Variable Fractional Delay FIR Filters Research Articles

Efficient Aperture Fill Time Correction for Wideband Sparse Array Using Improved Variable Fractional Delay Filters.

To solve the problem of aperture fill time (AFT) for wideband sparse arrays, variable fractional delay (VFD) FIR filters are applied to eliminate linear coupling between spatial and time domains. However, the large dimensions of the filter coefficient matrix result in high system complexity. To alleviate the computational burden of solving VFD filter coefficients, a novel multi-regultion minimax (MRMM) model utilizing the sparse representation technique has been presented. The error function is constrained by the introduction of L2-norm and L1-norm regularizations within the minimax criterion. The L2-norm effectively resolves the problems of overfitting and non-unique solutions that arise in the sparse optimization of traditional minimax (MM) models. Meanwhile, the use of multiple L1-norms enables the optimal design of the smallest sub-filter number and order of the VFD filter. To solve the established nonconvex model, an improved sequential-alternating direction method of multipliers (S-ADMM) algorithm for filter coefficients is proposed, which utilizes sequential alternation to iteratively update multiple soft-thresholding problems. The experimental results show that the optimized VFD filter reduces system complexity significantly and corrects AFT effectively in a wideband sparse array.

Design of Even-Order Variable Fractional Delay FIR Filters based on Pivot Element Weighting Iterative Algorithm

Design of Even-Order Variable Fractional Delay FIR Filters based on Pivot Element Weighting Iterative Algorithm

A Complex Exponential Structure for Low-Complexity Variable Fractional Delay FIR Filters

Variable fractional delay (VFD) filters generally require significantly greater computational resources for implementation than static filters. The motivation of this work is therefore to develop ways for implementation complexity reduction. The Farrow structure is adopted by most of the variable fractional delay (VFD) filters due to their effectiveness. This structure essentially assumes an algebraic polynomial approximation to the continuously varying impulse response. In this paper, we dispense with the polynomial function and instead propose the use of complex exponential functions for approximation. This new approach leads to a Farrow-like implementation structure with complex coefficient subfilters and a shape parameter. The complex exponential (CE) VFD filter is analyzed, and various types of symmetry properties are derived. Accordingly, a simplified implementation structure is obtained. The complexity analysis shows that the design and implementation complexities can be reduced to something comparable to the classical real algebraic polynomial (AP) VFD filters with the same number of subfilters and filter order. However, the CE filters can achieve a better approximation accuracy to the desired fractional delay characteristics, compared to the AP filters. Moreover, this superiority becomes significant, when only a few subfilters are used, or when the filter orders are high. Therefore, the reduction in complexity can be achieved for the same design specification. The design algorithm is developed using a weighted least-squares formulation. Using a matrix-based approach, the closed-form solution is derived firstly for a fixed shape parameter. This is then followed by optimization of the shape parameter using the Golden Cut method. Comprehensive design examples will be presented that compare the CE filters with the AP filters.

Matrix-Based Algorithms for the Optimal Design of Variable Fractional Delay FIR Filters

This paper investigates the weighted least squares (WLS) and minimax design problems for variable fractional delay (VFD) FIR filters. For the WLS design, the general case of incorporating arbitrary nonnegative weighting functions is considered. The optimal solution is characterized by two matrix equations. An efficient algorithm using conjugate gradient (CG) techniques is proposed to solve the WLS solution. The proposed algorithm is guaranteed to converge to the optimal solution in a finite number of iterations. Moreover, an iterative reweighted least squares (IRLS) algorithm that uses the proposed CG algorithm as its iteration core is developed for the minimax design problem. In both of the algorithms, the filter coefficients are arranged as matrices, achieving a great saving in computations and memory space. The associated computational complexity is analyzed. Some design examples are provided and comparisons with existing methods show that the proposed ones are either computationally more efficient or can obtain the better filter performance, or both.

Digital Self-Interference Cancellation With Variable Fractional Delay FIR Filter for Full-Duplex Radios

In full-duplex radios, delay alignment errors result in random mismatch between the self-interference (SI) signal and the reconstructed cancellation signal, leading to the performance degradation of the SI cancellation. In this letter, a variable fractional delay (VFD) FIR filter with more precise delay alignment is exploited to suppress the SI in digital domain. The closed-form expressions of the VFD FIR filter coefficients are reformulated by minimizing the residual SI power. Based on the obtained SI cancellation capability, the influence of the filter order on the SI cancellation is then characterized. Simulations show that the deployment of the VFD FIR filter significantly enhances the SI cancellation performance. Specifically, when the SI signal bandwidths are 0.5 and 10 MHz, the cancellation capability of employing the VFD FIR filter based on the proposed coefficients outperforms the VFD FIR filter under the LS criterion by 6.5 and 3.7 dB, respectively, which presents a potential contribution to enhance the SI cancellation performance.

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 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.

A method for the design of Farrow-structure based variable fractional-delay FIR filters

This paper proposes a method to design variable fractional-delay (FD) filters using the Farrow structure. In the transfer function of the Farrow structure, different subfilters are weighted by different powers of the FD value. As both the FD value and its powers are smaller than 0.5, our proposed method uses them as diminishing weighting functions. The approximation error, for each subfilter, is then increased in proportion to the power of the FD value. This gives a new distribution for the orders of the Farrow subfilters which has not been utilized before. This paper also includes these diminishing weighting functions in the filter design so as to obtain their optimal values, iteratively. We consider subfilters of both even and odd orders. Examples illustrate our proposed method and comparisons, to various earlier designs, show a reduction of the arithmetic complexity.

A generalized approach to the design of variable fractional-delay FIR digital filters

In this paper, a complex-oriented weighted least-squares approach is proposed for the design of arbitrary variable fractional-delay FIR filters. The objective error function is formulated in a quadratic form, such that the filter coefficients can be obtained by proper matrix/vector operation, including matrix inversion. However, all elements of related matrices and vectors can be represented in closed forms, and the matrix to be inversed is a positive-definite Hermitian symmetric matrix, which can be decomposed by the Cholesky factorization, such that computation time can be effectively reduced and the ill condition can be avoided. Comparing with the existing methods, the proposed method can be applied to design arbitrary complex coefficient or real coefficient variable fractional-delay filters. Several examples are presented to demonstrate the flexibility and effectiveness of the proposed method.

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.

Eigenfilter approach for the design of variable fractional delay FIR and all-pass filters

An eigenfilter approach is presented for designing 1-D and 2-D variable fractional delay FIR and all-pass filters. First, the coefficients of filters are expressed as a polynomial of the fractional delay parameter. Then, the optimal polynomial coefficients are obtained from the elements of the eigenvector corresponding to the minimum eigenvalue of a real, symmetric and positive definite matrix. Finally, several design examples of 1-D and 2-D variable fractional delay FIR and all-pass filters are used to illustrate the effectiveness of the eigenfilter approach.

An improved weighted least-squares design for variable fractional delay FIR filters

Digital filters capable of changing their frequency response characteristics are often referred to as variable digital filters (VDFs) and have been found useful in a number of digital signal processing applications. An important class of VDFs is the class of digital filters with variable fractional delay. This paper describes an enhanced weighted least-squares design for variable-fractional-delay finite-impulse response filters, which offers improved performance of the filters obtained with considerably reduced computational complexity compared to a recently proposed weighted least-squares (WLS) design method. The design enhancement is achieved by deriving a closed-form formula for evaluating the WLS objective function. The formula facilitates accurate and efficient function evaluations as compared to summing up a large number of discrete terms, which would be time consuming and inevitably introduce additional errors into the design.