Terhardt, E. (2000). Linear model of peripheral-ear transduction (PET). In: Auditory Worlds - Sensory Analysis and Perception in Animals and Man (G.A. Manley, H. Fastl, M. Kssl, H. Oeckinghaus, G. Klump, eds.). Wiley-VCH, Weinheim, p. 81-89


A system of linear filters (Peripheral-Ear Transduction, PET) was designed for modeling auditory preprocessing of sound (cf. [103] ). The system consists of 1) a linear filter that accounts for the ear-canal resonances (ECR) at about 3.3 and 10 kHz; 2) a bank of linear filters each of which accounts for the transfer function from outer ear to a particular place of the cochlear partition (cochlear transfer function, CTF).

The ECR filter internally consists of two chained filters of the type

Hr(s) = 1+a s

s2+2as+a2+w02

,

where s = jw; a is a real constant, a a real damping coefficient, and w0 is the filter's eigenfrequency. For all frequencies the absolute magnitude of Hr is greater than 1 but approaches this value asymptotically on both sides of the resonance frequency.

For modeling cochlear transmission an appropriate type of filter was chosen, i.e. of the type

Hn(s) = 
  sns*n

(s-sn)(s-s*n)

 
k
 
.

Here the index n denotes a particular cochlear channel and CTF filter, respectively. Each of the CTF filters is designed as a chain of k so-called singular filters of the type which in the above formula is defined by the expression inside the power function.

For a particular channel the paramater an is determined by the height of the resonance maximum. The 45-bandwidth of the singular CTF filter turns out to be Bn = an/p. When for a particular characteristic frequency the height of resonance is prescribed, the effective filter bandwidth is determined as well. If the maximum is high, bandwidth is low, and vice versa. For a cascaded CTF filter, i.e., a filter that consists of k identical singular filters, the quantitative relationship between resonance height and bandwidth depends on k. By choosing k appropriately, nearly any resonance height can be combined with any bandwidth.

The temporal behavior of the CTF filters can be characterized by the effective duration of the impulse responses, which is expressed analytically. For a CTF filter with fc = 1000Hz, k = 5, and a resonance height of 70 dB the effective time-window length is about 8 ms.

The majority of parameters included in the PET system can be deduced from the absolute threshold of hearing. In addition, the ear's temporal behavior and dynamic range provide essential criteria. When the an are chosen such that the bandwidths are a constant fraction of critical bandwidth or equivalent rectangular bandwidth (ERB), it turns out that the height of resonance peaks as a function of characteristic frequency approximately takes on the inverse shape of the threshold of hearing - save the contribution of ear-canal resonances. This implies that aurally adequate bandwidths of the filters are automatically obtained when one sets the an such that the height of resonance maxima equals the difference between the threshold of hearing and a constant absolute SPL - the PET system's reference level, Lr.

The value of the reference level Lr is obtained from physiological and psychoacoustical data on the ear's dynamic range. In particular, it can be assessed by analysis of tuning curves and narrow-band masked thresholds. In the present PET system Lr = 70 dB and k = 5 were chosen.

Tuning curves are simulated by calculating the SPL which is required to produce the internal threshold SPL for any given channel. An algorithm is described by which scaling of channel frequencies is obtained such that overlap of frequnency responses is one and the same in all channels.

Both the ECR- and the CTF-filters can be digitally computed by means of one and the same recursive algorithm. The algorithms are described in detail.

The relationship between the CTF filterbank and the gammatone filterbank is discussed. It is shown that for k = 1 the two types of filter are largely equivalent but are different for k > 1.


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