Linear-phase Filters Research Articles

Design of two-channel perfect-reconstruction linear-phase finite-duration impulse response filter banks for subband image coding by the genetic algorithm approach

In this paper, a powerful approach is proposed for the design of two-dimensional two-channel perfect-reconstruction finite-duration impulse response 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 in 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 pass band and stop band while imposing the spatial-domain constraint. One numerical design example is given to demonstrate the perfect reconstruction for the proposed system.

Design of two-channel perfect-reconstruction linear-phase finite-duration impulse response filter banks for subband image coding by the genetic algorithm approach

In this paper, a powerful approach is proposed for the design of two-dimensional two-channel perfect-reconstruction finite-duration impulse response 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 in 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 pass band and stop band while imposing the spatial-domain constraint. One numerical design example is given to demonstrate the perfect reconstruction for the proposed system.

Optimum equalization of multicarrier systems: a unified geometric approach

This paper presents a new iterative equalization algorithm that maximizes the capacity for discrete multitone (DMT) systems. The research modifies a previously proposed criterion and applies an appropriate transformation to map the objective function and the constraint set into a canonical region. The resulting constraint set exhibits an identifiable geometric characteristic. Using the gradient projection method in conjunction with projection onto convex sets (POCS) provides us with an iterative search algorithm that facilitates the gradient descent method. We also generalize the approach to two important subclasses of equalizers, namely linear phase and unit tap filters. We also derive a fundamental limit on the performance of the proposed approach. In comparison with the previous methods, the proposed equalization algorithm is less computationally complex and more geometrically intuitive. Simulation experiments confirm the validity of the proposed method for equalization of DMT systems.

A new algorithm for linear-phase paraunitary filter banks with pairwise mirror-image frequency responses

Subband filter banks have attracted much attention during the past few years. In this paper, an efficient design algorithm, which leads to linear-phase paraunitary filter banks with pairwise mirror-image frequency responses, is revisited and further studied. New lattice structures are presented to extend the algorithm to the case where the number ( M) of channels is odd. Design examples of filter banks with 3 and 5 channels are presented.

Fast initialisation of linear-phase paraunitaryfilter banksusing adjacent near-power-complementary constraints

An efficient initialisation algorithm is presented for linear-phase paraunitary filter banks by incorporating near-power-complementary constraints on spectrally adjacent filters. Experiments show that the proposed initialisation helps the design parameters in the lattice structures quickly converge to a set of initial values to be used in further re-optimisation of filter banks.

Design of low-delay FIR half-band filters with arbitrary flatness and its application to filter banks

Half-band filters are a class of important filters among digital filters, and have been widely used in many applications such as filter banks and wavelets. The conventional methods are mainly concerned with FIR half-band filters with exactly linear phase. However, the exactly linear phase filters have a drawback of large group delay when high-order filters are required. In this paper, the design of FIR half-band filters with a lower group delay is considered. In some applications of filter banks and wavelets, a flat magnitude response is required for the half-band filters. Therefore, a new method for designing low-delay FIR half-band filters with arbitrary flatness is proposed. In the proposed method, while taking the specified flatness condition into account, the design problem is formulated by using the complex Remez exchange algorithm in the stopband. Then, a set of filter coefficients can be easily obtained by solving a simple system of linear equations. The optimal solution with an equiripple response in the stopband is obtained by applying an iteration process. Finally, the proposed method is applied to the design of two-channel perfect reconstruction filter banks with low group delay to demonstrate its effectiveness. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(10): 1–9, 2000

Perfect reconstruction nonuniform filter banks with linear phase

A nonuniform filter bank divides the band nonuniformly according to the properties of the signal, which is expected to be more effective than a uniform division filter bank in applications such as subband coding. The structure of the nonuniform filter bank, however, is complex. In particular, no investigation has been made of the linear-phase nonuniform filter bank. From such a viewpoint, this paper considers a perfect reconstruction nonuniform filter bank with linear phase, and derives the necessary and sufficient conditions for a perfect reconstruction system as well as for a linear-phase filter. Then, a design method is presented based on the amplitude distortion in the frequency domain and the elimination of aliasing. By the proposed method, the nonuniform filter bank can be designed directly without using the equivalent transformation to the uniform filter bank. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(8): 103–114, 2000

Minimax design of 2-D linear-phase FIR filters with continuous and powers-of-two coefficients

This paper deals with the minimax design problem of two-dimensional (2-D) linear-phase FIR digital filters with continuous and powers-of-two (POT) coefficients. First, the minimax continuous-coefficient design problem is formulated as a linear programming problem with inequality constraints. We present a method based on a variant of Karmarkar's algorithm to solve the resulting design problem. During each iteration, the main computing cost required for finding the filter coefficients is to solve a set of linear equations. Based on the obtained continuous coefficients, an efficient method is presented for designing a minimax 2-D FIR filter with POT coefficients in the spatial domain. Simulation results are provided for illustration and comparison.

Processing arbitrary-length signals with linear-phase cosine-modulated filter banks

This paper presents methods for the decomposition of arbitrary-length signals with linear-phase cosine-modulated filter banks. The analysis filters of this filter bank are divided into two sets having different centers of symmetry. This prohibits the use of standard extension methods as described in the literature. Nevertheless, we show that symmetric extension can be adapted in such a way that the filter bank is support-preservative. Methods for dealing with arbitrary length of both signals and filters are presented. Finally, applications in image and audio coding are outlined. In audio coding the proposed processing methods allow to efficiently avoid pre-echoes.

Universal maximally flat lowpass FIR systems

The family of FIR digital filters with maximally flat magnitude and group delay response is considered. The filters were proposed by Baher (1982), who furnished them with an analytic procedure for derivation of their transfer function. The contributions of this paper are the following. A simplified formula is presented for the transfer function of the filters. The equivalence of the novel formula with a formula that is derived from Baher's analytical procedure is proved using a modern method for automatic proof of identities involving binomial coefficients. The universality of Baher's filters is then established by proving that they include linear-phase filters, generalized half-band filters, and fractional delay systems. In this way, several classes of maximally flat filters are unified under a single formula. The generating function of the filters is also derived. This enables us to develop multiplierless cellular array structures for exact realization of a subset of the filters. The subset that enjoys such multiplierless realizations includes linear-phase filters, some nonsymmetric filters, and generalized halfband filters. A procedure for designing the cellular array structures is also presented.

Complex-valued linear-phase paraunitary filterbanks

Complex-valued linear-phase paraunitary filter banks An algorithm for complex-valued paraunitary filter banks with conjugate symmetry is proposed which leads to the filters having linear-phase frequency responses. Lattice structures are proposed that can be treated as a generalisation of the factorisation of real-valued linear-phase filter banks.

The theory and implementation of arbitrary-length linear-phase cosine-modulated filter bank

In this paper, we propose an arbitrary-length linear-phase cosine-modulated filter bank (CMFB). It can be considered as an extension of the existing linear-phase CMFBs to arbitrary length. It is shown that the 2 M-channel arbitrary-length linear-phase CMFB needs to satisfy the same conditions as the conventional M-channel arbitrary-length CMFB does for perfect reconstruction. By using the linear-phase property of a prototype filter, we obtain an efficient implementation structure in which 2×2 lossless lattices are used instead of 2×1 ones in the traditional implementation of the existing linear-phase CMFBs. The total number of the lattices is reduced by half.

Time-domain design and lattice structure of FIR paraunitary filter banks with linear phase

In this paper, we present a novel way to design biorthogonal and paraunitary linear phase filter banks. The square error of the time-domain constraints on the perfect reconstruction filter bank is expressed in a quadratic form with respect to the filter coefficients. This cost function is minimized by solving a set of linear equations iteratively. This means one can design biorthogonal filter banks without using nonlinear programming techniques. With some modifications, the method can be extended to the design of paraunitary filter banks. In addition, the lattice structure of paraunitary filter banks with filter length KM+ β is also derived. Finally, some design examples are presented to validate the proposed method.

直線位相フィルタ型離散時間修正繰り返し制御とむだ時間補正型離散時間修正繰り返し制御の特性比較

The modified repetitive control system is designed using a modified repetitive controller with a low-pass filter incorporated into the positive feedback loop for the time-delay element. The modified repetitive controller, however, functions only for the gain determined by the low-pass filter at the basic frequency as the desired value and the frequency obtained by multiplying it by integers. This problem can be solved by two methods: one is by using a modified repetitive controller with linear phase filter and the other by using a modified repetitive controller with corrected dead time.This paper describes the results of our study about the above two methods to identify the differences between them in their control characteristics and formulate the guidelines for their use as summarized below:(1) The modified repetitive controller with linear phase filter can basically function as well as the modified repetitive controller with corrected dead time only if its filter is of higher order. However, if the spectrum of the desired value is distributed over a wide frequency range, the use of a higher-order filter in the controller to increase its gain conversely results in narrowed frequency bandwidth, not providing so much improvement in its control performance as expected, and frequently presenting the practical problem of increasing the time required for the computation. Notwithstanding this, if the spectrum is locally distributed in a narrowed frequency range close to the basic frequency, an increase in the order of the filter used in the controller as allowed by the computation time enables its control performance to be improved.(2) The modified repetitive controller with corrected dead time, even if its filter is of first order, can perform its control function comparable to that of the modified repetitive controller with higher-order linear phase filter. This controller is a practical control method, particularly having the great advantage of increasing its gain to∞for direct current.

Structures, factorizations, and design criteria for oversampled paraunitary filterbanks yielding linear-phase filters

We present here a special class of oversampled filterbanks (FBs), namely, paraunitary FBs with linear-phase filters. We propose some necessary conditions for the existence of such banks, based on the repartition between type I/II and type II/IV linear-phase filters in the bank. For a subset of these FBs, we develop a factorization that leads to a minimal implementation, as well as a direct parameterization of the FBs in terms of elementary rotation angles. This factorization is applied to some design examples, with two different optimization criteria: coding gain and reconstructibility of lost coefficients.

Design of two-channel linear-phase FIR PR filter banks with even length filters using convolution matrices

In this paper, the design problem of two-channel linear-phase finite-impulse response perfect reconstruction fitter banks in which both the filters are of even length (which are known as type A filter banks) is addressed. The condition on the determinant of the polyphase matrix is translated in terms of well-known convolution matrices. The perfect reconstruction condition is obtained as a system of linear equations. Using the derived condition, an algorithm for the design of type A filter bank by searching the entire class in a sequential approach for both equal and unequal length cases is presented. In this approach, the first analysis filter is chosen such that there exists a filter forming a type A system. The second analysis filter is then chosen using the remaining degrees of freedom. These two steps are iterated, leading to joint optimization of both the filters.

Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding

A lattice structure for an M-channel linear-phase perfect reconstruction filter bank (LPPRFB) based on the singular value decomposition (SVD) is introduced. The lattice can be proven to use a minimal number of delay elements and to completely span a large class of LPPRFBs: all analysis and synthesis filters have the same FIR length, sharing the same center of symmetry. The lattice also structurally enforces both linear-phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform (GLBT) with an arbitrary number of channels M and arbitrarily large overlap. The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions, which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the potential of the new transforms.

High-speed recursive digital filters based on the frequency-response masking approach

The frequency-response masking approach for highspeed recursive infinite-impulse response (IIR) digital filters is introduced. In this approach, the overall filter consists of a periodic model filter, its power-complementary periodic filter, and two masking filters. The model filters are composed of two all-pass filters in parallel, whereas the masking filters are linear-phase finite-impulse response (FIR) filters. The transfer functions of the all-pass filters are functions of z/sup M/, which implies that the maximal sample frequency for the overall filter is M times that of the corresponding conventional IIR filter. The maximal sample frequency can be increased to an arbitrary level for arbitrary bandwidths. The overall filter can be designed by separately optimizing the model and masking filters with the aid of conventional approximation techniques. The obtained overall filter also serves as a good initial filter for further optimization. Both nonlinear-phase and approximately linear-phase filters are considered. By using the new approach, the potential problems of pole-zero cancellations, which are inherent in algorithm transformation techniques, are avoided. Further, robust filters under finite-arithmetic conditions can always be obtained by using wave-digital all-pass filters and nonrecursive FIR filters. Several design examples are included illustrating the properties of the new filters.

Compactly Supported Wavelets Based on Almost Interpolating and Nearly Linear Phase Filters (Coiflets)

New compactly supported wavelets for which both the scaling and wavelet functions have a high number of vanishing moments are presented. Such wavelets are a generalization of the so-called coiflets and they are useful in applications where interpolation and linear phase are of importance. The new approach is to parameterize coiflets by the first moment of the scaling function. By allowing noninteger values for this parameter, the interpolation and linear phase properties of coiflets are optimized. Besides giving a new definition for coiflets, a new system for the filter coefficients is introduced. This system has a minimal set of defining equations and can be solved with algebraic or numerical methods. Examples are given of the various types of coiflets that can be obtained from such systems. The corresponding filter coefficients are listed and their properties are illustrated.

Filter structures composed of all-pass and FIR filters for interpolation and decimation by a factor of two

This paper introduces filter structures for interpolation and decimation by a factor of two. The structures are derived by using the frequency-response masking approach, in which the overall filter makes use of a periodic model filter, its complementary filter, and two masking filters. The periodic model filters are obtained by replacing each delay element in a model filter with M delay elements in cascade. The model filter is a half-band infinite-impulse response (IIR) filter composed of two all-pass filters in parallel, whereas the masking filters are linear-phase finite-impulse response (FIR) filters. In the final interpolator and decimator structures the filtering takes plate at the lowest of the two sampling rates involved. The corresponding overall filter can be designed by separately optimizing a half-band IIR filter and a linear-phase FIR filter. Both nonlinear-phase and approximately linear-phase filters are considered. One advantage of the proposed filter structures over conventional half-band IIR filter structures is that their maximal sample frequency is M times higher, which may be utilized to increase the speed in an implementation and/or to reduce the power consumption via power supply voltage scaling techniques. In the case of approximately linear-phase filters, the computational complexity can be reduced as well. Several design examples are included demonstrating the properties and advantages of the proposed filter structures.