Stability of sinusoidal responses of marginally stable bandpass sigma delta modulators

In this paper, we analyze the stability of the sinusoidal responses of second order interpolative marginally stable bandpass sigma delta modulators (SDMs) with the sum of the numerator and denominator polynomials equal to one and explore new results on the more general second order interpolative marginally stable bandpass SDMs. These results can be further extended to the high order interpolative marginally stable bandpass SDMs

Estimation of an initial condition of sigma-delta modulators via projection onto convex sets

Abstract—In this paper, an initial condition of strictly causal rational interpolative sigma-delta modulators (SDMs) is estimated based on quantizer output bit streams and an input signal. A set of initial conditions generating bounded trajectories is characterized. It is found that a set of initial conditions generating bounded trajectories but not necessarily corresponding to quantizer output bit streams is convex. Also, it is found that a set of initial conditions corresponding to quantizer output bit streams but not necessarily generating bounded trajectories is convex too. Moreover, it is found that an initial condition both corresponding to quantizer output bit streams and generating bounded trajectories is uniquely defined if the loop filter is unstable (Here, an unstable loop filter refers to that with at least one of its poles being strictly outside the unit circle). To estimate that unique initial condition, a projection onto convex set approach is employed. Numerical computer simulations show that the employed method can estimate the initial condition effectively

Global stability, limit cycles and chaotic behaviors of second order interpolative sigma delta modulators

It is well known that second order lowpass interpolative sigma delta modulators (SDMs) may suffer from instability and limit cycle problems when the magnitudes of the input signals are at large and at intermediate levels, respectively. In order to solve these problems, we propose to replace the second order lowpass interpolative SDMs to a specific class of second order bandpass interpolative SDMs with the natural frequencies of the loop filters very close to zero. The global stability property of this class of second order bandpass interpolative SDMs is characterized and some interesting phenomena are discussed. Besides, conditions for the occurrence of limit cycle and fractal behaviors are also derived, so that these unwanted behaviors will not happen or can be avoided. Moreover, it is found that these bandpass SDMs may exhibit irregular and conical-like chaotic patterns on the phase plane. By utilizing these chaotic behaviors, these bandpass SDMs can achieve higher signal-to-noise ratio (SNR) and tonal suppression than those of the original lowpass SDMs

Analysis of Nonlinear Behaviors, Design and Control of Sigma Delta Modulators

M PhilSigma delta modulators (SDMs) have been widely applied in analogue-to-digital (A/D) conversion for many years. SDMs are becoming more and more popular in power electronic circuits because it can be viewed and applied as oversampled A/D converters with low resolution quantizers. The basic structure of an SDM under analytical investigation consists of a loop filter and a low bit quantizer connected by a negative feedback loop. Although there are numerous advantages of SDMs over other A/D converters, the application of SDMs is limited by the unboundedness of the system states and their nonlinear behaviors. It was found that complex dynamical behaviors exist in low bit SDMs, and for a bandpass SDM, the state space dynamics can be represented by elliptic fractal patterns confined within two trapezoidal regions. In all, there are three types of nonlinear behaviors, namely fixed point, limit cycle and chaotic behaviors. Related to the unboundedness issue, divergent behavior of system states is also a commonly discovered phenomenon. Consequently, how to design and control the SDM so that the system states are bounded and the unwanted nonlinear behaviors are avoided is a hot research topic worthy of investigated. In our investigation, we perform analysis on such complex behaviors and determine a control strategy to maintain the boundedness of the system states and avoid the occurrence of limit cycle behavior. For the design problem, we impose constraints based on the performance of an SDM and determine an optimal design for the SDM. The results are significantly better than the existing approaches