Variable Fractional Delay Filter Research Articles

Variable fractional delay filter design with discrete coefficients

This paper investigates the optimal design of variable fractionaldelay (VFD) filter with discrete coefficients as a means ofachieving low complexity and efficient hardware implementation. Thefilter coefficients are expressed as the sum of signed power-of-two(SPT) terms with a restriction on the total number of power-of-twoterms. An optimization problem with least squares criterion isformulated as a mixed-integer programming problem. An optimalscaling factor quantization scheme is applied to the problemresulting in an optimal scaling factor quantized solution. Thissolution is then improved further by applying a discrete filledfunction, that has been extended for a mixed integer optimizationproblem. To apply the discrete filled function method, it requiresmultiple calculations of the objective function around theneighborhood of a searched point. Thus, an updating scheme isdeveloped to efficiently calculate the objective function in aneighborhood of a point. Design examples demonstrate theeffectiveness of the proposed optimization approach.

Design of a Variable Fractional Delay Filter Using Comprehensive Least Square Method Encompassing All Delay Values

The work presented in this paper is concerned with the designing of a variable fractional delay (VFD) FIR filter using least square method encompassing all delay values p. First, the frequency response, e-jωp, of the VFD filter is approximated using power series expansion of an exponential function and then, it is represented in terms of frequency response of the differentiator. Next, the proposed least square method adopts a novel cost function which includes all the fractional delay values. Finally, design and application examples are given to illustrate that the proposed least square method have better design accuracy and significantly reduction in magnitude error with the same degree of complexity than existing least square methods.

Variable Ratio Sample Rate Conversion Based on Fractional Delay Filter

Abstract In this paper a sample rate conversion algorithm which allows for continuously changing resampling ratio has been presented. The proposed implementation is based on a variable fractional delay filter which is implemented by means of a Farrow structure. Coefficients of this structure are computed on the basis of fractional delay filters which are designed using the offset window method. The proposed approach allows us to freely change the instantaneous resampling ratio during processing. Using such an algorithm we can simulate recording of audio on magnetic tape with nonuniform velocity as well as remove such distortions. We have demonstrated capabilities of the proposed approach based on the example of speech signal processing with a resampling ratio which was computed on the basis of estimated fundamental frequency of voiced speech segments.

Design of Wideband Digital Array Beamformer Using the Variable Fractional Delay Allpass Filter

To solve the space dispersion (SD) problem which makes the beam formed deviate from its expected direction in wideband digital array beamforming, a new wideband digital array beamformer (WDBF) is designed by introducing an identical variable fractional delay (VFD) filter on each sensor. The VFD filter is an all-pass filter with an infinite impulse response, thus it requires less filter order compared with finite impulse response filter. Since the VFD designed is suitable within the whole fractional delay range of-0.5 to 0.5 in sampling interval, the new beamformer designed is suitable for any direction without redesigning the fractional delay filter. In this way, the real-time property of the new beamformer is guaranteed. Simulation results show that the VFD deigned in this paper is high in precision. Applying it in the new WDBF, the SD problem is solved and the beam formed approximates the ideal beam very well.

Design of Variable Fractional Delay Filter with Fractional Delay Constraints

This letter develops an efficient computational procedure for the design of an odd-order variable fractional delay (VFD) digital filter with minimum peak variable fractional delay error subject to restrictions on the peak variable frequency response error. An iterative algorithm is developed to solve the formulated non-linear optimization problem where a first order linear approximation is employed for the variable fractional delay error. This results in a second-order cone programming (SOCP) problem with linear constraints. Design examples show that the peak fractional delay error can be reduced significantly from the minimax solution while maintaining approximately the same peak frequency response error. In addition, for a fixed level of peak variable frequency response error, approximately 1.61-2.09 dB improvement for the peak variable delay error can be obtained over existing methods.

Variable Fractional Delay Filter Design Using a Symmetric Window

In this paper a numerically efficient method for designing a nearly optimal variable fractional delay (VFD) filter based on a simple and well-known window method is presented. In the proposed method a single window extracted from the optimal filter with fixed fractional delay (FD) is divided into even and odd part. Subsequently, the odd part is discarded and symmetric even part of the extracted window is used to design a family of nearly optimal filters with varying FD. In addition to window extraction, the proposed approach requires filter gain correction which is dependent on the desired FD. Optimum values of the gain correction factor as well as the extracted window can be computed beforehand, which allows us to design a nearly optimal FD filter with arbitrary FD at low numerical costs during runtime. On the basis of the proposed filter design method, the universal structure of VFD filter allowing for change of filter type and length has been proposed. In the paper, three FD filter optimality criteria are considered, which are maximal flatness, Chebyshev (minimax), and least squares.

Using 1-D Variable Fractional-Delay Filters to Reduce the Computational Complexity of 3-D Broadband Multibeam Beamformers

A multibeam filter bank (MBFB) is proposed for recovering the broadband far-field signals received from multiple directions while rejecting interfering signals and noise. The MBFB consists of two sections. The first section employs variable fractional delays and independently steers the directions of the multiple beams. It achieves low complexity by using the Farrow structure. The second section achieves low complexity by maximal decimation methods and by exploiting the spatial symmetry of a 3-D double-frustum-shaped passband. Superior performance and lower complexity compared with previous methods are numerically confirmed for a typical narrow-angle broadband multibeam beamformer of the type that could be used in radio astronomy.

Variable Fractional Delay FIR Filter Design with a Bicriteria and Coefficient Relationship

This brief investigates a tradeoff between the integral squared error and the peak deviation error for a variable fractional delay (VFD) filter with a coefficient relationship. The integral squared error is minimized subject to additional constraints on the peak deviation error. The problem is solved by utilizing second-order cone programming. In addition, the performance of the VFD filter with discrete coefficients is investigated, in which the filter coefficients are expressed as the sum of power-of-two terms to reduce the filter operations to shifts and adds. Design examples show that the peak deviation error can be significantly reduced from the least squares solution while maintaining approximately the same integral squared error. Similarly, the integral squared error can be significantly reduced from the minimax solution while maintaining approximately the same peak deviation error. Furthermore, the tradeoff filters are less sensitive with respect to quantization than the least squares and minimax solutions.

Improved Filter Bank Approach for the Design of Variable Bandedge and Fractional Delay Filters

The paper proposes an optimization technique for the design of variable digital filters with simultaneously tunable bandedge and fractional delay using a fast filter bank (FFB) approach. In the FFB approach, full band signals are split into multibands, and each band is multiplied by a proper phase shift to realize the variable fractional delay. In the proposed technique, in the formulation of the optimization of the 0th stage prototype filter of the FFB, the ripples of the filters in the subsequent stages are all taken into consideration. In addition, a shaping filter is applied to the last retained band of the FFB to form the transition band of the variable filter, such that the transition width of each band in the FFB can be relaxed to reduce the computational complexity. In total three shaping filters, constructed from a prototype filter, can be shared by different bands, so that the extra cost incurred due to the shaping filter is low.

Design of allpass variable fractional delay filter with signed powers-of-two coefficients

This paper investigates the optimal design of allpass variable fractional delay (VFD) filters with coefficients expressed as sums of signed powers-of-two terms, where the weighted integral squared error is the cost function to be minimized. The design can be classified as an integer programming problem. To solve this problem, a new procedure is proposed to generate a reduced discrete search region to decrease the computational complexity. A new exact penalty function method is developed to solve the optimal design problem for allpass VFD filter with signed powers-of-two coefficients. Design examples show that the proposed method can achieve a higher accuracy when compared with the quantization method.

Investigation on the Optimization Criteria for the Design of Variable Fractional Delay Filters

In recent research on the optimization of variable fractional delay (VFD) filters, group delay errors have been paid additional attention besides the conventional criterion of frequency response errors. While group delay physically measures the delay of the magnitude envelop of a signal, an important application of the VFD filters is to estimate the instantaneous delayed samples, such as in the applications of interpolation. In view of this, the impact of the group delay error on the estimation of instantaneous samples is investigated in this brief. Analysis and simulation results show that in such applications, the traditional criterion of frequency response errors reflects the time-domain errors and group delay errors do not show a slight impact on the time-domain errors.

FIR Filter Design Using Distributed Maximal Flatness Method

Abstract In the paper a novel method for filter design based on the distributed maximal flatness method is presented. The proposed approach is based on the method used to design the most common FIR fractional delay filter - the maximally flat filter. The MF filter demonstrates excellent performance but only in a relatively narrow frequency range around zero frequency but its magnitude response is no greater than one. This ,,passiveness” is the reason why despite of its narrow band of accurate approximation, the maximally flat filter is widely used in applications in which the adjustable delay is required in feedback loop. In the proposed method the maximal flatness conditions forced in standard approach at zero frequency are spread over the desired band of interest. In the result FIR filters are designed with width of the approximation band adjusted according to needs of the designer. Moreover a weighting function can be applied to the error function allowing for designs differing in error characteristics. Apart from the design of fractional delay filters the method is presented on the example of differentiator, raised cosine and square root raised cosine FIR filters. Additionally, the proposed method can be readily adapted for variable fractional delay filter design regardless of the filter type.

Two-Rate Based Low-Complexity Variable Fractional-Delay FIR Filter Structures

This paper considers two-rate based structures for variable fractional-delay (VFD) finite-length impulse response (FIR) filters. They are single-rate structures but derived through a two-rate approach. The basic structure considered hitherto utilizes a regular half-band (HB) linear-phase filter and the Farrow structure with linear-phase subfilters. Especially for wide-band specifications, this structure is computationally efficient because most of the overall arithmetic complexity is due to the HB filter which is common to all Farrow-structure subfilters. This paper extends and generalizes existing results. Firstly, frequency-response masking (FRM) HB filters are utilized which offer further complexity reductions. Secondly, both linear-phase and low-delay subfilters are treated and combined which offers trade-offs between the complexity, delay, and magnitude response overshoot which is typical for low-delay filters. Thirdly, the HB filter is replaced by a general filter which enables additional frequency-response constraints in the upper frequency band which normally is treated as a don't-care band. Wide-band design examples (90, 95, and 98% of the Nyquist band) reveal arithmetic complexity savings between some 20 and 85% compared with other structures, including infinite-length impulse response structures. Hence, the VFD filter structures proposed in this paper exhibit the lowest arithmetic complexity among all hitherto published VFD filter structures.

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.

Design of Fractional Delay Filter Using Hermite Interpolation Method

In this paper, the design of fractional delay (FD) filters using the Hermite interpolation is investigated. First, the transfer function of fixed FD filters is obtained from the Hermite interpolation formula by using variable substitution. Then, two implementation methods for the Hermite-based FD filters are presented. One is the derivative sampling scheme using an analog differentiator, the other is the Shannon sampling scheme with an auxiliary digital differentiator. Next, the proposed method is applied to design wide-range variable fractional delay (VFD) filters and some extensions are made. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed Hermite interpolation design method.

Closed Form Variable Fractional Time Delay Using FFT

In this letter, a novel tunable frequency response is proposed for variable fractional delay filter with careful consideration on conjugate symmetry. This proposed fractional delay method can be efficiently realized by FFT using the tunable fractional delay parameter. Its equivalent closed form to windowing method is derived and presented. Several comparisons to other windowing methods are also demonstrated, such as the magnitude response, group delay, and approximation error between ideal frequency response.

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.

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.