Closed-Form Design Of Variable Fractional-Delay Fir Filters With Low Or Middle Cutoff Frequencies

The variable fractional delay technique has been widely used in a lot of fields such as sampling rate conversion,filter design,radar,multi-carrier communication,software radio,Image interpolation,array signal processing,speech processing and so on,since it has flexible fractional delay value.The most classical filtering structure in the design of variable fractional delay filters is Farrow structure.Once the Farrow structure is selected,the subfilter's coefficients configuration will become the key problem.Therefore,it is an urgent task to develop an efficient coefficient configurating method that can ensure an excellent transfer characteristic of the variable fractional delay filter.This paper will introduce the all-phase filter design method to implement the subfilter's coefficients configuration of the Farrow structure.The main work focus on enhancing both filter's transfer characteristics and design efficiency.The coefficients' closed-form formula of the all-phase FIR filter provides a straightforward and effective way of fast subfilter's coefficients configuration of the Farrow structure.Based on the absorption of the existing literatures,this paper deeply explored the theoretical and property of the all-phase digital filter.Further,a new efficient design method for FIR filters is proposed.This method is based on the natural optimization,and it doesn't involved any iteration or matrix inversion operations,so the computational complexity and hardware cost is very low.What's more,the method uses cyclic-shift technique,on the basis of cyclic shift graph,a novel three-steps design method is summarized,then the corresponding closed-form formula of all-phase FIR filter can be derived,which provides a technical support for the filter's efficient design,fast coefficients configuration and engineering application.Based on the full analysis of theoretical properties and design methods of all-phase FIR digital filters,the all-phase FIR filter analysis design,the cubic spline interpolation and the Taylor series expansion technology are organic combined in this paper,thereby deriving a fast configuration procedure for all tap coefficients in the Farrow structure.Meanwhile,both the fractional delay and the cutoff frequency can be flexibly adjusted by this procedure.Then analytical expressions for the sub filter's coefficients of Farrow structure under two different delay conditions are derived respectively.In the future design,once the corresponding parameters(such as fractional delay,filter length,cutoff frequency,etc.)are substituted into the analytic configuration formula,the required sub filter's coefficients can be immediately calculated.Hence,the design of variable fractional delay filter can be realized in a efficient and analytical way.Finally,the computational complexity and transmission performance of the variable fractional delay filter are analyzed in this paper,both the amplitude-frequency curve and phase characteristic curve can be directly observed by a series of experimental simulation.Compared the delay error and time complexity of our proposed design with the simultaneously variable WLS design,It can be concluded that our proposed design is an efficient closed-form design method for designing variable fractional delay filters with medium or low cut-off frequencies.