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

Periodic input response of a second-order digital filter with two’s complement arithmetic

The dynamic behaviors of a nonlinear second-order digital filter with two’s complement arithmetic under periodic inputs are explored. The conditions for avoiding overflow are derived. Various dynamic periodic responses are analyzed, accompanied by numerous simulation examples

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

Step response of a second-order digital filter with two’s complement arithmetic

It is well known that the autonomous response of a second-order digital filter with two’s complement arithmetic may exhibit chaotic behaviors [1]. In this paper, results of the step response case are presented. Even though in the presence of the overflow nonlinearity, it is found that the step response behaviors can be related to some corresponding autonomous response behaviors by means of an appropriate affine transformation. Based on this method, some differences between the step response and the autonomous response are explored. The effects of the filter parameter and input step size on the trajectory behaviors are presented. Some previous necessary conditions for the trajectory behaviors, initial conditions and symbolic sequences are extended and strengthened to become necessary and sufficient conditions. Based on these necessary and sufficient conditions, some counter-intuitive results are reported. For example, it is found that for some sets of filter parameter values, the system may exhibit the type I trajectory even when a large input step size is applied and overflow occurs. On the other hand, for some sets of filter parameter values, the system will not give the type I trajectory for any small input step size, no matter what the initial conditions are

Finite-Time Convergent Algorithms for Time-Varying Distributed Optimization

This paper focuses on finite-time (FT) convergent distributed algorithms for solving time-varying distributed optimization (TVDO). The objective is to minimize the sum of local time-varying cost functions subject to the possible time-varying constraints by the coordination of multiple agents in finite time. We first provide a unified approach for designing finite/fixed-time convergent algorithms to solve centralized time-varying optimization, where an auxiliary dynamics is introduced to achieve prescribed performance. Then, two classes of TVDO are investigated included unconstrained distributed consensus optimization and distributed optimal resource allocation problems (DORAP) with both time-varying cost functions and coupled equation constraints. For the previous one, based on nonsmooth analysis, a continuous-time distributed discontinuous dynamics with FT convergence is proposed based on an extended zero-gradient-sum method with a local auxiliary subsystem. Different from the existing methods, the proposed algorithm does not require the initial state of each agent to be the optimizer of the local cost function. Moreover, the provided algorithm has a simpler structure without estimating the global information and can be used for TVDO with nonidentical Hessians. Then, an FT convergent distributed dynamics is further obtained for time-varying DORAP by dual transformation. Particularly, the inverse of Hessians is not required from a dual perspective, which reduces the computation complexity significantly. Finally, two numerical examples are conducted to verify the proposed algorithms

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

Occurence of elliptical fractal patterns in multi-bit bandpass sigma delta modulators

It has been established that the class of bandpass sigma delta modulators (SDMs) with single bit quantizers could exhibit state space dynamics represented by elliptic or fractal patterns confined within trapezoidal regions. In this letter, we find that elliptical fractal patterns may also occur in bandpass SDMs with multibit quantizers, even for the case when the saturation regions of the multibit quantizers are not activated and a large number of bits are used for the implementation of the quantizers. Moreover, the fractal pattern may occur for low bit quantizers, and the visual appearance of the phase portraits between the infinite state machine and the finite state machine with high bit quantizers is different. These phenomena are different from those previously reported for the digital filter with two’s complement arithmetic. Furthermore, some interesting phenomena are found. A bit change of the quantizer can result in a dramatic change in the fractal patterns. When the trajectories of the corresponding linear systems converge to a fixed point, the regions of the elliptical fractal patterns diminish in size as the number of bits of the quantizers increases