Capacity of Two-User Wireless Systems Aided by Movable Signals

Movable signals have emerged as a third approach to enable smart radio environments (SREs), complementing reconfigurable intelligent surfaces (RISs) and flexible antennas. This paper investigates their potential to enhance multi-user wireless systems. Focusing on two-user systems, we characterize the capacity regions of the multiple access channel (MAC) and broadcast channel (BC). Interestingly, movable signals can dynamically adjust the operating frequency to orthogonalize the user channels, thereby significantly expanding the capacity regions. We also study frequency optimization, constraining it in a limited frequency range, and show that movable signals provide up to 45% sum rate gain over fixed signals.

💡 Research Summary

This paper introduces “movable signals” as a third paradigm for smart radio environments (SREs), complementing reconfigurable intelligent surfaces (RIS) and flexible antennas. While RIS manipulates electromagnetic properties and flexible antennas adjust spatial distances, movable signals exploit the frequency domain: the carrier frequency can be dynamically shifted over a wide band to reshape the wireless channel. The authors focus on a two‑user scenario and rigorously characterize the capacity regions of both the multiple‑access channel (MAC) and the broadcast channel (BC) when the two users share the same frequency.

The system model assumes a line‑of‑sight (LoS) environment typical of millimeter‑wave and terahertz bands. A base station equipped with an N‑element uniform linear array (ULA) serves two single‑antenna users. The uplink channel vectors h_ul,1 and h_ul,2 depend on the wavelength λ = c/f, the inter‑element spacing d_A, the users’ distances d₁, d₂, and their angular positions θ₁, θ₂. The downlink channels are taken as the transpose of the uplink channels under reciprocity.

For the MAC, the authors first derive single‑user SIMO bounds R_k ≤ log₂(1+P_k N/σ²) and a sum‑rate bound based on a two‑antenna cooperative MIMO model. They show that the sum‑rate bound is maximized when the two channel vectors are orthogonal, i.e., h_ul,1ᴴ h_ul,2 = 0. By explicitly writing the inner product as a function of λ, they obtain the orthogonalization condition:

 λ* = N·d_A·|sinθ₁ – sinθ₂| / L, L ∈ ℕ

or equivalently

 f* = L·f_A·N·|sinθ₁ – sinθ₂|, f_A = c/d_A.

The condition requires θ₁ ≠ θ₂; if the users are perfectly aligned, orthogonalization is impossible regardless of frequency. When f* is selected, the two channels become perfectly orthogonal, and the MAC capacity region collapses to a rectangle bounded only by the single‑user SIMO limits. The authors propose a simple matched‑filter receiver (z = Hᴴ y) that achieves these rates.

For the BC, a similar analysis shows that with the same orthogonalizing frequency f*, the base station can allocate powers P₁ and P₂ (P₁+P₂ = P) to the two users, precoding each symbol with the conjugate of its channel (x_dl = (1/√N) Hᴴ s). Because the channels are orthogonal, each user experiences no inter‑user interference, and its achievable rate is R_k = log₂(1+P_k N/σ_k²). The boundary of the BC capacity region is therefore described by the pair of inequalities (25)–(26), which are tight under orthogonalization. Maximizing the sum rate under equal noise variances leads to the optimal power split P₁ = P₂ = P/2, yielding a sum‑rate C_dl = 2·log₂(1+P N/σ²).

Recognizing that practical systems cannot select an arbitrary frequency, the paper formulates a constrained optimization problem: choose f ∈