A bound on limit cycles in digital filters which exploits a particular structural property of the quantization

This paper presents an improved upper bound on the rms value of self-sustained limit cycles due to quantization in those implementations of digital filter sections where the quantization possesses a particular structural property. This structural property exists whenever the design utilizes a single quantizer, with the quantizer function restricted to the sector bounded by the 45° line and the horizontal axis. An important example of a case where this property exists is quantization by a single magnitude-truncation quantizer. The main result is a closed form bound for the important case of second-order sections. The bound also has the following geometrical interpretation. For given filter feedback coefficients
a
and
b
, define the complex function
\lambda(\psi) = 1 - ae^{-i\psi} - be^{-2i\psi}
and consider the "Nyquist locus" of points obtained in the complex plane by allowing
\psi
to range over
[0, 2\pi]
. The bound states that if the locus has no intersection with the Im-
\lambda
axis, then limit cycles do not exist. If an intersection exists, then limit cycles may exist and the rms bound is the inverse of the distance from the origin to the nearest point of intersection. It is proven that the bound given here always improves on the Sandberg-Kaiser bound.