Multiband least squares FIR filter design

This paper gives two methods for the least squares approximation design of FIR digital filters. They both allow multiple pass and stop bands and arbitrary transition bands, all with explicit control of band edges. The first method is as simple to use as a window method. This design method builds up an optimal multiband frequency response by sequentially adding and subtracting optimal lowpass filters with transition bands. However, it assumes the ideal response has spline transition functions and does not allow error weighting. The second method allows different weights in the multiple passbands, stopbands and transition bands but it requires numerical solution of simultaneous equations which may be ill-conditioned. These methods compliment the Parks-McClellan algorithm which minimizes the Chebyshev error     

最小二乘与加权最小二乘空域矩阵滤波器设计

空域矩阵滤波器是一种新的信号处理技术,通过一个滤波矩阵与接收到的阵列数据相乘,可实现保留通带目标信号,抑制阻带干扰的目的.本文主要研究了最小二乘和加权最小二乘两类的空域矩阵滤波器.给出了空域滤波器设计基本原理,通过最优化问题得出了最优解.最小二乘空域矩阵滤波器是加权系数为1的加权加权最小二乘空域矩阵滤波器的特列.由加权最小二乘迭代仿真结果可以看出,迭代次数的增加使滤波器阻带响应极大值逐渐变小,可实现恒定阻带抑制效果,设计效率较高.     

Simplified weighted least squares design of 2-D FIR digital filters

A simple easy to use and very flexible approach is presented for the design of 2-D fir digital filters. It is illustrated by two examples, a rectangular filter and a diamond filter. Appropriate selection of the weighting function which is used in the weighted least squares method can yield very diverse characteristics, like the possibility to meet specifications on both frequency response and step response.     

Improved weighted least squares algorithm for the design ofquadrature mirror filters

This paper presents an improved weighted least squares (WLS) algorithm for the design of quadrature mirror filters (QMFs), First, a new term is incorporated into the objective function that effectively prevents an optimization algorithm from producing suboptimal QMFs. These suboptimal QMFs exhibit a transition band anomaly; the frequency responses of the filters have large oscillatory components in the transition band. The new term can be applied to the WLS design of any FIR filter to prevent a similar transition band anomaly. Next, we present an algorithm to obtain the QMF coefficients that minimize the objective function incorporating the new term. The computational requirement of this algorithm is also briefly discussed. Last, we include a set of practical design rules for use with our algorithm. These rules simplify the design process by providing good estimation of the design parameters, such as the minimum filter length, to meet a given set of QMF specifications     

A constrained weighted least squares approach for time-frequencydistribution kernel design

In most applications of time-frequency (t-f) distributions, the t-f kernel is of finite extent and applied to discrete time signals. This paper introduces a matrix-based approach for t-f distribution kernel design. In this new approach, the optimum kernel is obtained as the solution of a linearly constrained weighted least squares minimization problem in which the kernel is vectorial and the constraints form a linear subspace. Similar to FIR temporal and spatial constrained least squares (LS) design methods, the passband, stopband, and transition band of an ideal kernel are first specified. The optimum kernel that best approximates the ideal kernel in the LS error sense, and simultaneously satisfies the multiple linear constraints, is then obtained using closed-form expressions. This proposed design method embodies a well-structured procedure for obtaining fixed and data-dependent kernels that are difficult to obtain using other design approaches     

Constrained least squares design of 2-D FIR filters

We consider the design of 2-D linear phase finite impulse response (FIR) filters according to the least squares (LS) error criterion subject to equality and/or inequality constraints. Since we use a frequency domain formulation, these constraints can be used to explicitly prescribe (frequency-dependent) error tolerances, the maximum, minimum, or fixed values of the frequency response at certain points and/or regions. Our method combines Lagrange multiplier and Kuhn-Tucker theory to solve a much wider class of problems than do standard methods. It allows arbitrary compromises between the LS and the equiripple design     

Robust least mean square adaptive FIR filter algorithm

The authors propose a new robust adaptive FIR filter algorithm for system identification applications based on a statistical approach named the M estimation. The proposed robust least mean square algorithm differs from the conventional one by the insertion of a suitably chosen nonlinear transformation of the prediction residuals. The effect of nonlinearity is to assign less weight to a small portion of large residuals so that the impulsive noise in the desired filter response will not greatly influence the final parameter estimates. The convergence of the parameter estimates is established theoretically using the ordinary differential equation approach. The feasibility of the approach is demonstrated with simulations     

A weighted least squares approach to the design of FIR filters synthesized using the modified frequency response masking structure

This paper presents an application of the weighted least squares (WLS) method to the design of sharp linear phase finite-impulse response (FIR) digital filters synthesized using a modified frequency-response masking (FRM) structure. In our approach, the original minimax design problem is converted into a WLS problem. The WLS problem is highly nonlinear with respect to the coefficients of the filter. However, it can be decomposed into four linear least squares (LS) problems, each of which can be solved analytically. The design problem is then solved iteratively by using an alternating variable approach. The effectiveness of the method is demonstrated through solving a low-pass linear phase sharp FIR digital filter example.     

一种改进两步加权最小二乘的TDOA定位算法

现有的两步加权最小二乘水下声源定位算法中,两步法处理均存在平方以及开方等非线性运算忽略了二阶误差项,导致定位精度下降。提出了一种改进的两步加权最小二乘算法,首先引入时差方程,构建新的定位方程,初步估计出声源位置;接着引入中间变量并进行泰勒展开,利用中间变量与声源位置误差之间的非线性关系构建定位方程,对声源位置初步估计值进行校正,有效避免了二阶误差项的丢失,得到了更加精确的估计结果。通过与克拉美罗下界进行比较,验证了算法性能的同时进一步提升了算法定位精度。     

Fast sequential least squares design of FIR filters with linearphase

The authors propose a fast adaptive least-squares algorithm for linear-phase finite-impulse-response (FIR) filters. The algorithm requires 10m multiplications per data point where m is the filter order. This reduced computation stems from the fast adaptive forward-backward least-squares method. Both linear-phase cases i.e. with constant phase delay and with constant group delay, are examined. Simulation results demonstrate that the proposed algorithm is superior to the least-mean-squares gradient algorithm and the averaging scheme used for the modified fast Kalman algorithm     

FIR filter design over discrete coefficients and least square error

The difference routing digital filter (DRDF) consists of an FIR filter followed by a first-order integrator. This structure with power-of-two coefficients has been studied as a means of achieving low complexity, high sampling rate filters which can be implemented efficiently in hardware. The optimisation of the coefficients has previously been based on a time-domain least-squares error criterion. A new design method is proposed that includes a frequency-domain least-squares criterion with arbitrary frequency weighting and an improved method for handling quantisation of the filter coefficients. Simulation studies show that the new approach yields an improvement of up to 7 dB over existing methods and that oversampling can be used to improve performance     

Efficient least squares adaptive algorithms for FIR transversalfiltering

A unified view of algorithms for adaptive transversal FIR filtering and system identification has been presented. Wiener filtering and stochastic approximation are the origins from which all the algorithms have been derived, via a suitable choice of iterative optimization schemes and appropriate design parameters. Following this philosophy, the LMS algorithm and its offspring have been presented and interpreted as stochastic approximations of iterative deterministic steepest descent optimization schemes. On the other hand, the RLS and the quasi-RLS algorithms, like the quasi-Newton, the FNTN, and the affine projection algorithm, have been derived as stochastic approximations of iterative deterministic Newton and quasi-Newton methods. Fast implementations of these methods have been discussed. Block-adaptive, and block-exact adaptive filtering have also been considered. The performance of the adaptive algorithms has been demonstrated by computer simulations     

Two-stage, least squares design of biorthogonal filter banks

A two-stage approach is employed for the design of a class of two-channel biorthogonal filter banks. The filter banks belong to the class HPFB (halfband pair filter bank) and are defined by two kernels. The parametric Bernstein polynomial is used to construct the kernels. The design of the free parameters of the Bernstein polynomial is achieved through a least squares method. In the first stage, the analysis low-pass filter is designed and in the second stage, the synthesis lowpass filter is designed. With the two-stage approach, the design process is efficient and involves solving linear equations. The design technique allows filters with different characteristics to be designed easily.     

FIR digital filter design techniques using weighted Chebyshev approximation

This paper discusses the various approaches to designing FIR digital filters using the theory of weighted Chebyshev approximation. The different design techniques are explained and compared on the basis of their capabilities and limitations. The relationships between filter parameters are briefly discussed for the case of low-pass filters. Extensions of the theory to the problems of magnitude and complex approximation are also included, as are some recent results on the design of two-dimensional FIR filters by transformation.     

多级方向加权最小二乘滤波器及其在多传感器图像融合中的应用

针对多尺度分解在图像融合领域中的广泛应用,本文提出了一种多级方向加权最小二乘滤波器图像多尺度几何分析方法。该方法利用加权最小二乘滤波器对图像进行多级边缘保持分解,得到一个近似图像和多个不同尺度上的细节图像,然后采用小尺寸方向剪切滤波器对细节图像进行方向分析,在不同尺度上生成多个方向细节图像。根据近似图像和方向细节图像所具有的不同物理意义,分别采用不同的融合策略对分解后的图像系数进行合并处理,最后应用多级方向加权最小二乘滤波器的逆变换得到融合图像。多组图像融合实验结果表明,在图像融合领域,本文提出的基于多级方向加权最小二乘滤波器的图像分解方法优于已有文献中的一些典型多尺度分解方法。     

A robust recursive least squares algorithm

A new algorithm is developed, which guarantees the normalized bias in the weight vector due to persistent and bounded data perturbations to be bounded. Robustness analysis for this algorithm has been presented. An approximate recursive implementation is also proposed. It is termed as the robust recursive least squares (RRLS) algorithm since it resembles the RLS algorithm in its structure and is robust with respect to persistent bounded data perturbation. Simulation results are presented to illustrate the efficacy of the RRLS algorithm     

Multiplier-less FIR filter design using a genetic algorithm

A new synthesis technique is described for multiplier-less FIR digital filters consisting of a cascade of primitive linear phase sections. For medium-order filters the search space for an optimal cascade is typically of the order of 10³⁰ and this can be examined in a computation efficient way using the parallel-search capability of a genetic algorithm (GA). A particular form of GA based upon multilevel or structured chromosomes has been developed for the primitive cascade problem. Initial results suggest that, for the cost of increased filter delay, a typical 2:1 advantage can be achieved in both VLSI chip area and clock rate compared to filters designed using the usual equiripple method     

Equiripple FIR filter design by the FFT algorithm

The fast Fourier transform (FFT) algorithm has been used in a variety of applications in signal and image processing. In this article, a simple procedure for designing finite-extent impulse response (FIR) discrete-time filters using the FFT algorithm is described. The zero-phase (or linear phase) FIR filter design problem is formulated to alternately satisfy the frequency domain constraints on the magnitude response bounds and time domain constraints on the impulse response support. The design scheme is iterative in which each iteration requires two FFT computations. The resultant filter is an equiripple approximation to the desired frequency response. The main advantage of the FFT-based design method is its implementational simplicity and versatility. Furthermore, the way the algorithm works is intuitive and any additional constraint can be incorporated in the iterations, as long as the convexity property of the overall operations is preserved. In one-dimensional cases, the most widely used equiripple FIR filter design algorithm is the Parks-McClellan algorithm (1972). This algorithm is based on linear programming, and it is computationally efficient. However, it cannot be generalized to higher dimensions. Extension of our design method to higher dimensions is straightforward. In this case two multidimensional FFT computations are needed in each iteration     

一种快速递归全局最小二乘算法

针对FIR系统输入和输出信号均被噪声干扰的情况,提出一种快速递归全局最小二乘(XS-RTLS)算法用于迭代计算全局最小二乘解,算法沿着输入数据的符号方向并采用著名的快速增益矢量,搜索约束瑞利商(c-RQ)的最小值得到系统参数估计。算法关于方向更新矢量的内积运算可通过加减运算实现,有效降低了计算复杂度;另外XS-RTLS算法没有进行相关矩阵求逆递归运算,因而具有长期稳定性,算法的全局收敛性通过Laslle不变性原理得到证明。最后通过仿真比较了XS-RTLS算法和递归最小二乘(RLS)算法在非时变系统和时变系统中的性能,验证了XS-RTLS算法的长期稳定性。     

A numerically stable fast Newton-type adaptive filter based on order recursive least squares algorithm

This paper presents a numerically stable fast Newton-type adaptive filter algorithm. Two problems are dealt with in the paper. First, we derive the proposed algorithm from an order-recursive least squares algorithm. The result of the proposed algorithm is equivalent to that of the fast Newton transversal filter (FNTF) algorithm. However, the derivation process is different. Instead of extending a covariance matrix of the input based on the min-max and the max-min criteria, the derivation shown in this paper is to solve an optimum extension problem of the gain vector based on the information of the Mth-order forward or backward predictor. The derivation provides an intuitive explanation of the FNTF algorithm, which may be easier to understand. Second, we present stability analysis of the proposed algorithm using a linear time-variant state-space method. We show that the proposed algorithm has a well-analyzable stability structure, which is indicated by a transition matrix. The eigenvalues of the ensemble average of the transition matrix are proved all to be asymptotically less than unity. This results in a much-improved numerical performance of the proposed algorithm compared with the combination of the stabilized fast recursive least squares (SFRLS) and the FNTF algorithms. Computer simulations implemented by using a finite-precision arithmetic have confirmed the validity of our analysis.