Multipath mitigation in spectrum estimation using ℓ1 minimization

We consider the problem of spectrum estimation of an Auto-Regressive (AR) process in a sparse multipath environment. The presence of even a small number of delayed and attenuated replica of the source signal in the received signal may severely degrade the performance of classical AR spectrum estimation methods. Dwelling on the sparsity of the multipath reflections, we propose an approach which looks for a Finite Impulse Response (FIR) filter which, when convolved with the received signal's autocorrelation sequence, yields the sparsest sequence. We show that under certain conditions such an approach provides a consistent estimate of the source's AR parameters if the ℓ
₀
norm is used as a measure of sparsity. However, To maintain computational feasibility, we use the ℓ
₁
norm instead. Significant performance improvement relative to the classical Yule-Walker (or Modified Yule-Walker) based estimates is demonstrated in simulation. We also consider the expansion of the method to the case of multiple sensors.