Compressive sensing (CS) has been proposed for signals with sparsity in a linear transform domain. We explore a signal dependent
unknown linear transform, namely the impulse response matrix operating on a sparse excitation, as in the linear model of speech production, for recovering compressive sensed speech. Since the linear transform is signal dependent and unknown, unlike the standard
CS formulation, a codebook of transfer functions is proposed in a
matching pursuit (MP) framework for CS recovery. It is found that
MP is efficient and effective to recover CS encoded speech as well
as jointly estimate the linear model. Moderate number of CS measurements and low order sparsity estimate will result in MP converge
to the same linear transform as direct VQ of the LP vector derived
from the original signal. There is also high positive correlation between signal domain approximation and CS measurement domain
approximation for a large variety of speech spectra.