Higher order 1-bit Sigma-Delta modulation on a circle

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model (circle). We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes would provide mismatches and large errors at the initialization point if the signal to be converted is defined on a circle. In this paper, our goal is to get around these problems for higher order Sigma-Delta schemes. Our results show how to design an update for the second and third order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.