Grid-Shaped Pilot Signals

• Grid-shaped pilot signals are structured, lattice-based reference patterns enabling precise estimation, synchronization, and sensing across various domains.
• They are mathematically constructed using Kronecker products and exhibit quasi-orthogonality to minimize interference and reduce pilot overhead.
• These signals deliver enhanced performance in OFDM, ISAC, and watermarking by optimizing channel estimation, improving spectral efficiency, and mitigating geometric distortions.

A grid-shaped pilot signal is a structured reference pattern embedded in a signal space—typically a two-dimensional coordinate domain such as time-frequency, delay-Doppler, or spatial grids—to enable robust estimation, synchronization, or sensing. The defining feature is a regular lattice or array, often with distinctive coding or quasi-orthogonality properties, designed to maximize estimation accuracy for channel parameters, phase, or geometric transformations while minimizing pilot overhead and interference with data.

1. Mathematical Construction and Lattice Geometry

Grid-shaped pilots exploit multidimensional lattice structures for pilot placement and coding. In OFDM and ISAC contexts, the grid may be defined on the time-frequency (TF) plane, the delay-Doppler (DD) plane, or both. The canonical construction uses one-dimensional sequences $\mathbf{a} \in \mathbb{C}^N$ and $\mathbf{b} \in \mathbb{C}^M$ with favorable cyclic-autocorrelation properties, forming the pilot matrix as a 2D Kronecker product:

$$\mathbf{P} = \mathbf{b} \otimes \mathbf{a} \in \mathbb{C}^{M \times N},$$

where $(m, n)$-grid points encode pilot values per the product rule $P[m, n] = b[m] a[n]$ (Yuan et al., 2023, Yuan, 2024). This structure extends readily to geometric pilot embedding in image coordinates, as in watermarking, where grid lines are inserted at regular intervals $\gamma$ in both horizontal and vertical directions, possibly with multilevel amplitude coding to resolve axis ambiguities (Kawano et al., 26 Jan 2026).

Diamond, hexagonal, rectangular, and "Cell" (rotated hexagonal) grid geometries have all been explored, with the Cell pattern shown to outperform others in channel estimation accuracy for OFDM due to minimized maximum and average pilot-to-data distances (Lee et al., 2016).

2. Embedding, Resource Mapping, and Orthogonality

Embedding of grid-shaped pilots involves allocation of regular lattice locations in the resource grid and modulation of pilot amplitudes according to desired coding. In image watermarking, pilots are inserted using quantization-index modulation (QIM) in chosen color-space components, e.g., U in YUV, with

$$U'(x, y) = \Delta \cdot \left\lfloor \frac{U(x, y)}{\Delta} - \frac{p + 1}{3} + 0.5 \right\rfloor + \Delta \cdot \frac{p+1}{3}$$

for multilevel $p \in \{-1, 0, +1\}$ (Kawano et al., 26 Jan 2026).

In OFDM/ISAC systems, DD-domain pilots are carried into the time-frequency plane via the inverse symplectic FFT (ISFFT):

$$\mathbf{X} = \mathbf{F}_M \mathbf{P} \mathbf{F}_N^H$$

with pilot occupancy and “comb” patterns determined by grid repetition and resource allocation (Yuan, 2024). Pilot signal design ensures quasi-orthogonality—auto-correlation peaks at true pilot shift and low cross-correlation with data—so that pilot power remains largely non-intrusive, allowing co-existence with user data and negligible BER impact.

3. Detection, Estimation, and Algorithmic Extraction

Detection and estimation with grid-shaped pilots rely on matched filtering, cyclic-shift correlation, or transform-domain analysis, leveraging the sharp autocorrelation peak of the pilot structure. For geometric estimation in watermarking, the distorted grid is detected via Radon transform, extracting dominant direction angles and spacings to invert for geometric transformation matrices:

$$R(\varphi, \rho) = \int_{-\infty}^{\infty} \hat{p}(u \cos \varphi - v \sin \varphi, u \sin \varphi + v \cos \varphi)\, dv$$

with the two variance-peak directions corresponding to transformed grid axes. Subsequent interval and orientation resolution allow closed-form recovery of affine transformation parameters (Kawano et al., 26 Jan 2026).

In OFDM/ISAC, detection is typically by 2D pilot-matched correlation over the DD grid, yielding a unique, sharp peak at the correct delay-Doppler offset. Fractional Doppler and delay estimation is refined by interpolating side-peak magnitudes, as described by weighted averaging across integer bins (Yuan, 2024).

In multi-channel optical communication, pilot distributions over the channel-time grid are optimized via genetic algorithms or grouped cyclic shifts to minimize the mean squared error (MSE) of extended Kalman smoother-based phase estimation (Alfredsson et al., 2020).

4. Impact on Estimation Accuracy and System Performance

Grid-shaped pilots fundamentally enhance system performance by providing high-resolution, low-latency, and low-overhead estimation of channel characteristics and transformations.

• Watermarking/geometric synchronization: Grid-shaped pilots permit recovery of affine transformation matrices with near-zero median Frobenius-norm error under composite geometric attacks (scaling, rotation, shearing, cropping) in high-resolution images, maintaining watermark BER below $0.1$ (Kawano et al., 26 Jan 2026).
• ISAC and OFDM sensing: Grid-shaped pilots enable sub-sample delay/Doppler estimation and maintain communication BER and channel-estimation NMSE virtually identical to data-only or reference-signal-only scenarios, even when pilot power fraction $\alpha$ approaches $0.2$ (Yuan et al., 2023, Yuan, 2024).
• MU-MIMO and UAV communications: Adaptive grid pilot patterns yield $7\%$–$12\%$ (and up to $16\%$) spectral-efficiency gain relative to worst-case fixed patterns, as pilot overhead is minimized and distributed according to user channel conditions, Doppler, and delay spread (Ksairi et al., 2016, Rao et al., 2018).
• Optical communications: Optimized grid pilot placement decreases phase-noise MSE by $77$–$90\%$ and boosts achievable information rate by up to $0.41$ bits/symbol at high modulation orders (Alfredsson et al., 2020).

5. Design Trade-offs: Density, Power, and Resource Allocation

Design of grid-shaped pilots involves complex trade-offs among pilot density, spacing, power allocation, and resource grid geometry.

• Density and spacing: Maximum-pilot spacing in time or frequency is set by channel statistics (Doppler, delay spread), e.g., $\Delta^s_g \leq 1/(4 f_g T_s)$ and $\Delta^{SC}_g \leq 1/(4 \tau_g \Delta f)$ (Ksairi et al., 2016). Grouping and per-resource block allocation enable minimal overhead.
• Power allocation: Pilot:data power ratios $\rho$ are optimized per channel and time window to maximize post-equalization SINR and throughput, with practical feedback overhead kept below $100$ bps for UAV channels (Rao et al., 2018).
• Comb patterns and resource sparsity: By repeating the pilot along one grid dimension and utilizing the Kronecker structure, TF resource occupancy can be reduced (e.g., pilots only on every $i$th delay/Doppler tap) (Yuan, 2024). This enables “woven” data overlay.
• Sequence orthogonality: m-sequences and other low-sidelobe sequences limit interference between pilot and data (Yuan et al., 2023, Yuan, 2024).

6. Interpolation, Reconstruction, and Application-Specific Algorithms

Intermediate pilot-to-data interpolation and estimation algorithms are tailored to the grid geometry.

• Distance-weighted interpolation: In rotated hexagonal/Cell pilot patterns, inverse-distance weights minimize interpolation bias and complexity, outperforming both traditional FIR and LMMSE schemes in BER—up to $10$ dB improvement in moderate-SNR regimes (Lee et al., 2016).
• Watermark relocalization: Pilot-based transformation estimation allows precise relocalization of embedded watermarks after geometric attacks, without exhaustive search or brute-force methods (Kawano et al., 26 Jan 2026).
• ISAC and sensing: DD-domain correlation and side-peak interpolation enable sub-sample accuracy, while pilot design supports flexible scalability for delay/Doppler resolution, resource allocation, and power trade-offs (Yuan, 2024, Yuan et al., 2023).

7. Representative Metrics and Comparative Tables

The following table organizes selected grid-shaped pilot signal metrics and pilot placement schemes from referenced works.

Application Domain	Grid Placement Principle	Key Metric (Reference)
Watermarking	3-level grid in image coordinates	Frobenius error $E(T̂;T)$, watermark BER ≤ 0.1 (Kawano et al., 26 Jan 2026)
OFDM/ISAC Sensing	2D DD grid, Kronecker coding	Sensing SINR $Z$, BER ≲ $10^{-4}$ @ pilot:data=0.2 (Yuan, 2024, Yuan et al., 2023)
Optical multichannel	Time×channel grid, cyclic interleaving	Phase-noise MSE, AIR gain up to 0.41 b/sym (Alfredsson et al., 2020)
MU-MIMO communications	Adaptive RB grouping, pilot-density	Spectral efficiency gain up to 16% (Ksairi et al., 2016)
UAV OFDM	Time-frequency regular lattice	Rate gain 9%–80%, feedback <100 bps (Rao et al., 2018)

Conclusion

Grid-shaped pilot signals are a central tool in multidimensional parameter estimation across watermarking, wireless communications, and optical links. Their structured, quasi-orthogonal design minimizes overhead, maximizes estimation accuracy, and adapts naturally to dynamic channel environments and geometric distortions. By exploiting the full lattice geometry and coding possibilities, grid-shaped pilots underpin high-performance sensing, synchronization, and channel state acquisition in modern signal-processing systems.