Sparse Reconstruction-Based Time-Delay Estimation Algorithm in the Exponential Correlation Domain

For the estimation problem of multiple adjacent targets, the traditional correlation function estimation method based on the matched filter (MF) is often limited by resolution, resulting in insufficient estimation accuracy. Subsequently, an exponential filter (EF) with high resolution was constructed by introducing a controllable exponent
$p$
into the frequency response function of the matched filter. In this paper, we propose a sparse reconstruction and exponential filter-based estimation algorithm to estimate the time delays of multiple targets. This algorithm uses the exponential autocorrelation function (the output of the exponential filter) as a template and constructs a sparse overcomplete model of the time delays of the received signal in the exponential correlation domain. An interior-point method based on
$l_{1}$
-norm regularization is then employed to solve the proposed overcomplete model. Notably, the proposed algorithm does not require prior knowledge of the target number. Theoretical analysis and Simulation results demonstrate that, the proposed algorithm can effectively reduce the coherence between adjacent targets and thus achieve high-accuracy time-delay estimation. When the input SNR is higher than −24 dB, it has a higher estimation accuracy for multi-target time delays (especially for adjacent targets) than the previous MF or EF based time-delay estimation algorithms. Particularly, at high signal-to-noise ratios, the estimation error of the proposed algorithm can approach the error bound of the time delays derived from the Cramér–Rao lower bound; when the input SNR is higher than −4 dB, it achieves completely accurate estimation.