A time-domain, level-dependent auditory filter: The gammachirp
Toshio Irinoa)
NTT Basic Research Laboratories, 3-1 Morinosato Wakamiya, Atsugi-shi, Kanagawa, 243-01 JapanRoy D. Pattersonb)
MRC Applied Psychology Unit, 15 Chaucer Road, Cambridge CB2 2EF, United Kingdom~Received 28 June 1996; accepted for publication 4 September 1996!
A frequency-modulation term has been added to the gammatone auditory filter to produce a filter
with an asymmetric amplitude spectrum. When the degree of asymmetry in this ‘‘gammachirp’’auditory filter is associated with stimulus level, the gammachirp is found to provide an excellent fit
to 12 sets of notched-noise masking data from three different studies. The gammachirp has awell-defined impulse response, unlike the conventional roex auditory filter, and so it is an excellent
candidate for an asymmetric, level-dependent auditory filterbank in time-domain models of auditory
processing. © 1997 Acoustical Society of America. @S0001-4966~97!02701-X#
PACS numbers: 43.66.Ba, 43.66.Dc @WJ#

INTRODUCTION

In time-domain auditory models, the spectral analysis
performed by the basilar membrane is often simulated by abank of gammatone auditory filters ~see, for example, Patterson et al., 1995!. The impulse response of the gammatone isg t ~ t ! 5at n21 exp~ 22 p b ERB~ f c ! t ! cos~ 2 p f c t1 f !
~ t.0 ! , ~1!

where a, b, n, f c , and f are parameters. ERB( f c ) is theequivalent rectangular bandwidth of the filter, and at moderate levels ERB( f c )524.710.108f c in Hz ~Glasberg andMoore, 1990!. The filter gets its name from the fact that theenvelope formed by the power function and the exponential
is a gamma distribution function, and the cosine carrier is atone when it is in the auditory range. The amplitude spectrum of the gammatone filter is essentially symmetric on alinear frequency scale.
The gammatone function was introduced by Johannesma~1972! to characterize impulse-response data gathered physiologically from primary auditory fibers in the cat ~see Carney
and Yin, 1988, for an overview!. The gammatone has also
been used to characterize spectral analysis in humans at
moderate levels where the amplitude characteristic of the auditory filter is nearly symmetric on a linear frequency scale~see Patterson 1994, for an overview!.
The use of the gammatone filter is limited, however, by
the repeated demonstration that, below its center frequency,
the skirt of the auditory filter broadens substantially withincreasing stimulus level, and above its center frequency theskirt sharpens a little with increasing level ~Lutfi and Patterson, 1984; Patterson and Moore, 1986; Moore and Glasberg,
1987!. The level dependence of the auditory filter has beenmodeled using the ‘‘roex’’ function ~Patterson et al., 1982;
Glasberg and Moore, 1990; Rosen and Baker, 1994!. But theroex auditory filter does not have a well-defined impulsea!

Electronicmail:irino@nttlab.brl.ntt.co.jp;
www.brl.ntt.co.jp/people/irino/index.html
Electronic mail: roy.patterson@mrc-apu.cam.ac.uk

b!

412

J. Acoust. Soc. Am. 101 (1), January 1997

WWW:

http://

response which largely precludes its use in auditory filterbanks. More physiological models of cochlear mechanics~for example, Giguère and Woodland, 1994! do not provide
good fits to human masking data; nor do they have sufficiently simple impulse responses for the traditional filterbank
architecture.
Irino ~1995, 1996! recently demonstrated that an analytic relative of the gammatone function, referred to as the‘‘gammachirp’’ function, is a theoretically optimum auditory
filter, in the sense that it leads to minimal uncertainty in ajoint time and scale representation of auditory signal analysis. The derivation of the gammachirp function is based on
operator methods ~Gabor, 1946; Cohen, 1991, 1993! involving the Mellin transform ~Titchmarsh, 1948!; it is summarized in Appendix A. The gammachirp auditory filter is thereal part of the analytic gammachirp function, Eq. ~A20!. It
has an asymmetric amplitude characteristic, and in the following we show that, when the asymmetry is associated withstimulus level, the gammachirp filter provides an excellent fit
to human masking data. The gammachirp has a well-defined
impulse response and, with only one parameter more than
the gammatone, it would appear to be an excellent candidate
for an asymmetric, level-dependent auditory filterbank.

I. METHOD
A. The power spectrum model with a gammachirp
filter

The impulse response of the gammachirp auditory filterisg c ~ t ! 5at n21 exp~ 22 p b ERB~ f r ! t !
3cos~ 2 p f r t1c ln t1 f !

~ t.0 ! .

~2!

The only difference between it and the impulse response of
the gammatone @Eq. ~1!# is the term c ln t; c is an additionalparameter, and ln is the natural logarithmic operator. The
filter has a monotonically frequency-modulated carrier ~achirp! with an envelope that is a gamma distribution func-

0001-4966/97/101(1)/412/8/$10.00© 1997 Acoustical Society of America

412

tion, and hence the name ‘‘gammachirp.’’ 1 We use f r instead of f c for the frequency parameter because the peakfrequency of the amplitude spectrum varies with c, and to alesser extent, b and n. The equivalent rectangular bandwidthof the filter varies with stimulus level, but for convenience,
we associate the parameter b with stimulus level so that thebasic formula for filter width, ERB( f r )524.710.108f r , isthe same as in Eq. ~1!.
The auditory filter shape is derived using the power
spectrum model of masking ~Fletcher, 1940; Patterson,
1976!. In the experiment, the listener is required to detect abrief sinusoidal signal, referred to as a ‘‘probe’’ tone, in thepresence of a masker which is a noise with a spectral notchin the frequency region of the probe tone. This ‘‘notchednoise’’ has a constant spectrum level N 0 in a band below thetone between f l min and f l max and in a band above the tone
between f u min and f u max. The level of the probe tone is varied
to determine the power required to make it just audible~probe ‘‘threshold’’!, as a function of the width of the notchin the noise. The details of the experiment and the criterion
for threshold are described in Patterson ~1976!. If the‘‘shape’’ of the auditory filter ~that is, its power spectrum! isrepresented by the weighting function, W( f ), then the
power spectrum model isP s 5K1N 0
110 log10

SE

fl

fl

max

min

W~ f !d f 1

E

fu

fu

max

W~ f !d f

min

D

, ~3!

where P s is the power of the probe tone at threshold in dB,
and K is a constant which is related to the efficiency of thedetection mechanism following the auditory filter. Following
Patterson et al. ~1982!, a parameter r is introduced to limit
the dynamic range of the filter. The weighting function isassociated with the power spectrum of the gammachirp,u G C ( f ) u 2, as follows:
W ~ f ! 5 ~ 12r ! •W om ~ f ! • u G C ~ f ! u 2 1r.

~4!

Here, W om ( f ) is the ‘‘ELC’’ correction recommended by
Glasberg and Moore ~1990! to simulate the effects of theouter and middle ears. The maximum absolute magnitude ofW( f ) is normalized to unity ~See Appendix B for the analytic form of the amplitude spectrum of the gammachirp.!

B. Parameters and fitting procedure

We characterize the level dependence of the auditory
filter shape in terms of the level dependence of the five parameters of the gammachirp: n, b, c, K, and r. The auditory
filter becomes broader on the low side and sharper on thehigh side as stimulus level increases ~Moore and Glasberg,
1987!. Changes in the parameters n and b have little effecton the asymmetry of the amplitude spectrum, and K and r do
not affect asymmetry since they are not filter parameters.
Thus, the degree of asymmetry is primarily determined by c.
Rosen and Baker ~1994! showed, in an analogous fit with theroex auditory filter, that the level dependence can be summa413

J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997

rized with a linear function. In our initial fit, then, we provide two coefficients for c and one coefficient for each of n,b, and K.
The parameters n and b affect bandwidth reciprocally;
the bandwidth of the filter decreases, either when n increases
or when b decreases, and vice versa. We are mainly concerned with the filter shape around the center frequency andin this case we can fix either n or b and let the other vary tomatch the auditory filter shape. Preliminary simulations applied to several data sets and previous work with the gammatone suggested that we begin by fixing n at the value 4.
The fitting procedure is broadly similar to the PolyFit
procedure of Rosen and Baker ~1994!. Thresholds were calculated for a range of filters with center frequencies aroundthe probe frequency. The value of the filter giving the highest
signal-to-noise ratio was chosen as the threshold estimate P s~Patterson and Nimmo-Smith, 1980!. We used theLevenberg–Marquardt method ~Press et al., 1988! to minimize the squared error between the data and P s ; this is astandard procedure for a nonlinear least-mean-square problem. This fitting procedure is referred to as the ‘‘gammachirp
fit’’ in the following.
C. Data sets

We began by applying the gammachirp fit to thenotched-noise masking data of Rosen and Baker ~1994! because the results can be compared directly to their resultswith the roex filter and the PolyFit procedure. We will
specify the data source in the following by the initials of thesubject and the probe frequency, for example, ‘‘LM at 2000
Hz’’ for this data set which contains 78 tone-in-noise thresholds. Rosen and Baker used total squared error in dB2 toevaluate alternative fits. We will use rms ~root-meansquared! error in dB; it is a more intuitive measure andmakes it easier to compare fits when the data sets have different numbers of thresholds.
The gammachirp was also fitted to subsets of thenotched-noise data reported in Lutfi and Patterson ~1984!
and Moore et al. ~1990!. The data from Lutfi and Patterson
are those of HM, RL, RM, and WW at 1000 and 4000 Hz.
Each set contains 39 data points distributed over three different noise levels, except for RM at 4000 Hz, where there
are 52 data points over four noise levels. The data of Moore
et al. ~1990! are those of CP at 200, 400, and 800 Hz. Eachset contains 75 data points distributed over five noise levels.
II. RESULTS
A. Rosen and Baker (1994)

Rosen and Baker ~1994! fitted a wide range of roex filter
models to a set of masking data gathered with both probefixed and masker-fixed conditions to maximize the range ofsignal levels represented by the data. They discuss a subsetof their fits for probe-dependent models with 24, 15, 10, 8, 7,
6, and 5 variable coefficients. The rms errors for the 78
thresholds are 1.15, 1.16, 1.19, 1.19, 1.19, 1.20, and 1.42
dB.2 The rms errors for masker-dependent models are muchgreater than these values, and this is why they restricted theirattention to probe-dependent models. The focus of their disT. Irino and R. D. Patterson: Gammachirp auditory filter

413

FIG. 1. The roex auditory filter shape as a function of probe level ~30–60
dB SPL in 10-dB steps! with six variable coefficients @adapted from Rosen
and Baker ~1994!#.

cussion is the trade-off between number of free parametersand goodness of fit, and they conclude that a roex( p,r)model with six coefficients is the most appropriate. Specifically, their fit employed one parameter for each of p u and k,
and two for each of p l and r l . The auditory filter shapes
produced by this fit are shown in Fig. 1 as a function ofprobe level. The lower side of the filter becomes considerably broader as level increases; the upper side is invariant.
Column 7 of Table I shows rms error values ~dB! obtained with the gammachirp fit using probe-dependent models with various numbers of coefficients. The integers in columns 2–6 show the number of coefficients used for the
gammachirp parameter in that column. Following Rosen and
Baker, we used the absolute threshold value ~22.7 dB! tolimit the minimum value of P s . We also investigatedmasker-dependent models, but found, like Rosen and Baker,
that the rms errors were always much greater than those withthe probe-dependent models. Consequently, we only consider probe-dependent models in what follows. The fit without the parameter c, i.e., the gammatone fit, is shown in thelast row of Table I. The rms error is about 30% greater thanthat of the gammachirp fits, indicating that the gammatone
filter is not suitable as an asymmetric, level-dependent filter.
The rms errors in column 7 are the same for gammachirp models with between 4 and 7 variable coefficients.
This indicates that the coefficients converge even with relatively few coefficients. The rms errors with the gammachirp
are greater than those for roex models with six, seven, andeight variable coefficients ~compare columns 7 and 8!, andsmaller for models with five variable coefficients. The gammachirp model with four variable coefficients, where n isfixed to 4, produces the same rms error as the model withfive variable coefficients, where the estimated value of n is
3.89. Accordingly, the model with four variable coefficientsseems sufficient to explain the masking data. Rosen and
Baker do not report results with a four-coefficient model.
The fixed-n model also has advantages when fitting smaller
data sets and when comparing coefficients obtained with different data sets.
The coefficients for the four-coefficient model are listed
in row ‘‘LM 2000’’ in Table II. The auditory filter shapes
produced by this fit are shown in Fig. 2 as a function ofprobe level. In the fitting process, the peak frequency of theamplitude spectrum varies with level, as described previously; for clarity, however, the peak frequency is normalized
to 2000 Hz in the figure by adjusting the value of f r in Eq.
~2!. Below its peak frequency, the skirt of the gammachirp
auditory filter broadens substantially with increasing stimulus level; above its peak frequency, the skirt sharpens a little
with increasing level. These shapes are quite similar to theroex filter shapes in Fig. 1, although there are small differ-

TABLE I. Relationship between the number of filter coefficients and rms error. Columns 7 and 8 show rms~root-mean-squared! errors in dB obtained with the gammachirp filter and the roex filter when fitting theprobe-dependent model with various numbers of coefficients to all 78 data points in Rosen and Baker ~1994!.
The rms errors in column 8 are calculated from the total squared errors in Rosen and Baker ~1994!. The integers
in the first column show the total number of variable coefficients for both the gammachirp and the roex. The
integers in other columns show the number of coefficients used for the gammachirp parameter in that column:‘‘1’’ indicates a filter parameter that is constant across signal level and ‘‘2’’ indicates a linear dependence of theparameter on signal level. The symbol ‘‘-’’ indicates an r value of 2100 dB ~practically zero in linear terms!;‘‘*’’ indicates an n value of 4; ‘‘—’’ indicates no model fitted at that value. The last row shows the results
without parameter c, i.e., the gammatone fit.

414

Gammachirp

Number ofcoefficients

n

b

c

K

r

rms errorroex
rms error

10
9
8

2
2
2

2
2
2

2
2
2

2
2
1

2
1
1

1.18
1.27
1.29

1.19
—
1.19

7
6
6
5
4

2
1
2
1
*

2
2
1
1
1

2
2
2
2
2

1
1
1
1
1

-

1.33
1.33
1.33
1.33
1.33

1.19
1.20
—
1.42
—

4

1

2

0

1

-

1.72

—

J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997

T. Irino and R. D. Patterson: Gammachirp auditory filter414

B. Other data sets

FIG. 2. The gammachirp auditory filter shape as a function of probe level~30–60 dB SPL in 10-dB steps! with four variable coefficients when applied
to the masking data of Rosen and Baker ~1994!. The peak frequency isnormalized to 2000 Hz.

ences in the flatness around the peak frequency and in thevariability of the upper skirt. Unlike the roex, the upper skirtof the gammachirp has the ‘‘backward S’’ shape observed in
the dense threshold functions in Patterson ~1976!. The gammachirp filter naturally introduces the physical constraints ofrealistic filters into the estimation of the auditory filter shape.
The derived filter shapes are also in agreement with thosereported in previous studies ~Lutfi and Patterson, 1984;
Patterson and Moore, 1986; Moore and Glasberg, 1987!.
It also appears that we do not need the parameter r when
fitting the data of LM; absolute threshold is sufficient limit to
the dynamic range of the fitting process. This is another advantage of using the gammachirp fit.

Following the results in the previous subsection, we applied several probe-dependent models to the notched-noisemasking data of Lutfi and Patterson ~1984! and Moore et al.
~1990!. The models with four, five, and six variable coefficients were fitted to each set of data. As before, the modelwith four variable coefficients proved most appropriate andso we begin with it. Given their limited size, each set ofmasking data was fitted with seven different sets of initialvalues; the set that produced coefficients giving minimum
mean squared error is listed with the rms error in Table II. It
is clear that the value of b converges between 1 and 2 andthat c is always negatively correlated with probe level P s , asin the previous fits for LM at 2000 Hz. The filter shapes are
similar to the shapes in Fig. 2 in terms of change in slopewith level, except for four conditions: the filter shape is almost level independent for CP at 200 Hz and RM at 4000
Hz; the upper slope changes as much as the lower slope doesfor HM at 1000 and 4000 Hz. The last two rows in Table IIshow the means and standard deviations of the parametervalues. Since the mean coefficients are close to those for LM,a ‘‘typical’’ auditory filter set resembles those shown in Fig.
2 when the peak frequencies are normalized to unity. Thus,
the gammachirp with four variable coefficients provides areasonable summary to the masking data in these data sets,
although the rms errors are larger than those for the data setof LM.
For completeness, we also performed the gammachirp fit
with five variable coefficients ~1 n, 1 b, 2 c’s, and 1 K! andsix variable coefficients ~1 b, 2 c’s, 1 K, and 2 r’s!. Only
two of the models with five variable coefficients reduced therms error more than 5%, reductions that are negligible whencompared with the variance in the data sets. Since absolute
threshold values were not included in these fits, the modelwith level-dependent r was also applied to each set ~i.e., six

TABLE II. Rms errors and coefficients obtained with the gammachirp auditory filter when fitting a probedependent model with four variable coefficients. The first column specifies the data source by the initials of thesubject. CP represents data from Moore et al. ~1990!; HM, RL, RM, and WW represent data from Lutfi andPatterson ~1984!; LM represents data from Rosen and Baker ~1994!. The second column is probe frequency in
Hz. The third column shows rms error in dB. The remaining columns show the best coefficients for b, c, andK with n54 and r52100 ~dB!. The last two rows show the means and standard deviations for b, c, and K.

415

Subject

Frequency

rms error

b

CP
CP
CP

200
400
800

4.72
2.92
2.65

1.19
1.43
1.75

20.59
2.64
2.16

20.0097P s
20.082P s
20.070P s

2.82
21.15
24.35

HM
RL
RM
WW

1000
1000
1000
1000

4.04
4.46
2.93
3.46

1.17
1.59
1.21
1.38

8.43
4.98
5.27
3.56

20.180P s
20.146P s
20.148P s
20.098P s

23.25
211.70
27.25
26.64

LM

2000

1.33

1.68

3.38

20.107P s

26.08

HM
RL
RM
WW

4000
4000
4000
4000

4.14
5.10
4.94
2.75

1.85
1.75
1.50
1.79

6.31
5.17
0.61
4.18

20.153P s
20.182P s
20.019P s
20.110P s

27.67
214.39
23.63
25.87

mean
s.d.

—
—

—
—

1.51
0.24

3.88
2.46

20.109P s
0.057

25.73
4.51

J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997cK

T. Irino and R. D. Patterson: Gammachirp auditory filter

415

variable coefficients!. All of the derived values for the parameter r were negatively correlated with probe level P s andwere smaller than 240 dB, even when the signal level was
30 dB SPL. That is, the values are less than absolute threshold, and so, absolute threshold is a more suitable limit to the
dynamic range when fitting these data. Moreover, the rms
error was reduced more than 10% in only three cases. Thus,
parameter r does not seem necessary to explain the generalform of the masking data, and the model with four variable
coefficients seems sufficient to explain these masking data,
as well as those of Rosen and Baker.
III. SUMMARYA ‘‘gammachirp’’ function derived as an optimum auditory filter ~Irino, 1995, 1996! is shown to have an asymmetric amplitude characteristic in frequency. Using the power
spectrum model of masking, and the assumption that theasymmetry is associated with stimulus level, the amplitude
spectrum of the gammachirp was fitted to notched-noisemasking data from 12 data sets reported in 3 different studies. A probe-dependent model with four variable coefficientsis shown to provide an excellent fit to the masking data. The
resultant gammachirp filter shape is similar to that obtainedwith a six-coefficient roex filter by Rosen and Baker ~1994!.
The gammachirp has a well-defined impulse response unlike
the roex auditory filter and, thus, it is an excellent candidate
for an asymmetric, level-dependent auditory filterbank intime-domain models of auditory processing.3
ACKNOWLEDGMENTS

The authors wish to thank Brian C. J. Moore and Brian
R. Glasberg of Cambridge University, and Stuart Rosen andRichard J. Baker of University College, London for providing their notched-noise masking data, as well as fruitful discussions. The first author wishes to thank Takesi Okadomeof NTT BRL for the suggestion to study optimality of theauditory filter.

frames with a Fourier transform. There is a trade-off between
the resolution of time and the resolution of frequency in thisrepresentation. The trade-off is known as the uncertainty
principle, and Gabor ~1946! showed that the function which
satisfies minimal uncertainty in the joint time-frequency representation is a complex sinusoidal carrier with a Gaussianenvelope. This ‘‘Gabor function’’ is symmetric in time andsymmetric in frequency; moreover, the frequency bands all
have the same width in this time-frequency representation.
The spectral analysis produced by auditory filtering differs significantly from that produced by the Fourier transform: The impulse response of the auditory filter is asymmetric in time with a fast rise and a slow decay ~de Boer andde Jongh, 1975; Carney and Yin, 1988!; the amplitude spectrum of the auditory filter is definitely not Gaussian ~Patterson, 1976!, and at high sound levels, it is asymmetric withthe lower skirt shallower than the upper skirt ~Glasberg andMoore, 1990!. The gammatone function @Eq. ~1!# provides amuch better fit to auditory filtering data than the Gabor function. It is clear, however, that to the extent that the gammatone differs from the Gabor function, it does not satisfy minimum uncertainty in a joint time-frequency representation ofsound. Moreover, the bandwidth of the auditory filter increases with center frequency; in the region above about 500
Hz, it is essentially a ‘‘constant-Q system,’’ that is, bandwidth is proportional to center frequency ~Greenwood, 1990;
Glasberg and Moore, 1990!.
It is possible that the auditory system is non-optimal
because it has to satisfy some mechanical or physiologicalconstraint that is not compatible with minimal uncertainty,
and which restricts the bandwidth to be a proportion of thecenter frequency. On the other hand, it seemed reasonable,on encountering the discrepancy between optimality and auditory filtering, to explore the possibility that the auditory
system is optimal, but optimal for a different representationof sound. It is this hypothesis that led to the ‘‘scale transform’’ and the derivation of the gammachirp function.
B. Scale analysis

APPENDIX A: THE DERIVATION OF THE
GAMMACHIRP FUNCTION

The gammachirp function arose from consideration of
the contrast between the traditional representation of sound,the spectrogram, and the representation produced by auditory
filterbanks designed to mimic the spectral processing of thecochlea. The contrast is set out in Sec. A of this Appendix. It
has led to the hypothesis that the time-frequency representation of sound observed at the output of the cochlea is anintervening representation produced by the auditory system
to support a subsequent ‘‘scale transform,’’ and that thefunction that minimizes uncertainty in the time-scale representation is the gammachirp. The scale transform and the
gammachirp function are the subjects of Secs. B and C, respectively.
A. The spectrogram and auditory filtering

The spectrogram is a typical example of a joint timefrequency representation of sound. It is produced by converting successive segments of the sound wave into spectral
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J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997

Cohen ~1991, 1993! has suggested that ‘‘scale’’ is aphysical attribute of a signal just like time and frequency,
and that a time-scale representation is more appropriate than
Fourier analysis for ‘‘scaled’’ signals. A ‘‘scaled’’ signal issimply one that is compressed or extended in time relative tothe original, as when a tape recording is replayed at a rate
faster or slower than that at which it was recorded. Cohen~1991! introduced a ‘‘scale transform’’ in the form of an
orthogonal Mellin transform ~Titchmarsh, 1948! to produce
the scale representation. It is described in Sec. C. The Mellintransform converts a scaled signal into ~a! an invariant absolute distribution in the scale representation, and ~b! a value
specifying the scale value of the signal. When the speed of arecording fluctuates on playback, it has a pronounced effecton the pitch we hear, but the source of the sound is not
perceived to change. This suggests that the ‘‘scale’’ value
and the ‘‘invariant distribution’’ of the Mellin transform may
be analogous to pitch and timbre in the auditory system.
Thus, the time-scale representation of sound could have distinct advantages when analyzing systems where a vibrating
T. Irino and R. D. Patterson: Gammachirp auditory filter416

source with variable rate excites a complex resonator.
‘‘Source-filter’’ models of this sort are commonly used to
explain the production of sound by the vocal tract ~Fant,
1970! and musical instruments ~Fletcher and Rossing, 1991!.
The scale transform can be applied to a sound wave
directly. It is clear, however, that in the auditory system, thescale transform would have to be applied after auditory filtering. The wavelet transform is similar to the auditory filterbank inasmuch as it is a ‘‘constant-Q’’ system; both theenvelope and the carrier of the impulse response scale withcenter frequency in these systems. The Mellin transform converts the individual wavelets into an invariant distribution in
the scale representation. Thus, with a wavelet filterbank,when a sound is scaled, its components shift to wavelet filters that have been scaled by the same amount. So, the outputs of the scaled filters are exactly the same as the scaled
versions of the outputs of the original filters. Both the scaled
and unscaled filter outputs are transformed into the same
distribution in the scale representation. Thus, the wavelet
filterbank is ‘‘transparent’’ to the scaling of sounds, and in
this sense, the wavelet transform is optimal as a preprocessor
for the Mellin transform. This implies that the auditory filterbank would be a near optimum preprocessor for the Mellin transform.
The optimal relationship between the Mellin transformand the wavelet transform does not uniquely determine theform of the wavelet that produces minimal uncertainty in ajoint time-scale representation; Clearly, the Gabor function
does not; After Cohen ~1991, 1993! adapted Klauder’s~1980! results on affine variables in quantum mechanics toproduce the scale representation for time domain functions,
he showed that the optimal function for minimal uncertainty
in a time-scale representation has a gamma envelope and amonotonically frequency-modulated carrier. In Cohen’s case,the instantaneous frequency of the carrier starts at infinity
and converges on zero as time proceeds. This solution is notsuitable for relatively narrow-band applications like auditory
filtering. This led Irino ~1995, 1996! to introduce a frequency
shift term that makes it possible to model the bandwidth/
center-frequency function of the auditory system, and produce a narrow-band filter centered on a specific frequency.
This, in turn, led to the derivation of the gammachirp function through the optimality constraint. This, then, is the logic
for the time-scale representation of sound and the gammachirp function.

C. Mathematical derivation

E

0

~A1!

s ~ t ! t p21 dt,

where p is a complex argument. One of the important properties is
if s ~ t ! ⇒S ~ p ! , then s ~ at ! ⇒a417

where j 5 A21, and exp and ln are the exponential and natural logarithmic operators. Since u a 2p S(p) u 5 u a 2 p r u • u S(p) u ,the absolute distribution u S(p) u is not affected by a scaling of
the signal, except for the constant that specifies the scale ofthe current signal; nor is it affected when the distribution isnormalized.
b. Minimal uncertainty and operator methods

With the Mellin transform, questions concerning minimal uncertainty in a joint representation are assessed with
operator methods. They were introduced into signal processing from quantum mechanics by Gabor ~1946! because of thesimilarity in mathematical formalism. The following is a tutorial on operator methods based on the derivation of theGabor function; it is adapted from Cohen ~1991, 1993!.
Time and frequency operators are defined as T 5t andW 52j(d/dt) in the time domain. When the operator W isapplied to the function Ae j v t , the result is

S D

W Ae j v t 5 2 jd
Ae j v t 5 v Ae j v t .
dt

~A4!

Thus, for a complex exponential, the operator W introduces
the frequency term v. This is the essence of operator methods. The commutator between these operators is again an
operator; namely,

S DS D@ T ,W # 5T W 2W T 5t 2 j

d
d
2 2j
t5 j.
dt
dt

~A5!

It is easy to prove by applying this operator to the function
Ae j v t . Since the commutator is not zero, time and frequency
do not commute. Thus, time and frequency cannot be measured independently and there is uncertainty between them,and in this case, it is
Dt•D v > 21 u ^ @ T ,W # & u 5 21 u ^ j & u 5 21 ,~A6!

where ~D.!, u.u, and ^.& denote the standard deviation, the absolute value, and the average, respectively. Functions which
satisfy minimal uncertainty are solutions to the equation
~A7!

where

The Mellin transform ~Titchmarsh, 1948! of a signal,s(t) ~t.0!, is defined as
S~ p !5a 2p 5a 2 ~ p r 1 j p i ! 5a 2p r a 2 j p i 5a 2p r exp~ 2 j ln p i ! ,~A3!

~ W 2 ^ W & ! s ~ t ! 5l ~ T 2 ^ T & ! s ~ t ! ,a. The Mellin transform`where the arrow ~⇒! indicates ‘‘is transformed into’’ and a
is a real dilation constant. That is, the distribution S(p) is
just multiplied with a constant a 2p when the function s(t) is
scaled in time. If p is denoted by p r 1 j p i ,2p

S~ p !,

J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997

~A2!

l5 ^ @ T ,W # & /2~ DT ! 2 5 j/2~ Dt ! 2 .

~A8!

Using T 5t, W 52j d/dt, ^T &5^ t & , and ^W &5^v&, Eq.
~A7! is expanded as follows:

S

2j

D

j

d
s ~ t ! 1lts ~ t ! 1 ~ ^ v & 2l ^ t & ! s ~ t ! 50;
dt

d2 ^ v & s ~ t ! 5l ~ t2 ^ t & ! s ~ t ! ;dt

T. Irino and R. D. Patterson: Gammachirp auditory filter

417

S

D

d
1
^t&
s~ t !1s ~ t ! 50.
2 ts ~ t ! 1 2 j ^ v & 2
dt
2 ~ Dt !
2 ~ Dt ! 2~A9!
The nontrivial solution is

S

1
^t&
2
s ~ t ! 5a exp 2
t1 j ^ v & t
2 t 1
4 ~ Dt !
2 ~ Dt ! 2

H

5a 8 exp 2

J

S D

~A10!

c. The Mellin operator

Cohen ~1991, 1993! introduced the concept of a scale
operator into signal processing in the form:
C 5 21 ~ T W 1W T ! 5T W 2 21 j.

~A11!

Previously it had been known as the operator representing anaffine variable in quantum mechanics ~Klauder, 1980!. The
corresponding transform, that is, ‘‘the scale transform’’ ~Cohen, 1993!, isD~ c !5

A2 p

E

0

f ~ t !t2 jc21/2

dt.

~A12!

The correspondence between the scale transform and theMellin transform is revealed by setting p52 jc1 21 in Eq.
~A1!. Thus, Cohen’s scale transform is the Mellin transformwith a specific argument. In Eq. ~A12!, the argument is restricted in range; we can, however, extend it to cover theentire complex plane by the introduction of two real constants c 0 and m as follows:
p52 j ~ c2c 0 ! 1 ~ m 1 21 ! .

s ~ t ! 5at a 2 1 jc 1 exp~ 2 a 1 t1 j v 0 t !
5at a 2 exp~ 2 a 1 t ! exp~ j v 0 t1 jc 1 ln t ! ,where a is a constant. The envelope t a 2 exp(2a 1 t)is a
gamma distribution function g(t). The instantaneous frequency is v01c 1 /t; that is, a fractional function of time.
When played as a sound, the carrier would be a chirp, andhence the name ‘‘gammachirp’’ function. When c 150, Eq.
~A20! becomes a gammatone function. Thus, the gammatone
function is a first order approximation to the gammachirp
function.
APPENDIX B: THE AMPLITUDE SPECTRUM OF THE
GAMMACHIRP FUNCTION

The Fourier spectrum of the gammachirp function canbe derived analytically. For convenience, we consider a simplified version of the complex form of the gammachirp filter
in Eq. ~2!.
g c ~ t ! 5at n21 exp~ 2b 8 t ! exp~ j v r t1 jc ln t !
5at n211 jc exp~ 2b 8 t1 j v r t !

418

J. Acoust. Soc. Am., Vol. 101, No. 1, January 1997

~ t.0 ! ,~B1!

where b 8 52 p b ERB( f r ), v r 52 p f r , and the phase term f
is ignored. The Laplace transform of Eq. ~B1! isG ~ n1 jc !
$ s2 ~ 2b 8 1 j v r ! % n1 jc

~A14!

5a

G ~ n1 jc !
u s2 ~ 2b 8 1 j v r ! u n1 jc e j u • ~ n1 jc !

5a

G ~ n1 jc !
,u s2 ~ 2b 8 1 j v r ! u e• u s2 ~ 2b 8 1 j v r ! u jc e jn u
~B2!

~A15!

~A17!

n 2c uwhere u5arg$ s2(2b 8 1 j v r ) % . Thus, the absolute value is
u G C~ s !u 5

u aG ~ n1 jc ! u.
u s2 ~ 2b 8 1 j v r ! u n e 2c u

~B3!

Substituting s5 j v 5 j2 p f into Eq. ~B3! to derive the amplitude of the Fourier spectrum of the gammachirp function,~A16!

The operators in Eqs. ~A14! and ~A15! are not Hermitian
except when m50; nevertheless, (C a2^C a&! is Hermitianand, thus, the eigenvalue is real. The function that satisfies
minimal uncertainty between time and the quantity represented by the operator in Eq. ~A14! is the solution to theequation~ C a 2 ^ C a & ! s ~ t ! 5l ~ T 2 ^ t & ! s ~ t ! ,~ t.0 !

G C ~ s ! 5a

The frequency-shift term can be removed later following
consideration of the fluctuation of components at the outputof the auditory filter ~Irino, 1996!. The commutator between
time and this operator is@ T ,C a # 5 @ T ,C m # 5 @ T ,C # 5 jT .

~A20!

~A13!

Since we are concerned with signal processing by an auditory filterbank, we introduce a ‘‘frequency-shift’’ term v0
into the operator to specify the individual filters. The form of
the operator becomesC a 5T ~ W 2 v 0 ! 1 $ c 0 1 j ~ m 2 21 ! % .

ds ~ t ! 2 ~ v 0 1 j a 1 ! ts ~ t ! 1 ~ 2c 1 1 j a 2 ! s ~ t ! 50,
dt
~A19!

where a15^ t & /2(Dt) 2, a25m2 212Im^ c a & 1 ^ t & 2 /2(Dt) 2 , andc 15Re^ c a & 2c 0 , Re and Im indicate the real and imaginaryparts. The solution is

The corresponding Mellin operator is
C m 5T W 1 $ c 0 1 j ~ m 2 21 ! % .

~A18!

Equation ~A15! expands toD1~ t2 ^ t & ! 2 exp~ j ^ v & t ! ,4 ~ Dt ! 2`

l5 ^ @ T ,C a # /2~ DT ! 2 & 5 j ^ t & /2~ Dt ! 2 .

t 2j

where a and a 8 are constants. This is the ‘‘Gabor function,’’and the example shows how it was derived using the constraint that the required function satisfy minimal uncertainty
in a joint time-frequency representation.

1

where

u G C~ f !u 5

u aG ~ n1 jc ! u•e c u ,u b 8 1 j2 p ~ f 2 f r ! u n~B4!

where u5arg$ b 8 1 j2 p ( f 2 f r ) % .
1

Several auditory filters with gamma distribution envelopes and monotonically frequency-modulated ~FM! carriers have appeared recently. First,
Lyon ~1996! has reported an ‘‘all-pole gammatone filter ~APGF!’’ based on
reduction of zeros from the Laplace transform of the gammatone filter inthe s plane. In this case, the intent was to simulate basilar partition motion
T. Irino and R. D. Patterson: Gammachirp auditory filter

418

‘as a function of stimulus level. Although the impulse response of the
APGF is not mathematically equivalent to the gammachirp function, it issimilar in having a monotonic FM carrier and a gamma distribution envelope. Second, in an attempt to produce an asymmetric gammatone filter,
Baker ~1995! replaced the pure-tone carrier with a monotonic FM carrier.
Again, the result is not strictly a gammachirp function, but the impulse
response has an FM carrier and a gamma distribution envelope. Finally,
Laine and Härmä ~1996! have suggested similar filters for an auditory
filterbank on the Bark scale. The gammachirp function can be viewed asproviding the theoretical background to the larger family of auditory filterswith gamma distribution envelopes and chirp carriers.
2
In their paper Rosen and Baker ~1994! presented total-squared-error valuesof 103.6, 105.0, 110.9, 110.9, 111.0, 111.9, and 158.1 dB2, respectively, forthese conditions.
3
To construct such a filterbank, we would need to develop a mechanism tomeasure the output level of each filter on a moment to moment basis tospecify the appropriate value of c, and thus the filter’s asymmetry, at any
given moment. A mechanism of this sort has been developed by Lyon
~1982! for a nonlinear filterbank simulating cochlear mechanisms. Thus, itwould not appear to be an insurmountable problem to develop one for a
gammachirp filterbank. It is, however, beyond the scope of the present
paper.
Baker, R. ~1995!. Personal communication.
Carney, L. H., and Yin, T. C. T. ~1988!. ‘‘Temporal coding of resonances by
low-frequency auditory nerve fibers: single-fiber responses and a population model,’’ J. Neurophysiol. 60, 1653–1677.
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L. Franklin, Proc. SPIE, 1566, 109–133.
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with the Mellin transform,’’ in Proc. IEEE Int. Conf. Acoust., Speech
Signal Processing ~ICASSP-96!, 2, 981–984, Atlanta, GA.
Johannesma, P. I. M. ~1972!. ‘‘The pre-response stimulus ensemble of neurons in the cochlear nucleus,’’ in Symposium on Hearing Theory ~IPO,
Eindhoven, Holland!, pp. 58–69.
Klauder, J. R. ~1980!. ‘‘Path integrals for affine variables,’’ in Functional
Integration: Theory and Applications, edited by J. P. Antoine and E. Tirapgui ~Plenum, New York!.
Laine, U. K., and Härmä, A. ~1996!. ‘‘On the design of Bark-FAMlet Filterbanks,’’ in Proc. Nordic Acoustical Meeting, Helsinki, Finland.
Lutfi, R. A., and Patterson, R. D. ~1984!. ‘‘On the growth of masking asymmetry with stimulus intensity,’’ J. Acoust. Soc. Am. 76, 739–745.
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