⚡ electrical engineering

Sub-Sampling for Positioning Privacy in ISAC: Deception by Aliasing via Sparse Arrays and Pilots

This paper proposes a sub-sampling framework for communication-centric ISAC systems that leverages sparse arrays and pilot allocations to induce controlled spatial and frequency aliasing, thereby deceiving unauthorized receivers with ghost targets while preserving legitimate sensing performance.

Original authors: L. Yashvanth, Christos Masouros, Suraj Srivastava, Aditya K. Jagannatham, Lajos Hanzo

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: L. Yashvanth, Christos Masouros, Suraj Srivastava, Aditya K. Jagannatham, Lajos Hanzo

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 the air around us is filled with invisible ripples, like the surface of a pond after a stone is thrown. In the world of wireless technology, we have learned to ride these ripples to do two things at once: talk to our phones and "see" the world around us. This dual superpower is called Integrated Sensing and Communications (ISAC). Think of it like a lighthouse that not only sends a bright beam to guide ships (communication) but also uses the reflection of that light off the water to map the coastline (sensing).

However, there is a catch. Just as a lighthouse beam is visible to anyone with eyes, these wireless ripples can be seen by anyone with a receiver. This creates a privacy problem. If a friendly lighthouse is trying to map a secret island, a sneaky pirate ship nearby could use the same light to figure out exactly where the island is, even if the lighthouse didn't mean for them to know. Scientists have been trying to figure out how to let the friendly ship see clearly while blinding the pirate, without turning off the light or making the friendly ship's view fuzzy. This paper explores a clever trick to solve that puzzle.

The Magic of "Ghost" Maps

The researchers in this paper propose a sneaky solution that turns a common mistake in physics into a superpower for privacy. They call it "deception by aliasing." To understand this, imagine you are trying to take a picture of a fast-moving car with a camera that takes photos very slowly. If the car moves too fast between shots, the camera might think the car is in a different place, or even that there are three cars where there is only one. In the world of signals, this is called "aliasing," and it usually causes blurry, confusing pictures.

Usually, engineers try to fix aliasing to make sure their maps are perfect. But this paper asks a different question: What if we wanted the map to be confusing for the bad guy?

The team suggests a system where the friendly base station (let's call it the "Hero") uses two specific tricks to create "ghost targets" for the "Villain" (an unauthorized listener).

Trick 1: The Sparse Array (The Spaced-Out Antennas)
Imagine the Hero has a row of 16 microphones to listen for echoes. Normally, these microphones are packed tightly together, like teeth in a comb, to catch every detail. The researchers suggest the Hero should unplug some of the microphones, leaving big gaps between the ones that are still on. This is called a "sparse array."

When the Villain tries to listen to the echoes with their own microphones, those big gaps make the sound waves look like they are coming from multiple directions at once. It's like looking at a reflection in a funhouse mirror that splits one person into three. The Villain sees the real target, but also sees two or three "ghost" targets floating in the air at different angles. They can't tell which one is real.

Trick 2: The Sparse Pilots (The Missing Rungs on the Ladder)
Now, imagine the Hero sends a signal that is like a ladder with many rungs (these are called subcarriers). To measure distance, the Hero usually uses every single rung. But the researchers suggest the Hero only uses a few rungs, spaced far apart, leaving the rest empty. This is called "sparse pilots."

When the Villain tries to measure how far away the target is, they only have a few rungs to stand on. Because the rungs are so far apart, the Villain's brain gets confused about how many rungs they actually climbed. They might think the target is 10 meters away, or 20 meters, or 30 meters. Just like with the angles, the distance measurement gets split into a "real" distance and several "ghost" distances.

The Grand Illusion: Hiding the Location

The real magic happens when you combine these two tricks. The Villain is trying to find the target's exact location on a map. To do this, they need to know both the direction (angle) and the distance.

In this new system, the Villain sees a real target at a specific angle and distance. But they also see a whole cloud of "ghost" targets.

One ghost is at the right angle but the wrong distance.
Another is at the right distance but the wrong angle.
A third is at a completely wrong angle and a wrong distance.

Because the Hero's system is designed so that the "ghost" angle and "ghost" distance can mix and match, the Villain ends up with a map full of possible locations. They might think the target is at point A, or point B, or point C. The paper shows that with the right amount of "sparseness" (leaving enough gaps), the Villain can never be sure which point is the real one. It's like trying to find a specific person in a crowd where everyone is wearing a mask and standing in a circle; you can't tell who is who.

The Best Part: The Hero Sees Everything Clearly

You might wonder, "If the Hero is sending these confusing signals, doesn't the Hero get confused too?"

This is the most important finding of the paper: No. The Hero (the authorized receiver) knows the secret code. They know exactly which microphones are on and which rungs of the ladder are missing. Because they are working together with the Hero, they can fill in the gaps and see the real target perfectly. The "ghosts" only appear for the Villain who is trying to guess the pattern without the key.

The researchers ran computer simulations to test this. They found that by using a sparse array with a spacing factor of 16 (meaning they left out a lot of antennas) and a pilot sparsity of 1.5% (meaning they used very few rungs), they could create a "privacy gap" of 80 meters. This means the Villain might think the target is 80 meters away from where it actually is.

What This Means for the Future

The paper proves that this method works without hurting the Hero's ability to talk or sense. The communication speed stays the same, and the Hero's sensing accuracy remains sharp. The only thing that gets blurry is the Villain's view.

The authors suggest that this turns a traditional weakness (aliasing) into a strength. Instead of fighting to make signals perfectly clear for everyone, we can intentionally make them "fuzzy" for the wrong people. It's a clever way to protect privacy in the wireless world, ensuring that while we can all share the same airwaves, only the right people get to see the full picture. The paper concludes that this approach is a promising new tool for keeping our future wireless systems safe, turning the "ghosts" in the machine into guardians of our secrets.

Technical Summary: Sub-Sampling for Positioning Privacy in ISAC

Problem Statement

Integrated Sensing and Communications (ISAC) systems utilize shared hardware and spectrum to simultaneously communicate with user equipment (UE) and sense the environment. While this improves efficiency, it introduces a critical security vulnerability: sensing privacy. Specifically, an unauthorized, passive receiver (Eve) can exploit the signals transmitted by a legitimate Integrated Sensing and Communications Base Station (ISAC-BS) to estimate target parameters (range, angle, and position) without authorization.

Conventional physical-layer security techniques, such as encryption or artificial noise, are ill-suited for this scenario. Encryption fails because Eve can still infer target parameters from the sensing signatures even without decoding the data. Artificial noise or jamming is difficult to implement against a passive, silent Eve whose location and channel characteristics are unknown. Furthermore, existing methods that distort waveforms to confuse Eve often degrade the legitimate sensing performance of the authorized receiver (Alice), creating an undesirable trade-off.

Methodology

The paper proposes a sub-sampling-based sensing-privacy framework for communication-centric (CC)-ISAC systems. Instead of actively distorting the signal or adding noise, the authors exploit the natural phenomenon of aliasing caused by controlled undersampling in the spatial and frequency domains.

Spatial Sub-Sampling (Sparse Arrays): The ISAC-BS employs a Uniform Sparse Array (USA) where the inter-antenna spacing is an integer multiple ( $\eta_s \ge 1$ ) of half the carrier wavelength. This effectively activates only a subset of antennas from a Nyquist-sampled array.
Frequency Sub-Sampling (Sparse Pilots): In the Orthogonal Frequency Division Multiplexing (OFDM) waveform, sensing pilots are allocated periodically across subcarriers with a spacing factor $\eta_f \ge 1$ . This creates a sparse pilot pattern compared to a dense pilot allocation.
Mechanism of Deception:
- For the Unauthorized Receiver (Eve): Eve is assumed to be passive and disconnected from the network. Consequently, she can only utilize the pilot subcarriers for sensing, as she cannot decode the random data payload. Due to the sparse array ( $\eta_s > 1$ ) and sparse pilots ( $\eta_f > 1$ ), Eve's observation of the signal constitutes an undersampled version of the true space-time-frequency signal. This induces controlled aliasing, resulting in "ghost targets" (false peaks) in the Range-Angle Ambiguity Function (AF).
- For the Authorized Receiver (Alice): Alice cooperates with the BS and has access to both the pilot and the data payload. She utilizes the full set of subcarriers (pilots and data) for sensing. Crucially, because Alice operates in a monostatic configuration (collocated with the BS), she estimates the target direction purely from the receive array processing. Since her receive array is a compact Uniform Linear Array (ULA) with standard half-wavelength spacing, her spatial sampling is not undersampled. Consequently, her Ambiguity Function remains unambiguous regardless of the transmit array's sparsity, allowing her to resolve the target parameters correctly.

Key Contributions

The paper provides a rigorous theoretical analysis and numerical verification of this framework:

Closed-Form Ambiguity Function (AF): The authors derive a closed-form expression for the Range-Angle MIMO Ambiguity Function at Eve. They demonstrate that the AF is separable into receive-array, transmit-array, and delay components.
Characterization of Ghost Targets:
- Spatial Domain: It is proven that a sparse transmit array ( $\eta_s > 1$ ) causes the spatial AF to exhibit multiple local maxima. This results in multiple "ghost" Direction of Departure (DoD) estimates for any true target angle.
- Range Domain: Similarly, periodic sparse pilots ( $\eta_f > 1$ ) cause the range AF to exhibit multiple local maxima, leading to multiple "ghost" Time-of-Flight (TOF) estimates.
Positioning Privacy Condition: The paper establishes a sufficient condition (Theorem 3) under which the combination of spatial and range ambiguities translates into positioning ambiguity. It proves that for sufficiently high sub-sampling factors, the intersection of ambiguous DoD rays and ambiguous bistatic range ellipses creates multiple feasible target locations, preventing Eve from uniquely determining the target's position.
Preservation of Legitimate Performance:
- Sensing: The framework preserves the native sensing performance of Alice. Because Alice uses the full dataset (pilot + data) and operates in a monostatic mode with a standard receive array, her AF remains unambiguous regardless of the sub-sampling factors.
- Communication: The paper analyzes the impact on communication throughput. It shows that while frequency sub-sampling reduces pilot overhead (increasing the pre-log factor), it may degrade channel estimation accuracy on data subcarriers. However, the authors argue that this trade-off is inherent to OFDM systems and is not an additional penalty introduced by the privacy mechanism. Spatial sub-sampling does not negatively impact the communication throughput of i.i.d. MIMO channels.

Numerical Results

Monte Carlo simulations validate the theoretical findings:

Ambiguity Generation: With $\eta_s = 3$ and $\eta_f = 5$ , the Ambiguity Functions at Eve exhibit multiple peaks of equal magnitude, making it impossible to distinguish the true target from ghost targets.
RMSE Analysis: The Root Mean Square Error (RMSE) for target estimation at Eve increases monotonically with $\eta_s$ and $\eta_f$ , eventually saturating at a high error floor. In contrast, Alice's RMSE remains low and unaffected by these factors.
Positioning Gap: The simulations demonstrate a significant "positioning privacy gap." For specific parameters (e.g., $\eta_s=6, \eta_f=64$ ), the system creates a positioning error gap of approximately 40 to 80 meters between the true target location and the estimates made by Eve, while Alice maintains accurate positioning.
Communication Throughput: The achievable sum-rate at the UE remains stable or slightly improves with moderate pilot sparsity due to reduced overhead, confirming that the privacy mechanism does not inherently degrade communication performance.

Significance and Claims

The paper claims to offer a novel approach to ISAC security that transforms a traditional limitation (aliasing due to undersampling) into a useful mechanism for physical-layer sensing privacy.

Deception by Aliasing: The core significance lies in the ability to "deceive" an unauthorized receiver by inducing controlled aliasing. This is achieved passively through waveform and array design rather than active interference.
No Performance Trade-off: A primary claim is that this privacy mechanism does not require a sacrifice in legitimate system performance. Alice's sensing and the system's communication capabilities are preserved, provided the sub-sampling factors are chosen within reasonable bounds.
Robustness: The solution is effective regardless of Eve's location or channel state, as it relies on the structural properties of the transmitted signal rather than knowledge of the adversary.
Fundamental Shift: The work suggests a paradigm shift where sparse arrays and sparse pilots, traditionally viewed as resources-saving techniques, can be strategically deployed to secure the sensing functionality of future wireless systems.