Linear-phase Filters Research Articles

General form for designing two-dimensional quadrantally symmetric linear-phase FIR digital filters by analytical least-squares method

The analytical least-squares method has been generalized and extended for designing 16 types of two-dimensional FIR filters with quadrantally symmetric or antisymmetric frequency responses. By means of a closed-form transformation matrix, this fast design method can determine the filter's coefficients very effectively without a recourse to an iterative optimization technique or matrix inversion. Design examples are presented to illustrate the simplicity and efficiency of the proposed method.

Transform domain adaptive linear phase filter

Describes a new adaptive linear-phase filter whose weights are updated by the normalized least-mean-square (LMS) algorithm in the transform domain. This algorithm provides a faster convergence rate compared with the time domain linear phase LMS algorithm. Various real-valued orthogonal transforms are investigated such as the discrete cosine transform (DCT), discrete Hartley transform (DHT), and power of two (PO2) transform, etc. By using the symmetry property of the transform matrix, an efficient implementation structure is proposed. A system identification example is presented to demonstrate its performance.

Optimization of filter banks using cyclostationary spectral analysis

This paper proposes a design method of optimal biorthogonal FIR filter banks that minimize the time-averaged mean squared error (TAMSE) when the high-frequency subband signal is dropped. To study filter banks from a statistical point of view, cyclostationary spectral analysis is used since the output of the filter bank for a wide-sense stationary input is cyclostationary. First, the cyclic spectral density of the output signal is derived, and an expression for the TAMSE is presented. Then, optimal filter banks are given by minimizing the TAMSE with respect to the coefficients of the filters under the biorthogonality condition. By imposing the additional constraints on the coefficients, the optimal biorthogonal linear phase filter bank can be obtained.

Relations between thek-dimensional unconstrained and linear phase FIR Wiener filters

In this note, we show that thek-dimensional linear phase FIR Wiener filter can be obtained from the unconstrained filter with a simple “reverse and add” operation. The relation between the linear phase and unconstrained phase filters is obtained also in the case of a multichannelk-dimensional system. In this case, the properties of the autocorrelation matrix do not allow the simplifications noted for the single channel case.

Split filter structures for LMS adaptive filtering

In this paper, the performance of a split adaptive filter in either a parallel or a serial form for non-white inputs is investigated. A parallel split structure is constructed by using two linear phase filters connected in parallel while for serial split it is configured as a cascade of two transversal filters. The adaptation characteristics of the well-known LMS algorithm has been shown to be governed by the eigenvalue spread of the input correlation matrix. By adopting the split structures, we illustrate that the eigenvalue ratios of the associated covariance matrices can be reduced thereby giving rise to a faster convergence speed. The parallel and serial split adaptive filters are examined for both joint process estimation and linear prediction. A new type of linear predictor is formed by combining both split methods and its performance for speech analysis is studied. Simulation results are included to validate the superiority of the proposed split filter structures in improving the rate of convergence for LMS adaptation.

Design of two-dimensional linear-phase nonrecursive filters using a least-squares criterion

A method is described which can be used to design two-dimensional nonrecursive linear-phase filters. The approach is based on formulating the mean-square error between the amplitude responses of the practical and ideal digital filters as a quadratic function. The coefficients of the filters are obtained by solving a set of linear equations. This method leads to a lower mean-square error and is computationally more efficient than the eigenfilter method.

Design of efficient M-band coders with linear-phase and perfect-reconstruction properties

This paper presents a new design technique for obtaining M-band orthogonal coders where M=2/sup i/. The structures obtained using the proposed technique have the perfect reconstruction property. Furthermore, all filters that constitute the subband coder are linear-phase FIR-type filters. In contrast with conventional design techniques that attempt to find a unitary alias-component matrix in the frequency domain, we carry out the design in the time domain, based on time-domain orthonormality constraints that the filters must satisfy. The M-band design problem is reduced to the problem of finding a suitable lowpass filter h/sub 0/(n). Once a suitable lowpass filter is found, the remaining (M-1) filters of the coder are obtained through the use of shuffling operators on the lowpass filter. This approach leads to a set of filters that use the same numerical coefficient values in different shift positions, allowing very efficient numerical implementation of the subband coder. In addition, by imposing further constraints on the lowpass branch impulse response h/sub 0/(n), we are able to construct continuous bases of M-channel wavelets with good regularity properties. Design examples are presented for four-, eight-, and 16-band coders, along with examples of continuous wavelet bases that they generate. >

On Orthogonal Transforms of Images Using Paraunitary Filter Banks

The use of a paraunitary filter bank (PUFB) for image processing requires a special treatment at image boundaries, to ensure perfect reconstruction (PR) of these regions, and periodic or symmetric extensions are commonly assumed. We reduced the analysis to a one-dimensional signal assuming separable processing. Unlike infinite-length signals, PR PUFBs applied to finite-length signals will not necessarily lead to an orthogonal system. For quantization/processing of the subbands, artifacts at the image boundaries can appear due to artificial discontinuities at borders, lack of orthogonality of the effective boundary filter banks, or improper reconstruction procedure. We will explore linear-phase and nonlinear-phase PUFBs and methods to obtain orthogonality from the boundary filter banks. We will show that for symmetric extensions, orthogonality is only possible for special PUFBs based on linear-phase filters. Using time-varying boundary filter banks, we will discuss a procedure that explores all degrees of freedom of the border filters in a method essentially independent of signal extensions, allowing us to design optimal boundary filter banks, while maintaining fast implementation algorithms.

An efficient weighted least-squares design of linear-phase nonrecursive filters

An accelerated procedure for the design of linear-phase nonrecursive filters using a weighted least-squares technique is described. This procedure is based on formulating the error reflecting the difference between the desired amplitude response and the amplitude response of the practical filter in a quadratic form. The coefficients of the filter are obtained by solving a system of linear equations involving a Toeplitz-plus-Hankel matrix. Such a system of linear equations can be solved by computationally efficient algorithms having only O(N/sup 2/) complexity. By choosing the appropriate frequency-dependent weighting function, a filter with either a least-squares or an equiripple error variation can be designed. >

Fixed point roundoff noise in frequency sampling filters

A frequency sampling filter which is a finite impulse response digital filter uses a recursive structure that requires exact pole zero cancellation on the unit circle. When implemented with digital technology, errors due to finite length registers and finite precision arithmetic can prevent exact pole zero cancellation making the filter unstable. To guarantee stability, the poles and zeros on the unit circle are moved to a circle of radius r where 0 < r < 1. Most frequency sampling filter design techniques determine optimal frequency responses for r = 1 and then chose a value of r, 0 < r < 1, near to 1 so that the filter's frequency response does not differ significantly. Recently, other techniques have been presented that determine optimal frequency responses for 0 < r < 1. In this paper, the fixed point roundoff noise of Type 1 and Type 2 frequency sampling filters is determined as a function of register length, time and the value of r. An example demonstrates that for a fixed register length and output noise level, a frequency sampling filter designed with 0 < r < 1 can approximate a linear phase filter better than a frequency sampling filter designed with r = 1.

Design of FIR filters with complex desired frequency response using a generalized Remez algorithm

Complex approximation with a generalized Remez algorithm is used to design FIR digital filters with nonconjugate symmetric frequency responses. The minimax criterion is used and the Chebychev approximation is posed as a linear optimization problem. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang. Optimal Chebychev real-coefficient FIR filters with group delay smaller than half the filter length can be designed with slightly better magnitude responses compared to linear-phase filters. Linear-phase filters can also be designed when the group delay is specified to be half the filter length. Most importantly, the design method is capable of producing filters with complex coefficients that approximate nonconjugate symmetric frequency responses.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On approximated linear-phase recursive digital filters

The recursive forms to approximate the causal linear-phase IIR digital filters are studied. Actual causal linear-phase IIR digital filters have been proposed (Clements and Pease 1989); however, the lack of rational expressions of the considered systems render them unrealizable. As a consequence, applications for this kind of filter do not exist. In this paper, based on the Pade approximation and the eigenfilter design, the approximations of linear-phase recursive representations of the filters are proposed for practical applications. Experimental results show good approximation on phase linearity. Finally, the magnitude response is designed to meet the specifications in the passband of the low-pass filter.

Symmetric extension methods for M-channel linear-phase perfect-reconstruction filter banks

The symmetric extension method has been shown to he an efficient way for subband processing of finite-length sequences. This paper presents an extension of this method to general linear-phase perfect-reconstruction filter banks. We derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm. In the algorithm, different symmetric sequences are formulated in a unified form based on the circular-symmetry framework. The length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated. The effect of shifting the input sequence is included. When the algorithm is applied to equal-length filter banks, we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach.

Constraints on the cutoff frequencies of Mth-band linear-phase FIR filters

Constraints are derived for the cutoff frequencies of linear-phase FIR Mth-band filters such that the filters have good passband and stopband characteristics, i.e. ones that very closely approximate an ordinary (non Mth-band) filter designed using some optimal method. Constraints on lowpass filters are first considered, and the results are extended to multiband filters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of two-channel perfect-reconstruction linear-phase filter banks for subband image coding by the Lagrange multiplier approach

In this paper, a new approach is proposed for the design of two-dimensional two-channel perfect-reconstruction FIR filter banks employing diamond-shaped linear-phase filters. When one of the analysis filter pair is properly designed, the condition of perfect-reconstruction is formulated into a spatial-domain constraint equation for designing the other. The method is based on minimizing the integrated square error for the frequency response over the passband and stopband while imposing the spatial-domain constraint. One design example is given to demonstrate the perfect-reconstruction for the proposed system. >

Synthesis of coherent two-port lattice-form optical delay-line circuit

A method is presented for synthesizing a coherent two-port lattice-form optical delay-line circuit that is composed of optical delay lines, directional couplers, and phase shifters. The two bases of the method are the use of a unimodulus para-unitary matrix as a transfer matrix and the division of the transfer matrix into basic component transfer matrices. We succeeded in obtaining a set of recurrent equations with which to calculate circuit parameters to use for designing an optical delay-line circuit with a desired cross-port (through-port) transfer function. In the developed method, it is shown that two-port optical delay-line circuits can have the same transmission characteristics as finite impulse response digital filters with complex expansion coefficients. Three synthesis examples for optical frequency filters are described: a linear-phase Chebyshev filter, a multi-channel selector, and a group-delay dispersion equalizer. It is confirmed that transmission characteristics with a maximum transmittance of 100% can be always synthesized. The allowable parameter error for the synthesized linear-phase Chebyshev filter is also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

General analysis of two-band QMF banks

The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions. >

Complex, linear-phase filters for efficient image coding

With the exception of the Haar basis, real-valued orthogonal wavelet filter banks with compact support lack symmetry and therefore do not possess linear phase. This has led to the use of biorthogonal filters for coding of images and other multidimensional data. There are, however, complex solutions permitting the construction of compactly supported, orthogonal linear phase QMF filter banks. By explicitly seeking solutions in which the imaginary part of the filter coefficients is small enough to be approximated to zero, real symmetric filters can be obtained that achieve excellent compression performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Similarities and differences between one-sided and two-sided linear prediction

Provides a comparison between one-sided linear prediction (OSP) and two-sided linear prediction (TSP) with respect to prediction error, relationships to AR modeling and to two-sided AR modeling, and the application to time series interpolation, linear-phase filter design, and spectral estimation. Although some of these results have appeared previously in scattered references, the authors present a unified framework for deriving all of these results, as well as new, additional results. New contributions of the paper include: (i) proof that TSP produces smaller, nonwhite residuals than OSP, extending previous results; (ii) specification of the frequency-domain error criterion minimized by TSP, and comparison with the analogous OSP criterion; (iii) demonstration that TSP and two-sided AR modeling are different problems, unlike OSP; (iv) interpretation of performance of TSP interference-rejection filters. >

Digital filter bank design quadratic-constrained formulation

Formulate the filter bank design problem as an quadratic-constrained least-squares minimization problem. The solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters. The formulations of the perfect-reconstruction cosine-modulated filter bank, of the near-perfect-reconstruction pseudo-QMF bank, and of the two-channel biorthogonal linear-phase filter bank are derived using the proposed approach. Compared with other design methods, the proposed technique yields PR filter banks with much higher stopband attenuation. The proposed technique can also be extended to design multidimensional filter banks. >