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

We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable r th order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 / λ r , and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an r th 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.