Variable Fractional Delay Filter 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.

A Variable Fractional Delay Filter Based True-Time-Delay Array Architecture for Wideband Beamforming

True-time delay (TTD) units are critical in Wideband Large-Scale Antenna Systems (LSASs) for resolving beam squint and frequency selective fading. In this paper, we propose a TTD array architecture for wideband beamforming that addresses the challenges posed by LSASs by utilizing variable fractional delay (VFD) filters and time delay estimations (TDEs). This study introduces a convex optimization problem to develop a novel method for designing VFD filters that can achieve high precision time delay resolution while maintaining an acceptable frequency response peak error. Using these VFD filters, we present a TDE approach with Newton-Raphson iteration update that corrects the difference in arrival time between sensors. Due to the implementation of VFD filters and fast TDE modules, the new architecture can complete beam steering and tracking in 10 ms with MATLAB simulation. Furthermore, an example was designed to demonstrate that the proposed architecture is capable of producing precise beam pointing while maintaining the bit error rate (BER) that is extremely close to the theoretical curve over the entire bandwidth.

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.

Sparsity‐optimised farrow structure variable fractional delay filter for wideband array

AbstractIn this paper, a new sparsity‐optimised Farrow structure variable fractional delay (SFS‐VFD) filter is proposed to address the aperture effect in wideband array. Our method is based on coefficient (anti‐)symmetry and optimises the number and orders of its sub‐filters, greatly reducing the non‐zero coefficients. The established cost function is formulated as a parametric minimisation problem with multiple regularisation constraints, and solved by the modified three‐block alternating direction multiplier method (MTB‐ADMM), which is improved by introducing core variable correction items to ensure stable and fast convergence. Experimental results show that the SFS‐VFD filter reduces the complexity of the system by decreasing the use of multipliers and adders while ensuring high delay accuracy. In wideband array, the SFS‐VFD filter effectively corrects the aperture effect and achieves precise beam pointing.

Efficient Digital Implementation of a Multirate-Based Variable Fractional Delay Filter for Wideband Beamforming

In this brief, we propose a FPGA implementation of a novel Variable Fractional Delay (VFD) filter architecture for digital wideband (hundred of MHz or more) beamforming applications. The proposed architecture is based on a multirate structure that allows reaching high performance in terms of amplitude and group delay with a significant reduction in design complexity. Usually, the antenna’s beams are modified by introducing precise delays that can be multiples or fractions of the sampling period. The most used structures for realizing this operation are Farrow filters but, for wideband applications, the classical Farrow structure may require a huge amount of resources. A common solution to simplify the Farrow filter architecture is interpolation. However, this solution requires an increase of the clock frequency which is not always possible. In this context, our multirate approach mitigates the limitations in terms of computational complexity avoiding the requirement to increase of the clock frequency, while still exhibiting great performance. A comparison between a reference design based on the proposed approach and the literature is also provided. The experimental results confirm that the proposed architecture, characterized by a computational complexity similar to other works, exhibits better performance without the need to increase the clock frequency.

High‐speed, high‐resolution methodology for portable universal radar target‐echo simulator

AbstractIn this paper, digital parallelization and fractional delay‐based novel methods are proposed for the realization of high bandwidth, high‐resolution Ku‐band radar target simulator with 2.5 GHz intermediate frequency. High bandwidth waveform from radar is sampled by high‐speed Analog to Digital Converter, and samples are parallelized in Field Programmable Gate Array (FPGA) to work at the nominal clock frequency. In Digital RF Memory‐based target simulator, for finer range resolution, the FPGA clock frequency needs to be increased, which leads to increased system design complexity. The finer range resolution is accomplished without altering the system clock frequency using variable fractional delay filters and the digital parallelization methodology is proposed in this paper. The maximum target range that can be simulated is 20 km. As a result, memory requirements, computational complexity, and power dissipation are reduced. Finally, simulation and implementation results are presented.

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.

Parallel simulation method of GNSS multipath signal based on GPU

In order to solve the problems of the traditional GNSS multipath signal simulation methods, such as the complexity increases linearly with the number of multipath signals, inconvenient parameter updating and long operation time, a GPU based GNSS multipath signal parallel simulation method is proposed. Based on Farrow variable fractional delay filter, an independent parallel computing model of multi-channel filter coefficients is established, and a large number of floating-point units in GPU are used for parallel computing and filtering. Aiming at the performance bottleneck of this method, the optimization strategy of asynchronous parallel execution is given. The computer simulation results show that the complexity and operation time of the proposed method will not increase with the increase of the number of multipath signals. For the GNSS direct signal with sampling rate of 75MHz and time length of 1s, the maximum time consuming of the proposed method to generate 50 ∼ 1000 multipath signals is 158.6ms. Compared with the method of serial simulation on CPU, the proposed method speeds up to about 9.5 times, and the real-time simulation of multipath signals in complex environment can be realized.

Multisegment optimization design of variable fractional-delay FIR filters

This paper presents a multisegment optimization design algorithm for both even- and odd-order variable fractional-delay filters. The proposed algorithm is based on segmenting the fractional delay and separately solving the filter coefficients in each segment. Moreover, the lower bound of the polynomial order and the upper bound of the subfilter design error are derived. These bounds enable us to prove that the proposed algorithm effectively reduces the implementation complexity at the expense of storage resources. Hence, a joint search algorithm is proposed to find the lowest polynomial orders and subfilter orders that meet the filter specifications. Simulations and comparisons are conducted to illustrate the superiority of the proposed algorithm in terms of the implementation complexity over existing methods under common specifications.

Fractional Repetitive Control of Nanopositioning Stages for High-Speed Scanning Using Low-Pass FIR Variable Fractional Delay Filter

The repetitive control (RC), capable of tracking periodic trajectories and rejecting periodic disturbances, is a promising technique to control the nanopositioning stages for high-speed raster scanning. In digital implementation of the RC scheme, the number of delay points has to be an integer, which implies that the sampling frequency should be an integer multiple of the desired tracking frequency. Clearly, this is a severe limitation on the range of the trajectory frequencies where the RC scheme can effectively be applied. To overcome this limitation, this article proposes a low-pass FIR variable fractional delay filter as an alternative to the conventional interpolating method employed in conventional fractional RC scheme. This filter combines the interpolating and the low-pass filtering that are employed in the fractional RC and its coefficients are analytically computed as a function of fractional delay; thereby, making it suitable for trajectories of all frequencies. The weighted-least-square method is employed to design the low-pass FIR variable fractional delay filter, where the weights are tuned to minimize the approximation errors within the bandwidth-of-interested. Experimental results are presented to demonstrate the advantages of the proposed method over the conventional RC scheme as well as the interpolating-based fractional RC scheme. These results show that the proposed RC scheme with low-pass FIR variable fractional delay filter improves the tracking performance of the nanopositioner significantly, especially for the trajectories with high-frequency.

Design of Variable Fractional Delay Filter using FIR Filter Approximation

Beamforming plays an important role in the field of wireless communication. Beamforming means combination of a radio frequency (RF) signals from multiple antennas to form a single direction beam. This technique improves the quality of communication and reduces the interference of signal. In beam forming technique, different phases signals can be achieved with different signals and the received phase delay signals are converted into same phase, multiply with weight factor and combined this signals to form a beam in desired direction. The required phase delays are generated by using a Variable fractional delay filter. Variable fractional delay filter is design by using a direct form of a FIR filter structure. Variable fractional delay filter is calculated by two different phase signals from digital antennas and those two different phase signals are converted to in- phase and added together to form a beam forming. As the order of the filter increases, the delay also increases. The filter coefficients of the variable fractional delay filter are calculated my using a Lagrange interpolation method. The variable fractional delay filter is designed by using software Xilinx version 14.3.

Design of Variable Fractional Delay Filter using FIR Filter Approximation

Beamforming plays an important role in the field of wireless communication. Beamforming means combination of a radio frequency (RF) signals from multiple antennas to form a single direction beam. This technique improves the quality of communication and reduces the interference of signal. In beam forming technique, different phases signals can be achieved with different signals and the received phase delay signals are converted into same phase, multiply with weight factor and combined this signals to form a beam in desired direction. The required phase delays are generated by using a Variable fractional delay filter. Variable fractional delay filter is design by using a direct form of a FIR filter structure. Variable fractional delay filter is calculated by two different phase signals from digital antennas and those two different phase signals are converted to in- phase and added together to form a beam forming. As the order of the filter increases, the delay also increases. The filter coefficients of the variable fractional delay filter are calculated my using a Lagrange interpolation method. The variable fractional delay filter is designed by using software Xilinx version 14.3

Convergence of the Newton Structure Transfer Function to the Ideal Fractional Delay Filter

This letter presents a rigorous demonstration of the convergence of the Newton fractional delay filter to the ideal fractional delay filter. The Newton structure is a very efficient implementation of Lagrange interpolation using a variable fractional delay filter structure. Through the developed demonstration, a new approach is proposed to define the ideal fractional delay filter as the limit of any digital filter implementing on Lagrange interpolation. This letter also proves that the Z -transform expression of the ideal fractional delay has a fully defined frequency response on the unit circle.

Maximally Flat and Least-Square Co-Design of Variable Fractional Delay Filters for Wideband Software-Defined Radio

This paper describes improvements in a Farrow-structured variable fractional delay (FD) Lagrange filter for all-pass FD interpolation. The main idea is to integrate the truncated sinc into the Farrow structure of a Lagrange filter, in order that a superior FD approximation in the least-square sense can be achieved. Its primary advantages are the lower level of mean-square-error (MSE) over the whole FD range and the reduced implementation cost. Extra design parameters are introduced for making the trade-off between MSE and maximal flatness under different design requirements. Design examples are included, illustrating an MSE reduction of [Formula: see text] compared to a classical Farrow-structured Lagrange interpolator while the implementation cost is reduced. This improved variable FD interpolation system is suitable for many applications, such as sample rate conversion, digital beamforming and timing synchronization in wideband software-defined radio (SDR) communications.

A stability controlling-based approach for designing 1-D variable fractional delay all-pass filters

PurposeVariable fractional delay filtering is an important technology in signal processing; the research shows that all-pass variable fractional delay (VFD) filters achieve higher design accuracy than FIR VFD filters; therefore, the design, analysis and implementation of all-pass VFD filters are of great importance.Design/methodology/approachIn this paper, a two-stage approach for the design of general 1-D stable VFD all-pass filters is proposed. The method takes the desired group delay range [N−1, N], where N is the filter order.FindingsThe design algorithm is decomposed into two design stages: first, a set of fixed delay all-pass filters are designed by minimizing a set of objective functions defined in terms of approximating error criterion and filter stability constraint. Then, the design result is determined by fitting each of the fixed delay all-pass filter coefficients as 1-D polynomials. A design example together with its comparisons with those of the recent literature studies is given to justify the effectiveness of the proposed design method.Originality/valueAn illustrating design example shows that the method proposed can achieve better filter performances than the existing ones.

WLS optimal design for variable FIR filters with continuous parameter of fractional delay

A spectral parameterization method with discretization parameters will meet a tradeoff between the design accuracy and computational effort when designing a variable digital filter. Thus, a continuous spectral parameterization method is presented for obtaining the WLS optimal coefficients of variable fractional delay filters. Separated non-negative weighting function and some characters of matrix and matrix operations are applied to derive a closed-form solution, and Cholesky factorization is adopted to avoid the numerical problem of computing the inverses of a matrix. Its high accuracy and considerably reduced computational complexity of the proposed algorithm are experimentally verified. The given FIR filters can found their applications in high accuracy dynamic time compensation.

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.

Wideband transmit beamforming using integer‐time‐delayed and phase‐shifted waveforms

A method for wideband transmit beamforming using integer-time-delayed and phase-shifted waveforms is proposed. Unlike traditional true-time-delayed digital methods, which employ variable fractional delay filters or discrete Fourier transforms to create fractional time delays, the proposed method uses only integer-time-delayed waveforms with phase shifts corresponding to fractional time delays. Furthermore, the shifted phase can be directly generated; this generation can be simply achieved by controlling the parameters of a direct digital synthesiser. A beam pattern simulation confirms that the integer-time-delayed and phase-shifted waveforms are capable of synthesising wideband transmitting beam patterns. Finally, a power synthesis efficiency analysis demonstrates that the synthesis loss caused by phase shifting is negligible.

Coefficient-relation development and low-complexity odd-order variable-fractional-delay filter design

Variable fractional-delay (VFD) filter may have either even or odd order. This paper focuses on the development of a useful coefficient-dependence relation that exists in the odd-order VFD (OO-VFD) filter, and exploits it in designing the minimax OO-VFD filter. The design problem is first described as a quadratic cone (Qcone) programming one, and then solved by applying a Qcone programming solver. Since this coefficient-dependence relation reveals the mutual dependence between the coefficients of each subfilter pair, only nearly half of the coefficients are independent when the paired subfilters are of the same order. To further reduce the filter complexity, we adopt an optimization strategy to optimize the filter orders and make the orders unequal, which leads to a further simplified filter structure. After optimizing the unequal orders, we optimize the independent coefficients to suppress the peak error of the filter׳s frequency response. We will use an OO-VFD filter example to show that the coefficient-dependence exploitation significantly reduces the design complexity and simplifies the implementation structure.

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.