Double-sided least-squares problem with diagonal constraints

The double-sided least-squares problem with a constrained parameter matrix is formulated and solved using multilinear products.     

Variable wordlengths in filters with least-squares error criterion

Variable coefficient wordlengths in combination with a multiplier reduces the computational complexity of fixed-point IIR digital filters with minimax error specification. When variable wordlengths are used for the least-squares error specification here, improved filters are achieved without resorting to discrete optimisation. Also. The wordlength allocation for the least squares criterion gives near-optimum results for filters with minimax specification     

Weighted least-squares approximation of FIR by IIR digital filters

This paper presents a method for the weighted least-squares approximation of finite impulse response (FIR) filters by infinite impulse response (IIR) filters. It is shown, how a solution to this approximation problem can be obtained by solving a related pure least-squares approximation problem. For the latter, we utilize a generalized version of a previously published technique with low computational complexity and guaranteed stability of the IIR filters. Unlike the well-established model-reduction approaches that are carried out in the state space, our method works directly with the numerator and denominator coefficients of the transfer functions. Thus, the influence of finite-precision arithmetic on the results is small. This makes our approach applicable for the approximation of large-order FIR filters and allows the usage of arbitrarily shaped weighting functions. It is shown that our method can successfully be employed to achieve a uniform approximation     

Iterative reweighted least-squares design of FIR filters

Develops a new iterative reweighted least squares algorithm for the design of optimal L_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_p optimization problem are given. The primary purpose of L_p approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained L₂ approximation filters. The new method can also be applied to the complex Chebyshev approximation problem and to the design of 2D FIR filters     

Weighted least-squares design of recursive allpass filters

A method for the design of allpass filters is described. In this method, an error reflecting the difference between the desired phase response and the phase response of the practical allpass filter is formulated in a quadratic form. The coefficients are obtained by solving a system of linear equations involving the sum of a Toeplitz and an Hankel matrix     

Design of FIR filters with extra constraints using modified LMSalgorithm

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     

The positive side of filters: a summary

The linear filters characterized by a state-variable realization given by matrices with nonnegative entries (called positive filters) are heavily restricted in their achievable performance. Nevertheless, such filters are the only choice when dealing with the charged coupled device MOS technology of charge routing networks (CRNs), because nonnegativity is a consequence of the underlying physical mechanism. In order to exploit the advantages offered by this technology, the authors try to overcome the above-mentioned limitation by realizing an arbitrary transfer function as a difference of two positive filters     

Lowpass delay filters with flat magnitude and group delay constraints

This paper describes the design of finite impulse response (FIR) delay filters that minimize a squared error and have prescribed number of zeros at /spl omega/=/spl pi/ and prescribed magnitude and group delay flatness at /spl omega/=0. An important special case is the design of least squared error lowpass filters with prescribed flatness constraints and zeros at /spl omega/=/spl pi/. Even though the flatness constraints are in general nonlinear functions of the filter coefficients, we show the remarkable fact that for a subclass of the filters a simple orthogonal projection of least squared error filters onto a special linear subspace determined via Baher (1982) filters gives the solution. The paper also introduces the notion of delay filters that are high-order approximations to the ideal delay and establishes their equivalence to Baher filters. This connection gives novel elementary derivations of Baher filters and their properties. Matlab programs are provided at the end of the paper for the design of filters described in this paper.     

Counterexample in the least-squares inverse stabilisation of 2D recursive filters

A counterexample is presented disproving the conjecture that the planar least-squares inverse of a 2-variable polynomial is devoid of zeros when both variables are inside or on the unit circle.     

On Chebyshev design of linear-phase FIR filters with frequency inequality constraints

The alternation theorem is the basis of the Remez algorithm for unconstrained Chebyshev design of finite-impulse response (FIR) filters. In this paper, we extend the alternation theorem to the inequality-constrained case and present an improved Remez algorithm for the design of minimax FIR filters with inequality constraints in frequency domain. Compared with existing algorithms, the presented algorithm has faster convergence rate and guaranteed optimal solutions.     

Fast and stable least-squares approach for the design of linearphase FIR filters

We present a new numerical algorithm for solving the normal equations associated with the least-squares design of linear phase FIR filters. The usual solution methods have a computational complexity of O(N³). Moreover, solving the normal equations with Gaussian elimination commonly yields numerical errors, especially when the filter is long. Here, we convert a least-squares method into the problem of constructing a system of orthonormal functions. The proposed design algorithm needs only O(N²) computations, and numerical errors can be reduced. Some examples are given to show the performance of the algorithm     

Design of FIR digital filters with flatness constraints for the error function

A method will be presented for the approximation of a desired frequency response by the frequency response of a FIR filter. It is possible to match the functional values and an arbitrary number of derivatives of both responses for zero frequency, thus making the error flat up to a desired degree. Remaining degrees of freedom are used for a weightedL ₂-approximation. Closed form design formulae will be given.     

Optimal design of finite precision FIR filters using linearprogramming with reduced constraints

An algorithm for the design of optimal one-dimensional (1-D) and two-dimensional (2-D) FIR filters over a discrete coefficient space is proposed. The algorithm is based on the observation that the equiripple frequencies of a subproblem (SP) in the branch and bound (BaB) algorithm are closely related to those of neighboring SPs. By using the relationship among the SPs, the proposed algorithm reduces the number of constraints required for solving each SP. Thus, the overall computational load for the design of FIR filters with discrete coefficients is significantly alleviated, compared with the conventional BaB algorithm     

A weighted least-squares method for the design of stable 1-D and2-D IIR digital filters

We present a new approach to the least-squares design of stable infinite impulse response (IIR) digital filters. The design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function, and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function. When the iteration converges, a stable and truly least-squares solution is obtained. The method is then extended to address the least-squares design of stable IIR two-dimensional (2-D) filters. Examples are included to illustrate the proposed design techniques     

Design of 2D FIR digital filters with flatness constraints for the error function

A method will be presented for the approximation of a desired two-dimensional frequency response by the frequency response of a two-dimensional finite-impulse-response digital filter. It is possible to match the functional values and an arbitrary number of derivatives of both responses for zero frequency, thus making the error flat up to a desired degree. Remaining degrees of freedom are used for anL ₂-approximation. Closed form design formulae will be given.     

Design of linear phase FIR filters using fractional derivative constraints

In this paper, the designs of linear phase FIR filters using fractional derivative constraints are investigated. First, the definition of fractional derivative is reviewed briefly. Then, the linear phase FIR filters are designed by minimizing integral squares error under the constraint that the ideal response and actual response have several same fractional derivatives at the prescribed frequency point. Next, the fractional maximally flat FIR filters are designed by letting the number of fractional derivative constraints be equal to the number of filter coefficients. Finally, numerical examples are demonstrated to show that the proposed method has larger design flexibility than the conventional integer derivative constrained methods.     

Design of highly selective two-dimensional recursive fan filters byrelaxing symmetry constraints

Shows for the design of quadrantally symmetric 2-D fan filters that it is unnecessarily restrictive to prescribe exact quadrantal symmetry, which requires that the denominator of the Z-transform transfer function be product-separable. Superior approximately symmetric fan filter designs can be achieved using nonseparable denominators     

Design of FIR digital filters with prescribed flatness and peak error constraints using second-order cone programming

This paper studies the design of digital finite impulse response (FIR) filters with prescribed flatness and peak design error constraints using second-order cone programming (SOCP). SOCP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. It is utilized in this study for the optimal minimax and least squares design of linear-phase and low-delay (LD) FIR filters with prescribed magnitude flatness and peak design error. The proposed approach offers more flexibility than traditional maximally-flat approach for the tradeoff between the approximation error and the degree of design freedom. Using these results, new LD specialized filters such as digital differentiators, Hilbert Transformers, Mth band filters and variable digital filters with prescribed magnitude flatness constraints can also be derived.     

A single-chip codec with switched-capacitor filters

A single-chip CMOS codec with filters has been developed using charge redistribution and switched-capacitor techniques. Its features are ~30 mm/SUP 2/ small chip area, 35 mW low power dissipation, and small 16 pin package. These are achieved with novel analog circuit techniques for A/D and D/A conversions and clock generation. Measured transmission characteristics meet the system requirements.     

Design of high-order Chebyshev FIR filters in the complex domainunder magnitude constraints

This article discusses the design of FIR filters that approximate a complex-valued target frequency response in a Chebyshev sense. Additionally, the required stopband attenuation can be specified. Solving the dual of a semi-infinite linear program is currently the most efficient way to design such filters, but numerical problems prevent the design of high-order FIR filters. Modifications are proposed to overcome this limitation. Furthermore, an efficient method is presented for generating starting values that are close to the optimal solution such that the number of iterations is considerably reduced. Examples of filters with a length up to 250 taps are included