Statistical Performance of the Memoryless Nonlinear Gradient Algorithm for the Constrained Adaptive IIR Notch Filter

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.     

Tracking properties of a gradient-based second-order adaptive IIRnotch filter with constrained poles and zeros

Gradient-type adaptive IIR notch filters have many attractive merits for various real-life applications since they require a small number of computations and yet demonstrate practical performance. However, it is generally quite difficult to assess their performance analytically. Their tracking properties, in particular, have not yet been investigated. In this paper, the tracking performance of a plain gradient (PG) algorithm is analyzed in detail for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a linear chirp signal as its input. First, two sets of difference equations for the frequency tracking error and mean square error (MSE) are established in the sense of convergence in the mean and convergence in the mean square, respectively. Closed-form expressions for the asymptotic tracking error and MSE are then derived from these difference equations. An optimum step-size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking MSE. It is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size, which is an interesting property for a real-valued adaptive filtering algorithm. Extensive simulations are performed to support the analytical findings     

Steady-state analysis of a complex adaptive notch filter using modified gradient algorithm

Based on power spectral density (PSD) analytical technique, mean square error (MSE) (or variance) of the frequency estimate of a first-order complex adaptive IIR notch filter (ANF) using modified complex plain gradient (MCPG) algorithm is investigated in this paper. The steady-state expression for MSE is derived in closed form. A quantitative analysis for the estimation MSE has been carried out. It has been revealed that the MSE of frequency estimate is independent of an input frequency of a complex sinusoid. In addition, computer simulations are treated to corroborate the theoretical analysis and the relationships between MSE and system parameters are shown.     

导航卫星接收机窄带干扰抑制算法研究

在GNSS服务过程中,地面接收到的卫星导航信号的功率较小,且易受到周围复杂电磁干扰影响,这将会导致卫星无法正常定位。针对常见的窄带干扰,介绍一种新型自适应全通IIR陷波器,自适应算法采用高斯-牛顿迭代法,通过跟踪窄带干扰频点,对其频点进行陷波。文中在其基础上进行改进,在每次迭代过程中采用最佳收敛因子,最大程度地减少均方误差。这种算法可导致更快的收敛速度和更高的估计精度。仿真结果表明,IIR滤波器可有效去除高达70 dB的窄带干扰。     

基于IIR滤波器的窄带干扰抑制技术仿真与实现

研究了直扩通信中二阶自适应格型IIR陷波滤波器在抗窄带干扰中的应用,推导了系统信噪比改善因子和误比特率的闭合表达式,分析了算法的性能,并进行了MATLAB仿真。基础上,设计了FPGA可实现的自适应解扩重扩滤波器,该滤波器实现简单,硬件测试表明,滤波器达到了预期的性能指标,有效可抗窄带干扰在大约21dB。     

Suppression of transients in variable recursive digital filterswith a novel and efficient cancellation method

A new method for suppressing transients in recursive infinite impulse response (IIR) digital filters is proposed. The technique is based on modifying the state (delay) variables of the filter when coefficients are changed so that the filter enters a new state smoothly without transient attacks, as originally proposed by Zetterberg and Zhang (1988). In this correspondence, we modify the Zetterberg-Zhang algorithm to render it feasible for efficient implementation. We define a mean square error (MSE) measure for transients and determine the optimal transient suppressor to cancel the transients down to a desired level at the minimum complexity of implementation. The application of the method to all-pole and direct-form II (DF II) IIR filter sections is studied in detail. Time-varying recursive filtering with transient elimination is illustrated for tunable fractional delay filters and variable-bandwidth lowpass filters     

A new adaptive recursive RLS-based fast-array IIR filter for active noise and vibration control systems

Infinite impulse response filters have not been used extensively in active noise and vibration control applications. The problems are mainly due to the multimodal error surface and instability of adaptive IIR filters used in such applications. Considering these, in this paper a new adaptive recursive RLS-based fast-array IIR filter for active noise and vibration control applications is proposed. At first an RLS-based adaptive IIR filter with computational complexity of order O(n²) is derived, and a sufficient condition for its stability is proposed by applying passivity theorem on the equivalent feedback representation of this adaptive algorithm. In the second step, to reduce the computational complexity of the algorithm to the order of O(n) as well as to improve its numerical stability, a fast array implementation of this adaptive IIR filter is derived. This is accomplished by extending the existing results of fast-array implementation of adaptive FIR filters to adaptive IIR filters. Comparison of the performance of the fast-array algorithm with that of Erikson’s FuLMS and SHARF algorithms confirms that the proposed algorithm has faster convergence rate and ability to reach a lower minimum mean square error which is of great importance in active noise and vibration control applications.     

A complex adaptive notch filter using modified gradient algorithm

A modified gradient algorithm is developed for improving the convergence speed of a first-order complex adaptive IIR notch filter, which is used for estimating an unknown frequency of a complex sinusoidal signal embedded in white Gaussian noise. The new cost function using new error criterion is presented and analyzed theoretically. The proposed technique can significantly improve the convergence speed as compared with a complex notch filter using plain gradient algorithm. The computer simulations are conducted to demonstrate the validity of the proposed complex adaptive notch filter.     

二阶自适应陷波器频率估计方法统计性能分析

为对如何提高自适应陷波器频率估计精度提供参考,通过评估自适应陷波器频率估计方法性能,对基于均方误差函数的自适应陷波器频率估计方法进行了统计性能分析。首先,根据误差函数的不同,将自适应陷波器划分为自适应FIR陷波器和自适应IIR陷波器。然后,将自适应FIR陷波器看作自适应IIR陷波器的特例,重点分析了自适应陷波器的误差函数及稳态下的频率估计统计性能,讨论了自适应陷波器参数对正弦信号频率估计精度和收敛速度的影响。最后,给出正弦信号的频率估计计算结果。结果表明,实际计算结果同理论计算结果一致,证明了统计性能分析的正确性。      

卫星导航接收机时域窄带干扰抑制滤波器设计与性能分析

针对卫星导航接收机时域窄带干扰的有效抑制问题,本文首先给出二阶格型IIR陷波器参数设计方法,通过陷波带宽的定量调整,既可以有效抑制窄带干扰,又可以降低卫星信号的失真;其次推导了二阶IIR格型陷波器相关输出信干噪比改善因子的闭合表达式,该表达式相比干扰抑制后信干噪比的改善,更为直观地反映了陷波器对卫星导航信号的影响。理论分析和仿真实验,二阶格型IIR陷波器相关输出信干噪比改善因子与陷波器的带宽参数有关,而与陷波频率无关,且二阶格型IIR陷波器的性能优于最优线性预测Wiener滤波器、最优线性插值Wiener滤波器、五系数FIR滤波器以及二阶直接型IIR陷波器。      

Tracking analysis of an adaptive notch filter with constrainedpoles and zeros

This paper analyzes the asymptotic tracking properties of an adaptive notch filter (ANF) with pole-zero constraints for the cancellation or retrieval of multiple time-varying sine waves in additive noise. The asymptotic mean square error (MSE) is analyzed using the methods of Ljung and Gunnarsson (1990) when the variations in the underlying frequencies are assumed to be sufficiently small. Closed-form expressions for the MSE are derived as functions of the tuning variables of the algorithm. The results give insight into the operational properties of the algorithm and are used in order to minimize the MSE with respect to the tuning variables. Computer simulations confirm the validity of the derived results     

Pole-zero decision feedback equalization with a rapidly convergingadaptive IIR algorithm

A decision feedback equalizer (DFE) containing a feedback filter with both poles and zeros is proposed for high-speed digital communications over the subscriber loop. The feedback filter is composed of a relatively short FIR filter that cancels the initial part of the channel impulse response, which may contain rapid variations due to bridge taps, and a pole-zero, or IIR, filter that cancels the smoothly decaying tail of the impulse response. Modifications of an adaptive IIR algorithm, based on the Steiglitz-McBride (1965) identification scheme, are proposed to adapt the feedback filter. A measured subscriber loop impulse response is used to compare the performance of the adaptive pole-zero DFE, assuming a two-pole feedback filter, with a conventional DFE having the same number of coefficients. Results show that the pole-zero DFE offers a significant improvement in mean squared error relative to the conventional DFE. The speed convergence of the adaptive pole-zero DFE is comparable to that of the conventional DFE using the standard least mean square (LMS) adaptive algorithm     

直扩系统中IIR格型滤波器抑制窄带干扰新方法与性能分析

该文提出一种直扩系统中基于IIR格型陷波器时域频域相结合的窄带干扰抑制新方法,该方法采用时域频域并行处理的结构,通过最优加窗和频谱校正技术在频域精确估计出干扰信号的频率,根据频率估计误差和干信比求出最优陷波带宽,由此自适应地调整时域IIR格型陷波器的参数。分析和实验表明,该方法较传统的自适应IIR窄带干扰抑制方法不仅具有更高的稳定性和实时性,而且在干扰得到有效抑制的情况下使有用信号的损失达到最小。     

Convergence analysis of adaptive filtering algorithms with singulardata covariance matrix

The paper provides a rigorous analysis of the behavior of adaptive filtering algorithms when the covariance matrix of the filter input is singular. The analysis is done in the context of adaptive plant identification. The considered algorithms are LMS, RLS, sign (SA), and signed regressor (SRA) algorithms. Both the signal and weight behavior of the algorithms are considered. The signal behavior is evaluated in terms of the moments of the excess output error of the filter. The weight behavior is evaluated in terms of the moments of the filter weight misalignment vector. It is found that the RLS and SRA diverge when the input covariance matrix is singular. The steady-state signal behavior of the LMS and SA can be made arbitrarily fine by using sufficiently small step sizes of the algorithms. Indeed, the long-term average of the mean square excess error of the LMS is proportional to the algorithm step size. The long-term average of the mean absolute excess error of the SA is proportional to the square root of the algorithm step size. On the other hand, the steady-state weight behavior of both the LMS and SA have biases that depend on the weight initialization. The analytical results of the paper are supported by simulations     

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

This paper presents two-step design methodologies and performance analyses of finite-impulse response (FIR), allpass, and infinite-impulse response (IIR) variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense and solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a global solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming problem and solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to nonvariable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency ( ges 0.9pi) . Filter examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.     

基于自适应IIR陷波滤波器的窄带干扰抑制技术

在存在窄带干扰时，扩频通信系统的性能可以通过使用各种不同的抗干扰滤波器来进一步提高。文中研究了二阶自适应格型IIR陷波滤波器在直扩通信中抗窄带干扰的应用，推导了系统信噪比改善因子和误比特率的闭合表达式。计算机仿真表明了该分析结果的正确性。     

Bias-remedy least mean square equation error algorithm for IIRparameter recursive estimation

In the area of infinite impulse response (IIR) system identification and adaptive filtering the equation error algorithms used for recursive estimation of the plant parameters are well known for their good convergence properties. However, these algorithms give biased parameter estimates in the presence of measurement noise. A new algorithm is proposed on the basis of the least mean square equation error (LMSEE) algorithm, which manages to remedy the bias while retaining the parameter stability. The so-called bias-remedy least mean square equation error (BRLE) algorithm has a simple form. The compatibility of the concept of bias remedy with the stability requirement for the convergence procedure is supported by a practically meaningful theorem. The behavior of the BRLE has been examined extensively in a series of computer simulations     

抗干扰用陷波滤波器

详细介绍了采用Ｒｅｍｅｚ算法和最小二乘法综合带陷波点频响滤波器的设计方法,并给出了设计制作的７０ＭＨｚ陷波滤波器结果。     

Design of three-dimensional adaptive IIR notch filters

This paper investigates a realization of a three-dimensional (3-D) adaptive notch filter. The procedures are mainly divided into two parts: frequency-detecting and sinusoidal interference removal. The detections are based on adaptive line enhancer on infinite impulse response (IIR) lattice structure. In the interference removal part, a non-separable version of a 3-D notch filter is effectively applied. The magnitude response of a 3-D adaptive IIR notch filter is illustrated. At the end of the paper, the implementation of an IIR notch filter on a 3-D image is also conducted in order to show how to remove a sinusoidal interference superimposed on a 3-D image.     

Adaptive digital notch filter design on the unit circle for theremoval of powerline noise from biomedical signals

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 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