Closed-Form Least-Squares Design of Fast-Convolution Based Variable-Bandwidth FIR Filters

Conventional algorithms for constrained least-squares (CLS) designs of two-dimensional (2-D) FIR filters vectorize the filters' impulse response coefficients. Recently, an efficient algorithm exploiting the matrix nature of the impulse response is proposed for the CLS design of quadrantally symmetric 2-D linear-phase FIR filters. This paper extends the matrix-based algorithm to the CLS design of 2-D nonlinear-phase FIR filters with low group delay. A design example is presented to illustrate the effectiveness and efficiency of the proposed algorithm.

It has been shown by some researchers that in a problem of weighted least-square (WLS) design of an FIR filter, most of the design computation pertains to the evaluation of the inverse of a matrix in order to solve a system of equations. A new iterative procedure is developed for the inversion of the matrices involved in the design. By expanding the inverse of a matrix as a convergent series, an updating formula for evaluating the inverse for each iteration is obtained, so that the proposed algorithm requires an inverse for only a few initial iterations but does not need any numerical operations for matrix inversion in succeeding iterations. It is also shown that the proposed iterative procedure is applicable for a wide range of weighting functions used for least-square designs.

Conventional algorithms for phase-error constrained minimax (PCMM) designs of two-dimensional (2-D) FIR filters vectorize the filter's impulse response coefficient. Recently, an efficient algorithm exploiting the matrix nature of the impulse response is proposed for the constrained least-squares (CLS) design of 2-D nonlinear-phase FIR filters. This paper devotes to transform the PCMM design into the same form as the CLS design. A 2-D sigmoid function is introduced to constrain the phase error, resulting in 2-D FIR filters with much smaller maximum group delay error than that obtained under corresponding constant phase-error constraints. Design example and comparisons demonstrate the effectiveness and high efficiency of the proposed method.

An adaptive approach to the design of linear phase low-pass FIR filters with extra constraints on filter coefficients is presented. In this approach, the procedures using the LMS adaptive algorithm are modified to include the constraints on the filter coefficients. Numerical examples are presented and compared to the results obtained using least-square design in the frequency domain in which the filter design problem is transformed into an equivalent nonlinear optimisation problem.

Least-square (LS) design of two-dimensional (2-D) FIR filters with prescribed magnitude ripples is formulated as a positive definite quadratic programming, for which a recently proposed projected least-square (PLS) algorithm has been shown to be much more efficient than commonly used active methods and popular interior-point methods. After some modification, a more efficient PLS algorithm is obtained in this paper. Accompanied with a binary search, the modified PLS algorithm has been applied to design of smallest size 2-D circular and diamond-shaped FIR filters with prescribed magnitude ripples. Design examples demonstrate the effectiveness and efficiency of the presented design algorithms

A computationally efficient iterative algorithm is developed in this paper for the weighted least-squares (WLS) design of two-dimensional (2-D) FIR filters. The new algorithm is based on the optimality condition and uses the matrix iterative technique, which retains the coefficients of 2-D filters in their natural matrix form, resulting in a considerable saving in the amount of computation and memory space. Moreover, two vectors are introduced into the iterative equation of the algorithm, aiming to reduce the number of iterations required for convergence. Some design examples are presented to illustrate the good performance of the proposed algorithm.

We study a class of composite FIR filters (C-filters), each is composed of a prototype filter and a shaping filter in cascade, where the shaping filter is constructed by cascading several complementary comb filters. In particular, the problems of designing C-filters that are optimal in least-squares, equiripple passband and lease-squares stopband, and minimax sense are formulated, and three algorithms for designing such linear-phase FIR C-filters are proposed. The algorithms are based on an alternating optimization strategy in that the prototype and shaping filters are optimized in separate steps, which are coupled and carried out in a sequential manner to yield a satisfactory design. Design examples are presented to illustrate the algorithms and demonstrate the performance of the C-filters relative to their conventional FIR counterparts.

The group delay of a linear-phase FIR filter is about half of its length. Existing methods for low group-delay FIR filter design usually result in large group delay error, noticeably towards the band edges. Recently, Lai [2] presented a constrained least-squares design method that can specify both magnitude-error and phase-error upper bounds. Reduced group-delay error can be obtained by using a sigmoid phase-error upper-bound function in the method. In this paper, we improve the method by using a modified complex approximation error, such that we can more flexibly specify the magnitude and phase errors, and smaller magnitude error can be obtained under the same phase-error constraint. A multiple-exchange algorithm is used to solve the complex-error and phase-error constrained least-squares design problem. Through design examples, the proposed design method is compared with several existing method. Design examples demonstrate the effectiveness the proposed method.

Develops a new iterative reweighted least squares algorithm for the design of optimal L/sub p/ approximation FIR filters. The algorithm combines a variable p technique with a Newton's method to give excellent robust initial convergence and quadratic final convergence. Details of the convergence properties when applied to the L/sub p/ optimization problem are given. The primary purpose of L/sub p/ approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained L/sub 2/ approximation filters. The new method can also be applied to the complex Chebyshev approximation problem and to the design of 2D FIR filters. >

In this paper, a new approach for weighted least square (WLS) design of FIR filters is presented, in which an iterative procedure is developed for the inversion of the matrix involved in the design. By imposing a mild constraint on the update factor of the weighting function, the inverse of a matrix is expanded as a convergent power series. By exploiting the properties of some of the matrices involved in the design, a modified version of the series that converges rapidly is then proposed to evaluate the inverse in each iteration. It is shown that due to the fast convergence of the power series, one has to evaluate only the first two or three terms of the series after a few initial iterations, implying that the conventional operation for matrix inversion is simplified significantly.

Design of optimal linear phase FIR filters by a semi-infinite programming technique

Digital filters capable of changing their frequency response characteristics are often referred to as variable digital filters (VDFs) and have been found useful in a number of digital signal processing applications. An important class of VDFs is the class of digital filters with variable fractional delay. This paper describes an enhanced weighted least-squares design for variable-fractional-delay finite-impulse response filters, which offers improved performance of the filters obtained with considerably reduced computational complexity compared to a recently proposed weighted least-squares (WLS) design method. The design enhancement is achieved by deriving a closed-form formula for evaluating the WLS objective function. The formula facilitates accurate and efficient function evaluations as compared to summing up a large number of discrete terms, which would be time consuming and inevitably introduce additional errors into the design.

This paper proposes a closed-form weighted least-squares solution for designing variable two-dimensional (2D) digital filters with continuously variable 2D fractional delays. First, the coefficients of the variable 2D FIR filter are represented by using the polynomials of a pair of fractional delays (p/sub 1/,p/sub 2/). Then the weighted squared-error function of the variable 2D frequency response is derived without sampling the two frequencies (w/sub 1/,w/sub 2/) and the two fractional delays (p/sub 1/, p/sub 2/), which leads to a significant reduction in computational complexity. With the assumption that the overall weighting function is separable and stepwise, the design problem is reduced to the minimization of the weighted squared-error function. Finally, the closed-form solutions for the optimal coefficient matrices of the variable 2D FIR filter are derived.

In this paper, the least-square design problem of complex FIR filters with arbitrary frequency response characteristics is investigated. A closed-form solution is obtained by solving the approximation problem analytically, enabling a very fast calculation of the impulse response of the filter to be designed. An application of the derived solution to QMF bank design is explored. By using the Butterworth approximation to obtain an IIR filter for the analysis part and by employing the closed-form solution to approximate a nonlinear-phase FIR filter for the synthesis part, a class of nearly perfect-reconstruction QMF bank is developed. It is shown that the mixed IIR/FIR filter bank not only has a very good reconstruction performance but also enjoys an extremely low design complexity.

Digital filters capable of changing their frequency response characteristics are often referred to as variable digital filters (VDFs) and have found uses in a number of digital signal processing applications. An important class of VDFs is the digital filters with variable fractional delay. This paper describes an improved WLS design for variable-fractional-delay FIR filters, which offers improved performance of the filters obtained with considerably reduced computational complexity compared to a recently proposed WLS design method. The design improvement is achieved by deriving a closed-form formula for evaluating the WLS objective function. The formula facilitates accurate and efficient function evaluations as compared to summing up a large number of discrete terms, which would be time consuming and inevitably introduce additional errors into the design.