A New and Interesting Class of Limit Cycles in Recursive Digital Filters

Abstract

Limit cycles oscillations often occur in recursive digital filters due to the quantization of products in the feedback section, A new and interesting class of limit cycles has been discovered and categorized for second-order sections that round either sign-magnitude or twoscomplement products. These limit cycles are named rolling-pin limit cycles. They are completely defined by three integers and a simple construction rule and exist for B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> pairs lying within small rectangular regions in the B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> (coefficient) plane. Each set of integers completely defines the peak amplitude, the length, and the region of existence. The amplitude of these limit cycles can be made close to but does not exceed three times the Jackson peak estimate. Rolling-pin limit cycles often occur in filters with high Q poles located near dc or half the sampling frequency. When these large amplitude limit cycles occur, the idle channel performance of a filter is often unacceptable. Specialized techniques, requiring extra circuitry, can be used to suppress them. Alternatively, it may be less costly and more efficient to avoid the small rectangular regions within which the rolling-pin limit cycles exist in the B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> − B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> coefficient plane.