Fractional Delay FIR Research Articles

Design of variable fractional delay FIR filter using the high-order derivative sampling method

In this study, a design algorithm for a variable fractional delay finite impulse response (FIR) filter based on piecewise osculating interpolation (POI) is proposed. The proposed algorithm can use higher-order derivative information of the input sampling sequence. To improve the performance of the proposed POI algorithm at high frequencies, two frequency-domain optimization algorithms (including derivative weighted least-squares and derivative minimax algorithm) are proposed. Meanwhile, a one-by-one increase scheme is introduced for optimizing the polynomial orders, such that a given bound is satisfied exactly. Additionally, through derivation of an analytical formula, a two-level Farrow structure is developed to implement the proposed algorithms. As shown, the implementation structure has fixed coefficients and only requires multiplication and addition operations. The proposed algorithms are validated and compared with existing algorithms, and it is shown that the proposed algorithms outperform existing algorithms.

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.

Optimal design of fractional delay FIR filter using cuckoo search algorithm

SummaryThe conventional gradient‐based optimization methods are not sufficient to optimize nonlinear, multimodal, and nonuniform objective functions of fractional delay FIR (FD‐FIR) filters, and the objective function cannot converge to the global minimum solution. So a population‐based meta‐heuristic optimization algorithm called as cuckoo search algorithm (CSA) has been implemented in the design of optimal FD‐FIR filter. Cuckoo search algorithm is based on lifestyle and unique parasitic behavior in egg laying and breeding of some cuckoo species along with Lévy flight behavior of some birds and fruit flies. To attain a balance between exploration and exploitation in the search space, different set of control parameters is tested by simulation. Extensive simulations were performed to ensure how CSA exploits in the design of optimal FD‐FIR filter. A quantitative assessment of the proposed CSA‐based method is accomplished using several performance metric such as magnitude error, convergence rate, and optimal solution. The simulation results reveal the advantages of the proposed FD filter using CSA compared with the FD filter designed using evolutionary algorithm like genetic algorithm and conventional FD filter design methods such as Lagrange, discrete Hartley transform, discrete Fourier transform, discrete cosine transform, and radial basis function methods.

Design of half sample delay recursive digital integrators using trapezoidal integration rule

This paper describes the design of half sample delay recursive digital integrators. For this, a half sample delay is applied on trapezoidal integration rule, then a modified FIR fractional delay filter is used to design recursive digital integrators. The modified FIR fractional delay filter is less complex and more efficient than original one. In this way lower order recursive digital integrators have been designed and compared with existing half sample delay and conventional recursive digital integrators. The results show the effectiveness of the proposed integrators with low percentage absolute magnitude relative error (PARE) and linear phase response over almost 0% to 80% of the entire Nyquist frequency range.

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.

A frequency domain proof for the equivalence of the maximally flat FIR fractional delay filter and the Lagrange interpolator

One of the most important properties of the maximally flat (MF) FIR fractional delay (FD) filter is its equivalence with the Lagrange interpolator for uniformly sampled signals. In this article, to provide the required background for the reader, we first propose a straightforward algebraic proof for this equivalence. This proof is given by simply demonstrating that the system of linear equations governing the maximally flatness property of this filter is the same as the one from which the coefficients of the Lagrange interpolator are computed. We then present the main contribution of the paper, which is a frequency domain proof for the same equivalence. In contrast with its classic counterparts which typically deploy the system of equations characterizing the maximally flatness property of the MFFD filter, the proof presented here is just based on the definition of this property. The aim of the article is to shed more light on the important equivalence it discusses by revisiting it from a new perspective.

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.

Maxflat Fractional Delay IIR Filter Design

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Fractional delay (FD) filters are an important class of digital filters and are useful in various signal processing applications. This paper discusses a design problem of FD infinite-impulse-response (IIR) filters with the maxflat frequency response in frequency domain. First, a flatness condition of FD filters at an arbitrarily specified frequency point is described, and then a system of linear equations is derived from the flatness condition. Therefore, a set of filter coefficients can be easily obtained by solving this system of linear equations. For a special case in which the frequency response is required to be maxflat at <formula formulatype="inline"><tex Notation="TeX">$\omega=0$</tex> </formula> or <formula formulatype="inline"><tex Notation="TeX">$\pi$</tex> </formula>, a closed-form expression for its filter coefficients is derived by solving a linear system of Vandermonde equations. It is also shown that the existing maxflat FD finite-impulse-response (FIR) and IIR filters are special cases of the FD IIR filters proposed in this paper. Finally, some examples are presented to demonstrate the effectiveness of the proposed filters. </para>

Digital IIR Integrator Design Using Richardson Extrapolation and Fractional Delay

In this paper, a new design of digital integrator is investigated. First, the trapezoidal integration rule and differential equation are applied to derive the transfer function of the digital integrator. The Richardson extrapolation is then used to generate high-accuracy results while using low-order formulas. Next, the conventional Lagrange finite-impulse response fractional delay filter is directly applied to implement the designed integrator. Two implementation structures are studied: direct substitution and polyphase decomposition. Finally, numerical comparisons with conventional digital integrators are made to demonstrate the effectiveness of this new design approach.

Half-band FIR fractional delay filters with closed-form coefficient formulas and modular implementation based on Lagrange interpolators

The Taylor series-based representation of the maximally flat (MF) FIR fractional delay (FD) filter is manipulated to obtain a FD filter with a wider band-width. The band-width of the proposed filter can approach π / 2 rads / s , which is 1.5 times that of the prototype one. The design has closed-form coefficient formulas as well as a modular implementation. These two properties make it a practically favourable one.

A simple and efficient design of variable fractional delay FIR filters

In this paper, a simple and efficient approach for designing one-dimensional variable fractional delay finite impulse response digital filters is proposed. Two matrix equations, based respectively on the weighted least-squares function of the optimum fixed fractional delay filter and the filter coefficient polynomial fitting, are formulated in tandem to form the design algorithm, which only has the computation complexity comparable with that of designing fixed finite impulse response digital filters. A design example is also given to justify the effectiveness and advantages of the proposed design method.

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.

DESIGN OF VARIABLE FRACTIONAL DELAY FIR FILTERS WITH CSD COEFFICIENTS USING GENETIC ALGORITHM

This paper presents a new method for the design of variable fractional delay (VFD) FIR digital filters using Genetic Algorithm (GA). Each sub-filter of Farrow structure is designed individually with defined accuracy and bandwidth. A variable mutation probability is also employed, which improves the accuracy of the solution. Compared with existing methods, it reduces the computational complexity and enhances the design flexibility. Furthermore, a simple GA is used to compute the filter coefficients in canonic singed digit (CSD) representations. Since this algorithm selects the initial population in CSD representation and searches for the digits between [-1 1], most of the digits become zero. The filter multipliers can be built with simple shift registers and adders, resulting in a high-speed and low-power filter implementation. Multiplier-free structure typically shows 11.4% saving in power and 17.2% reduction in area consumption.

AN ACCURATE FIR APPROXIMATION OF IDEAL FRACTIONAL DELAY FILTER WITH COMPLEX COEFFICIENTS IN HILBERT SPACE

This paper presents a low-order and accurate method for the design of FIR fractional delay (FD) filters with complex coefficients. This method employs least square technique in Hilbert space to approximate the ideal FD transfer function with a FIR filter and to calculate its coefficients. The main advantages of the resulting filter are: very good response at all frequencies compared to other FD filter design methods and a good method to create very small delay. Design examples are presented to illustrate the effectiveness of this new design approach.

Optimal synthesis of a fractional delay FIR filter in a reproducing kernel Hilbert space

Based on a bandlimited signal model, the optimal fractional delay finite impulse response (FIR) filter and corresponding interpolating error bound is derived in a reproducing kernel Hilbert space. The resulting optimal filtering is a projection onto a prescribed finite dimensional subspace. The connection of this filtering accuracy to the delay time and the filter order is investigated via error analysis.

Methods for the transfer function design of multiple narrow-band bandpass sigma–delta modulators

This paper presents the design analysis of novel tunable narrow-band bandpass sigma–delta modulators, which can achieve concurrent multiple noise-shaping for multi-tone input signals. Four different design methodologies based on the noise transfer functions of comb filters, slink filters, multi-notch filters and fractional delay comb filters are applied for the design of these multiple-band sigma–delta modulators. The latter approach utilises conventional comb filters in conjunction with FIR, or allpass IIR fractional delay filters, to deliver the desired nulls for the quantisation noise transfer function. Detailed simulation results show that FIR fractional delay comb filter-based sigma–delta modulators tune accurately to most centre frequencies, but suffer from degraded resolution at frequencies close to Nyquist. However, superior accuracies are obtained from their allpass IIR fractional delay counterpart at the expense of a slight shift in noise-shaping bands at very high frequencies. The merits and drawbacks of each technique for the various sigma–delta topologies are assessed in terms of in-band signal-to-noise ratios, accuracy of tunability and coefficient complexity for ease of implementation.

Variable Taylor realization and improved approximation of FIR fractional delay systems

A modular realization, based on the Taylor structure, is proposed for a maximally flat fractional delay FIR systems. The realization enjoys independence of the multiplier coefficients to changes of the system order, availability of closed-form formulas and recurrence for the multiplier coefficients, requirement of one less multiplier coefficient compared to the direct-form realizations, and existence of schemes for efficient variable delay and order implementations. It is shown that for a variable delay structure, the coefficient updating operation can be performed efficiently using a simple recurrence, and the transient effects can be mitigated to at most two samples. A modified form of the structure with enhanced modularity is also proposed for odd-order systems. Two variable delay schemes are proposed for this modified structure: A transient-free scheme and a scheme that provides to gradually improve the approximations during the transient phase. A method is also proposed for improved approximation under a fixed total number of variable multiplier coefficients. © 1998 John Wiley & Sons, Ltd.

Variable Taylor realization and improved approximation of FIR fractional delay systems

A modular realization, based on the Taylor structure, is proposed for a maximally flat fractional delay FIR systems. The realization enjoys independence of the multiplier coefficients to changes of the system order, availability of closed-form formulas and recurrence for the multiplier coefficients, requirement of one less multiplier coefficient compared to the direct-form realizations, and existence of schemes for efficient variable delay and order implementations. It is shown that for a variable delay structure, the coefficient updating operation can be performed efficiently using a simple recurrence, and the transient effects can be mitigated to at most two samples. A modified form of the structure with enhanced modularity is also proposed for odd-order systems. Two variable delay schemes are proposed for this modified structure: A transient-free scheme and a scheme that provides to gradually improve the approximations during the transient phase. A method is also proposed for improved approximation under a fixed total number of variable multiplier coefficients. © 1998 John Wiley & Sons, Ltd.

The maximally flat fir fractional sample delay filter revisited

The maximally flat fir fractional sample delay filter revisited