⚡ electrical engineering

Zak-OTFS ISAC with Bistatic Sensing via Semi-Blind Atomic Norm Denoising Scheme

This paper proposes a semi-blind atomic norm denoising scheme for Zak-OTFS-based ISAC systems that enables accurate bistatic sensing and robust communication in high-mobility, doubly dispersive environments by formulating joint channel estimation and data detection as a constrained optimization problem solved via a rigorously proven accelerated iterative algorithm.

Original authors: Kecheng Zhang, Weijie Yuan, Maria Sabrina Greco

Published 2026-08-05

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CC BY 4.0

Original authors: Kecheng Zhang, Weijie Yuan, Maria Sabrina Greco

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 trying to have a conversation with a friend while you are both speeding past each other on a rollercoaster, shouting over the roar of the wind. In the world of wireless communication, this is exactly what happens when data travels to fast-moving objects like cars or drones. The signals get stretched and squashed, turning clear messages into a garbled mess. For decades, engineers have used a technique called OFDM to keep these conversations clear, but it's like trying to catch a slippery fish with a net that has holes in it; when things move too fast, the signal slips right through. Enter a newer, more robust idea called OTFS (Orthogonal Time Frequency Space). Instead of sending messages in a straight line, OTFS maps them onto a grid that represents both time and speed (Doppler). It's like drawing your message on a rubber sheet; even if the sheet stretches and twists, the drawing stays intact. This makes OTFS perfect for high-speed scenarios, but it also opens up a new superpower: because the signal is so sensitive to movement, the same signal used for talking can also be used to "see" objects, acting like a radar. This dual-purpose setup is called Integrated Sensing and Communication (ISAC). However, there's a catch: if the object isn't sitting perfectly on a pre-drawn grid line, the signal blurs, making it hard to know exactly where the object is or what message it received.

This paper tackles that exact blur problem. The authors, working with a specific version of OTFS called Zak-OTFS, propose a clever new way to untangle the mess. They treat the problem like a game of "guess the hidden picture" where the picture is made of a few sharp points (the targets) and a few letters (the data), but the image is smeared by fractional shifts. Instead of guessing blindly, they use a mathematical trick called "atomic norm denoising." Imagine trying to clean a muddy window to see a few distinct stars behind it. Most methods would try to guess where the stars are on a fixed grid, which fails if the stars are slightly off-center. This paper's method is like having a flexible lens that can zoom in and find the stars exactly where they are, even if they are floating between the grid lines. They also have to figure out the secret message being sent at the same time. To do this, they use a "semi-blind" approach: they know the pilot signals (like a known watermark) but not the random data. They formulate a complex math puzzle that forces the solution to look like a valid message (using a "negative square penalty" to push the answer toward the correct shapes) while simultaneously sharpening the image of the targets.

The paper doesn't just propose this idea; they build a fast, iterative algorithm to solve it. Think of it as a team of detectives who take turns refining their theory: one detective sharpens the image of the targets, and the next uses that sharper image to decode the message, then they swap roles and repeat until the picture is crystal clear. The authors prove mathematically that this back-and-forth process will eventually settle on a stable answer. In their computer simulations, this new method works incredibly well. It achieves "super-resolution," meaning it can spot targets much closer together than older methods could, and it decodes messages with very few errors, performing almost as well as if the receiver already knew the exact channel conditions perfectly. They tested this with scenarios involving multiple targets and different types of data, showing that even when the targets are moving and the signal is messy, their method can separate the "who" (the target) from the "what" (the message) with high precision.

However, the paper is careful to note that this isn't a magic wand for every situation. The method relies on the fact that the environment is "sparse," meaning there are only a few dominant targets or paths, not a dense forest of reflections. If the scene is too crowded or the targets are too close together, the method might struggle to tell them apart. Also, the results shown are based on simulations, not real-world field tests yet. The authors also point out that while their method works great for certain types of simple message shapes (like PSK), it might need adjustments for more complex, high-definition message formats. But for the specific challenge of high-speed, dual-purpose communication and sensing, this "semi-blind" approach offers a significant step forward, turning a blurry, confusing signal into a clear, actionable picture.

Technical Summary: Zak-OTFS ISAC with Bistatic Sensing via Semi-Blind Atomic Norm Denoising

1. Problem Statement

This paper addresses the challenge of Integrated Sensing and Communication (ISAC) in high-mobility scenarios using Zak-transform-based Orthogonal Time Frequency Space (Zak-OTFS) modulation. In bistatic ISAC configurations, where the transmitter and receiver are spatially separated, accurate channel estimation is critical for both reliable data detection and precise target parameter estimation (delay, Doppler, and channel gain).

The primary difficulties identified are:

Fractional Delays and Doppler Shifts: In practical high-mobility scenarios, target parameters often do not align with the discrete grid of the Delay-Doppler (DD) domain. These "off-grid" shifts cause channel responses to spread across multiple DD bins, leading to grid mismatch errors.
Strong Coupling: The off-grid nature creates a strong coupling between channel estimation and data detection. Errors in symbol detection act as interference, degrading channel estimation, while inaccurate channel estimates impair symbol detection.
Limitations of Existing Methods:
- Two-step approaches (estimating the effective channel matrix first) rely on ideal bi-orthogonal pulses or shift-invariant structures that are not physically realizable with practical rectangular pulses.
- Direct parameter estimation methods often rely on discretized grids (e.g., Sparse Bayesian Learning), which suffer from resolution limits and high computational complexity when grid resolution is increased.
- Existing solutions generally fail to jointly perform off-grid parameter estimation and data detection without restrictive assumptions (e.g., integer delays or known path counts).

2. Methodology

A. System Model and Discrete-Time I/O Relation

The authors first derive a discrete-time input-output (I/O) matrix relationship for Zak-OTFS systems employing rectangular time and frequency windows.

Zak-OTFS Framework: The system utilizes the inverse Zak transform to map DD-domain symbols to the time domain and the Zak transform to recover the received DD-domain signal.
Twisted Convolution: The channel propagation is modeled as a twisted convolution. The authors demonstrate that even with practical rectangular pulses, the system retains the "predictability property," meaning all DD-domain symbols experience the same effective channel response.
Discrete Formulation: A discrete-time matrix I/O relation is established, bridging the gap between continuous-time theory and digital implementation. This model confirms that model-free channel estimation (using a single pilot) remains a valid initialization strategy.

B. Problem Formulation: Semi-Blind Atomic Norm Denoising

The joint estimation of off-grid channel parameters and unknown data symbols is formulated as an optimization problem.

Atomic Norm Regularization: To handle the off-grid nature of delays and Doppler shifts without discretization, the channel vector is regularized using the atomic norm. This exploits the inherent sparsity of the DD-domain channel.
Handling Discrete Constraints: The data symbols belong to a discrete constellation (e.g., PSK). To avoid the NP-hard nature of mixed-integer programming, the authors employ a penalty-based reformulation.
- The discrete constraint is relaxed to its convex hull.
- A negative square penalty ( $-\rho \|x\|_2^2$ ) is introduced to the objective function. This concave penalty encourages the solution to move toward the vertices of the convex hull (the actual discrete symbols) as the penalty parameter $\rho$ increases.
Objective: The resulting problem minimizes the reconstruction error, the negative square penalty for symbol discreteness, and the atomic norm for channel sparsity.

C. Accelerated Iterative Algorithm

To solve the non-convex, coupled optimization problem, an accelerated iterative algorithm is developed based on the Majorization-Minimization (MM) framework and coordinate descent.

Initialization: Uses model-free channel estimation (from the pilot) to initialize the channel and LMMSE for initial symbol detection.
Channel Update: An inexact accelerated proximal gradient method is used. Instead of solving the atomic norm subproblem via expensive Semi-Definite Programming (SDP), the algorithm utilizes a coordinate descent framework that exploits the origin-symmetry of the atomic set. This involves an oversampled grid search followed by Newton-based local refinement to find the optimal off-grid parameters.
Symbol Update: An accelerated projected gradient method updates the data symbols. The negative square penalty is handled via a majorization step, followed by projection onto the convex hull of the constellation.
Homotopy Technique: The penalty parameter $\rho$ is gradually increased (homotopy) to track the solution path from the relaxed convex problem to the discrete solution, ensuring convergence to a valid discrete constellation point.

D. Convergence Analysis

The paper provides a rigorous theoretical analysis proving that the proposed algorithm converges to an $\epsilon$ -stationary point with a sublinear stationarity rate of $O(1/\sqrt{t})$ .

3. Key Contributions

Discrete-Time I/O Model: Derivation of a discrete-time matrix I/O relation for Zak-OTFS with rectangular windows, proving the validity of the predictability property for initialization.
Unified Optimization Framework: A novel semi-blind formulation that jointly recovers off-grid channel parameters and PSK symbols using atomic norm denoising combined with a negative square penalty for discrete constraints.
Efficient Algorithm: Development of an accelerated iterative algorithm combining MM, inexact accelerated proximal gradient, and accelerated projected gradient methods, avoiding the high complexity of SDP-based solvers.
Convergence Proof: Establishment of theoretical convergence guarantees to an $\epsilon$ -stationary point.

4. Simulation Results

The proposed scheme was evaluated via Monte Carlo simulations in a bistatic ISAC scenario with multiple targets and no Line-of-Sight (LoS) link.

Convergence: The accelerated algorithm demonstrated significantly faster convergence and lower computational burden (fewer inner iterations) compared to ordinary (non-accelerated) schemes, particularly when penalty parameters were updated rapidly.
Communication Performance: The Bit Error Rate (BER) performance approached the theoretical lower bound of perfect Channel State Information (CSI). The method outperformed standard LMMSE detectors, especially in multi-target scenarios, by effectively exploiting DD-domain diversity.
Sensing Performance: The algorithm achieved super-resolution sensing, accurately estimating off-grid delay and Doppler parameters. It demonstrated reliable multi-target detection with high probability of detection and low false alarm rates at sufficient SNR.
Resolution Limits: The paper notes that while the method enables off-grid estimation, its ability to resolve arbitrarily close targets is limited by the number of targets and their minimum separation, as well as potential symbol detection errors in the semi-blind setting.

5. Significance and Claims

The paper claims to provide a unified framework for reliable OTFS-ISAC over doubly dispersive channels without relying on restrictive assumptions (such as integer delays or known path counts).

Significance: It bridges the gap between theoretical Zak-OTFS models and practical implementation by addressing the critical issues of fractional shifts and the coupling between sensing and communication.
Modest Claims: The authors explicitly state that the proposed algorithm is designed for PSK-modulated symbols and that extending it to high-order QAM constellations (where the negative square penalty may not be equivalent to the original problem) is left for future work. They also acknowledge that the method is most suitable for sparse ISAC scenarios with a few dominant targets, as computational complexity scales with the square of the number of active atoms.

In summary, the paper presents a mathematically rigorous and computationally efficient solution for joint sensing and communication in high-mobility environments, achieving performance close to perfect CSI benchmarks while enabling super-resolution parameter estimation.