# Adaptive Modulation Tracking for High-Precision Time-Delay Estimation in Multipath HF Channels
## 1 Introduction  
High-frequency (HF) communication, also known as short-wave communication, offers distinctive advantages including strong survivability, low power consumption, and wide coverage. Consequently, it plays an irreplaceable role in critical applications such as over-the-horizon positioning, time synchronization, and ionospheric sounding. Time-of-arrival (TOA) estimation constitutes a fundamental initial step in signal processing required to achieve high-precision positioning, time synchronization, and ionospheric sounding. Traditional TOA estimation relies on signal correlation methods, including the Generalized Cross-Correlation (GCC) algorithm [1]. GCC estimates the time delay by identifying the peak of a weighted cross-correlation function between transmitted and received signals. This approach provides highly accurate TOA estimates under specific conditions: stationary additive Gaussian white noise and the assumption that the time intervals betweenmultipath components exceed the correlation time of the transmitted signal. Alternatively, in scenarios characterized by a single signal component, the GCC method without weighting is equivalent to the maximum likelihood (ML) estimator, capable of delivering highly accurate TOA estimates. However, HF skywave signals propagate via the ionosphere, whose nonlinear time-varying characteristics induce detrimental effects such as multipath fading and Doppler frequency shifts. These effects significantly degrade the accuracy of HF signal delay parameter estimation. Algorithms designed for multipath delay estimation can be broadly categorized as follows: For scenarios with known source signals, the multipath delay estimation problem is decomposed into multiple one-dimensional optimization problems through dimensionality reduction techniques for parameter estimation, with representative examples including the Expectation-Maximization (EM) algorithm [2], rooted in Maximum Likelihood Estimation (MLE), and the Weighted Fourier Transform and Relaxation (WR) algorithm [3], based on the Least Squares Method (LSM). When the source signal is unknown, the problem is transformed into its frequency-domain equivalent, resembling a sinusoidal signal frequency estimation problem, and subsequently addressed using subspace parameter estimation algorithms. Prominent algorithms in this category include the Multiple Signal Classification (MUSIC) algorithm [4; 5; 6] and the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm [7; 8]. Leveraging the inherent sparsity of delay parameters induced by the multipath effect, sparse reconstruction algorithms are employed for delay estimation. Representative algorithms in this class include the Sparse Iterative Covariance-based Estimation (SPICE) algorithm [9] and the Orthogonal Matching Pursuit (OMP) algorithm [10]. Furthermore, algorithms grounded in fractional-order Fourier transform theory offer novel approaches for multipath delay estimation of Linear Frequency-Modulated (LFM) signals [11], while deep learning methods present an emerging research direction in this domain. However, existing research predominantly focuses on analyzing individual interference factors such as multipath effects or frequency shifts, lacking comprehensive models for composite interference mechanisms in HF channels. Moreover, subspace-based methods (e.g., MUSIC/ESPRIT) incur prohibitive computational complexity due to their reliance on matrix operations--specifically \(O(N^{3})\) covariance decomposition--severely limiting real-time processing capabilities. In contrast, MATE achieves superior computational efficiency by confining FFT/IFFT operations to initial parameter acquisition and executing only multiplication/addition operations during the tracking phase (\(O(N)\) complexity). Carrier synchronization errors and modulation scheme variations further challenge conventional approaches in high-frequency time-of-arrival (TOA) estimation scenarios.