Overcomplete expansions and robustness

The motivation for the development of the theory of time-frequency and time-scale expansions towards wavelet and Weyl-Heisenberg frames stems mainly from the design freedom which is usually attained with overcomplete expansions. Also, it has been observed that for a given accuracy of representation overcomplete expansions allow for a progressively coarser quantization provided that the redundancy is increased. Increased robustness of overcomplete expansions compared to nonredundant ones is manifested for two primary sources of degradation, white additive noise and quantization. Reconstruction from expansion coefficients adulterated by an additive noise reduces the noise effect by a factor proportional to the expansion redundancy. We conjecture that the effect of the quantization error can be reduced inversely to the square of the expansion redundancy and prove that result in two particular cases, Weyl-Heisenberg expansions and oversampled A/D conversion.