Variable Fractional-delay Digital Filters Research Articles

Delay-error-constrained minimax design of all-pass variable-fractional-delay digital filters

This paper first derives a simplified variable-fractional-delay (VFD) expression for the all-pass (AP) VFD digital filter, and then uses the simplified VFD expression to formulate the minimax AP-VFD filter design as a two-step linear-programming (LP) problem. To suppress the maximum error of the VFD response (VFD-peak-error), this minimax design minimizes the maximum error of the variable-frequency-response (VFR) subject to the VFD-peak-error constraint. Thus, this two-step design can minimize the VFR-peak-error with the VFD-peak-error suppressed below a prescribed upper bound. With the aid of the simplified VFD expression, the VFD-peak-error constraint can be approximately linearized as a linear one, and thus the minimax design can be solved by using the LP method. We will use an illustrative example to verify that the simplified VFD expression is almost the same as the true one, and that the proposed LP-based two-step minimax design can significantly suppress the VFD-peak-error.

Coefficient relation-based minimax design and low-complexity structure of variable fractional-delay digital filters

Although the coefficient-relation of the even-order sub-filters in the Farrow structure has been revealed and adopted in the weighted-least-squares (WLS) design of even-order finite-impulse-response (FIR) variable fractional-delay (VFD) digital filters in the literature, it has never been exploited in implementing low-complexity VFD filters. This paper aims to(i)propose a low-complexity implementation structure through exploiting the coefficient-relation such that the coefficient-relation is not only utilized in the design process, but also utilized in the low-complexity implementation;(ii)formulate the minimax design of an even-order VFD filter through exploiting the coefficient-relation;(iii)propose a new two-stage scheme called increase-then-decrease scheme for optimizing the sub-filter orders so as to minimize the VFD digital filter complexity.With the above three advances, an even-order VFD filter can not only be designed and but also be implemented efficiently by exploiting the coefficient-relation. We will use a design example to illustrate the low complexity and high design accuracy.

Closed-Form Mixed Design of High-Accuracy All-Pass Variable Fractional-Delay Digital Filters

This paper presents a closed-form method for minimizing the weighted squared error of variable fractional-delay (VFD) of an all-pass VFD digital filter under an equality constraint on its normalized root-mean-squared (NRMS) error of variable frequency response (VFR). The main purpose is to reduce the squared VFD error as much as possible while keeping its NRMS VFR error exactly at a predetermined value. We first prove that the linearized VFR error of an all-pass VFD filter is almost the same as its linearized phase error, and then convert the equality-constrained weighted-least-squares (WLS) design into an unconstrained optimization problem through the minimization of a mixed error function that mixes the weighted squared VFD error and squared VFR error. To reduce the computational complexity, we derive a closed-form mixed error function by utilizing Taylor series expansions of trigonometric functions. Therefore, the error functions can be efficiently computed without discretizing the design parameters (frequency ω and VFD parameter p). The closed-form mixed error function not only reduces the computational complexity, but also speeds up the design process as well guarantees the optimality of the final solution. Furthermore, a two-point search (dichotomous search) scheme is proposed for finding the optimal range p ∈ [pMin,pMax] of the VFD parameter p, and then the subfilter orders are optimized under a given filter complexity constraint (the number of all-pass VFD filter coefficients). This two-stage optimization process utilizes the NRMS VFD error as an error criterion. Design examples and comparisons are given to demonstrate that the closed-form mixed WLS method yields low-complexity all-pass VFD filters with a high-accuracy VFD response but without noticeably degrading its frequency response.

Generalized WLS Method for Designing All-Pass Variable Fractional-Delay Digital Filters

This paper presents an optimal weighted least squares (WLS) method for designing low-complexity all-pass variable fractional-delay (VFD) digital filters. Instead of using a fixed range for the VFD parameter p and same-order constant-coefficient filters (subfilters), both the VFD parameter range p isin [p Min,p Max] and subfilter orders are optimized such that a low-complexity all-pass VFD filter can be achieved for the LS design. To suppress the peak errors of variable frequency response, weighting functions are adopted and optimized such that the boundary peak errors can be further reduced but without noticeably increasing the total error energy (integral of squared error) of variable frequency response. After optimizing the variable range of the VFD parameter, weighting functions, and subfilter orders, an all-pass VFD filter can be designed by using a generalized noniterative WLS method, which yields a closed-form solution. Design examples are given to illustrate that utilizing different-order subfilters, along with the optimal range and optimal weighting functions, can yield an all-pass VFD filter with significantly reduced complexity and design errors as compared with existing ones.

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.

Minimax phase error design of allpass variable fractional-delay digital filters by iterative weighted least-squares method

In this paper, an iterative weighted least-squares (LS) method is proposed for the design of allpass variable fractional-delay (VFD) digital filters, so that the maximum absolute phase error can be minimized. Meanwhile, it is found that the range of variable parameter will affect overall performance of the designed filter, which will be analyzed and discussed in this paper through several examples designed by least-squares and iterative weighted least-squares methods. And the stability of the designed filters and the effectiveness of the proposed iterative method are also revealed by these presented examples.

Optimal Design of All-Pass Variable Fractional-Delay Digital Filters

This paper presents a computational method for the optimal design of all-pass variable fractional-delay (VFD) filters aiming to minimize the squared error of the fractional group delay subject to a low level of squared error in the phase response. The constrained optimization problem thus formulated is converted to an unconstrained least-squares (LS) optimization problem which is highly nonlinear. However, it can be approximated by a linear LS optimization problem which in turn simply requires the solution of a linear system. The proposed method can efficiently minimize the total error energy of the fractional group delay while maintaining constraints on the level of the error energy of the phase response. To make the error distribution as flat as possible, a weighted LS (WLS) design method is also developed. An error weighting function is obtained according to the solution of the previous constrained LS design. The maximum peak error is then further reduced by an iterative updating of the error weighting function. Numerical examples are included in order to compare the performance of the filters designed using the proposed methods with those designed by several existing methods.

A New Method for Designing Causal Stable IIR Variable Fractional Delay Digital Filters

This paper studies the design of causal stable Farrow-based infinite-impulse response (IIR) variable fractional delay digital filters (VFDDFs), whose subfilters have a common denominator. This structure has the advantages of reduced implementation complexity and avoiding undesirable transient response when tuning the spectral parameter in the Farrow structure. The design of such IIR VFDDFs is based on a new model reduction technique which is able to incorporate prescribed flatness and peak error constraints to the IIR VFDDF under the second order cone programming framework. Design example is given to demonstrate the effectiveness of the proposed approach.

Design and parallel implementation of FIR digital filters with simultaneously variable magnitude and non-integer phase-delay

Variable fractional-delay digital filters are useful in various signal processing applications. To perform both fractional-delay filtering and signal frequency selecting, variable fractional-delay filters must also have variable magnitude characteristics. This paper proposes a method for designing variable finite-impulse response (FIR) filters with both variable magnitude and variable noninteger phase-delay. First, the coefficients of a variable FIR filter are expressed as different 2-variable polynomials of a pair of spectral parameters; one is for varying magnitude response, and the other is for varying noninteger phase-delay. Then the optimal coefficients of the 2-variable polynomials are found by minimizing the total weighted squared error of the variable frequency response. Since the coefficients of the obtained variable FIR filter are the polynomials of the two spectral parameters, we can yield variable magnitude and variable noninteger phase-delay simultaneously or independently by substituting different spectral parameter values to the 2-variable polynomial coefficients. Finally, we show that the resulting variable FIR filter can be implemented in a parallel form, which is suitable for high-speed signal processing.