分数阶线性相位滤波器的设计与实现

Design and realization of fractional⁃order linear phase filter

文中讨论了一种基于内点算法的分数阶线性相位滤波器的设计方法。从分数阶全极点低通滤波器传递函数的一般形式出发,推导了(1+α)阶和(2+β)阶线性相位滤波器传递函数的合理结构,进而对分数阶滤波器的相频特性优化问题进行建模和求解,得出了一种具有线性相位特性的分数阶滤波器,并对滤波器进行了稳定性分析和频域分析。结果表明所分析的滤波器均稳定并且有良好的相频特性,此外相比于整数阶滤波器,分数阶滤波器群延时的步阶更为精细。基于分数阶电容还设计了(1+α)阶和(2+β)阶滤波器的有源电路,其中分数阶电容用Foster-I型RC网络进行近似。最后对(1+α)阶和(2+β)阶滤波器有源电路进行了实测,测试结果与电路理论上的幅相频率特性之间的误差在可接受范围内。

A design method of fractional⁃order linear phase filter based on interior point algorithmis discussed in this paper.Proceeding from the general form of the transfer function of fractional order all pole low⁃pass filter,the reasonable structures of the transfer functions of(1+α)⁃order and(2+β)⁃order linear phase filters are derived.Then,the phase frequency characteristic optimization problem of fractional order filter is modeled and solved,and a fractional⁃order filter with linear phase characteristic is obtained.The stability analysis and frequency domain analysis of the filter are carried out.The results show that the analyzed filters are stable and have good phase frequency characteristics.In addition,compared with the integer⁃order filter,the group delay step of thefractional order filter is more precise.The active circuits of the(1+α)⁃order and(2+β)⁃order filters are also designed on the basis of fractional order capacitor,in which the fractional order capacitor is approximated by the Foster⁃I RC network.The active circuits of the(1+α)⁃order and(2+β)⁃order filters were tested.The error between the tested results and the amplitude phase frequency characteristics in the circuit theory is acceptable.