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 lscr₀-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 lscr₀-norm minimization, it is often more convenient to resort to lscr₁-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 lscr₁-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.