On compressed blind de-convolution of filtered sparse processes

Suppose the signal x ∈ 葷ⁿ is realized by driving a k-sparse signal z ∈ 葷ⁿ 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 small set of measurements obtained through suitable random projections? 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. We develop a novel LP optimization algorithm and show that both the unknown filter H and the sparse input z can be reliably estimated.