Grid-Shaped Pilot Signals

• Grid-shaped pilot signals are multi-dimensional patterns arranged in 2D grids (e.g., time–frequency, delay–Doppler) that enable accurate estimation, synchronization, and sensing.
• They utilize optimized sequence design, grid mapping, and controlled pilot spacing to minimize estimation errors and manage trade-offs between resource overhead and ambiguity.
• Advanced detection algorithms, including 2D correlation, Kalman filtering, and convex optimization, harness these pilots to enhance performance in systems like ISAC, massive MIMO, and optical channels.

A grid-shaped pilot signal refers to any multi-dimensional (typically 2D) pattern of known symbols embedded in a transmission or sensing resource grid to facilitate tasks such as channel estimation, synchronization, sensing, or geometric parameter recovery. Such pilot signals are arranged on a regular or systematically optimized lattice across the underlying domain, be it time–frequency (OFDM, OTFS), channel–time (optical/multichannel systems), delay–Doppler (ISAC/OTFS), spatial–temporal grids (mmWave MIMO), or Cartesian pixel-planes (image watermarking). Their structure and placement critically influence estimation error, SINR, resource allocation, ambiguity properties, and fundamental limits such as Cramer–Rao bounds.

1. Construction Principles of Grid-Shaped Pilots

The construction of grid-shaped pilots depends on both the domain (e.g., delay–Doppler, time–frequency, channel–time) and the desired operational properties—autocorrelation, orthogonality, coverage, and hardware constraints.

• Kronecker Structure and Autocorrelation: In the delay–Doppler (DD) domain, “thumbtack” pilots are crafted as Kronecker products of sequences with nearly ideal cyclic autocorrelation (e.g., P = a ⊗ b, with a, b chosen for low off-peak correlation). Cyclic shifts of P yield nearly orthogonal sensing templates, ensuring sharp peaks in 2D correlation for precise delay/Doppler localization (Yuan et al., 2023, Yuan, 2024, Du et al., 31 Dec 2025).
• Resource Grid Mapping: These DD grids are mapped into the time–frequency (TF) plane via the inverse symplectic finite Fourier transform (ISFFT), resulting in a full- or sparsely-populated TF grid. The resulting pilot occupies all or selected REs, with adaptable density and cohabitation with data symbols (Yuan et al., 2023, Yuan, 2024, Du et al., 31 Dec 2025).
• Optimal Channel/Time Grids: In optical or multichannel systems, grid-shaped pilot distributions are optimized via dynamic programming or surrogates. Structure-aware pilot positions (e.g., combined staggered + cyclically shifted 2D grids) minimize phase-noise or channel estimation MSE by balancing spatial and temporal separation (Alfredsson et al., 2020, Zhu et al., 7 Apr 2026).
• Pilot Spacing and Scheduling: For OFDM in fast-varying A2G or doubly dispersive channels, grid-shaped pilots are placed on periodic or adaptively-optimized rectangular lattices, with the spacing in frequency and time (Δpf, Δpt) tightly controlled depending on Doppler/depth spread, SNR, and delay statistics (Rao et al., 2018, Yu et al., 4 May 2026, Ozturk et al., 26 Mar 2025).

2. Functional Mechanisms and Detection Algorithms

Grid-shaped pilots underpin a diverse set of estimation and detection algorithms, with architectures exploiting both their geometric grid structure and deterministic sequence properties.

• 2D Correlation for Sensing: In ISAC/OTFS, the receiver forms 2D correlations between the received DD grid and hypothesized pilot shifts (e.g., z(l, k) = ⟨R, Ξ* ⊙ P_{[l,k]}⟩), leading to unambiguous delay/Doppler peaks and robust target detection, even with shared resources and low pilot power (Yuan et al., 2023, Yuan, 2024).
• Carrier-Phase and Channel Tracking: In joint-channel CPE for optical links, grid-shaped pilot distributions enable Kalman smoothing and extended Kalman filtering for phase noise estimation, with heuristic #4 (staggered/cyclic) shown to approach numerically optimized performance (Alfredsson et al., 2020).
• A-Optimal LMMSE Channel Estimation: In finite OFDM blocks over doubly dispersive channels, grid-shaped pilot placement is framed as an A-optimal sensor selection, solved via convex relaxation + randomized rounding or greedy swap, minimizing MSE (tr A^{-1}) over the resource grid (Yu et al., 4 May 2026).
• Sparse Grids and Compressed Sensing: In TDD multi-user settings, pilots are allocated via mixed-integer programming that minimizes worst-case coverage radius and limits collinear pilot arrangements, improving latest-slot sparse DD recovery (Zhu et al., 7 Apr 2026).
• POMDP-Driven Sequential Design: For mmWave MIMO, pilot beams select grid columns according to belief-state policies to maximize sparse path discovery, efficiently exploring 2D angular space (Seo et al., 2014).

3. Trade-offs: Resource Allocation, Ambiguity, and Scalability

Designing grid-shaped pilots involves inherent trade-offs between estimation accuracy, spectral/temporal resource overhead, unambiguous operating regions, and hardware or protocol constraints.

• Pilot Power vs. Data Power: Underlaid or superimposed pilots typically carry a small fraction of power (e.g., P_p/P_d ≈ 0.2). Experimental and analytic results show that this is sufficient for high sensing SINR and negligible BER or channel MSE degradation in communication (Yuan et al., 2023, Yuan, 2024, Du et al., 31 Dec 2025).
• Pilot Density vs. CRB: For 2D time–frequency regular pilot grids, reducing pilot spacing in frequency (n_p) proportionally decreases range CRB as 1/n_p², and reducing time spacing (m_p) decreases velocity CRB as 1/m_p². The same total overhead can be flexibly allocated depending on which estimation metric dominates (Ozturk et al., 26 Mar 2025).
• Unambiguous Estimation Limits: The maximum resolvable range Rmax and velocity vmax scale inversely with the pilot spacing in frequency and time, respectively. Dense grids increase ambiguity windows, but incur higher overhead (Ozturk et al., 26 Mar 2025).

Pilot Spacing (n_p, m_p)	Relative CRB_range	Relative CRB_vel	R_max	v_max
(2, 2)	∝1/4	∝1/4	c/(2Δf)	c/(4f_c T_s)
(5, 2)	∝1/25	∝1/4	c/(5Δf)	c/(4f_c T_s)
(2, 5)	∝1/4	∝1/25	c/(2Δf)	c/(10f_c T_s)

• Scalability and Frame Structure: Grid pilots constructed in the DD domain can be scaled (by adjusting M, N) to achieve desired delay/Doppler resolutions without disturbing the underlying OFDM numerology or cyclic prefix structures. Sparsification further enables adaptation to coarser requirements, reducing PAPR and interference (Yuan et al., 2023, Yuan, 2024).

4. Domains of Application

Grid-shaped pilots are deployed in multiple domains, each with domain-specific motivations, constraints, and metrics:

• Integrated Sensing and Communication (ISAC): Joint radar-communications systems use DD–TF grid pilots to enable simultaneous sensing and data transfer. Pilots provide thumbtack ambiguity, O(MN) complexity 2D correlation detection, and serve as communication reference signals (Yuan et al., 2023, Yuan, 2024, Du et al., 31 Dec 2025).
• Massive MIMO/Beamspace/Spatial Grids: Pilot beam probing in mmWave exploits 2D angular grids, with sequential or adaptive grid beam allocation maximizing sparse virtual channel estimation (Seo et al., 2014).
• Optical Multichannel Transmission: Channel–time grid optimization in carrier-phase estimation yields large reductions in MSE/AIR if staggered, cyclic pilot grids are used versus time-aligned grids (Alfredsson et al., 2020).
• OFDM/OTFS over Doubly Dispersive Channels: Pilot pattern minimization for LMMSE estimation leverages time–frequency grid adaptation, convex and greedy algorithms leveraging channel covariance eigenstructure (Yu et al., 4 May 2026, Rao et al., 2018).
• Image Synchronization and Watermarking: In geometric watermark synchronization, a grid lattice (distinct vertical/horizontal encodings) is embedded, and geometric distortions are estimated analytically from Radon transform peaks, enabling robust matrix inversion even after cropping (Kawano et al., 26 Jan 2026).

5. Optimization and Algorithmic Design

Pilot allocation over a grid is fundamentally an optimization problem, with methodology adapted to system requirements, tractability, and operational domain.

• A-Optimality and Integer Programming: For LMMSE MSE minimization under channel covariance constraints, the pilot allocation is A-optimal sensor selection. Convex relaxation (SDP) with randomized rounding and local swap refinement yields patterns closely tracking channel eigenvalue structure and outperforming standard rectangular and diamond grids (Yu et al., 4 May 2026).
• Mixed-Integer MILP (MCC Formulation): To simultaneously minimize coverage radius and collinearity, pilots are assigned to maximize minimum grid coverage and minimize repeated modular line occupancy (coherence), subject to fairness and contiguous-subband restrictions. The MILP is numerically tractable for k ≤ 20 (Zhu et al., 7 Apr 2026).
• Adaptive and Feedback-Based Updates: For time-varying A2G channels, pilot spacings are dynamically optimized based on real-time Doppler/rms delay estimation, with low feedback and complexity overhead (Rao et al., 2018).
• POMDP in Beamspace: Sparse-path mmWave MIMO tracking is recast as a POMDP, with reward-maximizing beam probing sequences driven by Markov beliefs and optimized by DP or point-based approximation (Seo et al., 2014).

6. Performance Gains, Benchmarks, and Guidelines

Grid-shaped pilot design yields quantifiable performance gains across estimation error, achievable information rate, ambiguity suppression, and system resource efficiency.

• ISAC/OTFS: Concentrating pilot energy into a single DD-cell produces an impulse-like ambiguity function, maximizing channel estimation SINR and minimizing ISL, thereby enlarging the radar-communications performance region (ISL suppression ≥ 9.45 dB, SINR gain ≥ 4.82 dB over cluster/flat pilots) (Du et al., 31 Dec 2025).
• Carrier-Phase Estimation (Optical): Cyclically shifted, staggered pilot designs reduce MSE by up to 90% over uniform rectangular grids, and increase AIR by up to 0.41 bits/symbol for high-order QAM under typical linewidths (Alfredsson et al., 2020).
• Sliding-Window Sparse Recovery (TDD): Geometry-aware (MCC) pilot grids reduce worst-case coverage radius by 20% and limit collinearity, improving NMSE by 2–3 dB over conventional block-hopping or chirp patterns (Zhu et al., 7 Apr 2026).
• OFDM A2G/UAV: Adaptively optimized pilot lattices provide 9–80% average rate gain and up to 114% median instantaneous gain over fixed LTE pilots, at minimal feedback/complexity cost (Rao et al., 2018).
• Image Synchronization: Grid-shaped pilot embedding enables closed-form recovery of affine transformation matrices even after cropping, with Frobenius-norm errors near zero for most single/composite distortions, and negligible PSNR penalty (~1 dB) (Kawano et al., 26 Jan 2026).
• LMMSE Channel Estimation: Optimized pilot patterns provide 1–2 dB MSE gain over standard rectangular/diamond lattices, with adaptivity to SNR/channel spread and graceful performance at block edges (Yu et al., 4 May 2026).

7. Guidelines and Key Insights

• Uniform rectangular or time-aligned pilots are generally suboptimal except under highly constrained scenarios or uniform statistics. 2D cyclic shift, polarization staggering, and modular line-dispersed pilots produce near-optimal estimation performance with deterministic, low-complexity designs (Alfredsson et al., 2020, Zhu et al., 7 Apr 2026, Yuan, 2024).
• Pilot density in frequency should be prioritized for range estimation, and in time for velocity/Doppler estimation; balanced pilot spacing achieves Pareto-optimal trade-offs under mixed requirements (Ozturk et al., 26 Mar 2025, Rao et al., 2018).
• For sparsity-exploiting or block-based systems, grid coverage radius and difference collinearity are crucial geometric metrics—minimization delivers robust latest-slot or channel recovery (Zhu et al., 7 Apr 2026).
• Pilot pattern optimization is computationally feasible for moderate grid sizes via convex heuristics, integer programming, or dynamic programming. For larger-scale or low-latency adaptation, greedy or local-swap methods are effective (Yu et al., 4 May 2026).
• Embedding resource-orthogonality in the TF or DD domain (e.g., periodic repetition with zero-insertion) enables pilots to co-exist with data, providing reference signals without excessive overhead (Yuan, 2024, Yuan et al., 2023).

Grid-shaped pilot signals, across domains, are an essential tool in realizing efficient, robust, and high-resolution estimation strategies. Their design—spanning mathematical sequence selection, domain mapping, grid arrangement, and algorithmic optimization—directly sets system performance bounds and operational scalability.