Randomly dithered quantization and sigma–delta noise shaping for finite frames

The main objective of this paper is controlling the mean-square reconstruction error induced by applying randomly dithered quantization, a stochastic round-off prescription, to the frame coefficients of a vector in a finite-dimensional Hilbert space. We establish bounds and asymptotics for the mean-square error of dithered quantization with and without sigma–delta noise shaping. The use of a random dither eliminates the need for assuming the white-noise hypothesis to establish these results. Our estimates are valid for a uniform mid-tread quantizer operating in the no-overload regime. For a fixed family of frames obtained from regular sampling of a bounded, differentiable path in the Hilbert space which terminates in the zero vector, the dither-averaged square of the Euclidean reconstruction error is asymptotically inversely proportional to the cubed number of frame vectors. This estimate is uniform in the set of input vectors that do not lead to overload of the quantizer.