Arbitrary Periodic Signal Research Articles

Analysis of time‐domain response of series‐parallel fractional‐order RCα and RLβ circuits based on Cole distribution function

SummaryFractional‐order circuits, which contain fractance devices ( and ) described by fractional‐order differential equations, are widely used in interdisciplinary engineering research. The application of time‐domain response (TDR) measurement increases rapidly in analyzing the characteristics of fractance devices and fractional‐order circuits. However, the previous methods are mainly focused on deducing TDR expressions of a single fractance device or series fractional‐order circuits in some specific input signals, while overlook the TDR analysis of series‐parallel fractional‐order circuits (SPFOC) that are commonly applied in modeling electrochemical impedance or electrical bioimpedance. This paper proposes a novel method to derive TDR expressions of SPFOC in an arbitrary periodic signal. First, a discretization method to represent the SPFOC as the sum of first‐order circuits is proposed in Cole distribution function, which could approximate the transfer function of fractional‐order impedance/admittance. Second, the TDR expression of SPFOC in sinusoidal input signals is deduced in inverse Laplace transform, which can be applied as the components of a general TDR expression of SPFOC in an arbitrary periodic signal based on the Fourier series theory. Finally, both TDR simulations and experiments on Cole impedance model (a basic SPFOC) are conducted to verify the proposed method, which shows excellent agreement compared to theorical analysis. This paper presents a TDR solution to SPFOC that consist of a or element. In future, the theoretical and application extension of the proposed TDR analysis method is still necessary on analysis of more general fractional‐order circuits.

A new method for calculation of closed‐form response of linear time‐invariant systems to periodic input signals

AbstractA new method for finding closed‐form time‐domain solutions of linear time‐invariant (LTI) systems with arbitrary periodic input signals is presented. These solutions, unlike those obtained based on the conventional Fourier‐phasor method, have a finite number of terms in one period. To implement the proposed method, the following steps are carried out: (1) For a given system, represented by a transfer function, an impulse response, a block diagram etc., the governing differential equation relating the output of the system, , to its input, , is obtained. (2) An auxiliary differential equation is formed by simply replacing with and equating the input side to alone. The auxiliary differential equation is solved for each time segment of the input signal in one period, leaving the constant coefficients associated with the homogeneous solutions as unknowns. For an nth‐order system with an input signal consisting of q segments in one period, there are such unknown coefficients. (3) Continuity of and its derivatives, at the connection points of successive segments and the periodicity conditions for the beginning and end points of the period are implemented. (4) The outcome of step 3 is a system of equations in terms of unknown coefficients described in step 2. Solving this system of equations, the solution for in one period is obtained. (5) Finally, using the linearity and differentiation properties of the system and the coefficients of the input side of the differential equation of the system, the total response, , in one period is constructed in terms of and its derivatives. For stable LTI systems, the proposed method can be used without any limitations.

Application of the MatLab Opportunities During the Study of the Fourier Series by Future Border Guard Officers

The article is devoted to the analysis of the possibilities of using modern applications in the teaching of higher mathematics for future border guard officers of the State Border Guard Service of Ukraine. The article describes the possibilities of the software stand developed on the basis of the mathematical package of the MatLab system, which allows cadets to perform spectral analysis of signals, which is important for solving professional problems of future professional activity. In addition, the article describes the process of the stand and presents its program code. The interface of the developed software stand allows to choose a window for the analysis of a kind of pulses, to receive their images, corresponding radio pulses, and also their spectra. The obtained amplitude spectra allows to compare and state the similarities and differences in the radio signal and signal spectra. In addition, if it is necessary to demonstrate the form of an arbitrary (periodic or non-periodic) signal or a signal with noise, the presented software product allows the cadet to generate a signal depending on its characteristics. The article notes that conducting classes using this software product develops the motivation of cadets to study fundamental disciplines, promotes interest and increases their knowledge and skills in higher mathematics and related disciplines, as the classes have a purposeful informational and emotional impact on future officers, their analytical-synthetic skills and critical thinking develop. In addition, the article considers the features of experimental verification of the classes effectiveness using a program stand to improve the knowledge and skills of cadets in professionally-oriented disciplines, as well as the reliability of the results using mathematical statistics.

The impact of the conversion starting threshold on the estimation of the RMS value

This paper presents an optimal solution for the selection of the starting instance of the analog-to-digital conversion for the purpose of the estimation of the root mean square (RMS) value of an arbitrary periodic signal when the ratio of the sampling frequency and the signal's fundamental frequency value is low and a non-integer. This selection of the optimal starting threshold is discussed in the context of the application of the two most common numerical methods used in measurement instrumentation today: the rectangular method and the trapezoidal rule. The appropriate threshold selection results in a significantly reduced measurement uncertainty of the estimated RMS value and improved measurement performances without necessitating major modifications or upgrade of the existing hardware. All theoretical results have been experimentally validated. The research is an original contribution to the field of mathematical modeling of methods implemented in microprocessor used in contemporary instrumentation.

Generation of Tunable Microwave Frequency Comb Utilizing a Semiconductor Laser Subject to Optical Injection from an SFP Module Modulated by an Arbitrary Periodic Signal

In this paper, a microwave frequency comb (MFC) is generated from a semiconductor laser subject to optical injection from a commercial small form-factor pluggable (SFP) optical module which is modulated by an arbitrary periodic signal. A sinusoidal signal or square wave signal is employed as the arbitrary periodic signal instead of the electric pulse signal adopted in the former references. When the frequency of the modulated signal is 1 GHz, the MFC with a maximal bandwidth of 15 GHz can be obtained. In addition, taking a sinusoidal signal as an example, the influence of the injection optical power to the slave laser and the modulation frequency of the optical module on the generation of the MFC is analyzed in detail. Finally, the results of MFC generated with a square wave signal injection are presented. The experimental results of this paper provide an important reference for the practical applications of MFC.

The programmable CPG model based on Matsuoka oscillator and its application to robot locomotion

The central pattern generator (CPG) is an important functional unit in the spinal cord which can produce rhythmic signals to control locomotion. Recently, there has been a growing interest in programmable central pattern generators (PCPG). In this paper, a new PCPG oscillator and a generic PCPG model based on the Matsuoka oscillator are presented. The perturbation method is used to determine the convergence of the generic PCPG model. The sine signal, the mix signal and chaotic signals are provided as inputs to the model, and the simulation results show that the generic PCPG can learn arbitrary periodic signals. In this paper, the generic PCPGs are allocated at each joint of the compass-like and the three-link robots and their outputs are chosen as joint position commands. The simulations show that the generic PCPG can be used to control robot locomotion effectively. The contributions of this paper are as follows: (1) A new PCPG oscillator based on the Matsuoka oscillator is presented as a beneficial enhancement to the PCPG oscillators. (2) A generic PCPG model is built comprising three PCPG oscillators. It can learn any periodic input signal. These findings are a significant contribution to generic PCPG research.

Autocorrelation and limited sampling algorithm with fixed packet length in period measurement

This paper proposed a real time autocorrelation and limited sampling algorithm with fixed packet length based on Field Programmable Gate Array(FPGA) for the period estimation and significant data transmission for arbitrary periodic signal, which contributes to information transfer in the Internet of things for smart sensors, and has a wide applications such as radar, communication, sensors network and measurement. Firstly, an explanation of the algorithm is shown. Secondly, several simulation results are analyzed. Finally, experimental and hardware implementation results are shown.

Attenuation of Position-Dependent Periodic Disturbance for Rotary Machines by Improved Spatial Repetitive Control With Frequency Alignment

In many real-world rotary machines, there is an inherent need to attenuate time-varying but position-dependent periodic disturbances. This article proposes an improved repetitive controller, in which a spatial internal model, a spatial low-pass filter, and a frequency alignment are synthesized. The controller is digitally implemented in a uniform time-sampling system. Specifically, a universal internal model of arbitrary periodic signal in spatial domain is derived for the attenuation of the position-dependent disturbances in the rotary machine, and a spatial low-pass filter is designed for the stability of the closed-loop system. Furthermore, a frequency alignment strategy is designed to compensate the bias resulted from the filter. Moreover, the cascaded utilization of the proposed controllers is introduced to improve the performance of a position-feedback servo system. Experimental verification and comparisons validate the feasibility and effectiveness of the proposed method in both single and cascaded configurations.

Effect of Adsorbent Grain Size on the Pressure Swing Adsorption

An analytical solution is obtained for a system of differential equations consisting of the equation for diffusion and absorption of a component in adsorbent grains and the balance equation for this component moving in the adsorbent layer.The technique for calculation of steady-state periodic adsorption processes based on expansion of a concentration signal into Fourier series of eigenfrequencies of this adsorption process is proposed. After decomposition each of the eigenwaves independently of the others is passed through the adsorbent layer, and at the output from this layer all this solutions for individual eigenwaves are summed. The solution to a problem of passing an arbitrary periodic concentration signal through the adsorbent layer is obtained as a result of summing the solutions for individual eigenwaves.The proposed method takes into account nonstationary diffusion processes inside the adsorbent grains. It makes possible to analyze effect of the adsorbent grain sizes on a pressure swing adsorption processes.

Stability of horizontal viscous fluid layers in a vertical arbitrary time periodic electric field

The stability of a horizontal interface between two viscous fluids, one of which is conducting and the other is dielectric, acted upon by a vertical time-periodic electric field is considered theoretically. The two fluids are bounded by electrodes separated by a finite distance. For an applied ac electric field, the unstable interface deforms in a time periodic manner, owing to the time dependent Maxwell stress, and is characterized by the oscillation frequency which may or may not be the same as the frequency of the ac electric field. The stability curve, which relates the critical voltage, manifested through the Mason number—the ratio of normal electric stress and viscous stress, and the instability wavenumber at the onset of the instability, is obtained by means of the Floquet theory for a general arbitrary time periodic electric field. The limit of vanishing viscosities is shown to be in excellent agreement with the marginal stability curves predicted by means of a Mathieu equation. The influence of finite viscosity and electrode separation is discussed in relation to the ideal case of inviscid fluids. The methodology to obtain the marginal stability curves developed here is applicable to any arbitrary but time periodic signal, as demonstrated for the case of a signal with two different frequencies, and four different frequencies with a dc offset. The mode coupling in the interfacial normal stress leads to appearance of harmonic and subharmonic modes, characterized by the frequency of the oscillating interface at an integral or half-integral multiple of the applied frequency, respectively. This is in contrast to the application of a voltage with a single frequency which always leads to a harmonic mode oscillation of the interface. Whether a harmonic or subharmonic mode is the most unstable one depends on details of the excitation signal.

Digital repetitive controller design via sampled-data delayed signal reconstruction

In this paper, we propose a design method of digital repetitive controllers to track continuous-time periodic references based on sampled-data H∞ delayed signal reconstruction. It is well known that perfect tracking is not possible for arbitrary periodic signals with a fixed period if the plant to be controlled is strictly proper.For this reason, we adopt the internal model control (IMC), with which the repetitive control is cast into the framework of sampled-data H∞ delayed signal reconstruction problem. We also extend this formulation to multi-rate and robust control. Design examples show the effectiveness of our method.

Control approaches for robotic knee exoskeleton and their effects on human motion

In this paper we compare three noninvasive control methods for a robotic knee exoskeleton and asses their kinematic influences on the repetitive squatting motions of able-bodied human subjects. The motion of the subjects wearing the knee exoskeleton was also compared to the motion of the subjects performing the same task without using the assistance of the knee exoskeleton. We chose the squatting motion because it approximates common movements with high metabolic cost, such as standing up from a chair and ascending or descending the stairs. Beside the two classical robotic control approaches, i.e. the position control and the gravity compensation, we propose a method that is based on a single adaptive frequency oscillator combined with an adaptive Fourier series in a feedback loop. The method can extract frequency and phase of an arbitrary periodic signal in real-time. This method is particularly appropriate for controlling novel robotic assisting devices since it does not require complex signal sensing or user calibration. The results show that the total knee torque was increased while using the exoskeleton device compared to the squatting without the assistance of the exoskeleton device. In effect, there were significant kinematic adaptations observed when the exoskleton device assisted the motion of the subjects. However, no significant kinematic differences were found between different control methods. We conclude that an assistive device can augment the abilities of the able-bodied humans in the targeted joints (i.e. the joints provided with additional mechanical power) but, on the other hand, significantly alters whole body kinematics.

Design, implementation and analysis of an alternation-based Central Pattern Generator for multidimensional trajectory generation

In this paper, we introduce a multidimensional Central Pattern Generator (CPG) model with an explicit and defined basin of attraction for generating any arbitrary continuous periodic signal. Having a defined basin of attraction is highly desired, especially in robotic applications, as it provides tracking stability in addition to robustness against disturbances. The CPG model is composed of a set of phase-locked coordinated one-dimensional models; called ζ -models. The idea behind the ζ -model is generating any one-dimensional periodic signal by altering the behavior of an existing oscillator through two nonlinear maps. The mappings are designed in such a way that the Poincaré–Bendixson theorem is satisfied and, consequently, the desired basin of attraction is shaped. The proposed CPG model is extensively tested for generating multidimensional signals; including DC, triangular, and smooth wavy ones. The results show that the CPG model has a low tracking error in addition to being robust against disturbances within the designed basin of attraction. Finally, the proposed CPG model is successfully employed to generate the dancing motion of a situated robotic marionette.

Multiheterodyne Detection for Spectral Compression and Downconversion of Arbitrary Periodic Optical Signals

Optical frequency combs are used as local oscillators for the measurement and analysis of unknown optical waveforms with periodic time domain structures. Experimental results obtained by heterodyning pulsed and phase-modulated laser sources are presented. The analysis is then extended to the heterodyning and sampling of bandlimited incoherent light sources. It is shown theoretically and experimentally that the correlations between photodetected white light at different times can generate RF interference that is sensitive to the optical phase.

Zero-broadening slow and fast light using an optimized Brillouin comb gain for arbitrary periodic signals

Based on the theoretical analysis of the conditions of the fast or slow light without distortion for arbitrary signal, a universal formula of an optimized Brillouin comb gain is obtained in accordance with the spectrum characteristics of the periodic signal. According to the amount of delay or advancement required, each Brillouin gain intensity and frequency detuning between Brillouin gain peak and the corresponding harmonic frequency of periodic signal can be found conveniently using this universal formula, then the whole Brillouin comb gain is identified. Numerical calculation for sinusoidal signal, Gaussian periodic signal and square periodic signal is performed. The results show that zero-broadening slow and fast light is achieved using this optimized Brillouin comb gain for the three types of periodic signals, and the validity of the universal formula is proved. Moreover, the magnitude of the delay or advancement can be tuned continuously by adjusting the Brillouin gain intensity and frequency detuning.

Robust regulation of stable systems in the H ∞-algebra

In this paper, robust regulation of stable infinite-dimensional plants with transfer functions in the H ∞-algebra is considered. The reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis, which makes it possible to consider e.g. arbitrary periodic signals. A controller depending on a positive scalar ε and design parameters Kk is proposed. The design parameters are matrices over H ∞, and by the main result of the paper, if the design parameters satisfy a certain condition, the proposed controller is a robust regulator for sufficiently small values of ε. The only knowledge needed to verify this condition is the so-called -stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals. The analysis in this paper is in the frequency domain and it is based on the fractional representation approach.

Frequency shifting of microwave signals by use of a general temporal self-imaging (Talbot) effect in optical fibers

A simple and practical microwave frequency-shifting technique based on a general temporal self-imaging (GTSI) effect in optical fiber is proposed, formulated, and experimentally demonstrated. The proposed technique can be applied to an arbitrary periodic microwave signal (e.g., a microwave tone) and provides unparalleled design flexibility to increase the frequency of the input microwave signal up to the desired value (limited only by the photodetector's bandwidth). For instance, we demonstrate frequency upshifting of microwave tones from approximately 10 to approximately 50 GHz and from approximately 40 to approximately 354 GHz. These results also represent what is to the authors' knowledge the first experimental observation of GTSI phenomena.

Sinusoidal interference rejection analysis of an LMS adaptive feedforward controller with a noisy periodic reference

This paper applies a new vector subspace model to determine the non-Wiener solutions of the LMS algorithm when the reference input is an arbitrary noisy periodic signal. The LMS weights are modeled as a deterministic time-varying mean plus a zero-mean fluctuating part. For the mean weight, each harmonic component of the periodic reference is shown to excite only a two-dimensional (2-D) subspace of the N-dimensional tap weight space. The fluctuating part of the weight is due to the reference noise that excites the full N-dimensional space. The power spectrum of the error is computed using the correlation function of the weight fluctuations. The error is shown to be primarily the sum of a white noise and the canceled periodic residual for an independent noise vector model. When the effects of the tapped delay line are incorporated in the model, the noise becomes colored with a spectrum centered about the canceled sinusoidal frequency. This weight model significantly simplifies the understanding of the non-Wiener behavior and can be applied to active noise cancellation problems when filters appear in the cancellation loop.

Stochastic resonance in the Heaviside nonlinearity with white noise and arbitrary periodic signal.

A Heaviside nonlinearity with adjustable threshold is fed by an arbitrarily distributed white noise plus a periodic signal of arbitrary wave form. A general and exact treatment demonstrates that this system is capable of stochastic resonance in a large variety of conditions and offers a complete characterization of this property. In particular it gives the possibility of observing nonzero phase shifts with nondynamic stochastic resonance. \textcopyright{} 1996 The American Physical Society.

Repetitive and periodic controller design

Compensators can be designed to give zero steady-state error in tracking arbitrary periodic or repetitive reference signals. The previously presented technique for design of the compensator has a limited convergence rate for the decay of tracking errors and a limited region of stability. Several alternative design approaches are presented which yield improvements in high-frequency convergence rate while some designs also improve stability. The design approaches are in two classes, designs based on adding periodic correction to an existing controller and total designs for transient and periodic performance. The approaches include FIR and IIR filter design by pole placement and linear quadratic control design.