Iterative Reduced-Rank MMSE Estimation of Sparse Range Profiles from Non-Contiguous Radar Transmission Spectra

Abstract

Ongoing demand for radio spectrum by commercial wireless services has steadily increased pressure on the frequency bands traditionally reserved for radar. This paper addresses the joint problem of designing non-contiguous radar transmission spectra and estimating the range profile from the resulting reduced measurement set. Transmission spectra are constructed using a Marginal Fisher Information (MFI) criterion that removes blocks of frequencies contributing least to estimation accuracy. To process the underdetermined signals acquired from the resulting sparse measurement vector, an iterative Reduced-Rank Minimum Mean-Square Error (RRMMSE) estimator is proposed. The estimator starts with a single-target hypothesis and grows the active target subspace one range bin at a time, updating the a~priori target covariance matrix in each iteration using both the largest estimated reflection coefficient and its posterior error variance. This avoids inversion of the full $M{\times}M$ covariance matrix that would be required by a one-step MMSE and concentrates the rank of the estimator on the support of significant scatterers. The Bayesian Cramér--Rao Lower Bound (CRLB) on the per-bin reflection coefficient is derived for the non-contiguous spectrum measurement model, and the computational complexity of the proposed estimator is shown to scale as $\Order(G^2 M K^2)$, where $G$ is the number of detectable scatterers, $M$ is the number of range bins, and $K$ is the number of preserved spectral samples. Simulations using $50\%$ and $75\%$ spectrally occupied MFI-designed spectra confirm that the algorithm recovers sparse range profiles with Mean-Square Error (MSE) close to the fully filled baseline when the number of significant scatterers is not larger than the rank of the sparse sensing matrix.