Compressed sensing of autoregressive processes

Abstract

Suppose the signal x ∈ Rⁿ is realized by driving a d-sparse signal z ∈ Rⁿ through an arbitrary unknown stable discrete-linear time invariant system H, namely, x(t) = (h * z)(t), where h(·) is the impulse response of the operator H. Is x(·) compressible in the conventional sense of compressed sensing? Namely, can x(t) be reconstructed from sparse set of measurements. For the case when the unknown system H is auto-regressive (i.e. all pole) of a known order it turns out that x can indeed be reconstructed from O(k log(n)) measurements. The main idea is to pass x through a linear time invariant system G and collect O(k log(n)) sequential measurements. The filter G is chosen suitably, namely, its associated Toeplitz matrix satisfies the RIP property. We develop a novel LP optimization algorithm and show that both the unknown filter H and the sparse input z can be reliably estimated. These types of processes arise naturally in Reflection Seismology.