The double loop sigma delta modulator with unstable filter dynamics: stability analysis and tone behavior

Conventional ΣΔ modulators suffer from idle tones. In order to alleviate the tone problem, ΣΔ modulators with unstable filter dynamics have been proposed. In this paper we present an analysis of the saturation characteristics and tone behavior of the double loop ΣΔ modulator with unstable filter dynamics. We begin by deriving stability boundaries for the double loop ΣΔ modulator with unstable filter dynamics which yield bounds on maximum internal signal levels. We then show via simulations and steady state analysis that while tonal properties are improved by using unstable filter dynamics, idle tones are not completely removed. Specifically, we show that some unstable limit cycles have an attractor region in their neighborhood which results in tones in the spectrum corresponding to the fundamental or harmonics of these limit cycles