An exact formula for the half-plane pull-in range of a PLL

A second-order phase-lock loop (PLL) that is based on a triangular-characteristic phase detector and imperfect-integrator loop filter is found in many applications where simplicity and economics are major considerations. For many of these applications, digital-logic-compatible reference and VCO signals are used, an exclusive-OR gate implements the phase detector, and the loop filter is constructed from passive components. When designing these loops, the half-plane pull-in range Ω₂ is of interest. Until now, this important loop parameter could only be calculated by using a computer-based technique that numerically integrated the nonlinear differential equation that describes the PLL model. This requirement/limitation is removed here by the development of an exact closed-form formula for Ω₂, the main contribution of this paper. More generally, the value of Ω₂ is dependent on the PLL phase detector characteristic that is used, be it triangular, sinusoidal, or something else. With regard to the value of Ω₂ produced, a comparison is given of two PLLs, both described by the same linear model so that the comparison is meaningful. The first PLL is based on a triangular-characteristic phase detector; the second loop is based on a sinusoidal phase detector.     

Fast exact maximum likelihood estimation for mixture of language model

Language modeling is an effective and theoretically attractive probabilistic framework for text information retrieval. The basic idea of this approach is to estimate a language model of a given document (or document set), and then do retrieval or classification based on this model. A common language modeling approach assumes the data D is generated from a mixture of several language models. The core problem is to find the maximum likelihood estimation of one language model mixture, given the fixed mixture weights and the other language model mixture. The EM algorithm is usually used to find the solution.     

VCO sweep-rate limit for a phase-lock loop

Phase-lock loops (PLLs) serve important roles in phase-lock receivers, coherent transponders, and similar applications. For many of these uses, the bandwidth of the loop must be kept small to limit the detrimental influence of noise, and this requirement makes the natural PLL pull-in phenomenon too slow and/or unreliable. For each such case, the phase-lock acquisition process can be aided by the application of an external sweep voltage to the loop voltage controlled oscillators (VCOs). The goal is to have the applied sweep voltage rapidly decrease the closed-loop frequency error to a point where phase lock occurs quickly. For a second-order loop containing a perfect integrator loop filter, there is a maximum VCO sweep-rate magnitude, denoted here as R_m rad/s², for which phase lock is guaranteed. If the applied VCO sweep rate is less than R_m, the loop cannot sweep past a stable phase-lock point, and it will phase lock. On the other hand, for an applied sweep-rate magnitude that is greater than R_m, the PLL may sweep past a lock point and fail to phase lock. In the existing PLL literature, only a trial-and-error approach has been described for estimating R_m, given values of loop damping factor ζ and natural frequency ω_n. Furthermore, no plots exist of computed versus ζ and versus ζ (B_L denotes loop-noise bandwidth). These deficiencies are dealt with in this paper. A new iterative numerical technique is given that converges to the maximum sweep-rate magnitude R_m. It is used to generate data for plots of and versus ζ, the likes of which have never appeared before in the PLL literature.     

An algorithm for the calculation of exact term discrimination values

Term discrimination values have been suggested as an effective means for the selection and weighting of index terms in automatic document retrieval systems. This paper reports an algorithm for the calculation of term discrimination values that is sufficiently fast in operation to permit the use of exact values, rather than the approximate values studied in previous work. Evidence is presented to show that the relationship between term discrimination and term frequency is crucially dependent upon the type of inter-document similarity measure that is used for the calculation of the discrimination values.     

An exact closed-form formula of collision probability in diverse multiple access communication systems with frame slotted aloha protocol

For diverse multiple access communication systems based on frame slotted aloha (FSA) protocol, it is important to analyze collision probability for the system performance evaluation. As shown in the literature, for general settings, it is difficult to derive an exact and closed-form solution for collision probability without approximation. Recently, an exact solution based on generic analytical approach (GAA) [31] has been proposed, yet its numerical computation will become difficult when the number of slots is larger than 16. In this paper, we develop an exact closed-form formula (ECFF) for collision probability that can not only overcome the computational deficiency of GAA in the presence of a large number of slots, but also reduce the computation complexity of collision probability. Surprisingly, by introducing a differentiation operator to form a hybrid recursive equation and applying various algebraic properties of Laplace transform and Z transform, the final collision probability can be represented by a compact double summation. Accuracy of the ECFF and comparison with the GAA have been studied by Monte Carlo simulation.     

Frequency sweep rate dependence on the dielectrophoretic response of polystyrene beads and red blood cells

Alternating current (AC) dielectrophoresis (DEP) experiments for biological particles in microdevices are typically done at a fixed frequency. Reconstructing the DEP response curve from static frequency experiments is laborious, but essential to ascertain differences in dielectric properties of biological particles. Our lab explored the concept of sweeping the frequency as a function of time to rapidly determine the DEP response curve from fewer experiments. For the purpose of determining an ideal sweep rate, homogeneous 6.08 μm polystyrene (PS) beads were used as a model system. Translatability of the sweep rate approach to ∼7 μm red blood cells (RBC) was then verified. An Au/Ti quadrapole electrode microfluidic device was used to separately subject particles and cells to 10Vpp AC electric fields at frequencies ranging from 0.010 to 2.0 MHz over sweep rates from 0.00080 to 0.17 MHz/s. PS beads exhibited negative DEP assembly over the frequencies explored due to Maxwell-Wagner interfacial polarizations. Results demonstrate that frequency sweep rates must be slower than particle polarization timescales to achieve reliable incremental polarizations; sweep rates near 0.00080 MHz/s yielded DEP behaviors very consistent with static frequency DEP responses for both PS beads and RBCs.     

An accelerated exact distributed first-order algorithm for optimization over directed networks

Distributed optimization over networked agents has emerged as an advanced paradigm to address large-scale control, optimization, and signal-processing problems. In the last few years, the distributed first-order gradient methods have witnessed significant progress and enrichment due to the simplicity of using only the first derivatives of local functions. An exact first-order algorithm is developed in this work for distributed optimization over general directed networks with only row-stochastic weighted matrices. It employs the rescaling gradient method to address unbalanced information diffusion among agents, where the weights on the received information can be arbitrarily assigned. Moreover, uncoordinated step-sizes are employed to magnify the autonomy of agents, and an error compensation term and a heavy-ball momentum are incorporated to accelerate convergency. A linear convergence rate is rigorously proven for strongly-convex objective functions with Lipschitz continuous gradients. Explicit upper bounds of step-size and momentum parameter are provided. Finally, simulations illustrate the performance of the proposed algorithm.     

A formula for the potential of a polygon with varying charge density

Explicit formulae are derived giving the Newtonian potential and field of a polygon whose surface charge density is a linear or quadratic function. The formulae involve only the elementary functions, logarithm and arctangent. Special attention is given to triangles and rectangles because they appear in various finite element calculations. The formulae may be used, together with the Fredholm integral equation, to solve electric or magneticfield problems.     

A recursive formula for the correlations of Walsh functions

The correlation function of two Walsh functions appears in image processing, signal multiplexing, detection and spectral analysis using Walsh functions. Though Walsh functions are two-valued, their correlation functions are real-valued and rather difficult to evaluate. A recursive formula is developed in this paper to evaluate the correlation functions. Some other properties of the correlation function are also discussed.     

Slow,frictionless ring forming: further exact solutions for a wide class of ductile materials

The plastic flow of material in a slowly deforming ring under frictionless conditions is analyzed. Exact solutions are obtained without necessarily making explicit the choice of the yield criterion. The results are valid for rigid-perfectly plastic materials with a linear relation between the deviator of the stress tensor and the rate-of-strain tensor. The graphical presentation of the results is for the particular cases of materials obeying the maximum shear stress yield criterion, the constant energy of distortion yield criterion and two other yield criteria. The general behavior is the same for all the materials studied here, but significant numerical differences are observed for process parameters.     

Optofluidic membrane interferometer: An imaging method for measuring microfluidic pressure and flow rate simultaneously on a chip

We present a novel image-based method to measure the on-chip microfluidic pressure and flow rate simultaneously by using the integrated optofluidic membrane interferometers (OMIs). The device was constructed with two layers of structured polydimethylsiloxane (PDMS) on a glass substrate by multilayer soft lithography. The OMI consists of a flexible air-gap optical cavity which upon illumination by monochromatic light generates interference patterns that depends on the pressure. These interference patterns were captured with a microscope and analyzed by computer based on a pattern recognition algorithm. Compared with the previous techniques for pressure sensing, this method offers several advantages including low cost, simple fabrication, large dynamic range, and high sensitivity. For pressure sensing, we demonstrate a dynamic range of 0-10 psi with an accuracy of ±2% of full scale. Since multiple OMIs can be integrated into a single chip for detecting pressures at multiple locations simultaneously, we also demonstrated a microfluidic flow sensing by measuring the differential pressure along a channel. Thanks to the simple fabrication that is compatible with normal microfluidics, such OMIs can be easily integrated into other microfluidic systems for in situ fluid monitoring.     

采用DDS+PLL技术实现数字式信号源

目前实验室常用的信号源一般都是用模拟电路来产生所需的信号。模拟信号有着显示不直观、输出不稳定、低频信号差等缺点。本文介绍一种采用DDS PLL技术来实现数字式信号源的基本方法和原理,设计电路提供了数字式的直接输入,操作简单、显示直观,以满足实验室的实际需要。     

应用改进的(G/G′)展开法求Ostrovsky方程的精确解

应用改进的(G/G′)展开法构造出(1+1)维Ostrovsky方程的精确解,这些解包含双曲函数通解、三角函数通解和有理函数通解三种形式。当双曲函数通解中的参数取特殊值时,得到了孤立波解。当三角函数通解中引入一个参量后,可得到对应通解的周期波函数解。