Nonlinear Behaviors of Bandpass Sigma Delta Modulators with Stable System Matrices (original) (raw)
Related papers
Nonlinear Behaviors of Bandpass Sigma–Delta Modulators With Stable System Matrices
IEEE Transactions on Circuits and Systems II: Express Briefs, 2000
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 stable.
In this paper, we find that, by computing the difference between two consecutive state vectors of second-order double-loop sigma-delta modulators (SDMs) and plotting one component of the subtracted vectors against the other component, irregular chaotic patterns will become two vertical lines. By multiplying a matrix on the subtracted vectors, it can be further transformed to two fixed points. However, second-order interpolative bandpass SDMs still exhibit chaotic behaviors after applying the same transformations. Moreover, it is found that the Lyapunov exponent of state vectors of second-order double-loop SDMs is higher than that of second-order interpolative bandpass SDMs, whereas the Lyapunov exponent of transformed vectors becomes negative infinity for second-order double-loop SDMs and increases for second-order interpolative bandpass SDMs. Hence, by examining the occurrence of chaotic behaviors of the transformed vectors of these two SDMs, these two SDMs can be distinguished from their state vectors and their transformed vectors without solving the state equations and knowing the information of input signals.
Occurrence of Elliptical Fractal Patterns in Multi-Bit Bandpass Sigma Delta Modulators
International Journal of Bifurcation and Chaos, 2005
Abstract⎯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 multi-bit quantizers, even for the case when the saturation regions of the multi-bit 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
Chaos, Solitons & Fractals, 2007
In this paper, we find that, by computing the difference between two consecutive state vectors of second-order double-loop sigma-delta modulators (SDMs) and plotting one component of the subtracted vectors against the other component, irregular chaotic patterns will become two vertical lines. By multiplying a matrix on the subtracted vectors, it can be further transformed to two fixed points. However, second-order interpolative bandpass SDMs still exhibit chaotic behaviors after applying the same transformations. Moreover, it is found that the Lyapunov exponent of state vectors of second-order double-loop SDMs is higher than that of second-order interpolative bandpass SDMs, whereas the Lyapunov exponent of transformed vectors becomes negative infinity for second-order double-loop SDMs and increases for second-order interpolative bandpass SDMs. Hence, by examining the occurrence of chaotic behaviors of the transformed vectors of these two SDMs, these two SDMs can be distinguished from their state vectors and their transformed vectors without solving the state equations and knowing the information of input signals.
International Journal of Bifurcation and Chaos, 2011
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.
Boundedness And Aperiodicity Of Commercial Sigma Delta Modulators
Sigma delta modulation is a popular form of A/D and D/A conversion. This nonlinear device exhibits a high degree of complex nonlinear behaviour, including chaotic dynamics. One of the main unsolved problems in the theory of sigma delta modulation concerns the ability to analytically derive conditions for the boundedness of solutions of a high order sigma delta modulator (SDM). In this work, we describe how a sigma delta modulator may be rephrased within the context of systems theory. We present several theoretical results concerning bounded solutions of general high order SDMs, including necessary and sufficient conditions for the lack of a finite escape time, necessary conditions for bounded solutions based on the nature of the output sequences, and topological properties of the solutions, which are a precursor to the study of chaotic solutions of SDMs.
The benefits of multibitchaotic sigma delta modulators
Sigma delta modulation is a popular technique for high-resolution analog-to-digital conversion and digital-to-analog conversion. We investigate chaotic phenomena in multibit first-order sigma-delta modulators. Particular attention is placed on the occurrence of periodic orbits or limit cycles. These may result in idle tones audible to the listener when sigma-delta modulation is used for audio signal processing. One suggested method of eliminating idle tones is the operation of a sigma delta modulator in the chaotic regime. Unfortunately, chaotic modulation of a first order sigma delta modulator is a poor system for signal processing. We show that minor variations on a traditional first order sigma-delta modulator, together with a multibit implementation, may be used to produce an effective, stable chaotic modulator that accurately encodes the input and helps remove the presence of idle tones.
Stability Analysis of Limit Cycles in High Order Sigma Delta Modulators
We present a mathematical framework, based on state space modelling, for the description of limit cycles of Sigma Delta Modulators (SDMs). Using a dynamical systems approach, the authors treat sigma delta modulators as piecewise linear maps. This enables us to find all possible limit cycles that might exist in an arbitrary sigma delta modulator with predefined input. We then focus on a DC input analyse their stability and show exactly the amount of dither that is necessary to remove any given limit cycle. Using several different SDM designs, we locate and analyse the limit cycles and thus verify the results by simulation.
Stability of Sinusoidal Responses of Marginally Stable Bandpass Sigma Delta Modulators
In this paper, we analyse 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.
IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 1996
Conventional TA modulators suffer from idle tones. In order to alleviate the tone problem, T A 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 EA modulator with unstable filter dynamics. We begin by deriving stability boundaries for the double loop EA 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 resultsin tones in the spectrum corresponding to the fundamental or harmonics of these limit cycles. ' Asymptotically stable limit cycles are periodit: orbits about which all sufficiently small perturbations tend to zero. Note that, only asymptotically stablc limit cycles persist and arc observed in practice. Unstable limit cycles, on the other hand, do not persist since, infinitesimal perturbations of the state space trajectory result in divergence from the Perioliic orbit. 'Note that, XA modulators with unstable filter dynamics are modulators with one or more open loop poles outside the unit circle. paper was recommended by Associate Editor D. A. Johns.