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Cascades of two-pole–two-zero asymmetric resonators are good models of peripheral auditory function
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Image of FIG. 1.
FIG. 1.

Diagram of the motion of the poles of a PZFC or CAR-FAC stage in response to a gain-control feedback signal, and the effect on the resonator gain. The positions indicated by crosses in the s plane plot (left) correspond to pole damping ratios (ζ) of 0.1, 0.2, and 0.3, while the zero’s damping ratio remains fixes at 0.1. Corresponding transfer function gains (right) of this asymmetric resonator stage do not change at low frequencies but vary by several decibels near the pole frequency. The fact that the stage gain comes back up after the dip has little effect in the transfer function of a cascade of such stages.

Image of FIG. 2.
FIG. 2.

Adaptation of the overall filterbank response at each output tap. (Top) The initial response of the filterbank before adaptation. (Bottom) The response after adaptation to a human/a/vowel of 0.6 s duration. The plots show that the adaptation affects the peak gains (the upper envelope of the filter curves shown), while the tails, behaving linearly, remain fixed.

Image of FIG. 3.
FIG. 3.

Schematic of the CAR-FAC design. The cascaded filter stages (upper row) have variable peak gains, which are controlled by their damping ratios, set by feedback from the coupled AGC filters (lower row). The “control” signals can be fast-acting in response to an onset, but usually vary slowly. In the case of quasi-linear PZFC filter models, the control values are static but level-dependent.

Image of FIG. 4.
FIG. 4.

The “asymmetric notched noise” masking paradigm, and data from human listeners, were introduced with this figure that explains the significant shifts between the filter with best SNR and the filter with CF at the probe-tone frequency (Patterson and Nimmo-Smith, 1980). In each example, the filter with best probe-tone-to-masking-noise ratio in its output (solid curve) is near the filter with highest probe-tone output power (dashed curve, filter with peak at probe-tone frequency f 0) but shifted in the direction that reduces the noise power output (generally toward a point slightly to the right of the center of the notch).

Image of FIG. 5.
FIG. 5.

Parallel (top), cascade (middle), and feedback (bottom) structures for level-dependent auditory filter models. The PrlGC and CasGC models originally used the upper and middle structures as a way to achieve a controllable gain near the tip while keeping a stable low-frequency tail. In the case of the PrlGC model, following an older parallel roex structure, the adder is actually adding power levels (Unoki et al., 2006), not signals, so this model structure does not correspond to an actual filter.

Image of FIG. 6.
FIG. 6.

Threshold-prediction rms errors for various filter models, versus number of fitted parameters, on the combined dataset. The fit numbers are for reference only; different filter models are identified by different symbols, as shown in the legend. For each model type, only the fit with lowest error at each number of parameters is shown; the errors are monotonically decreasing, since adding a free parameter never increases the error. The PZFC5 variants (+), such as fit 625, are the PZFC modified to have the zeros move with level, parallel with the poles, as opposed to the original PZFC () for which the zeros are fixed.

Image of FIG. 7.
FIG. 7.

Auditory filter gain plots for the best of each of six model types. The frequency axes are on the ERB-rate scale. In each case, the curves represent filter gain when the tone detection thresholds are 30 dB (highest curves), 50 dB, and 70 dB (lowest curves). The curve spacing is related to the input–output compression: curves close together, as at 250 Hz, correspond to a response that is only slightly compressive, while curve tips 15 dB apart represent a 4:1 compressive response. The model ERBs range from approximately the nominal ERB to more than twice that.

Image of FIG. 8.
FIG. 8.

The two degenerate cases of the OZGF, the APGF (left) and the DAPGF (right), provide good fits with only 4 parameters (quadratic bandwidth, and a bandwidth-level- dependence coefficent). They differ from the better-fitting OZGFs (the ones with more parameters) in the low-frequency tails, especially in the differentiated case (the DAPGF, which has a zero at DC).

Image of FIG. 9.
FIG. 9.

The impulse responses for the 1 kHz channel of two versions of the PZFC, at three tone threshold levels. The large (off-scale) curves are for the noise level that leads to 30 dB SPL tone threshold, the medium (full-scale) curves for 50 dB, and the small curves for 70 dB. The PZFC5 variant is designed to have stable zero-crossing times; the difference is apparent in the plots.


Generic image for table

Acronyms for the different auditory filter models discussed are tabulated here for reference; they are ordered from simplest to most complex, or number of fitted parameters required, roughly.

Generic image for table

A PZFC model with 9 filter parameters (fit 530); the channel density is fixed at 2 and not counted. The pole damping b 2 is computed from the CF-dependent B 2 as modified by the output power level (in dB) times B 2 1. In this version of the model, the zeros do not move with level.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Cascades of two-pole–two-zero asymmetric resonators are good models of peripheral auditory function