This paper describes a novel speech coding concept created by introducing sparsity constraints in a linear prediction scheme both on
the residual and on the prediction vector. The residual is efficiently
encoded using well known multi-pulse excitation procedures due to
its sparsity. A robust statistical method for the joint estimation of the
short-term and long-term predictors is also provided by exploiting
the sparse characteristics of the predictor. Thus, the main purpose
of this work is showing that better statistical modeling in the context
of speech analysis creates an output that offers better coding properties. The proposed estimation method leads to a convex optimization problem, which can be solved efficiently using interior-point
methods. Its simplicity makes it an attractive alternative to common speech coders based on minimum variance linear prediction.