### abstract ###
Suppose the signal  SYMBOL  is realized by driving a  SYMBOL -sparse signal  SYMBOL  through an arbitrary unknown stable discrete-linear time invariant system  SYMBOL , namely,  SYMBOL , where  SYMBOL  is the impulse response of the operator  SYMBOL
These types of processes arise naturally in Reflection Seismology
In this paper we are interested in several problems: (a) Blind-Deconvolution: Can we recover both the filter  SYMBOL  and the sparse signal  SYMBOL  from noisy measurements (b) Compressive Sensing: Is  SYMBOL  compressible in the conventional sense of compressed sensing
Namely, can  SYMBOL  be reconstructed from a sparse set of measurements
We develop novel  SYMBOL  minimization methods to solve both cases and establish sufficient conditions for exact recovery for the case when the unknown system  SYMBOL  is auto-regressive (i e all pole) of a known order
In the compressed sensing/sampling setting it turns out that both  SYMBOL  and  SYMBOL  can be reconstructed from  SYMBOL  measurements under certain technical conditions on the support structure of  SYMBOL
Our main idea is to pass  SYMBOL  through a linear time invariant system  SYMBOL  and collect  SYMBOL  sequential measurements
The filter  SYMBOL  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  SYMBOL  and the sparse input  SYMBOL  can be reliably estimated
### introduction ###
In this paper we focus on blind de-convolution problems for filtered sparse processes
Specifically, a sparse input  SYMBOL  is filtered by an unknown infinite impulse response (IIR) discrete time stable linear filter  SYMBOL  and the resulting output  SYMBOL  SYMBOL y(t) = x(t) + n(t) SYMBOL t = 0,\,1,\,\ldots,N SYMBOL u(t) SYMBOL H SYMBOL x(t) SYMBOL u(t) SYMBOL H SYMBOL H SYMBOL H SYMBOL H SYMBOL z SYMBOL H SYMBOL \ell_1 SYMBOL H SYMBOL u SYMBOL H SYMBOL u SYMBOL x(t) SYMBOL x(t) SYMBOL u(t) SYMBOL H SYMBOL z SYMBOL O(k\log^3(n)) SYMBOL GH SYMBOL H SYMBOL Hu SYMBOL x SYMBOL 3 SYMBOL \alpha_1=0 9,\,\alpha_2=0 7 SYMBOL \alpha_3=0 2 SYMBOL x SYMBOL u(t)=x(t)-x(t-1) SYMBOL x SYMBOL H SYMBOL u SYMBOL x=Hu SYMBOL H SYMBOL x SYMBOL G SYMBOL g(t) SYMBOL  SYMBOL  SYMBOL g*h SYMBOL  SYMBOL  SYMBOL u = Gu SYMBOL G SYMBOL G SYMBOL u SYMBOL H SYMBOL H SYMBOL H SYMBOL u(t) SYMBOL u(t) SYMBOL u(t) SYMBOL \ell_1 SYMBOL 1\% SYMBOL 10\% SYMBOL \ell_1 SYMBOL u SYMBOL \ell_1 SYMBOL \ell_1 SYMBOL 1\% SYMBOL 10\% SYMBOL H SYMBOL u SYMBOL H SYMBOL H^{\perp} SYMBOL p SYMBOL g * h = h * g SYMBOL \ell_1 SYMBOL u SYMBOL p SYMBOL m SYMBOL O(m+p) SYMBOL O(mp) SYMBOL \ell_1$ minimization  algorithm and the main result of this paper (Theorem ) is stated in this section
The proof of Theorem  can be found in Section
To help the reader understand the main idea of the proof we first consider a very simple case and Section  provides the proof for the general case
Section  addresses the blind-deconvolution problem, which can be regarded as a noisy version of our problem
We use lasso to solve this problem and the detailed proof is provided in Section
In Section , we extend the our techniques to two related problem, namely, decoding of ARMA process and decoding of a non-causal AR process
Finally, simulation results are shown in Section
