Design Of Linear Filters Research Articles

Fast and stable least-squares approach for the design of linear phase 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(N/sup 3/). 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(N/sup 2/) computations, and numerical errors can be reduced. Some examples are given to show the performance of the algorithm.

Description of FIR digital filters in the form of parallelconnection of linear phase FIRs

The original paper (see ibid., vol32, no. 11 p. 979-80, 1996) showed that an Nth-order one-dimensional digital FIR filter may be represented as a parallel connection of [log/sub 2/(N + 2)] linear phase digital FIR filters. Iwaki and Ishii state that their linear phase decomposition is not unique and that several decompositions are possible. The authors also state that linear phase decompositions may be useful in decomposing a general filter design problem into several linear phase subproblems, so that well-developed linear phase filter design techniques could be applied to a general filter design problem. The linear phase decomposition given in this comment is based on a given theorem.

The design of linear phase FIR filters using the IDFT

The paper presents a comprehensive set of equations for specifying the transfer function of linear phase finite-impulse response (LPFIR) filters. A procedure for designing the eight possible LPFIR filters (symmetric and anti-symmetric, even and odd length, type 1 and type 2 filters), using the inverse discrete Fourier transform (IDFT) is described. This technique can be equally applied to the window method and the frequency sampling method.

Synthesis of low‐sensitivity orthogonal digital filters

A novel method is proposed for the synthesis of low-sensitivity digital filters meeting any prescribed transfer function. The method is based on extracting an orthogonal matrix from the filter state matrix, resulting in structures that need n Givens rotations and at most n + 1 multipliers. Thus the proposed realization is canonic in the sense that it has the same degrees of freedom as the original transfer function. It is also shown that when the filter transfer function is a reciprocal reactant bounded function, it can be decomposed into allpass functions that need only Givens rotations for their realization. As the basic module in the realization is the Givens rotation, the CORDIC computation algorithm can be applied directly. This means considerable savings in computation time and complexity. It also results in structures that are less sensitive to the effects of finite word length. Illustrative examples, including the design and synthesis of linear phase selective filters, are given to show the extremely low sensitivity with respect to finite word length of the resulting realizations when compared with other methods.

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

The topic of this paper is the design of optimal linear phase FIR filters by means of semi-infinite programming technique. The filters are specified primarily in the frequency domain and can be additionally constrained with respect to the time domain. The new method allows constrained and unconstrained minimax- and also least-squares designs. In this way, it completely substitutes several other current methods which allows the filter designer to develop a large variety of filter designs by using a single computer program. Highly accurate solutions with up to around 2000 filter coefficients demonstrate several of the abilities of the new method.

Entwurf minimalphasiger nichtrekursiver digitaler Filter mit Verfahren der nichtlinearen Optimierung

The design of minimum-phase nonrecursive digital filters in the Chebyshev sense with respect to prescriptions on the magnitude response has been frequently discussed already. This filters make the best of given tolerance schemes of magnitude response at minimum degree and offer minimum signal delay. Both characteristics let the minimum phase filters be superior to linear phase filters at some distinct applications. Commonly, for some special characteristics, the filters will be designed indirectly via the design of a related linear phase filter from which the desired minimum-phase filter will be separated. Alternatively, we propose nonlinear optimization which then allows solutions of more general and constrained design problems. In this connection, the design problems will be discussed in view of optimization. Three numerical examples demonstrate the advantages of optimization for the task of designing this interesting class of digital filters where we care much about the easy application of available general purpose optimization software.

Description of FIR digital filters in the form ofparallel connection of linear phase FIRs

The authors aim to show that FIR digital filters can be described in the form of parallel connection of linear phase FIR digital filters. This representation method may be applicable to FIR digital filter design problems, reducing it to linear phase FIR digital filter design problems.

Robust filter design for uncertain systems defined by both hard and soft bounds

A new approach to robust linear filter design is described that attempts to combine the advantages of H/sub /spl infin// robust linear synthesis with a probabilistic method of system and noise modeling. The signal and measurement noise model parameters are assumed to be subject to perturbations represented by random variables with known covariances. The system is represented in polynomial form, and the uncertainty can be described by both soft and hard bounds. An H/sub /spl infin// cost-function is minimized and averaged with respect to model errors in signal and noise descriptions. The polynomial solution is no more complicated than the usual H/sub /spl infin// optimal filter and involves averaged spectral factorizations and linear equations. Both usual and deconvolution filtering problems are considered.

Thoughts on least squared-error optimal windows

Recently, a simple and versatile method for the design of linear phaser FIR filters with spline transition bands and optimal in a least-squared sense was introduced. The following question is raised: Given an arbitrary window, say, for example, a Hamming window, does there exist a transition function (like the spline function above) such that the Hamming window is least-squares optimal? A related question is the following: Given a transition function, does there exist a window sequence w(n) such that the least squared optimal FIR filter is given by g(n)w(n)? This correspondence shows that all windows have associated transition functions that make them least-squared optimal. For every window, there exists a transition function that makes it superoptimal.

Design of linear phase FIR filters with a maximally flat passband

The design of linear phase FIR digital filters having symmetric or antisymmetric impulse response is formulated as a constrained minimization problem. The constraints express the maximal flatness of the frequency response at the origin in the case of a low-pass filter or at an arbitrary frequency in the passband in the case of a bandpass filter. The objective function, which is a quadratic form in the filter coefficients, is formed as a convex combination of two objective functions representing the energy of the error between the frequency response of the designed filter and a scaled version of the frequency response of the ideal filter in both the stop and passbands.

Exchange algorithms for the design of linear phase FIR filters and differentiators having flat monotonic passbands and equiripple stopbands

The authors describe a modification of a technique proposed by Vaidyanathan (1985) for the design of filters having flat passbands and equiripple stopbands. The modification ensures that the passband is monotonic and does so without the use of concavity constraints. Another modification described in this brief adapts the method of Vaidyanathan to the design of low-pass differentiators having a specified degree of tangency at /spl omega/=0.

Design of linear phase Nyquist filters

The design of a linear phase Nyquist filter is introduced. The technique is based on setting a target function with a strictly linear phase and attempting to approximate the filter transfer function to the target function using the least-mean-square method. Further, the analysis is extended to cover the practical case of a pulse response rather than the impulse response. In all cases excellent results have been obtained, which compare favourably with other results available in the literature.

The design and implementation of multiplierless nonrecursive linear phase discrete time filters

This paper develops techniques for the design and implementation of linear phase discrete time filters that are implemented using only delays, adds and subtracts. Sometimes called discrete time linear phase transport processors, these filters have finite length integer valued impulse responses. The design technique developed in this paper determines an optimal integer valued impulse response that minimizes the mean square error in the filter's passband and stopband while bounding the impulse response values to a specified value. The implementation technique developed in this paper generates a structure that can realize integer valued impulse responses using only delays, adds and substracts.

Trade-offs in linear filter design

This paper examines trade-offs arising in the design of linear filters. These show that there exist fundmental limits on filter performance—limits that are independent of the method of design. These limits may be employed to determine a priori whether desired design specifications are achievable. Analogous constraints are well known in the area of feedback control systems, but, to the best of our knowledge, they have not been previously articulated for the filtering problem.

Optical correlation of images with signal-dependent noise using constrained-modulation filter devices

Images with signal-dependent noise present challenges beyond those of images with additive white or colored signal-independent noise in terms of designing the optimal 4-ƒ correlation filter that maximizes correlation-peak signal-to-noise ratio, or combinations of correlation-peak metrics. Determining the proper design becomes more difficult when the filter is to be implemented on a constrained-modulation spatial light modulator device. The design issues involved for updatable optical filters for images with signal-dependent film-grain noise and speckle noise are examined. It is shown that although design of the optimal linear filter in the Fourier domain is impossible for images with signal-dependent noise, proper nonlinear preprocessing of the images allows the application of previously developed design rules for optimal filters to be implemented on constrained-modulation devices. Thus the nonlinear preprocessing becomes necessary for correlation in optical systems with current spatial light modulator technlogy. These results are illustrated with computer simulations of images with signal-dependent noise correlated with binary-phase-only filters and ternary-phase-amplitude filters.

Robust minimum variance filtering

This paper deals with the robust minimum variance filtering problem for linear systems subject to norm-bounded parameter uncertainty in both the state and the output matrices of the state-space model. The problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty. Two methods for designing robust filters are investigated. The first one deals with constant parameter uncertainty and focuses on the design of steady-state filters that yield an upper bound to the worst-case asymptotic error variance. This bound depends on an upper bound for the power spectrum density of a signal at a specific point in the system, and it can be made tighter if a tight bound on the latter power spectrum can be obtained. The second method allows for time-varying parameter uncertainty and for general time-varying systems and is more systematic. We develop filters with an optimized upper bound for the error variance for both finite and infinite horizon filtering problems.

Design of FIR filters with extra constraints using modified LMS algorithm

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.

Linear filtering of images based on properties of vision

The design of linear image filters based on properties of human visual perception has been shown to require the minimization of criterion functions in both the spatial and frequency domains. We extend this approach to continuous filters of infinite support. For lowpass filters, this leads to the concept of an ideal lowpass image filter that provides a response that is superior perceptually to that of the classical ideal lowpass filter.

The design of the maximal SNR linear phase FIR digital filters

The gradient algorithm for the design of the maximal signal-to-noise ratio (SNR) finite impulse response (FIR) digital filters is presented. It is noted that the eigenvalue gradient determination cannot lead to the global maximum from any initial point. The properties of the linear phase low-pass, high-pass and band-pass filters designed by the proposed method are considered. The maximal SNR filters provide obtaining a significant filter's profit even when the filter's length is rather small. The filters are preferable for processing a mix of a stationary signal and a stationary noise with the known power spectra.

Minimum error sound source localization

A novel approach to the problem of computing the direction of arrival (DOA) of a sound source using a two-element microphone array is presented. Typically, the DOA is computed by peak picking from the resulting cross-correlation function. In order to improve such estimates it is usually desirable to pre-filter the signals prior to cross correlation. However, proper selection of these pre-filters is often problematic. The proposed algorithm adapts the filters during an initial training session to directly minimize the number of location estimation errors. The only information provided is the signal DOA, no explicit knowledge of the signal or noise spectra are required. It is assumed however that the overall signal/noise statistics are long term stationary over the training and subsequent testing. Examples of both linear and nonlinear filter design are presented for a talker location identification task.