• Document: Implementing a GammaTone Filter Bank*
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Implementing a GammaTone Filter Bank* John Holdsworth Ian Nimmo-Smith Cambridge Electronic Design MRC Applied Psychology Unit Science Park 15 Chaucer Road Milton Road Cambridge CB2 2EF Cambridge Roy Patterson Peter Rice MRC Applied Psychology Unit Cambridge Electronic Design 15 Chaucer Road Science Park Cambridge CB2 2EF Milton Road Cambridge 26th February 1988 Introduction The purpose of this technical note is to provide a self-contained summary of properties of the GammaTone filter and of the manner in which it can be effectively implemented through a cascade of identical recursive digital filters. • Section 1 defines the GammaTone filter as an IIR filter in the time domain and describes its provenance and some of its elementary properties. • Section 2 examines the behaviour of the GammaTone filter in the frequency domain, and shows that its form can lead to an (approximate) representation as a cascade of identical first order filters. • Section 3 continues the analysis of Section 2 to provide a way of calculating the parameters needed for a GammaTone filter to have a specified equivalent rectangular bandwidth. • Section 4 discusses the issue of phase compensation for the GammaTone filter. • Section 5 describes the way in which these features have been exploited to achieve an efficient digital implementation of the GammaTone filter on a general purpose computer. 1 The GammaTone filter in the time domain Prompted by de Boer and Kuyper (1968), the GammaTone filter was introduced by Johannsma (1972) to describe the shape of the impulse response function of the auditory system as estimated by the reverse correlation function of neural firing times. This was subsequently developed by de Boer and de Jongh (1978) and de Boer and Kruidenier (1988). For further details of the comparison between GammaTone filters, rounded-exponential filter shapes and experimental evaluation of the shape of the human auditory filter, see Patterson et al. (1987). The GammaTone filter is defined in the time domain (impulse response function) as gt(t) oc t n - 1 exp( -27rbt) cos(27r Jot + ¢» (t 2: 0) (1) • Annex C of the SVOS Final Report (Part A: The Auditory Filter Bank) 1 It is thus a causal filter with an infinite response time. The form of this function is that of an amplitude modulated carrier tone of frequency foHz, with an envelope proportional to t n - 1 exp( -27rbt), which is the familiar Gamma distribution from statistics. These two compo- nents give rise to the name GammaTone (de Boer and de Jongh, 1978). The parameters of the GammaTone filter are n, the order, which (for fixed b) controls the relative shape of the envelope, becoming less skewed as n increases; b (in Hz) which (for fixed n) controls the duration of the impulse response function, increased b leading to shorter duration; fo (in Hz) which determines the frequency of the carrier; and <P (in radians), the carrier phase, which determines the relative position of the fine structure of the carrier to the envelope. All four parameters have corresponding effects on the frequency domain characteristics of the GammaTone filter. 2 The GammaTone filter in the frequency domain If GT(J) represents the GammaTone filter in the frequency domain (frequency response function) then GT(J) ex [1 + j(J - fo)/bj-n + [1 + j(J + fo)/br n (-00 < f < 00) (2) This can be derived either directly from the time domain definition above, by application of the Fourier transform, or by observing that the product in the time domain of a Gamma function and a cosine function will correspond to the convolution in the frequency domain of the Fourier transform (1 + j f /b )-n of the Gamma function with a two-point distribution at ±fo. Here for simplicity we have set 1> = 0, as it has no important effect on the frequency domain characteristics of the filter. The role of the parameters in the frequency domain is apparent from the above formula. fo is the centre frequency of the filter; for fixed order n, b acts as a scaling parameter such that the bandwidth of the filter increases with b; the order parameter n controls the overall shape of the filter. For fixed b, the bandwidth decreases as n increases. GT(J) is approximately symmetric about fo on a linear frequency scale. As stated by de Boer and Krudenier (1988), the second term in Eqn.(2) can be ignored wh

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