Abstract

It has been established that the class of bandpass sigma delta modulators (SDMs) with single
bit quantizers could exhibit state space dynamics represented by elliptic or fractal patterns
confined within trapezoidal regions. In this letter, we find that elliptical fractal patterns may
also occur in bandpass SDMs with multibit quantizers, even for the case when the saturation
regions of the multibit quantizers are not activated and a large number of bits are used for the
implementation of the quantizers. Moreover, the fractal pattern may occur for low bit quantizers,
and the visual appearance of the phase portraits between the infinite state machine and the finite
state machine with high bit quantizers is different. These phenomena are different from those
previously reported for the digital filter with two’s complement arithmetic. Furthermore, some
interesting phenomena are found. A bit change of the quantizer can result in a dramatic change
in the fractal patterns. When the trajectories of the corresponding linear systems converge to a
fixed point, the regions of the elliptical fractal patterns diminish in size as the number of bits
of the quantizers increases.