Analysis of Decimation on Finite Frames with Sigma-Delta Quantization

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the $\mathrm{\Sigma }\mathrm{\Delta }$ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results have been proven for finite unitarily generated frames. We have devised a process called alternative decimation on finite frames that is compatible with $\mathrm{\Sigma }\mathrm{\Delta }$ quantization up to the second order. In both cases, decimation results in exponential error decay with respect to the bit usage.