We present filters with rational exponents in order to provide a continuum of filter behavior not classically achievable. We discuss their stability, the flexibility they afford, and various representations useful for analysis, design and implementations. We do this for a generalization of second order filters which we refer to as rational-exponent Generalized Auditory Filters/Filterbanks (GAFs) that are useful for a diverse array of applications. We present equivalent representations for rational-order GAFs in the time and frequency domains: transfer functions, impulse responses, and integral expressions - the last of which allows for efficient real-time processing without preprocessing requirements. Rational-exponent filters enable filter characteristics to be on a continuum rather than limiting them to discrete values thereby resulting in greater flexibility in the behavior of these filters. In the case of GAFs, this allows for having arbitrary continuous rather than discrete values for filter characteristics such as (1) the ratio of 3dB quality factor to maximum group delay - particularly important for filterbanks which have simultaneous requirements on frequency selectivity and synchronization; and (2) the ratio of 3dB to 15dB quality factors that dictates the shape of the frequency response magnitude.

翻译：本文提出有理数指数滤波器，以提供传统方法无法实现的连续滤波器行为谱。我们讨论了其稳定性、所提供的灵活性，以及适用于分析、设计与实现的各种表示形式。这些研究基于对二阶滤波器的推广，我们称之为有理数指数广义听觉滤波器/滤波器组（GAFs），其在多种应用场景中具有实用价值。我们给出了有理数阶GAF在时域和频域的等效表示形式：传递函数、脉冲响应及积分表达式——其中最后一种形式允许无需预处理的实时高效处理。有理数指数滤波器使滤波器特性能够处于连续谱上，而非局限于离散值，从而显著增强了这些滤波器行为的灵活性。对于GAF而言，这使得滤波器特性（如以下两个关键参数）可采用任意连续值而非离散值：（1）3dB品质因数与最大群延迟之比——这对同时具有频率选择性和同步性要求的滤波器组尤为重要；（2）决定频率响应幅度形状的3dB与15dB品质因数之比。