Allpass Fractional Delay Filters Research Articles

A stability controlling-based approach for designing 1-D variable fractional delay all-pass filters

PurposeVariable fractional delay filtering is an important technology in signal processing; the research shows that all-pass variable fractional delay (VFD) filters achieve higher design accuracy than FIR VFD filters; therefore, the design, analysis and implementation of all-pass VFD filters are of great importance.Design/methodology/approachIn this paper, a two-stage approach for the design of general 1-D stable VFD all-pass filters is proposed. The method takes the desired group delay range [N−1, N], where N is the filter order.FindingsThe design algorithm is decomposed into two design stages: first, a set of fixed delay all-pass filters are designed by minimizing a set of objective functions defined in terms of approximating error criterion and filter stability constraint. Then, the design result is determined by fitting each of the fixed delay all-pass filter coefficients as 1-D polynomials. A design example together with its comparisons with those of the recent literature studies is given to justify the effectiveness of the proposed design method.Originality/valueAn illustrating design example shows that the method proposed can achieve better filter performances than the existing ones.

A Simple Ladder Realization of Maximally Flat Allpass Fractional Delay Filters

This brief proposes a new ladder structure for the Thiran fractional delay filter (i.e., maximally flat allpass fractional delay filter given by the Thiran approximation). The proposed ladder structure is based on a continued fraction representation. Although there exists a similar approach that was proposed by Tassart and Depalle, their structure is not realizable because of delay-free loops. On the other hand, we show that the proposed method avoids generating delay-free loops and thus successfully yields a realizable ladder structure for the Thiran fractional delay filter in a very simple form.

Efficient Antialiasing Oscillator Algorithms Using Low-Order Fractional Delay Filters

One of the challenges in virtual analog synthesis is avoiding aliasing when generating classic waveforms such as sawtooth and square wave which have theoretically infinite bandwidth in their ideal forms. The human auditory system renders a certain amount of aliasing inaudible, which allows room for finding cost-effective algorithms. This paper suggests efficient algorithms to reduce the aliasing using low-order fractional delay filters in the framework of bandlimited impulse train (BLIT) synthesis. Examining Lagrange, B-spline interpolators and allpass fractional delay filters, optimized methods will be discussed for generating classic waveforms (sawtooth, square, and triangle). Techniques for generating more complicated harmonics such as pulse width modulation, hard-sync, and super-saw are also presented. The perceptual evaluation is performed by comparing the threshold of hearing and masking curve of oscillators with their aliasing levels. The result shows that the BLIT using the computationally efficient third-order B-spline generates waveforms that are perceptually free of aliasing within practically used fundamental frequencies.

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

This paper presents two-step design methodologies and performance analyses of finite-impulse response (FIR), allpass, and infinite-impulse response (IIR) variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense and solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a global solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming problem and solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to nonvariable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency ( ges 0.9pi) . Filter examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.

Digital IIR integrator design using recursive Romberg integration rule and fractional sample delay

In this paper, a digital IIR integrator design based on the recursive Romberg integration rule and the fractional sample delay is investigated. Starting with trapezoidal integrators, the approximation error of the frequency response of digital integrators can be reduced recursively by using the Romberg rule. Although the proposed IIR digital integrators involve the implementation of the fractional sample delay, this problem is easily solved by applying the well-documented design techniques of the FIR Lagrange and IIR allpass fractional delay filters. These techniques lead to the transfer function of the IIR Romberg integrator without using the fractional sample delay. Several design examples are presented in order to demonstrate the effectiveness of the proposed method.

Design of digital differentiator using difference formula and Richardson extrapolation

The design of digital differentiators is investigated. First, the backward difference formula in numerical differentiation is applied to derive the transfer function of a digital differentiator. In this design, the Richardson extrapolation is used to generate high-accuracy results while using low-order formulas. Then, conventional Lagrange finite impulse response (FIR) and Thiran infinite impulse response (IIR) allpass fractional delay filters are directly applied to implement the designed differentiator. Next, the proposed method is extended to design digital differentiators using central difference formula. Finally, several numerical examples are illustrated to demonstrate the effectiveness of this new design approach.

Fast design of relatively wide band fractional delay all-pass filters

A fast and simple procedure for designing relatively wide band fractional delay (FD) all-pass filters is proposed and filters designed based on it are compared to those designed using Thiran's method, which is one of the simplest, fastest and widely used approaches in designing FD all-pass filters. It is shown that filters designed using the introduced method have wider fractional group delay (fgpd) bandwidths compared to those obtained via Thiran's technique, as well as some other desired properties. The designed filters are also demonstrated to be stable for several pairs of filter orders and fgpd values.

Closed-Form Design of Digital IIR Integrators Using Numerical Integration Rules and Fractional Sample Delays

In this paper, the numerical integration rules and fractional sample delays will be used to obtain the closed-form design of infinite-impulse response (IIR) digital integrators. There are two types of numerical integration rules to be investigated. One is Newton-Cotes quadrature rule, the other is Gauss-Legendre integration rule. Although the proposed IIR digital integrators will involve the implementation of fractional sample delays, this problem is easily solved by applying well-documented design techniques of the finite-impulse response Lagrange and IIR allpass fractional delay filters. Several design examples are illustrated to demonstrate the effectiveness of the proposed method

Design of a Sample-Rate Converter From CD to DAT Using Fractional Delay Allpass Filter

In sampling rate conversion between two different formats, the desired output sample values are the interpolated sample values at noninteger (or, integer) multiples of the original sampling period. In this brief, we present an efficient structure for sampling rate conversion from 44.1-kHz compact disc to 48-kHz digital audio tape formats. For efficient conversion, we propose a new infinite-impulse response (IIR) fractional delay filter based on Thiran-based IIR allpass filter. It is shown that the proposed method requires less hardware complexity and maintains high quality of signal-to-noise ratio compared with other methods

Digital integrator design using Simpson rule and fractional delay filter

The IIR digital integrator is designed by using the Simpson integration rule and fractional delay filter. To improve the design accuracy of a conventional Simpson IIR integrator at high frequency, the sampling interval is reduced from T to 0.5T. As a result, a fractional delay filter needed to be designed in the proposed Simpson integrator. However, this problem can be solved easily by applying well-documented design techniques of the FIR and all-pass fractional delay filters. Several design examples are illustrated to demonstrate the effectiveness of the proposed method.

Closed-Form Design of All-Pass Fractional Delay Filters

In this letter, we propose a novel all-pass (AP) fractional delay filter whose denominator polynomial is obtained by truncating the power series of a certain function. This function is derived from the frequency response of the denominator whose magnitude response is related to the desired phase response through the Hilbert transform since the denominator of a stable AP filter is of minimum phase. The target function and corresponding power series are calculated analytically and expressed in closed form. The closed-form expressions facilitate the analysis of stability. According to the properties for the coefficients of the denominator polynomial, we show that the proposed AP filter is stable for positive delay. Numerical examples indicate that the phase delays of the proposed filters are flat around DC.

The Double-Density Dual-Tree DWT

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.

Design of 1-D and 2-D variable fractional delay allpass filters using weighted least-squares method

In this paper, a weighted least-squares method is presented to design one-dimensional and two-dimensional variable fractional delay allpass filters. First, each coefficient of the variable allpass filter is expressed as the polynomial of the fractional delay parameter. Then, the nonlinear phase error is approximated by a weighted equation error such that the cost function can be converted into a quadratic form. Next, by minimizing the weighted equation error, the optimal polynomial coefficients can be obtained iteratively by solving a set of linear simultaneous equations at each iteration. Finally, the design examples are demonstrated to illustrate the effectiveness of the proposed approach.

Efficient tunable IIR and allpass filter structures

A new recursive filter structure is proposed which can be controlled on-line using a single parameter. The structure can be used for interpolation in timing synchronisation of digital communications receivers. The technique is illustrated with an example of the implementation of a tunable fractional delay allpass filter using the Thiran design technique.