Robust Matched Filtering in $\ell_{p}$-Space

A common approach to time-delay estimation (TDE) and joint delay-Doppler estimation (JDDE) for target localization is matched filtering, which is equivalent to the cross correlation of the received and transmitted signals. The conventional correlation in Hilbert space is statistically optimal in white Gaussian noise. However, its performance significantly degrades in the presence of non-Gaussian noise or clutter. In this paper, two new concepts called $\ell_{p}$-correlation and $\ell_{p}$-ambiguity functions, which generalize the conventional matched filtering from Hilbert space to $\ell_{p}$-space, are proposed. In addition, several important mathematical properties of the $\ell_{p}$-correlation and $\ell_{p}$-ambiguity functions are given and proved. Compared with conventional ambiguity function, the time-frequency concentration of the $\ell_{p}$-ambiguity function is significantly enhanced, i.e., the delay-Doppler resolution is improved and the sidelobes are suppressed. Based on these two newly defined functions, a family of TDE and JDDE algorithms that are robust against impulsive noise or clutter are developed. Furthermore, the Cramér–Rao bounds of TDE and JDDE for non-Gaussian noise with arbitrary probability distribution are derived. Simulation results under several impulsive noise models demonstrate that the performance in terms of robustness, resolution, and estimation accuracy is substantially improved when compared with the conventional matched filtering and fractional lower-order moment based methods.