Rational-Exponent Filters with Applications to Generalized Exponent Filters

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.