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Compressive Sensing (CS) signal recovery has been formulated for signals sparse in a known linear transform domain. We consider the scenario in which the transformation is unknown and the goal is to estimate the transform as well as the sparse signal from just the CS measurements. Specifically, we consider the speech signal as the output of a time-varying AR process, as in the linear system model of speech production, with the excitation being sparse. We propose an iterative algorithm to estimate both the system impulse response and the excitation signal from the CS measurements. We show that the proposed algorithm, in conjunction with a modified iterative hard thresholding, is able to estimate the signal adaptive transform accurately, leading to much higher quality signal reconstruction than the codebook based matching pursuit approach. The estimated time-varying transform is better than a 256 size codebook estimated from original speech. Thus, we are able to get near "toll quality" speech reconstruction from sub-Nyquist rate CS measurements.