Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [O...Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31（3）：260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308：3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the（Σ△D） formula class are given in the closing section.展开更多
This paper presents a fully integrated CMOS filterless class D amplifier that can directly hook up lithium battery in mobile application The proposed amplifier embodies a 2-order feedback path architecture instead of ...This paper presents a fully integrated CMOS filterless class D amplifier that can directly hook up lithium battery in mobile application The proposed amplifier embodies a 2-order feedback path architecture instead of direct feedback of output to input of the integrator to decrease the high frequency intermodulation distortion associated with direct feedback and eliminate the integrator input common mode disturbance from the output in ternary modulation.The prototype class D amplifier realized in 0.35μm digital technology achieves a THD+N of 0.02%when delivering 400 m W to an 8Ωload from V_(DD)=3.6 V.The PSRR of the prototype class D amplifier is 80 dB at217 Hz.Furthermore a filterless method that can eliminate the external LC filter is employed which offers great advantages of saving PCB space and lowering system cost.In addition the prototype class D amplifier can operate in large voltage range with V_(DD)range from 2.5 to 4.2V in mobile application.The total area of the amplifier is 1.7mm^2.展开更多
Undesirable audio click and pop may be generated in a speaker or headphone.Compared to linear（class A/B/AB） amplifiers,class D amplifiers that comprise of an input stage and a modulation stage are more prone to prod...Undesirable audio click and pop may be generated in a speaker or headphone.Compared to linear（class A/B/AB） amplifiers,class D amplifiers that comprise of an input stage and a modulation stage are more prone to producing click and pop.This article analyzes sources that generate click and pop in class D amplifiers,and corresponding ways to suppress them.For a class D amplifier with a single-ended input,click and pop is likely to be due to two factors.One is from a voltage difference（KDIF） between the voltage of an input capacitance（Vcin） and a reference voltage（Vref） of the input stage,and the other one is from the non-linear switching during the setting up of the bias and feedback voltages/currents（BFVC） of the modulation stage.In this article,a fast charging loop is introduced into the input stage to charge F_（CIN） to roughly near VREF.Then a correction loop further charges or discharges V_（CIN）,substantially equalizing it with KREF.Dummy switches are introduced into the modulation stage to provide switching signals for setting up BFVC,and the power switches are disabled until the BFVC are set up successfully.A two channel single-ended class D amplifier with the above features is fabricated with 0.5μm Bi-CMOS process.Road test and fast Fourier transform analysis indicate that there is no noticeable click and pop.展开更多
This paper investigates the noise sources in a single-ended class D amplifier（SECDA） and suggests corresponding ways to lower the noise.The total output noise could be expressed as a function of the gain and noises ...This paper investigates the noise sources in a single-ended class D amplifier（SECDA） and suggests corresponding ways to lower the noise.The total output noise could be expressed as a function of the gain and noises from different sources.According to the function,the bias voltage（V_B） is a primary noise source,especially for a SECDA with a large gain.A low noise SECDA is obtained by integrating a filter into the SECDA to lower the noise of the V_B.The filter utilizes an active resister and an 80 pF capacitance to get a 3 Hz pole.A noise test and fast Fourier transform analysis show that the noise performance of this SECDA is the same as that of a SECDA with an external filter.展开更多
In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By ap...In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the conver gence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak L∞-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.展开更多
Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asympto...Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.展开更多
A novel monolithic digitalized random carrier frequency modulation spread-spectrum clock generator （RCF-SSCG） is proposed.In this design,the output frequency of the proposed RCF-SSCG changes with the intensity of th...A novel monolithic digitalized random carrier frequency modulation spread-spectrum clock generator （RCF-SSCG） is proposed.In this design,the output frequency of the proposed RCF-SSCG changes with the intensity of the capacitive charge and discharge current.Its analytical model is induced and the effect of the modulation parameters on the spread spectrum is numerically simulated and discussed.Compared with other works,this design has the advantages of small size,low power consumption and good robustness.The circuit has been fabricated in a 0.5μm CMOS process and applied to a class D amplifier in which the proposed RCF-SSCG occupies an area of 0.112 mm^2 and consumes 9 mW.The experimental results confirm the theoretical analyses.展开更多
文摘Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31（3）：260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308：3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the（Σ△D） formula class are given in the closing section.
基金supported by the National Natural Science Foundation of China(Nos.61234002,61322405,61306044)
文摘This paper presents a fully integrated CMOS filterless class D amplifier that can directly hook up lithium battery in mobile application The proposed amplifier embodies a 2-order feedback path architecture instead of direct feedback of output to input of the integrator to decrease the high frequency intermodulation distortion associated with direct feedback and eliminate the integrator input common mode disturbance from the output in ternary modulation.The prototype class D amplifier realized in 0.35μm digital technology achieves a THD+N of 0.02%when delivering 400 m W to an 8Ωload from V_(DD)=3.6 V.The PSRR of the prototype class D amplifier is 80 dB at217 Hz.Furthermore a filterless method that can eliminate the external LC filter is employed which offers great advantages of saving PCB space and lowering system cost.In addition the prototype class D amplifier can operate in large voltage range with V_(DD)range from 2.5 to 4.2V in mobile application.The total area of the amplifier is 1.7mm^2.
文摘Undesirable audio click and pop may be generated in a speaker or headphone.Compared to linear（class A/B/AB） amplifiers,class D amplifiers that comprise of an input stage and a modulation stage are more prone to producing click and pop.This article analyzes sources that generate click and pop in class D amplifiers,and corresponding ways to suppress them.For a class D amplifier with a single-ended input,click and pop is likely to be due to two factors.One is from a voltage difference（KDIF） between the voltage of an input capacitance（Vcin） and a reference voltage（Vref） of the input stage,and the other one is from the non-linear switching during the setting up of the bias and feedback voltages/currents（BFVC） of the modulation stage.In this article,a fast charging loop is introduced into the input stage to charge F_（CIN） to roughly near VREF.Then a correction loop further charges or discharges V_（CIN）,substantially equalizing it with KREF.Dummy switches are introduced into the modulation stage to provide switching signals for setting up BFVC,and the power switches are disabled until the BFVC are set up successfully.A two channel single-ended class D amplifier with the above features is fabricated with 0.5μm Bi-CMOS process.Road test and fast Fourier transform analysis indicate that there is no noticeable click and pop.
文摘This paper investigates the noise sources in a single-ended class D amplifier（SECDA） and suggests corresponding ways to lower the noise.The total output noise could be expressed as a function of the gain and noises from different sources.According to the function,the bias voltage（V_B） is a primary noise source,especially for a SECDA with a large gain.A low noise SECDA is obtained by integrating a filter into the SECDA to lower the noise of the V_B.The filter utilizes an active resister and an 80 pF capacitance to get a 3 Hz pole.A noise test and fast Fourier transform analysis show that the noise performance of this SECDA is the same as that of a SECDA with an external filter.
文摘In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws,we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the conver gence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak L∞-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.
基金Supported by the National Natural Science Foundation of China(No.11071045,No.11171179,No.11201080,No.11301391)the Research Fund for the Doctoral Program of Higher Education of China(No.20133705110002)
文摘Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.
基金Project supported by the National Natural Science Foundation of China(No.60436030).
文摘A novel monolithic digitalized random carrier frequency modulation spread-spectrum clock generator （RCF-SSCG） is proposed.In this design,the output frequency of the proposed RCF-SSCG changes with the intensity of the capacitive charge and discharge current.Its analytical model is induced and the effect of the modulation parameters on the spread spectrum is numerically simulated and discussed.Compared with other works,this design has the advantages of small size,low power consumption and good robustness.The circuit has been fabricated in a 0.5μm CMOS process and applied to a class D amplifier in which the proposed RCF-SSCG occupies an area of 0.112 mm^2 and consumes 9 mW.The experimental results confirm the theoretical analyses.
