Adaptive notch filter by direct frequency estimation

This paper investigates adaptive digital notch filters for the elimination of powerline noise from biomedical signals. Since the distribution of the frequency variation of the powerline noise may or may not be centered at 60 Hz, three different adaptive digital notch filters are considered. For the first case, an adaptive FIR second-order digital notch filter is designed to track the center frequency variation. For the second case, the zeroes of an adaptive IIR second-order digital notch filter are fixed on the unit circle and the poles are adapted to find an optimum bandwidth to eliminate the noise to a pre-defined attenuation level. In the third case, both the poles and zeroes of the adaptive IIR second-order filter are adapted to track the center frequency variation within an optimum bandwidth. The adaptive process is considerably simplified by designing the notch filters by pole-zero placement on the unit circle using some suggested rules. A constrained least mean-squared (CLMS) algorithm is used for the adaptive process. To evaluate their performance, the three adaptive notch filters are applied to a powerline noise sample and to a noisy EEG as an illustration of a biomedical signal.

A narrowband active vibration control algorithm with frequency estimator based on lattice notch filter is proposed to suppress the primary signal with multi line spectra. A adaptive iir (infinite impulse response) lattice gradient notch filter was adapted to estimated the frequency of the multi-harmonics buried in the primary signal, then the reference signal is filtered by the adaptive band pass filter. With the reference signal filtered by the adaptive band pass filter, active control algorithm is implemented. Simulations of the frequency estimation of the multi-sinusoidal signal are conducted and simulation results confirm that the proposed algorithm could achieve effective performance when the frequency of the primary disturbance is unknown. Active control of vibration has finds a lot application such as fans, engines, and air conditioners in nowadays. The basic concept of active vibration control is to suppress the primary disturbance with the secondary force which is of same amplitude and opposite phase. Signals generated by the rotating machines such as fans, diesel engines, and air conditioners is characterized with discreet frequency which cause damage to operator in the working environment. Narrowband active vibration control (NAVC) is an effective method to suppress such vibration with multi-line spectra. The application of NAVC has been implemented taken into applications since 1970's. A lot of algorithms have been proposed for suppress the primary disturbance with multi-spectral. There exists two families of algorithm in AVC system, which composed of filtered-X least mean square (FXLMS) algorithm and filtered-X recursive least square algorithm (FXRLS). Due to its simplicity and implementation efficiency, FXLMS algorithm is the most popular adaptive algorithm to suppress the primary disturbance compared to FXRLS algorithms. In the application of NAVC systems, the performance of NAVC systems is directly correlated with the reference signal and the primary signal. NAVC algorithm could not achieve effective performance if the correlation between the primary signal and the reference signal is weak. Frequency estimation of the primary signal is an effective method to adapt to improve the relativity between the reference signal and the primary signal. To estimate the frequency of a sinusoidal signal, a number of algorithm has been proposed to exploit the predictable nature of sinusoidal primary disturbance and characterize the frequency of the sinusoidal primary disturbance, such the Kalman algorithm, music, FFT (fast Fourier transform), etc. such method could be capable of tracking nonstationary or slowly-varyin g primary signals. However, they have some shortcomings in the application of NAVC systems. a lot of Kalman-based algorithms have been proposed for the frequency estimation of the sinusoidal signal in many applications such as the electrical power system, however, it requires much more computation burdens which is much more complicated compared the other algorithms. FFT methods have been proposed in the past to estimated the frequency of the primary signal, but it is has been found that the estimated result could not be accessed effectively in real-time of AVC system. Adaptive notch filter has been proposed for the frequency estimation of the sinusoidal signal in a lot of applications. Adaptive lattice filter shows much more stability and performance compared to the bilinear adaptive notch filter. In this paper, the adaptive lattice notch filter is introduced into an adaptive vibration control system to estimate the frequency of synthesized desired signal. Then, the reference signal is generated to suppress the primary vibration in the active vibration control system.

In this paper, a method of reducing cochannel FM interference with adaptive tracking notch filters, first proposed by E. Baghdady (1959), is refined and extended. Analytical descriptions and computer simulations of a novel adaptive notch filter are presented. The new form of adaptive notch filter is useful for suppressing strong cochannel FM and CW signals and allows simple, practical realizations of very high and adaptive notch filters in either analog or digital technologies. The analysis demonstrates that when the instantaneous frequency of the stronger interfering FM carrier signal is known precisely, the stronger signal can be completely eliminated. In some cases, placing an additional adaptive notch filter in front of the stronger signal demodulator to suppress the weaker desired cochannel signal improves the estimate of the stronger signal.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper proposes a multiple adaptive notch filter with the p‐th power average of the estimated error E[ep(n)] ‖as the cost function. The adaptive filter consists of a cascade connection of second‐order adaptive notch filters composed of the allpass filters. In general, the elimination bandwidth of the notch filter is desirably small from the practical point of view. However, as the elimination bandwidth is set smaller, the convergence speed of the tap coefficients of the adaptive filter becomes much slower. From the point of view of convergence theory, the radius of the pole of the second‐order allpass filter needs to be close to 1, so that the elimination bandwidth must be extremely small. Convergence speed must be sacrificed due to the practical and theoretical points of view. In this paper, an adaptive notch filter is studied in which the elimination bandwidth of the notch filter is designed small while the gradient of the tap vector of the cost function used in the adaptive algorithm is made effectively equal to the case in which the elimination bandwidth is rather large. In this way, realization of a multiple adaptive notch filter with a high convergence is possible. The absolute value of the transfer function of the notch filter is close to 0 near the elimination frequency while it is close to 1 at other frequencies. Hence, by setting the power of the absolute value larger, the elimination bandwidth becomes larger and the smoothness near the elimination frequency is increased. By means of the gradient algorithm with a cost function E[ep(n)], this property can be utilized so that excellent convergence speed and steady‐state characteristics can be obtained. Finally, verification of the analytical results and comparison with the conventional method are carried out by computer simulation. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(8): 1–8, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10162

This paper proposes a new adaptive notch filter with high convergence speed and low steady state error. The new adaptive notch filter is based on an improved Simplified Lattice Algorithm (SLA) which has variable step sizes using piloted adaptive notch filters. Computer simulations demonstrate that the adaptive notch filter claims an improved convergence speed under the same mean square error condition in comparison to the conventional SLA.

The task of removing sinusoidal components from observed signals can be accomplished by using a notch filter with a specific attenuation at a particular frequency. In some applications, however, such as acoustic feedback control, the frequency at which attenuation is required is unknown and possibly time-varying, and hence an adaptive notch filter is a more appropriate solution. Transitioning from a fixed notch filter to an adaptive one is by no means trivial and involves the understanding of a range of digital signal processing (DSP) topics from pole-zero placement techniques for designing infinite impulse response filters to optimal and adaptive filtering algorithms. In the signal processing algorithms and implementation graduate course taught at KU Leuven (Belgium), we study the design of an adaptive notch filter, which is based on a constrained biquadratic IIR representation, andwhose parameters are updated using a least-mean-square algorithm. Students also have to implement the algorithm on a 16-bit DSP TMS320C5515. In this presentation, we will discuss the design and implementation challenges of this adaptive notch filter and how it serves as an illustrative example/homework problem where several aspects of DSP are interwoven.

SummarySeveral continuous‐time frequency estimators for a measured sinusoidal signal, which have been proposed in the literature, are reviewed, reinterpreted and compared both theoretically and by simulations. It is argued that adaptive notch filters are feedback algorithms that contain a local adaptive observer in the feedback loop. It is shown that the adaptive notch filter (ANF), which was originally conceived as a discrete‐time ANF, is basically equivalent to the recently proposed adapted frequency locked loop called orthogonal signal generator. They both require a sufficiently slow frequency estimation and can be interpreted as third‐order adaptive observers. They exhibit local convergence properties for the estimation errors, that is, the convergence to zero is guaranteed provided that their initial error is sufficiently small. Three adaptive observers, which were independently proposed in 2002, are third‐order frequency estimators whose estimation errors are exponentially convergent to zero from any initial condition and for any value of frequency, amplitude and phase in the measured sinusoidal signal. They have the additional advantage of not requiring the frequency estimation dynamics to be sufficiently slow. Conversely, they may be interpreted as adaptive notch filters. Second‐order frequency estimators have been proposed as well: they may be interpreted as adaptive reduced‐order observers. Copyright © 2015 John Wiley &amp; Sons, Ltd.

We investigate the problem of direct frequency estimation. A new adaptive algorithm is developed for constrained pole-zero notch filters. The basic characteristic of this algorithm is that the notch filter is cascaded and, hence, purely parametrized by the notching frequencies. This makes the algorithm much simpler and, hence, more efficient and more robust such that the poles and zeros are automatically located on the respective constrained circle, no matter if the algorithm is implemented with infinite or finite precision. The algorithm is analyzed and compared with some existing algorithms. Several numerical examples are given as well as the corresponding simulations.

This paper presents a 2-link, 2-DOF flexible manipulator control using an inverse feedforward controller and an adaptive notch filter with a direct strain feedback controller. In the flexible manipulator, transient and residue vibrations inhibit the full potential of the manipulator. Vibrations caused by abrupt changes in the direction of the links are referred to as transient vibrations, whereas residual vibrations occur when the arm takes too long to settle after engaging in the intended task. The feedforward adaptive notch filter will reduce transient vibration caused by the manipulator arm beginning and halting suddenly, while the strain feedback will assure the quick decay of leftover vibrations. Maple, Maplesim, and MATLAB tools were used to model the manipulator and create the inverse controller and adaptive notch filter. The experiments took place in the dSPACE control desk environment. The experimental results of the spectral power of strain resulting from the two strategies are compared. From the results, the adaptive notch filter control had over an 80% improvement in the reduction in resonant frequencies that contribute to vibration. The results confirmed the feasibility of the approach, characterized by very minimal transient vibrations and a quick settling of the end effector.

To detect interharmonics whose amplitudes are often small, a detection method based on ANF and interpolated FFT is proposed. In the method, the parallel structure of the ANF (adaptive notch filter) is firstly used to detect the fundamental and harmonic components, in which the fundamental frequency signal with high amplitude is detected with the main ANF and then the measured base frequency is output to sub-parallel ANF structure so as to extract the harmonic components. Then a separated interpolated fast Fourier transformation (FFT) analysis of interharmonic components will be effectively completed without the influence of the fundamental and harmonic components. A detection platform based on the proposed method is implemented in LabVIEW. The results of simulation and experiments show the effectiveness of the method.

This paper deals with a state-space approach for adaptive second-order IIR notch filters with constrained poles and zeros. A simplified adaptive iterative algorithm is derived from the gradient-descent method for minimizing the mean-squared output error of an adaptive notch filter. Stability and parameter estimation bias are analyzed for the adaptive iterative algorithm. A numerical example is presented to demonstrate the validity and effectiveness of the proposed adaptive state-space notch filter and parameter-estimation bias analysis.

The acceleration output of an electro-hydraulic servo system corresponding to a sinusoidal input contains higher harmonics besides the fundamental input, because of complex nonlinearities occurring in the system. This causes harmonic distortion of the acceleration signal. The method for harmonic elimination based on adaptive notch filter is proposed here. The task is accomplished by generating a reference signal with a frequency that should be eliminated from the output. The reference input is filtered in such a way that it closely matches the harmonic. The filtered reference signal is added to the fundamental signal such that the output harmonic is cancelled leaving the desired signal alone. The weights of the adaptive filter are adjusted by the error between the input and the feedback acceleration to eliminate acceleration harmonic, creating an adaptive notch filter. The above concept is used as a basis for the development of an acceleration harmonic cancellation algorithm. Results of simulation and experiment on an electro-hydraulic servo shaking table demonstrate the efficiency and validity of the proposed control scheme.

Adaptive notch filters have been the focus of intense research for more than three decades. Low computational requirements and good performance make them attractive for tracking frequency modulated signals. Despite the extensive literature on adaptive notch filters, algorithm extensions and new models for describing the dynamics of the notch frequency continue to be proposed. In this study, the equivalence between adaptive notch filters using a plain gradient (PG) algorithm and frequency lock loops (FLLs) with exponential filtering is established. FLL theory is then used to analyse the noise performance and signal tracking capabilities of adaptive notch filters. A linear model describing the dynamics of the filter adaptation process is derived and the concepts of loop and Doppler bandwidths are introduced. Criteria based on the loop and Doppler bandwidths are suggested to set the PG adaptation step. Finally, algorithm extensions based on the FLL theory are proposed. Theoretical results are supported by Monte Carlo simulations which show the validity of the analysis performed.

Gradient-type algorithms for the adaptive infinite-impulse response (IIR) notch filters are very attractive in terms of both performance and computational requirements for various real-life applications. This paper presents, in detail, a statistical analysis of the memoryless nonlinear gradient (MNG) algorithm applied to the well-known second-order adaptive IIR notch filter with constrained poles and zeros. This analysis is based on a proper use of Taylor series expansion and nonlinearization of output signals of the notch and gradient filters. Two difference equations are derived first for the convergence in the mean and mean square senses, respectively. Two closed-form expressions, one for the steady-state estimation bias and the other for the mean-square error, are then derived based on the difference equations, with the former valid for both fast and slow adaptations and the latter valid for slow adaptation only. A closed-form coarse stability bound for the step size parameter of the algorithm is also derived. Extensive simulations are performed to reveal the validity and limitations of the analytical findings. Comparisons between the MNG and the conventional plain gradient algorithm are also made.

In this paper, an adaptive notch filter is developed (employing a frequency domain and time domain analysis) for the enhancement and tracking of sinusoids in additive noise, colored or white. The notch filter is implemented as a constrained infinite impulse response filter with the constraint enforced by a single parameter termed the debiasing parameter. The resulting notch filter requires few parameters, facilitates the formation of the desired band rejection filter response, and also leads to various useful implementations (cascade, parallel). For the adaptation of the filter coefficients, the stochastic Gauss-Newton algorithm is used. The convergence of this updating procedure is established by studying the associated differential equation. Also, it is shown that the structure present in the problem enables truncation of the gradient, thereby reducing the complexity of adapting the filter coefficients. Simulation results are presented to substantiate the analysis, and to demonstrate the potential of the notch filtering technique.