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

Nonlinear behaviors of bandpass sigma delta modulators with stable system matrices

It has been established that a class of bandpass sigma-delta modulators may exhibit state space dynamics which are represented by elliptical or fractal patterns confined within trapezoidal regions when the system matrices are marginally stable. In this brief, it is found that fractal or irregular chaotic patterns may also be exhibited in the phase plane when the system matrices are strictly stabl

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

Nonlinear behaviors of bandpass sigma delta modulators with stable system matrices

It has been established that a class of bandpass sigma delta modulators (SDMs) may exhibit state space dynamics which are represented by elliptical or fractal patterns confined within trapezoidal regions when the system matrices are marginally stable. In this paper, it is found that fractal patterns may also be exhibited in the phase plane when the system matrices are strictly stable. This occurs when the sets of initial conditions corresponding to convergent or limit cycle behavior do not cover the whole phase plane. Based on the derived analytical results, some interesting results are found. If the bandpass SDM exhibits periodic output, then the period of the symbolic sequence must equal the limiting period of the state space variables. Second, if the state vector converges to some fixed points on the phase portrait, these fixed points do not depend directly on the initial conditions

A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators

Just as their single-bit counterparts, multi-bit sigma delta modulators exhibit nonlinear behavior due to the presence of the quantizer in the loop. In the multi-bit case this is caused by the fact that any quantizer has a limited output range and hence gives an implicit saturation effect. Due to this, any multi-bit modulator is prone to modulator overloading. Unfortunately, until now, designers had to rely on extensive time-domain simulations to predict the overloading level, because there is no adequate analytical theory to model this effect. In this work, we have developed such an analytical theory based on multiple input describing function analysis. This way, we obtained expressions for the signal gain, the noise gain and the variance of the quantization noise. Here, both the case of DC as well as sinusoidal signals was considered. These results were used for the stability analysis of multi-bit Sigma Delta modulators, which allows to predict the overloading level. Code implementing the proposed expressions is available for download at http://cas1.elis.ugent. be/cas/en/download

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

DC stability analysis of high-order, lowpass ΣΔ modulators with distinct unit circle NTF zeros

This paper presents an analytical approach to the investigation of the dc stability of high-order (order > 2), low-pass (LP) ΣΔ modulators with distinct noise transfer function (NTF) zeros on the unit circle. The techniques of state-space diagonalization and decomposition, continuous-time embedding and Poincaré map analysis are combined and extended. It is revealed that high-order ΣΔ modulators can be transformed and decomposed into second- and first-order subsystems. The investigation, coupled with efficient numerical methods, generalizes itself to different types of transition flow and provides theoretical insight into the state trajectory and limit cycle behavior. It is shown that estimation of dc input bounds based solely on the boundary transition flow is inadequate. A procedure utilizing the information from different transition flow assumptions and the discrete nature of a modulator is introduced for locating the stable dc input bounds of practical, discrete-time ΣΔ modulators.published_or_final_versio

Prediction of the Spectrum of a Digital Delta–Sigma Modulator Followed by a Polynomial Nonlinearity

This paper presents a mathematical analysis of the power spectral density of the output of a nonlinear block driven by a digital delta-sigma modulator. The nonlinearity is a memoryless third-order polynomial with real coefficients. The analysis yields expressions that predict the noise floor caused by the nonlinearity when the input is constant

Fast detection of instability in sigma-delta modulators based on unstable embedded limit cycles

As a sequel to a previous study (Wong and Ng) on the nonlinear dynamical behavior of low-pass, high-order (order > 2), single-bit ΣΔ modulators with distinct unit-circle noise transfer function zeros, this paper proposes a novel scheme for the fast detection of unstable operation in these modulators under general time-varying input. The scheme is based on the variation of unstable embedded limit-cycle fixed points (which form the bounds beyond which the modulator becomes unstable) versus modulator input amplitude. Deployment of the detection scheme requires simple analog components with possible simplification. The effectiveness of the scheme is demonstrated with numerical examples. © 2004 IEEE.published_or_final_versio