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

Difference between irregular chaotic patterns of second-order double-loop ΣΔ modulators and second-order interpolative bandpass ΣΔ modulators

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

Fuzzy impulsive control of high order interpolative lowpass sigma delta modulators

In this paper, a fuzzy impulsive control strategy is proposed. The state vectors that the impulsive controller resets to are determined so that the state vectors of interpolative low-pass sigma-delta modulators (SDMs) are bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are, the occurrence of limit cycle behaviors and the effect of audio clicks are minimized, as well as the state vectors are close to the invariant set if it exists. To work on this problem, first, the local stability criterion and the condition for the occurrence of limit cycle behaviors are derived. Second, based on the derived conditions, as well as a practical consideration based on the boundedness of the state variables and a heuristic measure on the strength of audio clicks, fuzzy membership functions and a fuzzy impulsive control law are formulated. The controlled state vectors are then determined by solving the fuzzy impulsive control law. One of the advantages of the fuzzy impulsive control strategy over the existing linear control strategies is the robustness to the input signal, the initial condition and the filter parameters, and that over the existing nonlinear control strategy are the efficiency and the effectiveness in terms of lower frequency of applying the control force and higher signal-to-noise ratio (SNR) performanc

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

Galaxy Cluster Environments of Radio Sources

Using the Sloan Digital Sky Survey (SDSS) and the FIRST (Faint Images of the Radio Sky at Twenty Centimeters) catalogs, we examined the optical environments around double-lobed radio sources. Previous studies have shown that multi-component radio sources exhibiting some degree of bending between components are likely to be found in galaxy clusters. Often this radio emission is associated with a cD-type galaxy at the center of a cluster. We cross-correlated the SDSS and FIRST catalogs and measured the richness of the cluster environments surrounding both bent and straight multi-component radio sources. This led to the discovery and classification of a large number of galaxy clusters out to a redshift of z ~ 0.5. We divided our sample into smaller subgroups based on their optical and radio properties. We find that FR I radio sources are more likely to be found in galaxy clusters than FR II sources. Further, we find that bent radio sources are more often found in galaxy clusters than non-bent radio sources. We also examined the environments around single-component radio sources and find that single-component radio sources are less likely to be associated with galaxy clusters than extended, multi-component radio sources. Bent, visually-selected sources are found in clusters or rich groups ~78% of the time. Those without optical hosts in SDSS are likely associated with clusters at even higher redshifts, most with redshifts of z > 0.7.Comment: 47 pages, 24 figures. Accepted by A

A genome-wide association study for diabetic nephropathy genes in African Americans

A genome-wide association study was performed using the Affymetrix 6.0 chip to identify genes associated with diabetic nephropathy in African Americans. Association analysis was performed adjusting for admixture in 965 type 2 diabetic African American patients with end-stage renal disease (ESRD) and in 1029 African Americans without type 2 diabetes or kidney disease as controls. The top 724 single nucleotide polymorphisms (SNPs) with evidence of association to diabetic nephropathy were then genotyped in a replication sample of an additional 709 type 2 diabetes-ESRD patients and 690 controls. SNPs with evidence of association in both the original and replication studies were tested in additional African American cohorts consisting of 1246 patients with type 2 diabetes without kidney disease and 1216 with non-diabetic ESRD to differentiate candidate loci for type 2 diabetes-ESRD, type 2 diabetes, and/or all-cause ESRD. Twenty-five SNPs were significantly associated with type 2 diabetes-ESRD in the genome-wide association and initial replication. Although genome-wide significance with type 2 diabetes was not found for any of these 25 SNPs, several genes, including RPS12, LIMK2, and SFI1 are strong candidates for diabetic nephropathy. A combined analysis of all 2890 patients with ESRD showed significant association SNPs in LIMK2 and SFI1 suggesting that they also contribute to all-cause ESRD. Thus, our results suggest that multiple loci underlie susceptibility to kidney disease in African Americans with type 2 diabetes and some may also contribute to all-cause ESRD