Design Of Fractional Delay Filters Research Articles

Optimized Design of a Variable Fractional Delay Filter With Delay Error Constraints

A novel algorithm for designing variable fractional delay (VFD) filter coefficients is proposed to improve time delay accuracy. The principle of this method is to model the coefficient design as an optimization problem with fractional delay error constraints (FDECs) and use a linear approach to derive a new FDEC expression for solving the problem. The solution of the FDEC expression can be efficiently obtained through linear programming (LP). Compared with existing algorithms, the design examples show the proposed method achieves a 100-fold improvement in delay accuracy while maintaining the frequency response (FR) error at an acceptable level, and yields a 5 dB reduction in FR error with the same level of delay accuracy.

A Resampling Method Based on Filter Designed by Window Function Considering Frequency Aliasing

Resampling technology is becoming increasingly critical, as different frequency components are required to be measured in one device. In this paper, a resampling method based on filter designed by window function is proposed to satisfy the requirements of different sampling rates. The frequency responses of the filter in the resampling model based on the Farrow structure are investigated, and the design criterion of the filter in the resampling model is proposed. A fractional delay filter design model based on window function method is established. A fractional delay filter coefficient matrix, which is expressed by a polynomial form, is constructed. Then, the expression related to the subfilter coefficients is obtained and subfilter coefficients are solved by the least-square method. The simulation results show that the filter designed by the proposed method has better performance when used for resampling as compared those designed by existing methods, and can efficiently prevent the occurrence of frequency aliasing and mirroring components in the process of sampling rate conversion. Therefore, the proposed method meets the conditions for application to existing measurement devices.

Optimal design of symmetric fractional delay filter using firefly algorithm

SummaryA fractional delay filter is used to increase the accuracy, preciseness, time synchronization, and stability of signal processing system. However, designing a fractional delay filter for a specified delay, without affecting spectral characteristics of the signal is challenging because of nondifferentiability and multimodal nature of its objective function. In this paper, a more accurate design technique has been proposed for designing fractional delay filters, based on a recently developed firefly algorithm and its improved version. The designed filters offer variable fractional delay. A novel symmetric structure of implementation has been used to design filters. The efficacy of the proposed technique is evaluated by considering a filter design example. The performance of the proposed technique is compared with the other exiting algorithm. The comparative analysis of finite impulse response (FIR) fractional delay filter design proves that the proposed algorithm has a smaller design error and an implementation complexity than the other reported existing algorithms. In addition to this, the designed FIR fractional delay filter is implemented on Xilinx Virtex‐7 for experimental validation.

Fractional-Order New Generation of nk ± m-Order Harmonic Repetitive Control for PWM Converters

In this paper, a fractional-order new generation of $nk \pm m$ -order harmonic repetitive controller (FO-NG- $nk\pm m$ RC), composed of a proposed new generation of $nk \pm m$ RC (NG- $nk\pm m$ RC) and a Taylor Series expansion based fractional delay (FD) filter using Lagrange interpolation with Farrow structure, is proposed. Compared with conventional $nk \pm m$ RC, NG- $nk\pm m$ RC has more advantages while achieving the same performance as the conventional $nk \pm m$ RC. Different FD approximation algorithms are compared for the first time from the perspective of controller computational burden. The inner relationship between Taylor Series expansion method and Farrow structure FD filter is explained, detailed mathematical derivation is provided, and a complete set of FD filter design methods is formed. When the fundamental frequency is not constant, the performance of $nk\pm m$ RC to track or eliminate any specific $nk \pm m$ -order harmonics will be seriously degraded. However, without changing the sampling rate, the proposed FO-NG- $nk\pm m$ RC can be used to improve the frequency adaptive performance. What’s more, FO-NG- $nk\pm m$ RC provides a unified framework for integer/fractional-order $nk\pm m$ RCs. Experimental results of FO-NG- $nk\pm m$ RC controlled three-phase PWM inverter system show the effectiveness and advantages of the proposed FO-NG- $nk\pm m$ RC scheme.

Low-complexity Design of Variable Fractional Delay Filters

In order to design variable fractional delay filters with low complexity and high accuracy, an efficient design method with controllable cut-off frequency is proposed, which integrates the analytic all-phase filter design, the cubic spline interpolation and Taylor series expansion. In the proposed design, not only the time delay of the filter can be precisely adjusted by setting the delay parameter, but also the tap coefficients of each subfilter in the Farrow structure can be rapidly configured via setting the cut-off frequency parameter, thus the cut-off frequency of the filter can be adjusted flexibly. Numerical simulations show that, the proposed method is especially suitable to design variable fractional delay filters with low or middle cut-off frequencies, and its computation complexity is one order of magnitude lower than that of the existing Weighted Least Squares (WLS) design.

Taylor series expansion based repetitive controllers for power converters, subject to fractional delays

Digital repetitive controllers are widely employed to track/reject the periodic signals with zero steady-state error. Their implementation involves the use of single or multiple digital delay elements. Practically, the delay element is implemented by the use of memory locations, where samples are held and released after a specific number of sampling periods, equivalent to the desired time delay. A problem arises when the desired time delay becomes a non-integer multiple of the sampling time. Such time delays can be accurately realized by employing a fractional delay filterThis paper presents a Taylor Series expansion based digital repetitive controller designed to implement any (integer, non-integer) delay in the control of power converters, occurring due to uncontrollable variations in the reference frequency. The T3644aylor Series expansion transforms the fractional delay filter design problem to a differentiator/sub-filter design. Finite impulse response (FIR) and infinite impulse response (IIR) fractional delay (FD) filter concepts can be applied to realize the required fractional delay. This structure provides efficient on-line tuning capabilities i.e. FD can easily generate any required fractional delay without redesigning the filter when the delay parameter varies. An example is demonstrated to show the effectiveness of this approach, for a single-phase power inverter feeding a passive load.

Implementation of wideband digital transmitting beamformer based on LFM waveforms

A wideband digital transmitting beamformer based on linear frequency modulation (LFM) signals is presented in this study. The wideband beamformer is realised as a combination of direct digital synthesisers and fractional delay (FD) filters in polyphase structure. By using coordinate rotation digital computer algorithm, high intermediate frequency wideband LFM signal is generated and phase compensation for beamforming is accomplished in field programmable gate array. The impact of different number of quantisation bits on signal generation is analysed. The results of waveform generation and FD filter design are given. At last, the transmitting beam pattern is simulated.

Design of Allpass Variable Fractional Delay Filter with Powers-of-Two Coefficients

In this letter, we develop an algorithm for the design of allpass variable fractional delay filter with powers-of-two coefficients as a means of achieving low complexity and efficient hardware implementation. The algorithm includes an efficient procedure to obtain the objective function in the neighborhood of a search point, which significantly reduces the complexity associated with the global optimization approach. Design examples show that significant improvement is obtained by using the proposed method over existing methods.

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 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.

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.

Comments on “Closed Form Variable Fractional Time Delay Using FFT”

In this letter drawbacks of the aforementioned paper are pointed out. The proposed approach is improved with minor modifications of the discrete frequency response. This allows for design of fractional delay filters which are close to optimal and can be efficiently implemented in the frequency domain using the sliding DFT based structure. Alternatively, the derived equivalent closed form formulae for offset windows can be used to design filters in the time domain.

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.

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.

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.

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.

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.

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.