  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 - the 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 provide mismatches and large errors at the
initialization point if the signal to be converted is defined on the circle. In
this paper, our goal is to get around these problems for Sigma-Delta schemes.
Our results show how to design an update for the first and the second order
schemes based on the reconstruction error analysis such that for the updated
scheme the reconstruction error is improved.
