  In this paper, we propose the family of Iterative Methods with Adaptive
Thresholding (IMAT) for sparsity promoting reconstruction of Delta Modulated
(DM) voice signals. We suggest a novel missing sampling approach to delta
modulation that facilitates sparsity promoting reconstruction of the original
signal from a subset of DM samples with less quantization noise. Utilizing our
proposed missing sampling approach to delta modulation, we provide an
analytical discussion on the convergence of IMAT for DM coding technique. We
also modify the basic IMAT algorithm and propose the Iterative Method with
Adaptive Thresholding for Delta Modulation (IMATDM) algorithm for improved
reconstruction performance for DM coded signals. Experimental results show that
in terms of the reconstruction SNR, this novel method outperforms the
conventional DM reconstruction techniques based on lowpass filtering. It is
observed that by migrating from the conventional low pass reconstruction
technique to the sparsity promoting reconstruction technique of IMATDM, the
reconstruction performance is improved by an average of 7.6 dBs. This is due to
the fact that the proposed IMATDM makes simultaneous use of both the sparse
signal assumption and the quantization noise suppression effects by smoothing.
The proposed IMATDM algorithm also outperforms some other sparsity promoting
reconstruction methods.
