Bilinear Integrator Research Articles

Research and practical application of a bilinear integration learning model for a college psychology course

Abstract Information technology in education has triggered a change in “psychology” learning, and dual-line integrated learning will become an important form of learning in the future. The impact of information technology on education has led to changes in psychological learning, and dual-line integrated learning will become a crucial form of learning in the future. Then, based on the FuzzyCDF model of fuzzy set theory, the students’ psychology cognition, learning ability, and knowledge mastery were tested to assess the student’s cognitive level and learning effectiveness of psychology courses. Then, we analyzed the effect of dual-line integration teaching, evaluated the teaching recognition and cognition of students in different grades, and explored in depth the effects of online and offline integration teaching on students’ learning attitude, learning anxiety, self-efficacy, and academic performance. Comparative analysis was used to analyze the impact of psychology course education on students’ mental health. The results showed that the student’s recognition of the dual-line integration learning mode was all above 3.7, the correlation of the online and offline learning mode on the dimensions of students’ learning attitude, self-efficacy, and academic achievement were all between (0.12,0.65), and the negative correlation on students’ learning anxiety. Psychoeducators in colleges and universities can benefit from this study in developing effective teaching strategies and improving the quality of psychology courses for college students.

A Low-Noise Instrumentation Amplifier With Built-in Anti-Aliasing for Hall Sensors

We present a compact, versatile Hall readout system with digital output, fully integrated in 180nm technology. The core of the system is an instrumentation amplifier architecture that provides inherent anti-aliasing filtering, where the anti-aliasing characteristic is locked into a shape that maximally prevents aliasing to low frequencies. The efficiency for blocking out-of-band white noise is comparable to that of a second-order filter, eliminating the need for an explicit anti-aliasing filter before the ADC. Chopping/spinning is applied for up-modulating offset and 1/f noise to just beyond the signal band. A mostly-digital ripple reduction loop (RRL) is added for mitigating offset-related dynamic range limitations. In this, a bilinear integrator is introduced for eliminating the impact of the RRL on the system's DC gain. Moreover, the resolution of the DAC generating the analog offset compensation is reduced significantly, and the effect thereof is eliminated by digital noise cancellation logic. The one-step amplification and the simple, low-resolution DAC for offset compensation both aid in keeping the area footprint low: the analog circuits (including DAC and ADC) only occupy 0.21mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Notable performance characteristics are an input-referred noise floor of 55nT/ √{Hz} within a 410kHz bandwidth, a current consumption of only 5.1mA, and a 47dB dynamic range. The amplifier architecture can be easily applied as an analog preconditioning circuit in other sensor readout situations as well.

Weighted estimates for bilinear square functions with non-smooth kernels and commutators

Under weaker conditions on the kernel functions, we discuss the boundedness of bilinear square functions associated with non-smooth kernels on the product of weighted Lebesgue spaces. Moreover, we investigate the weighted boundedness of the commutators of bilinear square functions (with symbols which are BMO functions and their weighted version, respectively) on the product of Lebesgue spaces. As an application, we deduce the corresponding boundedness of bilinear Marcinkiewicz integrals and bilinear Littlewood-Paley g-functions.

Bilinear Fractional Integral Along Homogeneous Curves

The boundedness of the bilinear fractional integrals along homogeneous curves \(\gamma (t)=(t^{\alpha _1},t^{\alpha _2})\) with \(\alpha _2>\alpha _1\ge 1\) is obtained. The authors extend the results of the bilinear fractional integrals of Kenig and Stein (Math Res Lett 6:1–15, 1999) and Grafakos and Kalton (Math Ann 319(1):151–180, 2001) to integrals along the curves.

Bilinear integration and applications to operator and scattering theory

We show how to integrate operator valued functions with respect to a spectral or orthogonally scattered measure. Such measures typically have a variation which has either the value zero or infinity on any set and cannot therefore be treated by the approaches of Bartle or Dobrakov. Bilinear integrals of this type arise from trace class operators between Banach function spaces and in the connection between stationary-state scattering theory and time-dependent scattering theory in Hilbert space.

Structure and realization of pole-shared switched-current complex wavelet filter

A pole-shared switched-current complex wavelet filter structure with follow-the-leader feedback configuration is proposed for synthesizing the real and imaginary approximation functions with the same poles. The double-sampling fully-balanced SI bilinear integrator and current mirror are employed as the building cells. By sharing the implementation circuit for approximation poles of the real and the imaginary part, the proposed structure only has the same circuit complexity as that of real-valued wavelet filter, which is very suitable for small-size and low-power application. The complex Morlet wavelet is selected as an example to elaborate the design procedure. Simulation results confirm that the presented complex wavelet filter structure can generate the real and imaginary coefficients of complex wavelet transform accurately with simple synthesis method and explicit design formulas.

Singular Bilinear Integrals in Quantum Physics

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

Switched-current filter structure for synthesizing arbitrary characteristics based on follow-the-leader feedback configuration

A switched-current (SI) filter structure with follow-the-leader feedback configuration has been proposed for synthesizing arbitrary transfer functions. The double-sampling fully-balanced SI bilinear integrator and current mirror are employed as the building cells in order to enhance overall performance and make circuit design more flexible. Also, explicit design formulas are derived by using coefficient matching. The third-order low-pass and band-pass filter are designed as the examples. Simulation results confirm that the presented filter structure is very suitable for the operation with high sampling-to-cutoff frequency ratio, and can synthesize arbitrary filter characteristics precisely and easily while keeping low sensitivity.

An application of bilinear integration to quantum scattering

Scattering theory has its origin in Quantum Mechanics. From the mathematical point of view it can be considered as a part of perturbation theory of self-adjoint operators on the absolutely continuous spectrum. In this work we deal with the passage from the time-dependent formalism to the stationary state scattering theory. This problem involves applying Fubini's Theorem to a spectral measure integral and a Lebesgue integral of functions that take values in spaces of operators. In our approach, we use bilinear integration in a tensor product of spaces of operators with suitable topologies and generalize the results previously stated in the literature.

Systematic selection of analogue redesign method for forward-type digital power converters

This article proposes a systematic approach to determine the most suitable analogue redesign method to be used for forward-type converters under digital voltage mode control. The focus of the method is to achieve the highest phase margin at the particular switching and crossover frequencies chosen by the designer. It is shown that at high crossover frequencies with respect to switching frequency, controllers designed using backward integration have the largest phase margin; whereas at low crossover frequencies with respect to switching frequency, controllers designed using bilinear integration with pre-warping have the largest phase margins. An algorithm has been developed to determine the frequency of the crossing point where the recommended discretisation method changes. An accurate model of the power stage is used for simulation and experimental results from a Buck converter are collected. The performance of the digital controllers is compared to that of the equivalent analogue controller both in simulation and experiment. Excellent closeness between the simulation and experimental results is presented. This work provides a concrete example to allow academics and engineers to systematically choose a discretisation method.

A low power 14 bit 51.2 kS/s double-sampling extended counting ADC with a class-AB OTA

A 14 bit 51.2 kS/s extended counting analog to digital converter (EC-ADC) is presented. Two techniques are utilized to reduce its power consumption. First, a double-sampling configuration based on a fully-floating bilinear integrator is proposed to reduce the clock frequency. Second, a class-AB operational transconductance amplifier (OTA) is designed to improve the power efficiency. In addition, the chopping technique is used to eliminate the OTA flicker noise effect. The proposed ADC is fabricated in 0.18 μm CMOS technology with a core area of 0.04 mm2. At a 51.2 kS/s conversion rate, it achieves a 94 dB SFDR and an 11.6 bit ENOB, while consuming only 77 μW from a 1.8 V power supply. The figure of merit is only 0.48 pJ/step.

A 8 mW 72 dB ΣΔ-modulator ADC with 2.4 MHz BW in 130 nm CMOS

A double-sampling split ΣΔ-ADC with bilinear integrators and a 7-level quantizer is presented. It achieves third order noise shaping with a second order modulator through quantization noise-coupling. The modulator is integrated in a 130 nm CMOS technology. For a clock frequency of 48 MHz and an oversampling ratio of 20 (2.4 MHz signal bandwidth), it achieves 72 dB DR and 68 dB SNR. The prototype consumes 8 mW from a 1.2 V voltage supply.

A Bilinear Fractional Integral on Compact Lie Groups

AbstractAs an analog of a well-known theoremon the bilinear fractional integral on ℝn by Kenig and Stein, we establish the similar boundedness property for a bilinear fractional integral on a compact Lie group. Our result is also a generalization of our recent theorem about the bilinear fractional integral on torus.

Bilinear integrals of the radiative transfer equation

It is shown that the group of problems in the theory of radiative transfer that are reducible to the sourcefree problem admits a class of integrals involving quadratic moments of the intensity of arbitrarily high orders. Based on a variational principle, it is found that these integrals, which include the R-integral, follow from the corresponding conservation laws. Some of the results are generalized to the case of anisotropic scattering.

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials.

Efficient floating double-sampling integrator for ADCs

A novel double-sampling integrator is proposed, which is insensitive to path mismatch, and does not require bilinear integration. It allows the use of efficient design methods for ΔΣ ADCs.

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T : L 2(0,∞) → L 2(0,∞) is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.

Completeness of non-normalizable modes

We establish the completeness of some characteristic sets of non-normalizable modes by constructing fully localized square steps out of them, with each such construction expressly displaying the Gibbs phenomenon associated with trying to use a complete basis of modes to fit functions with discontinuous edges. As well as being of interest in and of itself, our study is also of interest to the recently introduced large extra dimension brane-localized gravity program of Randall and Sundrum, since the particular non-normalizable mode bases that we consider (specifically the irregular Bessel functions and the associated Legendre functions of the second kind) are associated with the tensor gravitational fluctuations which occur in those specific brane worlds in which the embedding of a maximally four-symmetric brane in a five-dimensional anti-de Sitter bulk leads to a warp factor which is divergent. Since the brane-world massless four-dimensional graviton has a divergent wave function in these particular cases, its resulting lack of normalizability is thus not seen to be any impediment to its belonging to a complete basis of modes, and consequently its lack of normalizability should not be seen as a criterion for not including it in the spectrum of observable modes. Moreover, because the divergent modes we consider form complete bases, we can even construct propagators out of them in which these modes appear as poles with residues which are expressly finite. Thus even though normalizable modes appear in propagators with residues which are given as their finite normalization constants, non-normalizable modes can just as equally appear in propagators with finite residues too -- it is just that such residues will not be associated with bilinear integrals of the modes.

Modified Bilinear Integration Algorithm Based on the Modulated Sine Function

This study presents a novel discretization method for converting an analogue controller into its corresponding counterpart for a discrete-time system. The proposed method is appropriate for the natural response curve and, thus, is more effective for certain digitally redesigned systems. Based on the mapping region of the transformation using the proposed method from the s-domain to the z-domain, a transformation involving a modulated parameter n is used to generate a corresponding stable discrete-time controller for a stable continuous-time controller. As a suitable modulated parameter in the proposed transformation is selected to digitalize an analogue-controlled system, the new sampled-data-controlled system achieves a precise discrete-time response, and can also tolerate a large sampling period for sample-data implementation. A simulated example is used to demonstrate the results.

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If W: R n → [0, oo] is Borel measurable, define for σ-finite positive Borel measures μ, ν on R n the bilinear integral expression I(W; μ, ν):= ∫ R n∫ R W(x - y) dμ(x) dv(y) We give conditions on W such that there is a constant C > 0, independent of μ and ν, with I(W; μ, v) ≤ CI(W;μ, μ)I(W; v, ν). Our results apply to a much larger class of functions W than known before.