An Analysis/Synthesis Auditory Filterbank ATR Human ... · Analysis/Synthesis gammachirp filterbank - 3-J. Acoust. Soc. Jpn.(E), Vol. 20,No. 5, pp397-406, Nov. 1999 For simulation

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An Analysis/Synthesis Auditory Filterbank

Based on an IIR Implementation of the Gammachirp

Toshio Irino* and Masashi Unoki*,**

* ATR Human Information Processing Research Labs.

2-2 Hikaridai Seika-cho Soraku-gun Kyoto, 619-0288, JAPAN* * Japan Advanced Institute of Science and Technology

1-1 Asahidai Tatsunokuchi Nomi Ishikawa, 923-1292, JAPAN

Irino and Unoki:Analysis/Synthesis gammachirp filterbank

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J. Acoust. Soc. Jpn.(E),Vol. 20,No. 5, pp397-406, Nov. 1999

Abstract

This paper proposes a new auditory filterbank that enables signal resynthesis

from dynamic representations produced by a level-dependent auditory

filterbank. The filterbank is based on a new IIR implementation of the

gammachirp, which has been shown to be an excellent candidate for

asymmetric, level-dependent auditory filters. Initially, the gammachirp filter is

shown to be decomposed into a combination of a gammatone filter and an

asymmetric function. The asymmetric function is excellently simulated with a

minimum-phase IIR filter, named the “asymmetric compensation filter”. Then,

two filterbank structures are presented each based on the combination of a

gammatone filterbank and a bank of asymmetric compensation filters controlled

by a signal level estimation mechanism. The inverse filter of the asymmetric

compensation filter is always stable because the minimum-phase condition is

satisfied. When a bank of inverse filters is utilized after the gammachirp

analysis filterbank and the idea of wavelet transform is applied, it is possible to

resynthesize signals with small time-invariant errors and achieve a guaranteed

precision. This feature has never been accomplished by conventional active

auditory filterbanks. The proposed analysis/synthesis gammachirp filterbank is

expected to be useful in various applications where human auditory filtering has

to be modeled.

Keywords: Auditory filterbank, Level-dependent asymmetric spectrum,

Analysis/synthesis system, Wavelet, Gammatone

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1. INTRODUCTION

Intensive efforts have been made to introduce human auditory

characteristics into the signal processing for telecommunications systems

including a recent example in audio coding. A number of auditory models have

been proposed to simulate the peripheral auditory system (for a review, see

Giguère and Woodland, 1994), but none of them have been used as successfully

as linear predictive analysis and the Fourier transforms in such systems. One

obvious reason for this has been the processing speed; fast digital signal

processors should resolve this problem in the near future. One of the other

major reasons might be that no signal resynthesis procedure is provided with

any realistic auditory model.

Linear auditory filterbanks, or wavelet transforms, have been used for

signal resynthesis (Combes et. al, 1989; Yang et. al, 1992), but they are unable

to account for the dynamic characteristics of basilar membrane motion. Iterative

procedures to reconstruct signals from cochleagrams (i.e., short-time averaged

amplitude responses of basilar membrane motion without phase information)

(Irino and Kawahara, 1993; Slaney, 1995) are applicable to such nonlinear

filterbanks, but they do not guarantee the precision of the resynthesis due to

local minima. Thus, it would be desirable to have a dynamic auditory

filterbank that also provides a sound resynthesis procedure resulting in no

perceptual distortion. This paper shows that it is possible to derive such an


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analysis/synthesis filterbank with time-varying coefficients through a new

implementation of the "gammachirp" (Irino, 1995, 1996; Irino and Patterson,

1997).

The gammachirp was analytically derived as a function satisfying

minimal uncertainty in joint time-scale representations (Cohen, 1993; Irino,

1995, 1996). The gammachirp auditory filter is an extension of the popular

gammatone filter (for a review, see Patterson et. al 1995); it has an additional

frequency-modulation term to produce an asymmetric amplitude spectrum.

When the degree of asymmetry is associated with the stimulus level, the

gammachirp filter can provide an excellent fit to 12 sets of notched-noise

masking data from three different studies (Irino and Patterson, 1997). The

gammachirp has a much simpler impulse response than recent physiological

models of cochlear mechanics (Giguère and Woodland, 1994), which have not

provided a good fit to human masking data. Moreover, the chirp term in the

gammachirp is consistent with physiological observations on frequency-

modulations or frequency “glides” in mechanical responses of the basilar

membrane (Møller and Nilsson, 1979; de Boer and Nuttall, 1997; Recio et. al,

1998).

The gammachirp filter has been implemented as a finite impulse

response (FIR) filter because the gammachirp is defined as a time-domain

function. Application to an auditory filterbank, however, poses some problems.


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For simulation of the dynamic characteristics of the cochlea, for instance, the

filter coefficients have to be recalculated and applied to the signal for each

sample time. Unfortunately, the large number of FIR coefficients, especially at

low frequencies, precludes fast filtering. Moreover, this simulation becomes

unrealistic if the signals are stored until all FIR coefficients are recalculated for

every sample point. The calculation of the filter output and the update of the

filter coefficients should be performed almost simultaneously. Therefore, the

gammachirp filter needs to be implemented with a small number of filter

coefficients which dictates that it is an infinite impulse response (IIR) filter

(Irino and Unoki, 1997a,b).

IIR implementations of modified gammatone filters have been

developed to introduce asymmetry into auditory filter shapes, i.e., the All-Pole

Gammatone Filter (APGF) or the One-Zero Gammatone Filter (OZGF)

(Slaney, 1993; Lyon, 1996). The degree of filter asymmetry has been associated

with signal level using a level estimation circuit (Pflueger et. al, 1998). The

shapes of these filters, however, depend on the sampling rate of the system

(Irino and Unoki, 1997a) and have not been directly fitted to psychoacoustical

masking data. Moreover, it has not been demonstrated that signals are

resynthesized from the output of such nonlinear filterbanks with time-varying

asymmetric filters. These are main topics of this paper.


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2. IMPLEMENTATION OF GAMMACHIRP FILTER

2.1 Definition and Fourier transform of the gammachirp

The complex impulse response of the gammachirp (Irino, 1995, 1996;

Irino and Patterson, 1997) is given as

gc(t)= at n−1 exp −2πbERB( fr) t( ) exp j2πf rt + jc ln t + jφ( ), (1)

where time t>0, a is the amplitude, n and b are parameters defining the

envelope of the gamma distribution, and fr is the asymptotic frequency. c is a

parameter for the frequency modulation or the chirp rate, φ is the initial phase,

ln t is a natural logarithm of time, and ERB (fr) is the equivalent rectangular

bandwidth of an auditory filter at fr. At moderate levels, ERB(fr)=24.7+0.108fr

in Hz (Glasberg and Moore, 1990). When c=0, the chirp term, cln t, vanishes

and this equation represents the complex impulse response of the gammatone

defined by the envelope which is a gamma distribution function and the carrier

which is a sinusoid at frequency fr (Patterson et. al, 1995). Accordingly, the

gammachirp is an extension of the gammatone with a frequency modulation

term.

The Fourier transform of the gammachirp in Eq. (1) is derived as

follows.


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GC ( f) =aΓ(n + jc)e jφ

{2πbERB( fr ) + j2π( f − fr)}n + jc

=a

{2π b 2 + ( f − fr )2 ⋅ e jθ}n + jc

= a ⋅1

{2 π b 2 + ( f − fr)2 }n ⋅e jn θ

⋅1

{2π b 2 + ( f − f r)2 }jc ⋅ e−cθ

= a ⋅1

{2 π b 2 + ( f − fr)2 }n

⋅e− jnθ

⋅ e cθ ⋅ e− jc ln{2π b 2 +( f − fr ) 2 }

(2)

θ = arctanf − fr

b (3)

where a = aΓ(n + jc)e jφ and b = bERB( fr) . The first term a is a constant. The

second term is known as the Fourier spectrum of the gammatone,GT ( f ). The

third term represents an asymmetric function, HA( f ) , that is described in more

detail in the next subsection. When the amplitude is normalized (a = 1), the

frequency response of the gammachirp is

GC ( f) = GT( f ) ⋅ HA( f ) . (4)

The amplitude spectrum is

| GC( f ) |=| GT ( f ) |⋅ | HA ( f ) |=1

{2π b 2 + ( f − fr)2 }n

⋅e cθ . (5)

Obviously, when c=0, | HA ( f) |(=e cθ ) becomes unity and Eq. (5) represents the

amplitude spectrum of the gammatone, | GT( f ) |. Figure 1 shows the amplitude

spectra of (a) a gammachirp filter | GC( f ) |, (b) a gammatone filter | GT( f ) |,

and (c) an asymmetric function | HA ( f) | when the chirp parameter c=-2. The

amplitude of | HA ( f) | is biased by about -4 dB to normalize the peak of


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| Gc( f ) | to 0 dB. Since the amplitude spectrum of the gammatone filter

| GT( f ) | is symmetric on a linear frequency axis, the asymmetric function

| HA ( f) | introduces spectral asymmetry and a shift of the peak frequency into

the gammachirp spectrum | GC( f ) |.

--- Insert Figure 1 about here ---

The peak frequency fp in the amplitude spectrum can be obtained

analytically by setting the derivative of Eq. (5) to zero and solving the equation

for the frequency. The result is

fp = fr +c ⋅b

n= f r +

c ⋅bERB( fr )

n. (6)

Therefore, the size of the peak shift is proportional to the chirp parameter c and

the ratio of the envelope parameter b ERB(fr) to n.

2.2 Characteristics of the gammachirp and the asymmetric function

To describe the spectral characteristics of the gammachirp and the

asymmetric function precisely, Eq. (4) is rewritten in a form that explicitly uses

the relevant parameters; that is,

GC ( f;n,b,c, fr ) = GT ( f ;n,b, fr ) ⋅ HA( f ;b,c, fr ). (7)

The asymmetric function uses parameters b, c, and fr whereas the gammatone

uses parameters n, b, and fr.


Figure 2 shows the amplitude spectra of (a) the gammachirp

| GC( f ;n,b,c, f r) | and (b) the asymmetric function | HA ( f;b,c, fr ) | when the


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values of the chirp parameter c are integers between –3 and 3. Several

characteristics are derived from this figure and the equations described above.

(a) Figure 2(a) shows that the filter slope below the peak frequency is shallower

than the slope above it in the gammachirp when the parameter c is negative.

The situation is the reverse when the parameter c is positive. The filter shape

is symmetric when c is zero because it is the gammatone.

(b) The asymmetric function | HA ( f;b,c, fr ) | in Fig. 2(b) is an all-pass filter

when c=0. Using Eq. (2),

HA( f ;b,0, fr) = 1. (8)

| HA ( f;b,c, fr ) | is a high-pass filter when c>0, and a low-pass filter when

c<0. The slope and the range of the amplitude increase when the absolute

value of c increases. The filter shapes of the gammachirps in Fig. 2(a) reflect

these characteristics.

(c) | HA ( f;b,c, fr ) | changes monotonically in frequency. Neither a peak nor a

dip ever occurs in this function.

(d) The gain of the asymmetric function is anti-asymmetric. For an arbitrary

frequency f1,

| HA ( fr − f1; b,c , fr) |=| HA( fr + f1;b,c, fr) |−1. (9)

(e) With Eq. (2), this produces

HA( f ;b,c, fr ) = HA( f ;b,−c, f r)−1 . (10)

(f) For arbitrary chirp parameters c1 and c2,


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HA( f ;b,c1 + c2 , fr) = HA( f; b,c1, fr ) ⋅ HA( f ;b, c2 , fr ). (11)

(g) Using Eqs. (7), (10), and (11),

GC ( f;n,b,c1, f r) = GT ( f ;n, b, fr) ⋅ HA( f ;b,c1 , fr)

= GT ( f;n,b, f r) ⋅ HA( f ;b,c1 + c2 , fr) ⋅ HA ( f ;b,−c2, fr )

= GC( f ;n, b, c1 + c2, fr) ⋅ HA ( f ;b,−c2 , fr ) (12)

Equation (12) states that a gammachirp, with an arbitrary chirp parameter c1, is

a product of a gammachirp with a different chirp parameter c1 +c2, and an

asymmetric function having the difference between them, -c2. This is because

the asymmetric function HA( f ;b,c, fr ) is an exponential function in parameter

c.

These characteristics are necessary conditions for designing the

approximation filter in the next subsection, and they act as a guide for

establishing an analysis/synthesis filterbank in Section 3.

2.3 Asymmetric compensation filter

As shown by Eq. (4), a gammachirp filter can be implemented by

cascading a gammatone filter and an asymmetric filter. Since efficient

implementations of the gammatone are already known (Slaney, 1993; Patterson

et. al, 1995), this section concentrates on an approximation filter for the

asymmetric function described in the previous section. It is necessary to design

a filter satisfying the conditions (a) to (g) in the previous section. As a first

step, a filter satisfying condition (d) is considered because this characteristic

seems the most relevant for filter design purposes.


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IIR filters satisfying Eq. (9) have a pair of a pole and a zero

symmetrically located at fr+∆fk and fr−∆fk , and the number of pairs corresponds

to the number of ∆fk. This is so the absolute values, r, of the corresponding

poles and zeros are equal. They must be inside the unit circle so that IIR filters

converge; this is known as the minimum phase condition (Oppenheim and

Schafer, 1975). Since the bandwidth gets narrower when r gets closer to unity, r

is negatively correlated to the bandwidth parameter bERB(fr). Condition (b)

implies that ∆f is proportional to c and is positively correlated with bERB(fr). A

cascaded second-order digital filter satisfying these properties is

HC (z) = HCk(z)k =1

N

∏ , (13)

HCk (z) =(1− rke

jϕk z−1)(1− rke− jϕ kz−1)

(1− rkejφk z−1)(1− rke

− jφ k z−1 ), (14)

rk = exp{−k ⋅ p1 ⋅ 2π bERB(f r)/ fs} , (15)

φk = 2π{ fr + p0k −1 ⋅ p2 ⋅ c⋅ bERB( fr)}/ fs , (16)

ϕ k = 2π{fr − p0k− 1 ⋅ p2 ⋅c⋅ bERB( fr)}/ fs , (17)

where p0, p1, and p2 are positive coefficients and fs is the sampling rate. The

reason for cascading filters with slightly offset poles and zeros is to satisfy

condition (c) approximately. This filter is referred to as an "asymmetric

compensation (AC)" filter.


Figure 3 shows the amplitude spectra of this digital filter | HC( f ) |


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(dashed lines) and the asymmetric function | HA ( f) | (solid lines) in Eq. (5) as a

function of the chirp parameter c. The number of cascaded filters was four, the

amplitude was normalized at frequency fr, and the values of p0, p1, and p2 were

set properly (described in the next subsection). The dashed lines are very close

to the solid lines when the frequency is less than 3000 Hz. Above 3000 Hz,

however, the disparity gets larger. This, however, does not cause serious

problems because the asymmetric compensation filter is always accompanied

by the gammatone filter, which is a band-pass filter.

Actually, the results will show that four cascaded second-order filters

are sufficient when the parameter b is equal to or greater than unity and the

chirp parameter c is between -3 and 1 (see subsection 2.4.1). In this case, the

numbers of poles and zeros are 16 in total. Although it is possible to improve

the fitting by increasing the number of cascaded filters, a reasonable number of

stages is determined by considering the trade-off between the number of

coefficients and the degree of fitting.

2.4 Asymmetric compensation gammachirp

The asymmetric compensation filter cascaded to the gammatone filter

approximates the gammachirp filter. The amplitude spectrum of this filter is

found, by replacing | HA ( f) | with | HC( f ) | in Eq. (5),

| GCAC( f ) |=| GT ( f ) |⋅ | HC ( f) |. (18)

This filter GCAC ( f ) is referred to as an "Asymmetric Compensation -


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gammachirp" or "AC-gammachirp" filter until the end of Section 2, so as to

distinguish it from the original gammachirp defined by Eq. (1).


2.4.1 Comparison of the amplitude spectrum

Figure 4 shows the amplitude spectra of the gammachirp | GC( f ) | in Eq.

(5) (solid lines) and the AC-gammachirp | GCAC( f ) | in Eq. (18) (dashed lines).

The amplitude | GCAC( f ) | has been normalized to improve the fit. The

frequency for normalizing the amplitude of each second-order filter is closely

related to the peak shift in Eq. (6), and for the k-th filter,

f = f r + k ⋅ p3 ⋅c ⋅bERB( fr )/ n . (19)

The coefficients p0, p1, p2, and p3 are set heuristically as

p0 = 2, (20)

p1 = 1.35 - 0.19 |c|, (21)

p2 = 0.29 - 0.0040 |c|, (22)

p3 = 0.23 + 0.0072 |c|. (23)

The root-mean-squared (rms) error between the original gammachirp

filter and the AC-gammachirp filter is less than 0.41 dB in Fig. 4 in the range

where | GC( f ) |> −50 dB . The average rms error is only 0.63 dB for 90 sets of

parameter combinations {n = 4; b = 1.0, 1.35, and 1.7; c = 1, 0, -1, -2, and -3; fr

= 250, 500, 1000, 2000, 4000, and 8000 (Hz) }, i.e., about the range of

parameter values in a typical fit (Irino and Patterson, 1997). The rms error


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exceeds 2 dB only for three sets when fr = 8000 Hz and c = -3.

The fit improved only slightly when the coefficients in Eqs. (21), (22),

and (23) were optimized using an iterative least squared-error method. It would

be possible to improve the fit by changing the locations of the poles and zeros

defined in Eqs. (15), (16), and (17), but that is beyond the scope of this paper.


2.4.2 Comparison of the impulse response and the phase spectrum

Figure 5(a) shows an example of the impulse response of the

gammachirp defined in Eq. (1) (solid line) and the AC-gammachirp

corresponding to Eq. (18) (dashed line). The difference in the impulse response

between the original gammachirp and the AC-gammachirp is about -50 dB

SNR and therefore is negligible. Their phase spectra shown in Fig. 5(b) are very

close to each other. Therefore, the AC-gammachirp is able to provide an

excellent approximation to the original gammachirp in terms of phase

characteristics, i.e.,

GC ( f) ≅ GCAC( f ) = GT( f ) ⋅ HC( f ), (24)

gc(t) ≅ gcAC(t) = gt(t) *hc (t) , (25)

where * denotes the convolution.

2.4.3 Similarity between the AC-gammachirp and the original gammachirp

The similarity between the AC-gammachirp filter and the original

gammachirp filter is discussed in this subsection. The characteristics of the


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asymmetric function HA( f ) are listed as the conditions (b), (c), (d), (e), and (f)

for the design of the asymmetric compensation filter HC ( f) . For condition (b),

HC ( f) strictly satisfies Eq. (8) and the other conditions. When setting c to 0,

the phases of the poles and zeros in Eqs. (16) and (17) become the same, and

then, Eqs. (13) and (14) become unity. It is obvious from Fig. 3 that HC ( f) is

high-pass when c>0 and low-pass when c<0. For condition (c), the asymmetric

compensation filter HC ( f) approximately satisfies the condition in positive

frequencies when fr> 0. HC ( f) has slopes centered at +fr and -fr in the

amplitude spectrum, whereas HA( f ) has a slope centered at fr. This is

because HC (z) is designed to have real coefficients using conjugate pairs of

poles and zeros to accompany a gammatone filter with a real sinusoidal carrier.

For condition (d), HC ( f) strictly satisfies Eq. (9) because changing fr+f1 to fr-f1

simply replaces the denominator and the numerator of Eq. (14). HC ( f) strictly

satisfies Eq. (10) for condition (e). Changing the sign of c replaces the poles

and zeros in Eqs. (16) and (17), moreover, it is possible to derive a stable

inverse filter since the asymmetric compensation filter satisfies the minimum

phase condition. The inverse filter is always stable even if the parameter values

are time varying. Accordingly, it is possible to cancel out the forward filter with

the inverse filter. Then, the total response of the combination is a unit impulse.

This feature leads to an analysis/synthesis filterbank (described in subsection

3.3). For condition (f), HC ( f) approximately satisfies Eq. (11), as shown in


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Fig. 3.

Consequently, the AC-gammachirp filter follows condition (a) and

approximately satisfies condition (g) (Eq. (12)). Since the IIR asymmetric

compensation filter has few coefficients, fast level-dependent auditory filtering

can be performed by combining the compensation filter with a fast

implementation of the gammatone (Slaney, 1993; Patterson et. al, 1995).

3. GAMMACHIRP FILTERBANK

This section presents an analysis/synthesis gammachirp filterbank with

time-varying coefficients and a parameter controller based on sound level

estimation. Initially, we describe examples of the analysis filterbank to consider

a basic structure that establishes a synthesis procedure that should be

independent of the method of parameter control when used in various

applications.


3.1 Analysis filterbanks

Figure 6 shows an example of a gammachirp filterbank consisting of a

gammatone filterbank, a bank of asymmetric compensation filters, and a

parameter controller (Irino and Unoki, 1997a,b, 1998). It is a straightforward

implementation of Eqs. (24) and (25) for each filter. Since the auditory filter

shape is level-dependent (Lutfi and Patterson, 1984; Glasberg and Moore, 1990


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for a review; Irino and Patterson, 1997), the sound level of incoming signals is

estimated in the parameter controller using the output of the asymmetric

compensation filterbank. An example of the parameter controller is shown at

the right-bottom. The controller consists of a bank of rectifiers, leaky

integrators (LI) and level-to-parameter converters. It is possible to consider a

number of implementations for level estimation (for example, Giguère and

Woodland, 1994; Pflueger et. al, 1998), but they are basically similar in level

estimation at the output of bandpass filters (see discussion in Rosen and Baker,

1994). This filterbank has been applied to noise suppression (Irino, 1999); here,

we introduce physiological knowledge into the filterbank structure.

When the sound level is sufficiently high, the cochlear filter has a broad

bandwidth and behaves like a passive and linear filter. As the signal level

decreases, the filter gain increases and the bandwidth becomes narrower

because of the active processes (Pickles, 1988 for a review). This suggests that

a physiologically plausible auditory filter would be a combination of a linear

broadband filter and a nonlinear level-dependent filter that sharpens the filter

shape. Recent observations have shown that the frequency modulation or

“glide” persists even in post-mortem or at high sound pressure levels (Recio et.

al, 1998). Accordingly, the linear filter can be simulated with a broadband

gammachirp filter. As shown in Eq. (12), a gammachirp filter with an arbitrary

chirp parameter c can be produced with a combination of another gammachirp


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filter and an asymmetric function. Therefore, the second filter can be simulated

using a level-dependent asymmetric compensation filter as long as the total

filter response is simulated using the gammachirp.


Accordingly, a candidate filterbank structure is proposed in Fig. 7. It

consists of a linear gammatone filterbank, a linear asymmetric compensation

filterbank, and a level-dependent asymmetric compensation filterbank

controlled by a parameter controller. The output of the linear asymmetric

compensation filterbank is equivalent to the output of a linear gammachirp

filterbank (dashed box (c) at top). This output is fed into the asymmetric

compensation filterbank to obtain the total output (d). The parameter controller

is similar to that described above. The structure is based on a combination of a

linear filterbank with bandpass filters and a nonlinear asymmetric compensation

filterbank, as in the previous filterbank. For signal processing applications, this

filterbank structure is very important in facilitating the synthesis procedure.

However, to determine the parameters, it is necessary to wait for results on

gammachirp fits to psychoacoustical masking data across the full range of

center frequencies (for preliminary results, Irino and Patterson, 1999).


3.2 Analysis/synthesis filterbank

One of the most important features of the gammachirp filterbank is its

ability to establish an analysis/synthesis system as shown in Fig. 8. Moreover,


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this feature, as discussed in the following, is valid for any kind of parameter

controller. Initially, a signal (a) is filtered by a linear gammachirp filterbank

(A). When the chirp parameter c is set to zero for all channels, it is a

gammatone filterbank. The output of the linear filterbank (b) is converted into

the output of the level-dependent gammachirp filterbank (c) using a bank of

asymmetric compensation filters (B) controlled by the parameter controller (C).

Since the asymmetric compensation filter is an IIR minimum phase filter, it is

possible to make a bank of inverse asymmetric compensation filters (D) by

exchanging poles and zeros. When the time-varying coefficients for the

original and inverse filterbanks are always the same at each sampling point, the

output of the level-dependent gammachirp filterbank (c) is converted into a

representation (d), which is strictly the same as the output of the linear

gammachirp filterbank (b). The filterbank output is, then, equalized in phase

using the time-reversal gammachirp filterbank (E), which is the same as the

linear filterbank (A), except that the impulse response of each filter is reversed

in time. Finally, the output with phase equalization is summed with a

weighting function to reproduce the signal.

A combination of the linear analysis filterbank (A), the linear synthesis

filterbank (E), and the weighted sum (F) is almost equivalent to a linear,

wavelet, analysis/synthesis procedure (Combes et. al, 1989). Since the

combination of the asymmetric compensation filterbank (B) and its inverse


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filterbank (D) produces unit impulses for all channels, the error between the

original and synthetic signals is strictly determined by this linear

analysis/synthesis filterbank.


Figure 9 shows an example of analysis/synthesis frequency

characteristics for a level-dependent gammachirp filterbank with equally-

spaced filters for ERB rates between 100 and 6000 Hz using a gammatone

filterbank in (A) and (E), i.e., the gammachirp filterbank when c =0 for all

channels. Figure 9(b) shows the same graph with a magnified ordinate scale.

The maximum error is less than 0.01 dB with 100 channels and is only about

0.03 dB even with 50 channels. It appears that about 100 channels are sufficient

to minimize the errors. Moreover, the errors are completely independent of

parameter control. Consequently, the gammachirp filterbank is able to perform

signal resynthesis without producing any undesirable distortion.

The discussion above guarantees the minimum distortion of the

analysis/synthesis filterbank system. This filterbank is applicable to various

applications when inserting a modification block between the asymmetric

compensation filterbank (B) and its inverse filterbank (C). For example, it is

possible to construct a noise-suppression filterbank which does not produce any

musical noise that is perceptually undesirable (Irino, 1999).


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4. SUMMARY

This paper presents an analysis/synthesis auditory filterbank using the

gammachirp. Initially, the gammachirp function is analyzed to find

characteristics for effective digital filter simulation. The gammachirp filter is

shown to be approximated excellently by the combination of a gammatone filter

and an IIR asymmetric compensation filter. The new implementation reduces

the computational cost for time-varying filtering because both filters can be

implemented with only a few filter coefficients. Since the IIR asymmetric

compensation filter is a minimum phase filter, the inverse filter is also stable.

Two examples of gammachirp filterbanks are presented; each is a combination

of a linear gammachirp filterbank and a bank of level-dependent asymmetric

compensation filters, controlled by the signal-level estimation mechanism. A

synthesis procedure for such analysis filterbanks is proposed to accomplish

signal resynthesis with a guaranteed precision and no undesirable distortion.

This feature has never been accomplished with conventional auditory

filterbanks. The analysis/synthesis gammachirp filterbank with time-varying,

level-dependent coefficients is usable in various signal processing applications

requiring the modeling of human auditory filtering.

ACKNOWLEDGMENTS

The authors wish to thank Roy D. Patterson of Cambridge Univ., for his

continuous advice, Minoru Tsuzaki and Hani Yehia of ATR-HIP, and Malcolm


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Slaney of Interval Research for their valuable comments. A part of this work

was performed while the second author was a visiting student at ATR-HIP. The

authors also wish to thank Masato Akagi of JAIST, and Yoh'ichi Tohkura,

Hideki Kawahara, and Shigeru Katagiri of ATR-HIP for the arrangements. This

work was partially supported by CREST (Core Research for Evolutional

Science and Technology) of the Japan Science and Technology Corporation

(JST).

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Figure Captions

Figure 1. Amplitude spectra of (a) a gammachirp filter | Gc( f ) |, (b) a

gammatone filter | GT( f ) |, and (c) an asymmetric function | HA ( f) |, where

n=4, b=1.019, c=-2, and f r=2000 Hz.

Figure 2. Amplitude spectra of (a) a gammachirp filter | Gc( f ) | and (b) an

asymmetric compensation filter | HA ( f) | as a function of the chirp parameter

c where n=4, b=1.019, and fr=2000 Hz. The amplitude is normalized to 0 dB

at the peak frequency in panel (a) and at fr in panel (b).

Figure 3. Amplitude spectra of asymmetric functions | HA ( f) | (solid lines) and

asymmetric compensation filters | HC( f ) | (dashed lines) where n=4,

b=1.019, c is an integer between -3 and 3, and fr=2000 Hz. The amplitude is

normalized to 0 dB at fr.

Figure 4. Amplitude spectra of original FIR gammachirp filters | Gc( f ) | (solid

lines) and asymmetric compensation (AC) gammachirp filters | GCAC( f ) |

(dashed lines) where n=4, b=1.019, c=-1, and the values for fr are 250, 500,

1000, 2000, 4000, and 8000 Hz.

Figure 5. (a) Impulse responses and (b) phase spectra of an FIR gammachirp

filter (Eq. (1)) (solid lines) and an asymmetric compensation (AC)

gammachirp filter (dashed lines). The parameters are n=4, b=1.019, c=-1,


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and fr=2000 Hz.

Figure 6. Block diagram of a level-dependent gammachirp filterbank.

Figure 7. Block diagram of a gammachirp filterbank based on physiological

constraints.

Figure 8. Block diagram of a level-dependent, analysis/synthesis gammachirp

filterbank.

Figure 9. Frequency responses of the analysis/synthesis gammachirp filterbank

shown in Fig. 8 when the frequency range of the filterbank is between 100

and 6000 Hz and the number of channels is 50 (dashed lines) or 100 (solid

lines). Panel (b) is the magnified ordinate of panel (a).


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Fig. 1


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Fig. 2


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Fig. 3


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Fig. 4


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Fig. 5


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Fig. 6

Σ

Fromadjacentchannels

Activity to parameter

Parameter Control Unit

LI

LI

LI

LI

k-thchannel

k-th control-output

LI

SignalInput

GammatoneFilterbank

ParameterController

AsymmetricCompensation

Filterbank

GammatoneFilterbank

Output

GammachirpFilterbank

Output


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Fig. 7

(a) SignalInput

(c) LinearGammachirp

Filterbank Output

(d) GammachirpFilterbank Output

(C)Asymmetric

CompensationFilterbank

(D)ParameterController

(B)Linear


Filterbank

(b) LinearGammatone

Filterbank Output

(A)Linear

GammtoneFilterbank


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Fig. 8

(A)Linear

GammachirpFilterbank

(E)Time-reversal

LinearGammachirp

Filterbank

(D)Inverse


Filterbank

(a) SignalInput

SynthesisAnalysis

(C)ParameterController

(B)Asymmetric

CompensationFilterbank

(e) ResynthesizedSignal

(c) GammachirpFilterbank Output

(F)Weighted

Sum

Σ

(b) LinearGammachirp

Filterbank Output

(d) RecoveredLinear Gammachirp

Filterbank Output


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Fig. 9

An Analysis/Synthesis Auditory Filterbank ATR Human ... · Analysis/Synthesis gammachirp filterbank - 3-J. Acoust. Soc. Jpn.(E), Vol. 20,No. 5, pp397-406, Nov. 1999 For simulation

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