Alternative dual frames for digital-to-analog conversion in sigma-delta quantization

We design alternative dual frames for linearly reconstructing signals from sigma-delta (Sigma Delta) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order sigma-delta quantizer produces approximations where the approximation error is at most of order 1/lambda(r), and lambda > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for R-d when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order sigma - delta quantizer to achieve approximation error of order 1/N-r for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.