Noisy filtered sparse processes: Reconstruction and compression

In this paper we consider estimation and compression of filtered sparse processes. Specifically, the filtered sparse process is a signal x ∈ ℝⁿ obtained by driving a k-sparse signal u ∈ ℝⁿ through an arbitrary unknown stable discrete-linear time invariant system H of a known order. The signal x(t) is measured noisily. We consider estimation of x(t) from noisy measurements. We also consider compression of x(t) by means of random projections analogous to compressed sensing. For different cases including AR and MA systems we show 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 u can be reliably estimated.