The all-pole gammatone filter and auditory models

The All-Pole Gammatone Filter (APGF) is defined by discarding the zeros from a pole-zero decomposition of the Gamma-Tone Filter (GTF) that was popularized in auditory modeling by Johannesma, de Boer, Patterson, and others. Equivalently, the orderN APGF is the Nth power of a filter with a complex-conjugate pair of poles; the GTF has this same set of poles, but in addition has “spurious” zeros on the real axis that complicate its description and behavior. The One-Zero Gammatone Filter (OZGF) is also introduced, by differentiating, or adding one zero at DC to, the APGF. Order-3 GTF and OZGF were originally used by Flanagan in 1960 to model Basilar Membrane motion. The APGF does not have the simple time-domain description of the GTF (a gamma distribution times a tone), but is simpler and more well-behaved in other ways, and provides a more robust foundation for modeling and analyzing a variety of auditory data that involve filter asymmetry, frequencies well below CF, level dependence, etc. The GTF magnitude frequency response is approximately symmetric, while the APGF frequency response is inherently asymmetric, being exactly symmetric in frequency-squared space. The lowfrequency tail of the APGF is unaffected by the bandwidth parameter, unlike the awkward behavior of the GTF, making the APGF suitable for a range of nonlinear and parametric applications not well addressed by the GTF. Fixing the low-frequency tail gain, rather than the peak gain, as parameters are varied with level, allows better answers to questions about how the auditory system behaves across levels, emphasizing approximate linearity at very low frequencies; varying the damping parameter of the APGF poles can then be interpreted 2 in terms of automatic gain control (AGC) near CF. The APGF has a simple efficient implementation, a cascade of N identical two-pole filter stages, and is very closely related to the “system of nonlinear differential equations” of Kim et al. and to the even more efficient cascade filterbank auditory models of Lyon and Mead. The APGF provides an improved time-domain match to Basilar Membrane mechanical impulse response measurements, with initial zero-crossing intervals being stretched out. The APGF is offered as a link between physiological, psychological, and mechanical models of auditory filtering. Introduction The Gamma-Tone Filter (GTF), introduced by Johannesma to describe cochlear nucleus response in 1972 [J72] and previously used as a basilar membrane model by Flanagan in 1960 [F60], has been adopted as the basis of a number of successful auditory modeling efforts [F62a, S85, P94, MH91, AS89]. Besides its catchy name, due to Aertsen and Johannesma [AJ80], three factors account for the success and popularity of the GTF: • It provides an appropriately shaped “pseudo-resonant” frequency transfer function with a simple parameterization, making it easy to reasonably well match measured responses • It has a very simple description in terms of its time-domain impulse response—a gamma-tone: a gamma distribution times a sinusoidal tone • It provides the possibility of an efficient digital or analog filter implementation. Two key limitations of the GTF are: • It is inherently nearly symmetric, while measurements show a significant asymmetry in the auditory filter • It is not easy to use the parameterization of the GTF to realistically model leveldependent changes in the auditory filter We introduce a close relative of the GTF, which we call the All-Pole Gammatone Filter (APGF) to highlight its similarity to and distinction from the GTF (and in the choice of acronym, we specifically demote the Tone). The APGF can be defined by discarding the zeros from a pole-zero decomposition of the GTF—all that remains is a complex conjugate pair of Nth -order poles. We argue that the zeros of the GTF are “spurious,” and that they limit the usefulness of the GTF in modeling low-frequency tails, impulse response chirping, and level-dependent nonlinearities such as bandwidth, gain, and delay variation. 3 The APGF was originally introduced by Slaney [S93] as an “All-Pole Gammatone Approximation,” an efficient approximate implementation of the GTF, rather than as an important filter in its own right. In this paper we also introduce the One-Zero Gammatone Filter (OZGF), a better approximation to the GTF which inherits all the advantages of the APGF. The OZGF is defined by differentiating the APGF to introduce one zero at DC (i.e., at s = 0 in the Laplace domain). An order-3 OZGF was first used to model Basilar Membrane motion by Flanagan [F60], as an alternative to the order-3 GTF. See the section “History of Gamma-Tone Development” for more details. The APGF and OZGF have several properties that make them particularly attractive for applications in auditory modeling: • They exhibit a realistic asymmetry in the frequency domain, providing a potentially better match to psychoacoustic data than the GTF • They have a simpler parameterization than the GTF and its asymmetric variants • With a single level-dependent parameter, they exhibit reasonable bandwidth and center frequency variation, while maintaining a linear low-frequency tail • They are even more efficiently implemented than the GTF • They provide a logical link to filter cascade models of the underlying cochlear mechanics These points are supported in the paper by way of examples, illustrations, and mathematical derivations. The OZGF is proposed as the better filter for most auditory filter applications, due to its sloping tail, but the closely related APGF is explored in more depth due to its analytic simplicity. In the short main body of the paper we define the APGF and OZGF, and detail the history of the GTF, especially in relationship to the properties of auditory filters that have not been well modeled to date. Supporting analysis is all deferred to the appendices, as we did not want to discourage the general reader. Defining the All-Pole Gammatone Filter We define the APGF in terms of its Laplace transform H(s) . For conciseness, let the s -plane pole positions be given by the complex number p and its conjugate p. Then the APGF is given by: 4 H(s) = K [(s − p)(s − p)] where K is a constant that will normally be adjusted to give unity gain at DC: H(0) = 1. In terms of the Cartesian parameterization of complex pole position, p = −b + iωr , and using the denominator form used by Flanagan [F60], the APGF becomes H(s) = K [(s + b) + ωr 2 ] with K = b + ωr 2 The OZGF adds “one zero” at DC, a differentiator, or a factor of s (some other rationale for choosing K is needed, since the DC gain is 0 ): H (s) = sK [(s + b) + ωr 2 ] At very low frequencies, the APGF gain is nearly constant (near unity), so we say it has a “flat” low-frequency tail. The gain of the high-frequency tail, by contrast, falls very rapidly with frequency. The OZGF, due to the differentiator, has zero gain at D.C. and a slope of 6 dB per octave at very low frequencies, so is said to have a “sloping” low-frequency tail. Its passband and high-frequency tail are very much like the APGF’s. The filter shape, gain, bandwidth, delay, dispersion, impulse response, etc. of the APGF and OZGF are analyzed, illustrated, and compared with the GTF in the appendices, starting with and building on the complete analysis of the N = 1 degenerate case. In addition to the Cartesian pole position parameterization mentioned here, we introduce several other parameterizations that are about equally simple; some parameterizations are easier than others for certain analyses. There are several reasonable options for how to couple a single parameter to be level dependent to extend the filter models to include nonlinear behaviors. In the Cartesian parameterization, for example, the b parameter, or real part of the pole position, related to bandwidth, could depend on level. Figure 1 shows a range of shapes of the GTF, APGF, and OZGF in the frequency domain as the b parameter is varied, while Figure 2 shows corresponding impulse responses. In the appendices we discuss a “theta parameterization” that has several advantages over the Cartesian parameterization, including relating the value of θ conceptually to the degree 5 of active amplification due to outer hair cells. Dropping θ to zero changes the APGF from a band-pass active filter to a low-pass passive filter. 0.1 1.0 -40 -30 -20 -10 0 10 20 30 Magnitude Gain (dB)