown channel parameters, so it cannot identify the full cascaded channel.Adding more pilot subcarriers within the same configuration does not remove the RIS configuration’s non-invertible impact.
  • Channel estimation: N OFDM blocks with different RIS configurations can produce MN linearly independent observations when the configuration matrix is invertible.A DFT matrix is one example of a suitable configuration sequence.
  • Channel estimation: The resulting linear model permits classical channel-estimation methods, with least squares as the direct estimate and prior structure enabling shorter pilots.Fading distributions or spatial-temporal sparsity can improve estimation efficiency.
  • Operational protocol: RIS switching should occur in silent guard intervals because the nonlinear operation can modulate reflected signals into other frequency bands.After estimation, the selected configuration can remain active while the channel is static.
  • Operational protocol: Progressive reconfiguration handles mobility by continuously transmitting data while periodically sending pilots to re-estimate the channel.Only M of K subcarriers are used for pilots, leaving the remainder for data.

IV. RIS-AIDED LOCALIZATION AND SENSING

Localization and sensing infer geometry from channel parameters rather than using an unstructured channel only for communication. RIS control adds geometric measurements and can make single-BS localization identifiable while improving accuracy and coverage.

  • Signal-based localization: Localization and sensing estimate geometric parameters such as delays, angles, and frequency shifts, then invert their relation to recover locations and object states.Tracking methods recursively update these estimates over time.
  • 4G and 5G: 4G localization relies on multiple synchronized base stations and time-difference-of-arrival measurements, while multipath limits accuracy to tens of meters.At least four LOS base stations provide three TDOA measurements for 3D positioning.
  • 4G and 5G: 5G adds angle-of-arrival and angle-of-departure measurements, enabling localization from the AOD of two base stations and environment sensing from a single snapshot.These angle measurements reduce infrastructure needs and support SLAM.
  • RIS-aided localization: A single base station and RIS can localize a user and provide partial map information when the controllable RIS path is exploited.The RIS acts as an additional synchronized reference with a phased path.
  • Practical limitations: Dense multipath at lower frequencies makes individual paths difficult to resolve, limiting conventional approaches to favorable environments or data-driven fingerprinting.Nuisance parameters and data associations also create high-dimensional nonlinear localization and sensing problems.
  • RIS-aided localization: RIS inclusion creates new synchronized, configurable reference opportunities and leads to additional geometric measurements that improve localization accuracy and coverage.The paper describes these opportunities under far-field LOS assumptions for the RIS links.

RIS configuration encoding:

RIS-aided localization uses time-varying configurations and balanced codes to resolve otherwise indistinguishable paths, separate controllable and uncontrollable channels, and quantify location information with the FIM and SPEB.

  • Similar and correlated path delays or angles can remain unresolved, producing large localization biases.
  • Time-varying RIS configurations create new dimensions that make correlated paths resolvable.
  • A small set of RIS configurations is assigned unique temporally balanced codes, with switching between OFDM blocks to avoid interference between configurations.
  • Grouping observations by configuration and correlating them with the corresponding code separates controllable and uncontrollable channels while reducing RIS complexity and storage.
  • The Fisher information matrix is transformed through a Jacobian and Schur complement to obtain location information, whose inverse bounds unbiased location-error covariance; SPEB provides a scalar design metric.

1) Offline design for optimized coverage:

Offline RIS design selects placement and configurations to improve localization coverage across a deployment region, while online design adapts configurations to likely user locations and channel conditions.

  • 1) Offline design for optimized coverage:: Offline design can maximize the fraction of the deployment region whose SPEB is below a required accuracy ε.
  • 1) Offline design for optimized coverage:: The offline placement problem can be solved by exhaustive search over a finite design set while ignoring the uncontrollable channel except for the LOS path.
  • 1) Offline design for optimized coverage:: Online design minimizes worst-case localization performance over a high-probability region of user locations and can be convex when the design variables enter the FIM linearly.
  • 1) Offline design for optimized coverage:: Localization-optimized RIS designs generally differ from communication-centric designs such as capacity, despite both improving with higher SNR.
  • 4) Localization and sensing:: Channel estimation separates uncontrollable and controllable channels, after which geometric localization uses TDOA hyperbolas and RIS AOD bearing lines.
  • 4) Localization and sensing:: With two RISs, delay measurement is unnecessary for accurate localization over narrowband channels, and likelihood-based refinement can improve the estimates.

1) FIM analysis:

FIM analysis shows that RIS localization requires balancing configurations for time-of-arrival and angle-of-departure information, and that placement and configuration strongly affect coverage.

  • 1) FIM analysis:: The RIS contributes two fundamental Fisher-information directions: uBS − uRIS from path-delay accuracy and uRIS with reduced intensity.
  • 1) FIM analysis:: Optimal TOA and AOD estimation require conflicting RIS configurations, so a compromise allocates transmissions between direct-beam and derivative-beam configurations.
  • 1) FIM analysis:: The RIS can behave like an additional synchronized phased-array base station when configurations balance the TOA and AOD information terms.
  • 1) FIM analysis:: Three RISs provide acceptable localization performance throughout the deployment region, whereas a single RIS can produce infinite PEB in geometrically degenerate or blocked regions.
  • 1) FIM analysis:: Random configurations perform worse than optimally designed configurations when user-location information is available.
  • 1) FIM analysis:: For the illustrated single-RIS location, localization is best at T1/T ≈ 0.63, while T1 = 0 and T1 = T cause the PEB to diverge.

4) Localization and sensing:

RIS-aided localization combines controllable-path geometry with channel estimation to locate users and support sensing, while its design requirements differ from communication objectives and depend on propagation regime.

  • 4) Localization and sensing:: Localization intersects a TDOA hyperbola from LOS and RIS delays with an RIS AOD bearing line to estimate the user position.
  • 4) Localization and sensing:: The estimated user location and uncontrollable-channel TOA define a TSOA ellipse constraining the sensed object’s location.
  • 4) Localization and sensing:: A snapshot cannot determine the sensing-object location, but sufficient user movement and appropriate data association can uniquely identify it.
  • F. Conclusions from RIS-aided localization: Communication and localization favor different RIS configurations, and localization can require both LOS and RIS paths even when communication gains are limited by LOS.
  • F. Conclusions from RIS-aided localization: Localization needs large bandwidth, whereas the RIS can be small because it primarily adds dimensions that resolve identifiability issues; more elements can reduce the bandwidth requirement.
  • 4) Localization and sensing:: RIS sensing is mainly indirect because the controllable channel does not depend on the uncontrollable channel; improved user localization provides the principal benefit.
  • F. Conclusions from RIS-aided localization: Near-field propagation yields finite SNR limits while enabling point focusing and potentially improving localization beyond flat-mirror behavior.
  • F. Conclusions from RIS-aided localization: RIS channel modeling remains in its infancy with limited experimental validation, and interference and electromagnetic-noise interactions remain open modeling problems.

C. Non-linear RIS operation

The article examines RIS operation beyond fixed, piecewise-constant configurations, including time-varying control that can create nonlinear channels and new frequency components. It also highlights mutual coupling as a modeling and algorithm-design challenge with implications for communication, localization, and sensing.

  • Non-linear RIS operation: For piecewise-constant configurations, the RIS can be modeled as a linear filter, while LTV system theory addresses Doppler mitigation under mobility.The article focuses on piecewise-constant configurations but also describes time-varying-system analysis for mobility-related effects.
  • Non-linear RIS operation: Continuously varying the RIS configuration during transmission modulates the signal before reradiation, creating a nonlinear end-to-end channel.The received signal can contain a wider range of frequencies than the transmitted signal.
  • Mutual coupling: Mutual coupling makes each element’s frequency response depend on neighboring-element configurations rather than behaving independently.The mutual impedance depends on element properties and may require lengthy full-wave simulations for each configuration.
  • Mutual coupling: The CMS approach does not capture the desired RIS operation, whereas patch or slot antennas can decouple reflected-wave amplitude and phase for full 2π phase control.Without accurate mutual-impedance modeling, the complexity–performance trade-off of densifying the RIS cannot be evaluated.
  • Mutual coupling: Mutual coupling affects algorithmic design and communication or localization performance, motivating circuit-theory models and possible machine-learning-based system identification.The article characterizes this modeling research as still being in its infancy.
  • Applications and open problems: RIS system models support communication, localization, and sensing, but preferred implementations differ in bandwidth, surface dimensions, and optimal configuration.The article identifies refined electromagnetic models, experimental validation, and realistic applications as open signal-processing problems.
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