Rational-Exponent Filters with Applications to Generalized Exponent Filters

TL;DR

This work introduces rational-exponent Generalized Exponent Filters (GEFs) by raising a base second-order all-pole filter to a rational power $B_u$, yielding a continuum of filter behaviors unattainable with integer exponents. It derives and analyzes multiple representations—transfer functions, impulse responses, ODEs/state-space, and integral expressions—covering both integer and rational (including half-integer) $B_u$, and demonstrates stability and causality inherited from the base filter. The paper highlights the practical advantages of these representations for design, implementation, and real-time processing, and shows how non-integer exponents provide richer tuning of peak frequency, quality factors, and group delay, with near-term applicability to cochlear-inspired and filterbank systems. It also supplies approximation strategies (e.g., extrapolated GTFs) and discusses future directions including extensions to other base filters and hardware realizations, supported by open-source code. Overall, rational-exponent GEFs offer a flexible, realizable framework for advanced signal processing in auditory-like and other all-pole filter applications, while preserving straightforward stability analysis.

Abstract

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.

Figures (9)

• Figure 1: Allowing for non-integer $B_u$ enables greater variety of behavior: Figure shows Bode plots of GEFs with integer and half-integer $B_u$. The two sets of figures are generated with different values for the filter constant $A_p$ and different ranges for $B_u$. The left set of figures is more sensitive to changes in $B_u$ in non-integer increments, illustrating that the flexibility in signal processing behavior afforded by relieving the integer constraint depends on the values of $A_p$ and $B_u$. In these plots, the magnitude (in dB) is plotted in reference to the magnitude at the peak, and the phase (in cycles) is in reference to the first phase shown.
• Figure 2: Allowing for rational, rather than only integer, $B_u$ provides greater flexibility in behavior: The figures show the continuum of behavior (quantified by TF characteristics) that may be accessed once we allow for rational-$B_u$(lines), as opposed to the discrete behavior accessible via the restricted integer-$B_u$ cases (markers). The left figure shows the continuum of behavior as a function of (practically) continuous $B_u$ for simple filter characteristics ($N, Q_{erb}, Q_{10}, Q_3, Q_{15}$) while holding $A_p$ constant. The right figure shows the continuum of behavior seen from the perspective of compound characteristics that practically only depend on $B_u$ ($\frac{Q_{erb}}{N}, \frac{Q_{erb}}{Q_{10}}, \frac{Q_3}{Q_{15}}$ ).
• Figure 3: Software implementations of the non-integer-$B_u$ TF representation can be used to process signals: The output (bottom panel) of a filter with peak frequency $\text{CF}(x) =$ 2 kHz is computed using a software implementation of the TF representation using a non-integer-$B_u$ value of $2.5$. The output is generated in response to a input signal (top panel) which is composed of tone pips and is of the form $v_{stapes} = \sum\limits_{i=0}^3 e^{-(t-t_i)^2/T^2} sin(2\pi f_i t)$, where $T=5, f_0 = \textrm{CF}, t_0 = 20, f_1 = 5 \textrm{CF}, t_1 = 50, f_2 = \frac{7}{8}\textrm{CF}, t_2 = 70, f_3 = \frac{1}{5}\textrm{CF}, t_3 = 40$ with frequency in kHz, and time in ms. As expected, the filter responds maximally to frequency components closest to its $\text{CF}(x)$. This figure illustrates realizability of software implementations using the TF representation with non-integer-$B_u$.
• Figure 4: Normalized impulse responses as a function of normalized time $\mathbf{\tilde{t} = 2\pi \text{CF}(x) t}$ with $\mathbf{b_p=1}$. Impulse responses are plotted as normalized to the maximum absolute value. Top: shows the dependence of the impulse response (table \ref{['tab:ImpulseResponseIntBuExpression']}) behavior on the filter constants. For instance, delay increases with increasing $B_u$ and decreasing $A_p$. The figure is generated by varying $A_p$ and integer$B_u$ (which limits the oscillatory component of $h(\tilde{t})$ to a sin or cos). Bottom: shows that $h(\tilde{t}$) for integer and half-integer $B_u$ (table \ref{['tab:ImpulseResponseHalfIntBuExpression']} or equation \ref{['eq:hIntegerHalfinteger']}) allows for greater control and variation of the phase of the oscillatory component rather than it being restricted to sin and cos as is the case for integer $B_u$.
• Figure 5: Comparison between the exact expression for impulse responses and its gammatone approximation: For various values of filter constants, the figure shows the comparison between $h(\tilde{t})$ (solid blue lines) as a function of normalized time $\tilde{t} = 2\pi \text{CF}(x) t$ with $b_p=1$ and its highest order term approximation, $h_{GTF}$ of equation \ref{['eq:hGTFapprox']} (dotted red lines), which is simply a single gammatone filter (GTF) extrapolated to allow for half-integer $B_u$ and with a tonal component $\cos(\tilde{t}_b - B_u \frac{\pi}{2})$. The top figure panels is for integer $B_u$, and the bottom panels are for half-integer-$B_u$. As demonstrated in the figures, the impulse response is well-approximated by the highest-order term for small $A_p$ except at the very earliest times.