Stability Analysis of a Single-Bit Infinite Impulse Response Sigma-Delta Modulator

Abstract

Although sigma-delta modulators have been applied as re-modulator blocks in single-bit filter systems, their inherently non-linear behaviour leads to stability concerns. When applied in simple systems such as a first order infinite impulse response (IIR) filter loop, positive feedback within the loop causes the system to become unstable after just a few input samples. In this paper, a new model is proposed that takes into account widely accepted input and quantiser gain limits of the second-order sigma delta modulator inside the IIR filter. A linear analysis of the limits to the quantiser gain and feedback parameter is presented. The proposed method gives sufficient control over the input to the sigma delta modulator and the quantiser gain while improving the acceptable limits of IIR filter feedback parameter. The new model requires fewer data bits to achieve an equivalent signal-to-noise ratio, reducing hardware costs and potentially improving performance, especially when mapping the hardware to a field programmable gate array.