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 Exponent Filters (GEFs) that are useful for a diverse array of applications. We present equivalent representations for rational-exponent GEFs 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 without additional complexity in causality and stability analyses compared with classical filters. In the case of GEFs, 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：本文提出具有有理数指数的滤波器，以提供传统方法无法实现的连续滤波行为。我们讨论了其稳定性、所提供的灵活性，以及适用于分析、设计和实现的各种表示形式。这项工作针对二阶滤波器的广义化形式展开，我们称之为有理数指数广义指数滤波器（GEFs），其在多种应用场景中具有实用价值。我们给出了有理数指数GEF在时域和频域的等效表示：传递函数、脉冲响应和积分表达式——其中最后一种表示方式无需预处理即可实现高效的实时处理。有理数指数滤波器使滤波特性能够连续变化，而非局限于离散值，从而在不增加因果性和稳定性分析复杂度的前提下，相比传统滤波器获得了更大的行为灵活性。对于GEF而言，这使得滤波特性参数可以取任意连续值而非离散值，例如：(1) 3dB品质因数与最大群延迟的比值——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品质因数比值。