Difference Between Irregular Chaotic Patterns of Second-Order Double-Loop Sigma Delta Modulators and Second-Order Interpolative Bandpass Sigma Delta Modulators (original) (raw)
Related papers
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.
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 stable.
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.
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.
Multibit Chaotic Sigma Delta Modulation
When sigma delta modulation is used for audio signal processing, limit cycles in the output may result in idle tones that are audible to the listener. We show that a multibit implementation of a modified first order sigmadelta modulator may produce an effective, stable chaotic modulator that accurately encodes the input and prevents idle tones.
Exploiting Chaos in Multibit Sigma Delta Modulation
2001
When sigma-delta modulation is used for audio signal processing, limit cycles in the output may result in idle tones that are audible to the listener. We show that a multibit implementation of a modified first order sigma-delta modulator may be used to produce an effective, stable chaotic modulator that accurately encodes the input and helps remove the presence of idle tones.
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.
Linearising sigma-delta modulators using dither and chaos
Circuits and Systems, 1995. ISCAS'95., …, 1995
- Recent work has shown that high-order single-bit sigma-delta modulators suffer from low-level artifacts such as idle tones and noise modulation. Techniques that have been proposed to reduce or eliminate these errors incllude the application of dither inside the one-bit ...
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