An adaptive notch filter for frequency estimation of a periodic signal

Online frequency estimation of a pure sinusoidal signal is a classical problem that has many practical applications. Recently an ANF with global convergence property has been developed for this purpose. There exist some practical applications in which signals are not pure sinusoidal and contain harmonics. Therefore, online frequency estimation of periodic but not necessarily sinusoidal signals espoused by such applications becomes quite important. This note presents an alternative stability analysis for a modified ANF that permits the presence of harmonics in the incoming signal. Also, this stability analysis is simpler and alleviates the problem complexity even in the case of pure sinusoidal signal. Simulation results confirm theoretical issues.     

归一化频率自适应梳状滤波器的稳定性分析

采用多个归一化频率估计器并联形成梳状滤波器, 以跟踪和检测平稳概周期信号各正弦成分的未知频率和未知幅值. 滤波器包括相互耦合的状态估计和频率估计两个非线性微分方程. 运用慢积分流形实现两个微分方程之间的解耦, 获得关于多个频率估计值的概周期非线性动力系统, 再应用平均方法导出估计频率的非线性自治方程. 分析了自治系统的三种局部稳定性: 孤立平衡点的指数稳定性, 中心流形存在性与半稳定性以及结构扰动下的有界性. 说明幅值估计与信号跟随的收敛性和有界性. 给出滤波器参数对频率跟踪和幅值估计的暂态和稳态性能的影响. 算法实现了在给定频率区间而不是给定数值条件下的正弦分量及其幅值的准确跟随, 并且响应速度不受正弦分量幅值大小的影响. 通过仿真验证了算法的有效性.     

Reduction of quantization noise via periodic code for oversampled input signals and the corresponding optimal code design

This paper proposes to reduce the quantization noise using a periodic code, derives a condition for achieving an improvement on the signal to noise ratio (SNR) performance, and proposes an optimal design for the periodic code. To reduce the quantization noise, oversampled input signals are first multiplied by the periodic code and then quantized via a quantizer. The signals are reconstructed via multiplying the quantized signals by the same periodic code and then passing through an ideal lowpass filter. To derive the condition for achieving an improvement on the SNR performance, first the quantization operator is modeled by a deterministic polynomial function. The coefficients in the polynomial function are defined in such a way that the total energy difference between the quantization function and the polynomial function is minimized subject to a specification on the upper bound of the absolute difference. This problem is actually a semi-infinite programming problem and our recently proposed dual parameterization method is employed for finding the globally optimal solution. Second, the condition for improving the SNR performance is derived via a frequency domain formulation. To optimally design the periodic code such that the SNR performance is maximized, a modified gradient descent method that can avoid the obtained solution to be trapped in a locally optimal point and guarantee its convergence is proposed. Computer numerical simulation results show that the proposed system could achieve a significant improvement compared to existing systems such as the conventional system without multiplying to the periodic code, the system with an additive dithering and a first order sigma delta modulator.     

Asymptotic rejection of asymmetric periodic disturbances in output-feedback nonlinear systems

This paper deals with asymptotic rejection of periodic disturbances which may have asymmetric basic wave patterns. This class of disturbances covers asymmetric wave forms in the half-period such as alternating sawtooth wave form, some disturbances which are generated from nonlinear oscillation such as Van de Pol oscillators, as well as disturbances with symmetric half-period wave forms such as sinusoidal disturbances and triangular disturbances etc. The systems considered in this paper can be transformed to the nonlinear output feedback form. The amplitude and phase of the disturbances are unknown. The novel concept of integral phase shift is introduced together with the newly introduced half-period integration operator to investigate the invariant properties of asymmetric periodic disturbances. They are used for the estimation of unknown disturbances in the systems, together with observer design techniques to deal with nonlinearity. The proposed control design with the disturbance estimation asymptotically rejects the unknown disturbance, and ensures the overall stability of the system.     

Peak-to-peak filtering for periodic piecewise linear polytopic systems

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques.     

采样正弦信号的时延估计

提出了一种正弦信号的时延估计算法,它涉及采样和多重相关。该方法可以大大降低时延估计的复杂程度,提高估计的范围和估计的灵活性。并且它可以减小噪声干扰,适用于高频信号,有利于高频数字接收机的实现。仿真显示该算法的性能和优势非常明显。     

一类周期切换控制稳定的算法设计

利用切换控制及凸组合弱化控制系统稳定性条件,使组合和满足Hurwitz条件而实现切换稳定。该文对切换系统的表达形式、切换信号及切换路径、切换收敛、稳定性判断准则及周期切换特征进行了分析。针对无扰动或扰动切换系统进行周期稳定性设计,提出3种切换控制率及对应的切换算法。通过对交通实例进行仿真设计,比对切换实验数据,反映出不同切换率稳定和收敛情况,通过上述算法实现系统的周期稳定。     

Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency

Asymptotically stable, observable linear systems of order n which are not required to be minimum phase and are affected by an additive noisy biased sinusoidal disturbance with unknown bias, magnitude, phase and frequency are considered. The problem of designing an output feedback compensator which regulates the output to zero for any initial condition and for any biased sinusoidal disturbance with no noise is addressed, under the assumption that the system parameters are known. This problem is solved by a (2n+6)-order compensator which generates asymptotically convergent estimates of the biased sinusoidal disturbance and of its parameters, including frequency. The robustness of the closed loop system with respect to sufficiently small additive unmodelled noise is characterized in terms of input-to-state stability.     

神经网络正弦信号发生器

针对正弦信号发生器设计中，直接数字频率合成技术存在相位截断误差的问题，以神经网络为技术基础，以FPGA为硬件核心，提出了一种新型的高频正弦信号发生器设计方案，有效克服了上述问题。阐述了这种方案的工作原理、电路结构以及设计思路和方法。经过设计和仿真测试，系统的主时钟频率可以达到95 MHz且不占用ROM存储空间，输出的正弦信号为2.5 MHz时，输出信号的杂散抑制为80 dB，可见该方案资源占用率低，无相位截断，输出信号杂散小且输出频率较高。     

Hybrid dead-beat observers for a class of nonlinear systems

This paper studies both the full order and the reduced order dead-beat observer problem for a class of nonlinear systems, linear in the unmeasured states. A novel hybrid observer design strategy is proposed, with the help of the notion of strong observability in finite time. The proposed methodology is applied for the estimation of the frequency of a sinusoidal signal. The results show that accurate estimation can be provided even if the signal is corrupted by high frequency noise. A brief discussion of the robustness properties of the proposed observer with respect to measurement errors is also provided.     

基于对偶观测器的谐波提取方法

提出一种基于对偶观测器的标准正弦信号提取方法.将受到谐波干扰的信号通过一种辅助滤波器得到与谐波频率相关的可测变量;然后利用对偶关系将受干扰信号分解为已知函数和受标准正弦信号驱动的待估项;进而构造观测器估计未知参数,直接对标准正弦信号进行重构.该方法可以将谐波信号表示为关于辅助变量的参数形式,直接建立频率与干扰谐波之间关系,实现标准信号与干扰谐波的分离.由于只需要估计受标准正弦信号驱动的未知参数,很大程度上可以降低计算复杂度.利用李雅普诺夫稳定性理论证明:当不存在有界噪声时,这种观测器可以渐近跟踪标准频率信号;在多源干扰环境下,观测器误差动态满足一致最终有界特性.仿真结果验证了该对偶观测结构的有效性.     

A magnitude/phase-locked loop approach to parameter estimation of periodic signals

Presents a parameter estimator that is designed to estimate the frequencies, magnitudes, and phases of the components of a periodic signal. The structure of the algorithm is reminiscent of a phase-locked loop, although significant differences can be observed. The performance of the estimator is analyzed, and useful design guidelines are provided. A version of the algorithm is presented that combines different components of the signal and/or signals from multiple sensors in order to estimate the fundamental frequency. In this manner, the algorithm is able to maintain tracking of the fundamental frequency despite changes in signal characteristics. The results are verified in simulations and the algorithms are found to be simple and effective for estimation and tracking of time-varying parameters. Experimental results are reported where periodic signals are collected from an active noise control testbed.     

A globally convergent frequency estimator

Online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications. In this paper the authors provide a solution to the problem of ensuring a globally convergent estimation. More specifically, they propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features: 1) all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values; 2) the authors obtain a simple tuning procedure for the estimator design parameters, which trades-off the adaptation tracking capabilities with noise sensitivity, ensuring (exponential) stability of the desired orbit; and 3) transient performance is considerably enhanced, even for small or large frequencies, as witnessed by extensive simulations. To reveal some of the stability-instability mechanisms of the existing algorithms and motivate our modifications the authors make appeal to a novel nonlinear (state-dependent) time scaling. The main advantage of working in the new time scale is that they remove the coupling between the parameter update law and the filter itself, decomposing the system into a feedback form where the required modifications to ensure stability become apparent. Even though they limit their attention here to the simplest case of a single constant frequency without noise the algorithm is able to track time-varying frequencies, preserve local stability in the presence of multiple sinusoids, and is robust with respect to noise     

Asymptotic rejection of finite frequency modes of general periodic disturbances in output feedback nonlinear systems

This paper deals with asymptotic rejection of individual frequency modes in nonlinear systems under general periodic disturbances. An iterative observer design is proposed for disturbance estimation, and it is shown that the proposed observer design successfully extracts individual frequency modes from general periodic disturbances. The proposed iterative observer design is then applied to asymptotic rejection of nonlinear systems under general periodic disturbances.     

A globally convergent estimator for n-frequencies

In this paper, we propose a solution to the well-known problem of ensuring a simultaneous globally convergent online estimation of the state and the frequencies of a sinusoidal signal composed of n sinusoidal terms. We present an estimator which guarantees global boundedness and convergence of both the state estimation and the frequency estimation for all initial conditions and frequency values     

Multi-channel active noise control for periodic sources—indirect approach

This paper presents a multi-channel active noise control algorithm that is designed to reject periodic signals of unknown frequency. It is based on a so-called indirect approach, where the frequency of the disturbance is estimated in real time, and the estimate is used in a disturbance rejection scheme designed for a known frequency. Improvements over an earlier algorithm include an extension to multi-channel systems, a better frequency estimation algorithm, and a thorough experimental evaluation. For disturbance rejection, a so-called inverse G algorithm is proposed and its properties are compared through analysis and experiments to those of a gradient algorithm. A new frequency estimator is also considered that is simple and flexible in design, and is able to use multiple harmonics or multiple signals in order to estimate the fundamental frequency of the noise source. In this manner, the algorithm maintains tracking of the fundamental frequency despite significant changes in signal characteristics. The ability of the indirect approach to reject periodic noise with fixed or time-varying frequency and amplitudes is demonstrated in active noise control experiments. The algorithm may also be useful in other control applications where periodic disturbances of unknown frequency must be rejected.     

Phase noise analysis of oscillators with Sylvester representation for periodic time-varying modulus matrix by regular perturbations

Phase noise analysis of an oscillator is implemented with its periodic time-varying small signal state equations by perturbing the autonomous large signal state equations of the oscillator. In this paper, the time domain steady solutions of os- cillators are perturbed with traditional regular method; the periodic time-varying Jocobian modulus matrices are decomposed with Sylvester theorem, and on the resulting space spanned by periodic vectors, the conditions under which the os- cillator holds periodic steady states with any perturbations are analyzed. In this paper, stochastic calculus is applied to disclose the generation process of phase noise and calculate the phase jitter of the oscillator by injecting a pseudo sinusoi- dal signal in frequency domain, representing the white noise, and a δ correlation signal in time domain into the oscillator. Applying the principle of frequency modulation, we learned how the power-law and the Lorentzian spectrums are formed. Their relations and the Lorentzian spectrums of harmonics are also worked out. Based on the periodic Jacobian modulus matrix, the simple algorithms for Floquet exponents and phase noise are constructed, as well as a simple case is demonstrated. The analysis difficulties and the future directions for the phase noise of oscillators are also pointed out at the end.     

H∞ filtering for linear periodic systems with parameter uncertainty

This paper focuses on H_∞ filtering for a class of linear periodic systems with a certain type of norm-bounded time-varying parameter uncertainty which appears in both the state and output matrices. The problem addressed is the design of a linear periodic estimator that guarantees both the quadratic stability and and prescribed H_∞ performance on infinite horizon for the estimation error for all admissible parameter uncertainties. A solution to this problem is obtained via a Riccati equation approach.     

Nonlinear Least-Squares Approach to Frequency Estimation and Detection for Sinusoidal Signals with Arbitrary Envelope

In this paper, we consider the problem of estimating the frequency of a sinusoidal signal whose amplitude could be either constant or time-varying. We present a nonlinear least-squares (NLS) approach when the envelope is time-varying. It is shown that the NLS criterion can be explicitly minimized with respect to the signal amplitude and envelope values, leaving out a periodogram-like function whose peak location gives the frequency estimate. This result is a significant generalization of a similar result on the NLS frequency estimation for a sinusoidal signal with constant amplitude. A statistical analysis shows that the NLS frequency estimator is asymptotically statistically efficient. The problem of detecting amplitude time variations is next addressed. A statistical test is formulated, based on the statistics of the difference between two frequency estimates. The test is computationally efficient and yields as a by-product consistent frequency estimates under either hypothesis (i.e., constant or time-varying amplitude). Numerical examples are included to show the performance both in terms of estimation and detection.     

Design of Asymptotically Convergent Frequency Estimator Using Contraction Theory

The problem of estimating unknown frequencies of a sinusoidal signal simultaneously is a classical problem in signal and system theory. Many approaches and algorithms are proposed in literature to develop estimators for a measurable sinusoidal signal having multiple sinusoids with unknown amplitudes, frequencies and phases. In this work, an asymptotically convergent frequency estimator is given for estimation of n-unknown frequencies of a measurable sinusoidal signal. The contraction theory approach is adopted to show the asymptotic convergence of the proposed estimator in quite a simplified manner. Approach given here exploits the results of contraction theory related to semi-contracting systems. The nonlinear estimator based on dynamic system approach, guarantees global boundedness and convergence of the state and frequency estimation for all initial conditions and frequency values. It further ensures simultaneous globally convergent estimation of states and frequencies of a sinusoid involving multiple frequencies. Numerical simulations are presented for different combinations of frequencies to justify the claim.