On the consistency of ℓ₁-norm based ar parameters estimation in a sparse multipath environment

When an autoregressive (AR) process is observed through a sparse multipath environment, its AR parameters may be estimated by searching for a symmetric Finite Impulse Response (FIR) filter, which, when convolved with the observed signal's autocorrelation sequence, yields the sparsest output. The zeros of that filter would then correspond to the poles of the AR process. When the ℓ₀-norm of the output is used as a measure of its sparsity, consistency of the resulting estimate (under some simple conditions) is readily obtained. However, due to problematic aspects of ℓ₀-norm minimization, it is often more convenient to resort to ℓ₀-norm minimization. A question of major interest in this context is whether (and if so, under what conditions) consistency of the resulting estimate is maintained. By analyzing the perturbations of the ℓ₁-norm about the desired solution, we derive (and illustrate) specific conditions for consistency. We show that when the multipath reflections are sufficiently sparse, consistency is guaranteed for a very wide range of AR parameters and reflection gains.