⚡ electrical engineering

Capacity of Two-User Wireless Systems Aided by Movable Signals

This paper investigates movable signals as a third approach for smart radio environments, demonstrating that in two-user line-of-sight systems, they can dynamically adjust operating frequencies to orthogonalize channels and significantly expand capacity regions, achieving up to a 45% sum rate gain.

Original authors: Matteo Nerini, Bruno Clerckx

Published 2026-07-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matteo Nerini, Bruno Clerckx

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to shout a secret message to a friend across a noisy, crowded room. In the world of wireless communication, this "room" is the air around us, and the "shouting" is radio signals. For decades, engineers have tried to make these signals clearer by building smarter walls (like Reconfigurable Intelligent Surfaces) or by wiggling the antennas around (like Flexible Antennas). These methods try to shape the room itself or move the speaker. But there is a third, slightly magical way to do it: instead of moving the speaker or the room, you simply change the pitch of your voice.

This is the world of "movable signals." Think of it like a radio station that can instantly slide its frequency up or down the dial, not just to find a quiet spot, but to make its voice sound completely different to different people. If two people are trying to listen to the same broadcast, usually their voices get mixed up, creating a messy jumble of interference. But if you can tune the signal's "pitch" just right, you can make the signal sound like a clear, distinct note to one person while sounding like a completely separate, non-interfering path to the other. This paper explores whether this "pitch-shifting" trick can let two people talk to a base station at the same time without stepping on each other's toes, even when they are in a direct line of sight.

The Magic of the Moving Pitch

In this paper, researchers Matteo Nerini and Bruno Clerckx investigate a clever idea: what if we could make wireless channels "orthogonal" just by changing the frequency? In plain English, "orthogonal" means "at right angles." If two signals are orthogonal, they don't interfere with each other at all, like two people walking on perpendicular paths who never bump into one another.

Usually, when two users try to talk to a base station at the same time (a setup called a Multiple Access Channel, or MAC), or when a base station talks to two users (a Broadcast Channel, or BC), their signals get tangled. The base station has to work hard to untangle them, often losing speed in the process. The authors show that by using "movable signals"—where the operating frequency can be dynamically adjusted within a wide range—we can tune the system so that the two users' channels become perfectly orthogonal.

The Big Discovery: Tuning the Frequency
The paper proves that if you pick the exact right frequency, you can make the signal for User 1 completely non-interfering to User 2, and vice versa. It's like having a magic tuning fork that, when struck at a specific note, makes a glass sing for one person but ensures the sound travels on a path that doesn't disturb the other person.

The authors derived a specific formula for this "magic frequency." It depends on the angle of the users relative to the base station and the spacing of the antennas. If the two users are in different directions (which is usually the case), there is a specific frequency (or a set of frequencies) where the math works out perfectly: the signals are arranged so they cancel out interference for the "wrong" user and amplify for the "right" one. When this happens, the system reaches its maximum possible speed, known as the "capacity region."

The Uplink and Downlink
The paper looks at two scenarios:

The Uplink (MAC): Two users sending data to the base station. The authors show that by shifting the frequency, the base station can separate the two signals perfectly, achieving the highest possible total speed.
The Downlink (BC): The base station sending data to two users. Here, the base station uses a technique called "matched beamforming" (aiming the signal like a laser) combined with the special frequency. This ensures User 1 gets their message loud and clear, while User 2 receives their own message without any interference from User 1's part of the signal.

What Happens When We Can't Tune Freely?
In the real world, we can't just pick any frequency; we are usually stuck within a specific band, like a radio dial that only goes from 100 to 180. The paper asks: "What if we can't find the perfect 'magic' frequency?"

The researchers simulated what happens when the frequency is constrained to a range where the perfect cancellation isn't possible. They found that even in this imperfect situation, movable signals are still a game-changer. By picking the "least bad" frequency within the allowed range, the system still performs much better than if the frequency were fixed.

The Numbers
The simulations in the paper show some impressive gains. When the signal-to-noise ratio is 10 dB and there are 2 antennas at the base station, using movable signals with a constrained frequency range improves the total data speed by 31% compared to using a fixed frequency. If we could use the perfect, unconstrained frequency, the speed would be 45% higher than the fixed system.

What This Means
The paper concludes that movable signals offer a powerful third way to build "Smart Radio Environments." Unlike other methods that require moving physical hardware or changing the properties of walls, this approach just requires the software to change the frequency. It's a lightweight, flexible way to make wireless networks faster and cleaner. The authors suggest that while this works great for two users in a direct line of sight, future work will need to see if it holds up in more complex, crowded, or non-line-of-sight scenarios. But for now, it's a strong proof that sometimes, the best way to fix a noisy connection is to just change your tune.

Technical Summary: Capacity of Two-User Wireless Systems Aided by Movable Signals

Problem Statement
Smart Radio Environments (SREs) aim to treat the wireless channel as an optimization variable to enhance communication performance. While Reconfigurable Intelligent Surfaces (RISs) manipulate the electromagnetic (EM) properties of the environment and flexible antennas adjust spatial distances, a third approach utilizing the frequency domain has emerged: movable signals. Unlike conventional systems where frequency is selected within a narrow range, movable signals allow the operating frequency to be reconfigured across a wide spectrum (e.g., $f_{max} = 1.8f_{min}$ ) to reshape the channel.

This paper addresses a gap in the literature regarding the fundamental limits of two-user wireless systems aided by movable signals. While previous works demonstrated that movable signals could steer beams for single receivers or assign distinct frequencies to users, the potential of serving multiple users on the same frequency by dynamically reconfiguring that frequency to orthogonalize channels remains under-explored. The specific problem investigated is the characterization of the capacity regions for the Multiple Access Channel (MAC) and Broadcast Channel (BC) in Line-of-Sight (LoS) scenarios, where a multi-antenna base station serves two single-antenna users.

Methodology
The authors model a system where a Uniform Linear Array (ULA) at the base station communicates with two users in LoS. The channel response depends on the signal wavelength $\lambda$ (and thus frequency $f$ ), the antenna spacing $d_A$ , and the users' angles of arrival/departure $\theta_k$ .

Channel Orthogonalization Analysis: The core theoretical contribution involves deriving the specific frequency conditions under which the channel vectors of the two users, $\mathbf{h}_{ul,1}$ and $\mathbf{h}_{ul,2}$ , become orthogonal ( $\mathbf{h}_{ul,2}^H \mathbf{h}_{ul,1} = 0$ ). By modeling the channel as a geometric series, the authors derive a closed-form expression for the optimal frequency $f^\star$ that satisfies this orthogonality condition.
Capacity Region Characterization:
- Uplink (MAC): The capacity region is bounded by individual user rates and the sum rate. The authors demonstrate that if the frequency is set to $f^\star$ such that channels are orthogonal, the sum rate constraint becomes equivalent to the sum of individual single-user SIMO capacities, effectively eliminating inter-user interference.
- Downlink (BC): Similarly, for the broadcast channel, the authors show that orthogonal channels allow for matched beamforming (precoding) where each user receives their signal without interference from the other stream. The capacity region boundary is achieved by varying power allocation between users while operating at the orthogonalizing frequency.
Constrained Frequency Optimization: Recognizing that practical systems operate within a limited frequency range $[f_{min}, f_{max}]$ , the paper formulates an optimization problem to minimize the cosine similarity (interference) between channels within this interval. The solution involves checking if an ideal orthogonalizing frequency exists within the range; if not, the optimal frequency is selected from the boundaries ( $f_{min}$ or $f_{max}$ ) based on which minimizes the interference metric $|\sin(Nf\beta)/\sin(f\beta)|$ .

Key Contributions

Fundamental Limits: The paper establishes the capacity regions for two-user MAC and BC systems aided by movable signals in LoS environments.
Orthogonalization Mechanism: It proves that movable signals can dynamically adjust the operating frequency to mathematically orthogonalize user channels, a capability not present in conventional fixed-frequency systems.
Achievable Strategies: The authors provide specific transmission strategies (matched filtering for uplink, matched beamforming for downlink) that allow the system to exactly achieve the derived capacity region boundaries.
Practical Constraints: The study extends theoretical results to practical scenarios with limited frequency ranges, providing an optimization algorithm to select the best available frequency.

Results
Numerical results, based on a BC scenario with $d_1=d_2=10$ m and angles uniformly distributed, highlight the following:

Capacity Gains: Movable signals significantly expand the capacity region compared to fixed signals. In a scenario with $N=2$ antennas and an SNR of 10 dB, movable signals with a constrained frequency range ( $f_{min}$ to $1.8f_{min}$ ) achieve a sum rate 31% higher than fixed signals.
Approaching Capacity: Even with the frequency constraint, movable signals nearly achieve the theoretical sum-rate capacity (which assumes unconstrained frequency). The unconstrained capacity is approximately 45% higher than the sum rate of fixed signals in the same scenario.
Antenna Scaling: The performance gain of movable signals diminishes as the number of base station antennas ( $N$ ) increases. This is attributed to the fact that with a large number of antennas, user channels tend to orthogonalize naturally due to spatial separation, reducing the relative benefit of frequency-based orthogonalization.
SNR Dependence: The gain provided by movable signals increases with SNR, as their interference suppression capabilities are most effective in interference-limited regimes.
Computational Cost: The frequency reconfiguration strategy requires negligible computational overhead as it relies on closed-form expressions.

Significance and Claims
The paper claims that movable signals offer a distinct and appealing third approach to enabling Smart Radio Environments. Unlike RISs, they do not require reconfiguring the EM properties of surrounding surfaces, and unlike flexible antennas, they do not require physical repositioning of hardware. Instead, they exploit the frequency selectivity of the environment.

The authors emphasize that while conventional frequency allocation over a narrow range cannot meaningfully alter LoS channels, movable signals enable channel orthogonalization and substantial capacity gains. The work fills a critical gap by showing how to simultaneously serve multiple users on the same frequency by treating the frequency itself as a dynamic optimization variable, thereby transforming the channel characteristics to eliminate interference. Future work is suggested to extend these findings to non-LoS scenarios, more than two users, and systems with hardware impairments.

The Magic of the Moving Pitch

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