Approximating a bandlimited function using very coarsely quantized data: Improved error estimates in sigma-delta modulation

Sigma-delta quantization is a method of representing bandlimited signals by $0{-}1$ sequences that are computed from regularly spaced samples of these signals; as the sampling density $\lambda \to \infty$, convolving these one-bit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. We present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigma-delta quantization.