Variable Fractional Delay Research Articles

Design of variable comb filter using FIR variable fractional delay element

In this paper, a new approach is proposed for the design of infinite-impulse-response (IIR) comb filter. The problem of the comb filter design is transformed into that of fractional delay filter design. Using the proposed method, the multiplier inside the proposed structure can be fixed regardless the value of fundamental frequency of harmonic interferences. This improvement makes the proposed structure be possible for the design of variable cases. To realize variable comb (VC) filter, the variable fractional-delay (VFD) filter is designed in weighted least-squares (WLS) sense first. Then, by embedding the designed VFD filter with a switch, the variable comb filter could be implemented in a proposed structure. Several experimental results and examples are presented to demonstrate the effectiveness of the proposed method.

Mixed-Radix Fast Filter Bank Approach for the Design of Variable Digital Filters With Simultaneously Tunable Bandedge and Fractional Delay

A popular technique to design variable fractional delay (VFD) filters or variable bandedge filters is the polynomial based finite impulse response (FIR) filters, where each filter coefficient is approximated as a polynomial of the variable parameter. However, if the filter is required to have both VFD and variable bandedge, the computational complexity becomes very high, because two-dimensional polynomials have to be used, provided that the same polynomial idea is followed. In this paper, a filter bank approach is proposed for the design of VFD filters. The basic idea of the approach is to split the full band signals into subbands and each subband is, respectively, shifted by a phase, which is determined by the required fractional delay. The overall fractional delay is achieved by combining all subbands. Using this technique, the variable bandedge could be incorporated into the VFD filter with trivial extra computations. Design examples show that the proposed technique achieves high performance with low computational complexity.

Hybrid parallel/cascade structure for designing variable fractional-delay filters

There are two different ways to implement variable fractional-delay (VFD) filters. One is the Farrow structure, which is constructed by connecting differentiators with various orders in parallel, the other one is the first-order differentiator structure, which is constructed by cascading first-order differentiators. The parallel Farrow structure requires a large memory to store filter coefficients but possesses low filter delay, and the cascade first-order differentiator structure requires less memory but results in long filter delay. In this study, a hybrid structure is presented to implement the VFD filter, which considers the parallel and cascade connections of differentiators simultaneously such that the trade-off between memory requirement and filter delay can be made. Some numerical examples are demonstrated to show the effectiveness of the proposed structure.

A Nearly Optimal Fractional Delay Filter Design Using an Asymmetric Window

A Nearly Optimal Fractional Delay Filter Design Using an Asymmetric WindowIn this paper a numerically efficient filter design method suitable for variable fractional delay (VFD) filter implementation is investigated. We propose to use a well known window method with an asymmetric window extracted from optimal filter designed beforehand. As we will demonstrate, such an approach, if additional gain correction is applied, allows for nearly optimal VFD filter design. Thus, the proposed approach combines window method simplicity with performance comparable to that of optimal filters. Efficiency of the presented technique makes it suitable for designing filters with varying delay in real time.

Design of Allpass Variable Fractional Delay Filter

This correspondence investigates the least squares and minimax design problems for allpass variable fractional delay (VFD) filters. A two stage optimization approach is proposed to solve the resulting minimax optimization problem. This approach includes a combination of a one-dimensional global search method and an adaptive scheme to refine the discretization points. In addition, the paper investigates the design of allpass VFD filters which minimizes the weighted integral squared error subject to constraints on peak error deviation from the desired response. By using approximations, the design problem can be formulated as a quadratic optimization problem. Design examples show that a tradeoff between the weighted integral squared error and the peak error deviation can be achieved. In addition, the integral squared error can be reduced significantly from the minimax solution while maintaining approximately the same peak error deviation. Similarly, the peak error deviation can be significantly reduced from the least squares solution while maintaining approximately the same integral squared error.

Decoupling Minimax Design of Low-Complexity Variable Fractional-Delay FIR Digital Filters

This paper presents a simple linear programming (LP) technique for designing high-accuracy low-complexity finite-impulse-response (FIR) variable fractional-delay (VFD) digital filters in the minimax error sense. The objective of the minimax design is to minimize the maximum absolute error of the variable frequency response (VFR) of an FIR VFD filter, which is a nonlinear problem and difficult to solve. This paper shows that the minimax design can be approximately decomposed into a pair of separate LP subproblems by decoupling the minimization of the real-part VFR error from that of the imaginary-part error. As a result, the original nonlinear minimax design problem can be easily solved by solving the two LP subproblems separately. To reduce the VFD filter complexity, we also propose a one-by-one increase scheme for optimizing the subfilter orders in the Farrow structure such that a given design specification (maximum absolute error of VFR) can be exactly satisfied. Both even-order and odd-order design examples are given to illustrate that the decoupling minimax method is not only simple, but also can achieve excellent high-accuracy low-complexity FIR VFD filters.

Minimax Design of Low-Complexity Even-Order Variable Fractional-Delay Filters Using Second-Order Cone Programming

This brief proposes an optimal minimax method for designing even-order finite-impulse-response variable fractional-delay (VFD) digital filters by using the second-order cone programming (SOCP), which minimizes the peak error of the variable-frequency response and yields a true minimax design. To minimize the VFD filter complexity, we also present an algorithm for jointly optimizing all the subfilter orders in the Farrow structure such that all the subfilter orders can be simultaneously optimized for exactly meeting a given upper error bound. A design example is given to demonstrate the high performance and low complexity of the SOCP-based VFD filter.

Efficient Sampling of Band-Limited Signals From Sine Wave Crossings

This correspondence presents an efficient method for reconstructing a band-limited signal in the discrete domain from its crossings with a sine wave. The method makes it possible to design A/D converters that only deliver the crossing timings, which are then used to interpolate the input signal at arbitrary instants. Potentially, it may allow for reductions in power consumption and complexity in these converters. The reconstruction in the discrete domain is based on a recently-proposed modification of the Lagrange interpolator, which is readily implementable with linear complexity and efficiently, given that it re-uses known schemes for variable fractional-delay (VFD) filters. As a spin-off, the method allows one to perform spectral analysis from sine wave crossings with the complexity of the FFT. Finally, the results in the correspondence are validated in several numerical examples.

Efficient Design and Implementation of Variable Fractional Delay Filters Using Differentiators

In this paper, the power series expansion of exponential function is used to transform the design problem of variable fractional delay (VFD) filter into the designs of digital differentiators with various orders such that conventional digital differentiators can be applied to implement VFD filter efficiently. The proposed method is flexible because the VFD filter can be designed by making the trade-off among the storage requirement of filter coefficients, computational complexity and delay of filter. Finally, some numerical examples are demonstrated to show the effectiveness and flexibility of the proposed design methods.

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.

A New Method for Least-Squares and Minimax Group-Delay Error Design of Allpass Variable Fractional-Delay Digital Filters

A double-loop iterative method is proposed to design allpass variable fractional-delay (VFD) digital filters basing on the minimization of root-mean-squared group-delay error. In the inner loop, an iterative quadratic optimization is proposed to replace the original nonlinear optimization for the minimization of root-mean-squared group-delay error, while an iterative weighting-updated technique is applied in the outer loop to further reduce the maximum group-delay error. Several examples will be presented to demonstrate the effectiveness and good convergence of the proposed method.

Pitch Shifting by H∞-Optimal Variable Fractional Delay Filters

AbstractFractional delay filters are those that are designed to delay the input samples by a fractional amount of the sampling period. Since the delay is fractional, the intersample behavior of original analog signals becomes crucial. For this, we propose an optimal design via sampled-data H∞ control theory. By this theory, our design problem is equivalently reduced to a discrete-time H∞ optimization, and, an analytical solution is obtained under an assumption on the original analog signals. Using this analytical solution, we propose a sampling rate conversion with arbitrary conversion rate. This conversion is much faster than conventional methods using an upsampler, a digital filter, and a downsampler. We also show an application of the proposed sampling rate conversion to pitch shifting of guitar sounds. A design example is shown to illustrate the advantage of the proposed method.

Minimax Design of Low-Complexity Allpass Variable Fractional-Delay Digital Filters

This paper proposes noniterative and iterative linear programming (LP) procedures for designing low-complexity allpass variable fractional-delay (VFD) digital filters in the minimax sense. Expressing each coefficient of an allpass VFD filter as a polynomial in the VFD parameter p, we show that the frequency response error of an allpass VFD filter can be written as a pure imaginary part divided by its denominator. Thus, the minimax design can be approximately formulated as an LP problem through neglecting the denominator, which leads to a noniterative minimax design. To improve the minimax design accuracy, we propose an iterative LP procedure that utilizes the denominator from the preceding iteration as a known. The iterative LP minimization is repeated until it converges to the minimax solution. Moreover, we also present a two-stage algorithm for optimizing the optimal variable range p ∈ [pMin, pMax] of the VFD parameter p and successively reducing the subfilter orders. Design examples are given to show that both noniterative and iterative LP methods can achieve much better minimax designs (smaller peak errors) than the existing iterative weighted-least-squares (WLS) approaches, which aim to minimize the peak errors of VFD response and variable phase response, respectively. Moreover, the resulting allpass VFD filters have lower complexities than those from the iterative WLS approaches.

Allpass VFD Filter Design

This correspondence proposes a general design for allpass variable fractional delay (VFD) digital filters with minimum weighted integral squared error subject to constraints on maximum error deviation from the desired response. The resulting optimization problem is nonlinear and nonconvex with a nonlinear continuous inequality constraint. Stability of the designed filters are discussed. An effective procedure is proposed for solving the optimization problem. Firstly, a constraint transcription method and a smoothing technique are employed to transform the continuous inequality constraint into one equality constraint. Then, by using the concept of a penalty function, the transformed constraint is incorporated into the cost function to form a new cost function. The nonlinear optimization problem subject to continuous inequality constraints is then approximated by a sequence of unconstraint optimization problems. Finally, a global optimization method using a filled function is employed to solve the unconstraint optimization problem. Design example shows that a trade-off can be achieved between the integral squared error and the maximum error deviation for the design of allpass VFD filters.

Design of Wideband Fractional Delay Filters Using Derivative Sampling Method

In this paper, the design of wideband fractional delay filter is investigated. First, the reconstruction formula of derivative sampling method is applied to design wideband fractional delay filter by using index substitution and window method. The filter coefficients are easily computed because closed-form design is obtained. Then, the weighted least squares method is used to design fixed and variable wideband fractional delay filters. Finally, numerical examples are demonstrated to show that the proposed method has smaller design error than the conventional fractional delay filter without sampling the derivative of signal.

Hybrid Structures for Low-Complexity Variable Fractional-Delay FIR Filters

This paper proposes a pair of new structures for implementing low-complexity odd-order and even-order variable fractional-delay (VFD) FIR filters using hybrid structures with both even-order and odd-order subfilters. For odd-order VFD filter design, the main idea is to replace the conventional even-symmetric odd-order (type-2) subfilters with the cascade of a type-2 half-delay filter and even-symmetric even-order (type-1) subfilters. This is because the magnitude responses of type-2 subfilters are zero at frequency ? = ?, which causes large approximation errors, especially for wideband VFD filter design. Similarly, for even-order VFD filter design, the conventional odd-symmetric even-order (type-3) subfilters are replaced with the cascade of a type-2 half-advance filter and odd-symmetric odd-order (type-4) subfilters. This replacement avoids the zero magnitude responses at frequency ? = ? of type-3 subfilters and makes low-complexity wideband VFD filter design possible. As compared with the existing Farrow structures with either odd-order subfilters (odd-order design) or even-order subfilters (even-order design), the hybrid structures consist of three types of FIR filters (type-1 and type-4 subfilters, as well as type-2 half-delay or half-advance filter). Replacing type-2 subfilters with type-1 ones (odd-order design) and replacing type-3 subfilters with type-4 ones (even-order design) can significantly reduce the VFD filter complexity for wideband designs. Design examples are given to illustrate the effectiveness of both odd-order and even-order hybrid structures.

Allpass variable fractional delay filters by pole loci interpolation

A simple, accurate and efficient variable fractional delay (VFD) IIR filter is presented. The method is based on the interpolation of pole loci of maximally-flat allpass fractional-delay filters. The responses of the proposed VFD filter for different fractional delay values are simulated and the approximation errors are shown to be very small.

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.

Robust Structure Transformation for Causal Lagrange-Type Variable Fractional-Delay Filters

We have rigorously proved that an odd-order causal Lagrange-type variable fractional-delay (VFD) digital filter can be implemented as the Farrow structure with symmetric or antisymmetric subfilters through utilizing matrix transformation. This paper reveals a numerical problem that occurs in the matrix transformation due to the so-called catastrophic cancellation in numerical computation. Our computer simulations have verified that utilizing the matrix transformation can get numerically stable solutions for the VFD filter whose order N is below about 20, but fails for higher order cases. To solve this problem, we present a robust approach to the structure transformation for both even- and odd-order causal Lagrange-type VFD filters, which does not rely on matrix operations and, thus, can yield numerically stable solutions even for high-order cases. Moreover, the symmetry and antisymmetry of the subfilter coefficients after the robust structure transformation are also rigorously proved. Numerical examples are included to illustrate the robustness of the proposed structure transformation and show the coefficient symmetries.