Wideband RIS: Frequency-Selective Phase Response

Why Wideband RIS Is Hard

Chapters 1–15 assumed a narrowband RIS: each element's phase-shift ΞΈn\theta_n is a scalar, frequency-independent. In reality, the phase-shift element is a tuned resonant circuit (PIN diode, varactor); its reflection coefficient depends on frequency: Ξ“n(f)=βˆ£Ξ“n(f)∣ejΟ•n(f)\Gamma_n(f) = |\Gamma_n(f)| e^{j\phi_n(f)}. When the signal bandwidth BB spans much of the element's resonance bandwidth (e.g., >5%> 5\%), the phase response becomes frequency-selective β€” destroying the coherent combining at most subcarriers.

Definition:

Frequency-Selective RIS Phase Response

The frequency-selective RIS matrix is Ξ¦(f)=diag(Ξ“1(f),Ξ“2(f),…,Ξ“N(f)).\boldsymbol{\Phi}(f) = \text{diag}\left(\Gamma_1(f), \Gamma_2(f), \ldots, \Gamma_N(f)\right). Each Ξ“n(f)\Gamma_n(f) is characterized by its element bandwidth BnB_n (typically 5–15% fractional bandwidth). For a signal of bandwidth BB centered at fcf_c:

  • If Bβ‰ͺBnB \ll B_n: narrowband model holds; Ξ¦(f)β‰ˆΞ¦(fc)\boldsymbol{\Phi}(f) \approx \boldsymbol{\Phi}(f_c).
  • If B∼BnB \sim B_n: frequency selectivity is significant; per- subcarrier optimization or codebook-based design required.
  • If B>BnB > B_n: the RIS is effectively uncorrelated across the band; no coherent combining possible.

Theorem: Frequency-Selective Combining Loss

Consider an RIS with element phase response Ο•n(f)=Ο•n0+Ξ±n(fβˆ’fc)\phi_n(f) = \phi_n^0 + \alpha_n (f - f_c) (linear in ff). An OFDM signal with NfN_f subcarriers indexed by k=0,…,Nfβˆ’1k = 0, \ldots, N_f-1 at frequencies fk=fc+kB/Nff_k = f_c + k B/N_f. The RIS optimized for fcf_c yields per-subcarrier gain Gk=N+βˆ‘nβ‰ mNej(Ξ±nβˆ’Ξ±m)(fkβˆ’fc)G_k = N + \sum_{n \neq m}^N e^{j(\alpha_n - \alpha_m)(f_k - f_c)} which reduces to N2N^2 at k=0k = 0 (center) and falls off as the subcarrier offset grows.

Proof
Signal model

At subcarrier kk, the RIS applies phase Ο•n(fk)=Ο•n0+Ξ±n(fkβˆ’fc)\phi_n(f_k) = \phi_n^0 + \alpha_n (f_k - f_c). Optimal at fcf_c: Ο•n0=βˆ’arg⁑[effectiveΒ channelΒ atΒ fc]\phi_n^0 = -\arg[\text{effective channel at } f_c].

Combining

The signal amplitude at subcarrier kk is Ak=βˆ‘n∣hn(fk)∣ejΟ•n(fk)βˆ’jarg⁑hn(fk)A_k = \sum_n |h_n(f_k)| e^{j\phi_n(f_k) - j\arg h_n(f_k)}. If the channel is approximately flat (narrowband): Akβ‰ˆβˆ‘nejΞ±n(fkβˆ’fc)A_k \approx \sum_n e^{j\alpha_n (f_k - f_c)} β€” a random walk of phasors with angle Ξ±n(fkβˆ’fc)\alpha_n (f_k - f_c).

Power

∣Ak∣2=N+βˆ‘nβ‰ mcos⁑[(Ξ±nβˆ’Ξ±m)(fkβˆ’fc)]|A_k|^2 = N + \sum_{n \neq m} \cos[(\alpha_n - \alpha_m)(f_k - f_c)]. For independent Ξ±n∼Uniform(βˆ’A,A)\alpha_n \sim \text{Uniform}(-A, A): E[∣Ak∣2]=N+N(Nβˆ’1)sinc2(A(fkβˆ’fc))\mathbb{E}[|A_k|^2] = N + N(N-1) \mathrm{sinc}^2(A(f_k - f_c)). At fcf_c: N2N^2. At large offsets: β†’N\to N. β– \blacksquare

Example: 5.8 GHz RIS with 400 MHz Signal Bandwidth

A 5.8 GHz RIS with element bandwidth Bn=300B_n = 300 MHz (5% fractional) is used for a 400 MHz 5G signal. The per-element phase is linear with slope Ξ±n\alpha_n uniformly distributed in [βˆ’Ο€/(Bn/2),Ο€/(Bn/2)][-\pi/(B_n/2), \pi/(B_n/2)]. What's the edge-subcarrier combining loss?

Solution
Parameters

Ξ±max⁑=Ο€/(Bn/2)=2Ο€/Bn=2Ο€/3Γ—108=2.09Γ—10βˆ’8\alpha_{\max} = \pi/(B_n/2) = 2\pi/B_n = 2\pi / 3 \times 10^8 = 2.09 \times 10^{-8} rad/Hz. Edge offset: 200200 MHz. Ξ±max⁑⋅200\alpha_{\max} \cdot 200 MHz =4.18= 4.18 rad.

Combining efficiency

From Theorem 1 (sincΒ² with argument 4.184.18): sinc2(4.18/Ο€)=sinc2(1.33)β‰ˆ0.04\mathrm{sinc}^2(4.18/\pi) = \mathrm{sinc}^2(1.33) \approx 0.04. At edge: Gedge=N+0.04N(Nβˆ’1)β‰ˆ0.04N2G_{\text{edge}} = N + 0.04 N(N-1) \approx 0.04 N^2.

dB loss

10log⁑10(0.04)=βˆ’1410 \log_{10}(0.04) = -14 dB. The edge subcarriers lose 14 dB relative to the center. This is unacceptable for OFDM β€” a wideband RIS design is needed: true-time-delay, switched true-time-delay, or per-subcarrier codebooks.

Definition:

True-Time-Delay (TTD) RIS

A true-time-delay RIS introduces a frequency-flat delay Ο„n\tau_n at each element, so the reflection coefficient is Ξ“n(f)=eβˆ’j2Ο€fΟ„n\Gamma_n(f) = e^{-j 2\pi f \tau_n}, giving Ο•n(f)=βˆ’2Ο€fΟ„n\phi_n(f) = -2\pi f \tau_n, which is linear in ff with slope βˆ’2πτn-2\pi \tau_n. By choosing Ο„n\tau_n to match the channel's differential delay, the RIS achieves coherent combining across a large bandwidth. TTD elements are harder to fabricate (require long transmission lines or high-Q tunable filters), increasing BOM cost by 33-5Γ—5\times.

RIS Beamforming Gain vs. Bandwidth

Plot the RIS-aided SNR as a function of the signal bandwidth BB, for a fixed element bandwidth BnB_n. Compare (i) phase-only narrowband RIS, (ii) TTD-based wideband RIS, (iii) per-subcarrier optimized RIS.

Parameters
256
28
0.05
0.05

Three Strategies for Wideband RIS

  1. TTD elements: expensive per-element but correct physical solution. Used in high-end prototypes.
  2. Per-subcarrier phase optimization: keep narrowband elements but choose NfN_f different Ξ¦\boldsymbol{\Phi} matrices, one per subcarrier. Only possible if subcarriers are independently controllable β€” not the case for most hardware, but digitally-controlled reflectarrays allow it.
  3. Frequency-aware codebook: design a single Ξ¦\boldsymbol{\Phi} that optimizes average subcarrier performance (waterfilling over subcarriers). A compromise between (i) and (ii).
⚠️Engineering Note

Published Wideband RIS Benchmarks

  • 2.4 GHz / 80 MHz (3.3% frac.): phase-only RIS, N=144N=144, flatness within 1 dB across band.
  • 5.8 GHz / 200 MHz (3.4% frac.): phase-only, 2 dB flatness.
  • 28 GHz / 400 MHz (1.4% frac.): phase-only, 1 dB flatness.
  • 28 GHz / 800 MHz (2.9% frac.): requires TTD, 1.5 dB flatness.
  • 100 GHz / 8 GHz (8% frac.): requires TTD, 3-4 dB flatness (state-of-the-art, lab only).

Common Mistake: Don't Assume Narrowband for mmWave and Above

Mistake:

Applying narrowband RIS optimization to a 400 MHz 28 GHz 5G signal.

Correction:

The 400 MHz / 28 GHz = 1.4% fractional bandwidth is mild, but still causes 1-2 dB edge loss. For 800 MHz signals (2.9%), phase-only RIS is no longer sufficient. Always check B/fcB/f_c against the element bandwidth and, if needed, explicitly design for wideband response (TTD or codebook).

Prerequisites & NotationRIS Channel Aging and Control-Loop Design