设计非90°移相希尔伯特变换器的新方法

New method for designing fractional Hilbert transformer

提出了一种实现非90°移相希尔伯特变换器的新方法,基本思想是利用一个指定相位要求的全通滤波器来实现非90°移相器。众所周知,对于一个平稳的全通滤波器,其分母多项式必定是一个最小相位系统。根据最小相位系统的相位函数与其复倒谱序列之间的关系,该方法通过构造一个纯虚奇对称相位函数,同时利用傅里叶变换的共轭对称性,得到分母系数所对应的复倒谱序列的解析表达式;然后,依据复倒谱的基本理论,通过求解一个非线性递归差分方程得到分母多项式系数。给出了逼近理想特性和非理想特性的实现方法,设计实例表明该方法很好地近似了设计要求,是一种简单、高效、无限精度的实现方法。

A new method for designing Fractional Hilbert Transformer( FHT) was proposed. The basic idea is to realize the FHT to design the allpass filter with desired phase characteristic. It is well known that the denominator polynomial of a stable allpass filter must be a minimum phase system. By constructing a pure imaginary odd symmetry phase function, and using symmetry properties of the Fourier transform, this method could obtain the cepstral sequence of the denominator polynomial using the relationship between cepstral sequence and phase function of a minimum phase system. Then, from the cepstral spectrum theory, the denominator polynomial coefficients could be determined through a nonlinear recursive difference equation. Approximated ideal and non-ideal characteristic methods were given. Design examples indicate that the proposed filters exhibit good approximation to the desired phase response, and have the advantage of simple, efficient and infinite precision.