Causal Compensation for Erasures in Frame Representations

In a variety of signal processing and communications contexts, erasures occur inadvertently or can be intentionally introduced as part of a data reduction strategy. This paper discusses causal compensation for erasures in frame representations of signals. The approach described assumes linear synthesis of the signal using a prespecified frame but no specific generation mechanism for the coefficients. Under this assumption, it is demonstrated that erasures can be compensated for using low-complexity causal systems. If the transmitter is aware of the occurrence of the erasure, an optimal compensation is to project the erasure error to the remaining coefficients. It is demonstrated that the same compensation can be executed using a transmitter/receiver combination in which the transmitter is not aware of the erasure occurrence. The transmitter precompensates using projections, as if assuming erasures will occur. The receiver undoes the compensation for the coefficients that have not been erased, thus maintaining the compensation only of the erased coefficients. The stability of the resulting systems is explored, and stability conditions are derived. It is shown that stability for any erasure pattern can be enforced by optimizing a constrained quadratic program at the system design stage. The paper concludes with examples and simulations that verify the theoretical results and illustrate key issues in the algorithms.