Optimal Pilot Pattern Design for LMMSE Channel Estimation in OFDM Systems with Finite Block Size over Doubly Dispersive Channels

Pilot pattern design over doubly dispersive channels has regained significant research interest, driven by emerging high-mobility applications in 5G-Advanced and 6G systems, as well as recent developments in Orthogonal Time Frequency Space (OTFS) modulation. This paper addresses the design of LMMSE-optimal pilot patterns for OFDM systems over doubly dispersive channels with finite time-frequency grids. We formulate the problem as A-optimal sensor selection and propose two heuristic algorithms, both combining an initialization stage with local swap refinement. The first employs convex relaxation with randomized rounding, while the second uses greedy selection. Simulations on practical resource block dimensions demonstrate that the proposed designs consistently outperform conventional rectangular and diamond lattice patterns.

Characterizing ISCI in Multi-carrier ISAC Systems over Doubly Dispersive Channel: Joint Sensing and Communication Performance Analysis

This paper presents a systematic analysis of inter-symbol and inter-carrier interference (ISCI) modeling in doubly dispersive channels for integrated sensing and communication (ISAC) systems. We propose a generalized OFDM (Weyl-Heisenberg) framework to evaluate four ISCI treatment approaches: (1) explicit estimation and compensation, (2) complete ignorance, (3) uncorrelated colored noise approximation, and (4) correlated colored noise modeling. Through continuous delay-Doppler channel characterization, we derive LMMSE channel estimators and corresponding estimation errors (as sensing metrics) for both pilot-assisted and fully-known symbol scenarios. The communication performance is quantified via ergodic capacity bounds under imperfect CSI. Our theoretical analysis and numerical results reveal fundamental performance-complexity trade-offs, providing insights for practical ISAC waveform and receiver design in doubly dispersive channels.

LMMSE-Optimal Pilot Pattern Design Based on Covariance Matrix Approximation for OFDM Channel Estimation in Doubly Dispersive Channel

This paper investigates the optimal pilot pattern design, in the linear minimum mean square error (LMMSE) estimator sense, for OFDM systems in doubly dispersive channels. To enable analytical tractability, the channel covariance matrix is decomposed into the Kronecker product of two Hermitian Toeplitz matrices corresponding to the delay and Doppler domains. By invoking the Szegö limit theorem, these matrices are shown to be approximately diagonalizable by discrete Fourier transform (DFT) matrices. Based on this structure, the LMMSE channel estimation error is reformulated into a compact analytical form, from which a closed-form lower bound is derived. Furthermore, we establish the condition under which this bound is achieved by a lattice-based pilot pattern. Numerical results verify that the proposed matrix approximation introduces negligible error and examples of the proposed lattice design are given.

Low-Cost Infrastructure-Free 3D Relative Localization with Sub-Meter Accuracy in Near Field

Relative localization in the near-field scenario is critically important for unmanned vehicle (UxV) applications. Although related works addressing 2D relative localization problem have been widely studied for unmanned ground vehicles (UGVs), the problem in 3D scenarios for unmanned aerial vehicles (UAVs) involves more uncertainties and remains to be investigated. Inspired by the phenomenon that animals can achieve swarm behaviors solely based on individual perception of relative information, this study proposes an infrastructure-free 3D relative localization framework that relies exclusively on onboard ultra-wideband (UWB) sensors. Leveraging 2D relative positioning research, we conducted feasibility analysis, system modeling, simulations, performance evaluation, and field tests using UWB sensors. The key contributions of this work include: derivation of the Cram\'er-Rao lower bound (CRLB) and geometric dilution of precision (GDOP) for near-field scenarios; development of two localization algorithms -- one based on Euclidean distance matrix (EDM) and another employing maximum likelihood estimation (MLE); comprehensive performance comparison and computational complexity analysis against state-of-the-art methods; simulation studies and field experiments; a novel sensor deployment strategy inspired by animal behavior, enabling single-sensor implementation within the proposed framework for UxV applications. The theoretical, simulation, and experimental results demonstrate strong generalizability to other 3D near-field localization tasks, with significant potential for a cost-effective cross-platform UxV collaborative system.

Channel Estimation, Interpolation and Extrapolation in Doubly-dispersive Channels

The OTFS (Orthogonal Time Frequency Space) is widely acknowledged for its ability to combat Doppler spread in time-varying channels. In this paper, another advantage of OTFS over OFDM (Orthogonal Frequency Division Multiplexing) will be demonstrated: much reduced channel training overhead. Specifically, the sparsity of the channel in delay-Doppler (D-D) domain implies strong correlation of channel gains in time-frequency (T-F) domain, which can be harnessed to reduce channel training overhead through interpolation. An immediate question is how much training overhead is needed in doubly-dispersive channels? A conventional belief is that the overhead is only dependent on the product of delay and Doppler spreads, but we will show that it's also dependent on the T-F window size. The finite T-F window leads to infinite spreading in D-D domain, and aliasing will be inevitable after sampling in T-F domain. Two direct consequences of the aliasing are increased channel training overhead and interference. Another factor contributing to channel estimation error is the inter-symbol-carrier-interference (ISCI), resulting from the uncertainty principle. Both aliasing and ISCI are considered in channel modelling, a low-complexity algorithm is proposed for channel estimation and interpolation through FFT. A large T-F window is necessary for reduced channel training overhead and aliasing, but increases processing delay. Fortunately, we show that the proposed algorithm can be implemented in a pipeline fashion. Further more, we showed that data-aided channel tracking is possible in D-D domain to further reduce the channel estimation frequency, i.e., channel extrapolation. The impacts of aliasing and ISCI on channel interpolation error are analyzed.

Fast Signal Interpolation Through Zero-padding and FFT/IFFT

Based on the sampling theorem, interpolation should be conducted by employing the sinc functions as the kernels. Inspired by the fact that the discrete Fourier transform (DFT) is sampled from the discrete time Fourier transform, a fast signal interpolation algorithm based on zero-padding and fast Fourier transform (FFT) and inverse FFT (IFFT) is presented. This algorithm gives a good approximate of the ideal interpolation, in spite of the windowing effect. The fundamental difference of this algorithm and the ideal sinc interpolation is unveiled, and shown to be deeply rooted in the connection of the sinc function and the Dirichlet function.