Perceptually motivated wavelet packet transform for bioacousticsignal enhancement
Yao Ren,a� Michael T. Johnson, and Jidong TaoSpeech and Signal Processing Laboratory, Marquette University, P.O. Box 1881, Milwaukee,Wisconsin 53233-1881
�Received 14 December 2007; revised 18 April 2008; accepted 25 April 2008�
The presence of background noise and interfering sig-nals is a fundamental problem in the collection and analysisof bioacoustic data, regardless of the specific species understudy or the type of environment. This noise takes a varietyof forms, including ambient background noise due toweather conditions, continuous interference from nearby ve-hicular or boat traffic, or the presence of numerous nontargetvocalizations from other species and individuals. Since thedistance from the acoustic recording device to the individu-als under study can be quite large leading to significant sig-nal attenuation, interfering noise can create a substantial ob-stacle to analysis and understanding of the desiredvocalization patterns.
Common techniques to reduce noise artifacts in bioa-coustic signals include basic bandpass filters and relatedfrequency-based methods for spectrogram filtering andequalization, often incorporated directly into acquisition andanalysis tools �Mellinger, 2002�. Other approaches in recentyears have included spectral subtraction �Liu et al., 2003�,minimum mean-squared error �MMSE� estimation �Álvarezand García, 2004�, adaptive line enhancement �Yan et al.,2005; Yan et al., 2006�, and denoising using wavelets �Gurand Niezrecki, 2007�.
In comparison, there are a wide variety of advancedtechniques used for human speech enhancement, some ofwhich form the basis for the more recent bioacoustic en-hancement methods cited above. Historically the most com-mon approaches for speech enhancement have focused on
316 J. Acoust. Soc. Am. 124 �1�, July 2008 0001-4966/2008/1
spectral subtraction �Boll, 1979�, Wiener filtering �Lim andOppenheim, 1978�, and MMSE and log-MMSE estimationsusing Ephraim–Malah �EM� filtering �Ephraim and Malah,1984; 1985�. Added to this in recent years are newer methodsbased on subspace estimation and filtering �Ephraim andTrees, 1995� and wavelet decomposition �Johnson et al.,2007�.
In this paper, we introduce a new bioacoustic signal en-hancement technique which is based on a perceptually scaledwavelet packet decomposition, using spectral estimationmethods similar to those used for human speech enhance-ment. The underlying goal is to obtain higher quality andmore intelligible enhanced signals through the use of moreperceptually meaningful frequency representations. Thismethod is robust across a wide range of species, needingonly fmin and fmax frequency boundary parameters to gener-alize for application to a new species of interest.
The new method is compared to a variety of other en-hancement and denoising techniques, including simple band-pass filtering, spectral subtraction, Wiener filtering, and theEM log-MMSE estimation. To evaluate and compare its ap-plicability across a variety of species, the method is appliedto the animals of the order Passeriformes �ortolan bunting�,Primates �rhesus monkey�, and Cetaceans �humpbackwhale�. Evaluation is done by using both signal-to-noise ra-tio �SNR� and segmental SNR �SSNR�, which is known to bea more perceptually relevant quality measure for humanspeech �Deller et al., 2000�.
Bandpass filtering removes signal energy outside of aspecified frequency range. This can be applied in either thetime domain or the frequency domain �e.g., applied to aspectrogram� and is effective primarily in cases where sig-nals are predominately narrow band and are well separatedfrom the noise spectrum.
B. Spectral subtraction
Spectral subtraction �Boll, 1979� was one of the firstalgorithms applied to the problem of speech enhancement. Itis based directly on the additive noise model:
y�n� = x�n� + d�n� , �1�
where y�n�, x�n�, and d�n� denote the noise-corrupted inputsignal, clean signal, and additive noise signal, respectively.The noise spectrum is estimated from the Fourier transformmagnitude of a silence region in the wave form, so that foreach frame of the signal, an estimate for the clean signal inthe frequency domain can be given directly as
X��� = ��Y���� − �D�����ej�y���, �2�
where �y��� is the phase component of the noisy signal,used under the assumption that the spectral phase is muchless important than the spectral magnitude for reconstruction.
Note that application of Eq. �2� may result in negativemagnitude values, which are typically set to zero. This oftenresults in some processing artifacts that are usually describedby listeners as “musical tones.” The presence of such arti-facts is one disadvantage of the spectral subtraction ap-proach.
C. Wiener filtering
Wiener filtering is conceptually similar to spectral sub-traction but replaces the direct subtraction with a mathemati-cally optimal estimate for the signal spectrum in a MMSEsense �Lim and Oppenheim, 1978�.
The frequency domain formulation of the Wiener filter isgiven as
H��� =Sxx���
Sxx��� + Sdd���, �3�
where H��� is the desired filter response and Sxx��� andSdd��� are power spectral densities �PSDs� of the desiredclean signal and noise. Since these two PSDs are unknown,this filter cannot be determined directly and instead needs tobe realized in an iterative fashion. In particular, Sdd��� isestimated from a silence region and Sxx��� is initialized fromthe noisy wave form and then updated from the output of thefilter after each iteration. This process is repeated either afixed number of times or until a convergence criterion is
reached.
J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
D. Ephraim–Malah filtering
The Wiener filter is an optimal linear estimator of theclean signal spectrum in a MMSE sense. Ephraim and Malahextended this idea by deriving an optimal nonlinear estimatorof the clean spectral amplitude. This estimator assumes thatthe real and imaginary parts of the spectral magnitude have azero-mean Gaussian probability density distribution and arestatistically independent. Under this statistical model, a shorttime spectral amplitude estimator was derived by using theMMSE optimization criteria �Ephraim and Malah, 1984�.This work was then modified to use log spectral amplitude�LSA� rather than spectra as an optimization criterion�Ephraim and Malah, 1985� since the log spectral distance isa more perceptually relevant distortion criteria, resulting inimproved overall enhancement results. This estimator,known as the EM filter, can be summarized by using thefollowing estimation formula for the clean signal Fourier
transform coefficient Ak in each frequency bin:
Ak =�k
1 + �ke�1/2���k
� �e−i/t�dtRk, �4�
In this equation, �k=�x�k� /�d�k�, �k= ��k / �1+�k���k, and�k=Rk
2 /�d�k�, where Rk is the noisy speech Fourier transformmagnitude in the kth frequency bin, and �d�k� and �x�k� arethe average noise and signal powers in each bin. Similar tothe spectral subtraction method, the noise power is estimatedfrom silence regions in the wave form, while �x�k� is a mov-ing average of spectrally subtracted noisy spectra �Rk
2
−�d�k��. The a priori SNR �k is estimated via the EM well-known “decision-directed method,” which is updated fromthe previous amplitude estimate using a forgetting factor � asfollows:
�k�n� = �Ak
2�n − 1��d�k,n − 1�
+ �1 − ��P��k�n� − 1� , �5�
where the indicator function P is given by
P��k�n� − 1� � ��k�n� − 1, �k�n� − 1 0
0 otherwise.� �6�
The key characteristics of this estimator are that it tendsto do less enhancement �i.e., less change to the noisy signalspectrum� when the SNR is high, and that musical noiseartifacts are significantly reduced.
E. Wavelet denoising
Spectral subtraction, Wiener filtering, and EM filteringare all based on the same mathematical tool, the short timeFourier transform �STFT�, with the waveform divided intoshort frames during which the signal is assumed to be sta-tionary. The STFT is a compromise between time resolutionand frequency resolution: a shorter frame length results in abetter time resolution but poorer frequency resolution. Thewavelet transform �WT� by comparison has the advantage ofimplicitly using a variable window size for different fre-quency components. This often results in better handling ofbroadband nonstationary signals, including speech and bioa-
coustic data.
Ren et al.: Bioacoustic signal enhancement 317
avel
Whereas the STFT is a function of frequency for eachindividual signal frame, the WT is a function of two vari-ables, time and scale. Scale is used rather than frequencybecause depending on the wavelet basis being used, eachscale may actually represent information across a range offrequencies. Like the Fourier transform, the WT has bothcontinuous WT and discrete WT �DWT� implementations. ADWT can be efficiently implemented by using a quadraturemirror filter decomposition, resulting in scales that are pow-ers of 2, called a dyadic transform. A further generalizationof the DWT is the wavelet packet transform �WPT�. In theWPT, the filtering process is iterated on both the low fre-quency and high frequency components, whereas the DWTiterates only on the low frequency components. Filter de-composition structures for the DWT and WPT are shown inFig. 1. In the decomposition tree, each node is labeled �l ,n�,where l is the decomposition level and n represents a sub-band node index. The root of the tree, �l ,n�= �0,0�, refers tothe entire signal space. The left and right branches denotelow-pass and high-pass filterings followed by 2:1 downsam-pling, respectively.
The application of wavelets for signal enhancement,sometimes referred to as denoising, is a three step procedureinvolving wavelet decomposition, wavelet coefficient thresh-olding, and wavelet reconstruction. Given an appropriatechoice of the wavelet basis function, the signal energy willbe concentrated in a small number of relatively large coeffi-cients while ambient noise will be spread out, allowing co-efficients to be thresholded.
Threshold selection and implementation are two factorswhich significantly impact wavelet denoising methods. Com-mon methods include hard, soft, and nonlinear thresholdingapproaches. Hard thresholding sets all coefficient values be-neath the threshold to zero, leaving the others unchanged�Jansen, 2001�; soft thresholding additionally reduces all co-efficient values to maintain continuity; while nonlinearthresholding typically enforces a smoothness constraint onthe coefficient mapping function as well. Typical thresholdselection methods include universal thresholding and theStein unbiased risk estimator �Donoho, 1995�, both imple-mented by using soft thresholding.
Recently, the EM suppression rule �Ephraim and Malah,
FIG. 1. �a� Discrete WT. �b� W
1984� for speech enhancement has been applied to the wave-
318 J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
let domain as a more advanced time-varying thresholdingapproach �Cohen, 2001�. This method helps reduce the “mu-sical noise” artifacts caused by uniformly applied thresholds.
III. PROPOSED METHOD
The method introduced here is based on a modifiedwavelet packet decomposition using a MMSE coefficient es-timation for thresholding. The key element of the techniqueis the use of the Greenwood warping function to determinethe WPT decomposition structure based on a perceptuallymotivated frequency axis.
Greenwood �1961� has shown that many land andaquatic mammals perceived frequency on a logarithmic scalealong the cochlea, which corresponds to a nonuniform fre-quency resolution. This relationship can be modeled by theequation
A�10�x − k� , �7�
where �, A, and k are species-specific constants and x is thecochlea position. Transformation between true frequency fand perceived frequency fp can be obtained through the fol-lowing equation pair:
Fp�f� = �1/��log10�f/A + k� , �8�
Fp−1�fp� = A�10�fp − k� . �9�
The constants �, A, and k can be found if frequency-cochlear position data are available. However, since cochlearinformation has never been measured for many species, anapproximate solution is needed. Lepage �2003� has shownthat k can be estimated as 0.88 based on both theoreticaljustification and experimental data acquired on a number ofmammalian species. By assuming this value for k, � and Acan be solved for a given approximate hearing range,fmin– fmax, of the species �Clemins, 2005; Clemins andJohnson, 2006; Clemins et al., 2006�:
A =fmin
1 − k, �10�
� = log10 fmax + k . �11�
et packet decomposition tree.
A
Ren et al.: Bioacoustic signal enhancement
Thus, a frequency warping function can be constructedby using the species-specific values of fmin and fmax.
A perceptually motivated WT can be designed to mimicthe auditory frequency scale by using decomposition criticalbands. This implementation was originally proposed byBlack for coding �Black and Zeytinoglu, 1995� and has beenwidely used for perceptual speech enhancement �Cohen,2001; Fu and Wan, 2003; Shao and Chang, 2006�. To gener-alize this technique to bioacoustic signal enhancement, wepropose to decompose a wavelet packet tree into the criticalbands with respect to the species-specific Greenwood fre-quency warping curve.
Figure 2 shows an approximation of the Greenwoodscale by critical-band WPD for three distinct species: ortolanbunting �Emberiza hortulana� downsampled to 20 kHz,rhesus monkey �Macaca mulatta� downsampled to 20 kHz,and the humpback whale �Megaptera novaeanglia� sampledat 4 kHz. The corresponding decomposition trees are illus-trated in Fig. 3. The perceptual WPD splits the frequency
0 5 10 15 20 25 30 35 400
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Critical band
Ce
nte
rfr
eq
ue
ncy
(Hz)
(a)
0 5 100
200
400
600
800
1000
1200
1400
1600
1800
2000
C
Ce
nte
rfr
eq
ue
ncy
(Hz)
FIG. 2. Center frequencies of the Greenwood scale �solid line� and WPD
range corresponding to different species data into critical
J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
bands: ortolan bunting, 0 Hz–10 kHz, 36 critical bands;rhesus monkey, 0 Hz–10 kHz, 30 critical bands; humpbackwhale, 0 Hz–2 kHz, 31 critical bands. The bands are estab-lished automatically by optimally matching the subband cen-ter frequencies to the perceptual scale curve in the meanerror sense. For the Greenwood scale calculation, the fmin
and fmax used in Eqs. �10� and �11� are 400 and 7200 Hz forthe ortolan bunting �Edward, 1943�, 20 and 42 000 Hz forthe rhesus monkey �Heffner, 2004�, and 2 and 6000 Hz forthe humpback whale �Helweg, 2000�.
Given this perceptual decomposition structure, a MMSEestimator for performing thresholding can be derived in thewavelet domain �Cohen, 2001; Cohen and Berdugo, 2001�.Using an additive time-domain model, the resulting waveletdomain model is
Yl,n�k� = Xl,n�k� + Dl,n�k� , �12�
where Yl,n= �y ,l,n,k�, Xl,n�k�= �x ,l,n,k�, Dl,n�k�= �d ,l,n,k�, k
is the index of the coefficients in each subband, l is the
Ren et al.: Bioacoustic signal enhancement 319
decomposition level, n is the node index, and l,n,k is thescaled and shifted mother wavelet. The notation �x ,� repre-sents the WT of signal x by using as the mother wavelet.
The optimally modified LSA estimator �Cohen and Ber-dugo, 2001� is used to perform wavelet denoising. Under thisapproach, the clean speech wavelet packet coefficients areestimated by using a MMSE criterion under the assumptionsthat both speech and noise are complex Gaussian variables.Speech presence uncertainty is also incorporated by usingthe hypothesis testing framework given by
sy signals, and signals enhanced by bandpass filtering, spectral subtraction,column is for white noise and the right is for environment noise�.
noi
proaches were applied to ortolan bunting �Emberiza hortu-
Ren et al.: Bioacoustic signal enhancement 321
left
lana�, rhesus monkey �Macaca mulatta� and humpbackwhale �Megaptera novaeanglia�. Norwegian ortolan buntingvocalization data were collected from County Hedmark, Nor-way in May of 2001 and 2002 �Osiejuk et al., 2003�. Rhesusdata were recorded on the island of Cayo Santiago, PuertoRico by Joseph Solitis and John D. Newman �Li et al.,2007�. Humpback whale data �Payne and McVay, 1971� wasprovided by MobySound �Mellinger and Clark, 2006�, a da-tabase for research in automatic recognition of marine animalcalls. These data were collected in March 1994 off the northcoast of the island of Kauai, HI. Ten clean vocalizations fromeach species were segmented from the original recording
FIG. 5. Spectrograms of rhesus monkey signals: Clean signal, −10 dB SNRWiener filtering, EM log-MMSE filtering, and perceptual WPT filtering �the
data.
322 J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
Both white noise and true environment noise were addedto the clean data at SNR levels of −15, −10, −5, 0, +5, and+10 dB. The environment noise came from ambient noiseregions of appropriate domain recordings for each species,spectrally flattened with a low order filter to preserve thebasic noise characteristics while ensuring that the energy isspread through the entire frequency band. For the rhesusmonkey vocalizations, background noise was taken from aVervet monkey data set �Seyfarth and Cheney, 2004�. For theortolan bunting vocalizations, background noise came di-rectly from the data set. For the humpback whale, marinenoise was taken from a Beluga whale vocalization data set
y signals, and signals enhanced by bandpass filtering, spectral subtraction,column is for white noise and the right is for environment noise�.
nois
�Scheifele et al., 2005�, downsampled to 4000 Hz.
Ren et al.: Bioacoustic signal enhancement
lumn
Based on visual examination of the clean data from Figs.4–6, tight passbands are chosen around the vocalizations.Selected ranges are 2600–5600, 1000–10 000, and200–2000 Hz for the ortolan bunting, rhesus monkey, andhumpback whale data, respectively. For the spectral subtrac-tion, Wiener filter, and EM filter approaches, the signal isdivided into 32 ms windows with 75% overlap betweenframes. This frame length was chosen empirically, as it issufficiently long for good spectral estimation in each framebut not so long as to affect temporal change in the signals,and adjustments to this value cause only minor changes tothe overall enhancement results. Frequency analysis is doneusing a Hanning window and noise estimation is accom-
FIG. 6. Spectrograms of humpback whale: Clean signal, −10 dB SNR noisyfiltering, EM log-MMSE filtering, and perceptual WPT Filtering �the left co
plished using the first three frames of the signal. For wavelet
J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
analysis, the discrete Meyer wavelet is used as the motherwavelet, which was chosen to provide good separation ofsubbands due to their regularity property �Cohen, 2001�. Thedecomposition was done as illustrated in Fig. 3. The forget-ting factor � used in Eqs. �5� and �18� isset to 0.98 for the EM filter and 0.92 for the wavelet denois-ing.
SNR and SSNR are used as objective measurement cri-teria for all sets of experiments. SSNR is computed by cal-culating the SNR on a frame-by-frame basis over the signaland averaging these values. This permits the measure to as-sign equal weights to the loud and soft portions of the signal,
als, and signals enhanced by bandpass filtering, spectral subtraction, Wieneris for white noise and the right is for environment noise�.
sign
Ren et al.: Bioacoustic signal enhancement 323
e at −
which has been shown to have a higher correlation with per-ceived quality in human speech evaluation �Deller et al.,2000�. The formulas for SNR and SSNR are
SNR = 10 log10�nx2�n�
ˆ 2 , �19�
-15 -10 -5 0 5 10-10
-5
0
5
10
15
20
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Ortolan Bunting
Band-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
-15 -10 -5 0 5 10-10
-5
0
5
10
15
20
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Rhesus Monkey
Band-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
-15 -10 -5 0 5 10-10
-5
0
5
10
15
20
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Humpback Whale
Band-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
FIG. 7. SNR and SSNR results for white nois
�n�x�n� − x�n��
324 J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
SSNR =1
M�j=0
M−1
10 log10� �n=Nj+1
N�j+1�x2�n�
�x�n� − x�n��2� , �20�
where M is the number of frames, each of length N, and x�n�ˆ
and x�n� are the original and enhanced signals, respectively.
Ren et al.: Bioacoustic signal enhancement
oise
For visualization, spectrograms of the enhanced signalsfor the white noise and environment noise conditions at−10 dB SNR can be seen in Figs. 4–6.
SNR and SSNR results for the white noise and environ-
-15 -10 -5 0 5 10-10
-5
0
5
10
15
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Ortolan BuntingBand-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
-15 -10 -5 0 5 10-10
-5
0
5
10
15
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Rhesus MonkeyBand-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
-15 -10 -5 0 5 10-10
-5
0
5
10
15
20
Input SNR (dB)
SN
RIm
pro
ve
me
nt(d
B)
Humpback Whale
Band-pass filter
Spectral Subtraction
Wiener Filter
Ephraim Malah
U-PWT
P-PWT
FIG. 8. SNR and SSNR results for environment n
ment noise are shown in Figs. 7 and 8. The SNR and SSNR
J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
values are given as amount of improvement over the originalinput noisy values. The methods shown in these figures in-clude bandpass filtering, spectral subtraction, Wiener filter-ing, EM filtering, the proposed perceptual wavelet packet
transform �P-WPT�, as well as a uniform band wavelet
Ren et al.: Bioacoustic signal enhancement 325
packet transform �U-WPT�, which is identical to the pro-posed method except that it utilizes uniformly spaced fre-quency bands rather than the perceptual scaling.
From reviewing the spectrograms and the SNR andSSNR plots, several conclusions can be drawn. It is clear thatthe proposed perceptual wavelet denoising method and theEM filtering method have the best overall performance inboth the white noise and the environment noise conditions.The proposed method shows better enhancement perfor-mance for the higher noise �lower original SNR� cases, inparticular. By comparing the SNR improvement to the SSNRimprovement in Figs. 7 and 8, it can be seen that the SSNR,which is generally considered to be a more perceptuallymeaningful metric, shows greater superiority for the pro-posed method over the other methods than does SNR.Wiener filtering and spectral subtraction have moderate en-hancement performance overall, while bandpass filtering re-sults are a little sporadic, giving generally moderate resultswith good results in a few specific environment cases. Spe-cifically, as expected, bandpass filtering works relatively wellin the ortolan case where the vocalization frequency range isnarrow and has limited overlap with the environment noisespectrum. By comparing the P-WPT and U-WPT results, itcan be seen that the use of the perceptual scale has littleoverall impact. In the white noise case, the SNR is slightlyhigher for the uniform scaling, and SSNR measures showlittle difference. For environmental noise, the SNR is againslightly higher for the uniform scaling, and SSNR is againsimilar, showing a slight benefit for the perceptual scaling intwo of the three examples. Under the noisiest conditions, thetwo wavelet-based enhancement techniques significantly out-perform all of the baseline methods.
One interesting thing to note is that each of the differentenhancement methods has unique characteristics, as seen inthe spectrograms of Figs. 4–6. Bandpass filtering has theexpected look, keeping all noise in the target range andeliminating nearly everything out of band. Spectral subtrac-tion shows some temporal streaking due to the fact that thenoise spectrum being removed is fixed. Wiener filtering andEM filtering have similar looks, except that the EM providesbetter overall results. The proposed method has the bestnoise removal but can also be seen to possess an artifact�most noticeable in Fig. 5�, seen as a faint reflection of theprimary signal. This artifact, which is not audible and doesnot contain enough energy to significantly impact the SNR orSSNR metrics, illustrates some of the processing differencesbetween a frequency domain approach such as the EM and awavelet domain approach such as the proposed method. Be-cause the mother wavelet used for analysis is somewhatbroadband, each of the nodes in the decomposition treesshown in Fig. 3 contains more than a single frequency com-ponent. Thus the nodes that are given primary emphasis forreconstruction have energy at more than one frequency.However, since the nature of this wavelet representation isalso more compact, coefficients not given primary emphasiscan be more strongly thresholded, yielding less energythroughout the entire background frequency range, as canalso be seen in the spectrograms. The selection of the mother
wavelet also impacts the degree of this artifact. The overall
326 J. Acoust. Soc. Am., Vol. 124, No. 1, July 2008
effect is that while the residual noise for the EM and percep-tual wavelet approaches have similar total energy �with theperceptual wavelet having a little less in high noise situa-tions�, this residual noise in the EM approach is spread moreevenly across the frequency range, while in the perceptualwavelet approach, it is more concentrated.
V. CONCLUSIONS
Enhancement techniques taken from the field of speechprocessing have been generalized and applied to noise reduc-tion of bioacoustic vocalizations. Four baseline methods, in-cluding spectral subtraction, Wiener filtering, and EM filter-ing, as well as simple bandpass filtering, were compared to anew technique based on perceptual wavelet decomposition.Results indicate improved performance of the new method,particularly for the most noisy conditions. The new approachcan be easily applied to any species, requiring only upperand lower frequency limits for the species to create the ap-propriate Greenwood function frequency warping curve.
ACKNOWLEDGMENTS
This material is based on work supported by NationalScience Foundation under Grant No. IIS-0326395. The au-thors also want to express their thanks to Joseph Solitis andJohn D. Newman for providing the rhesus monkey vocaliza-tions, T. S. Osiejuk for providing the ortolan bunting vocal-izations, and Mobysound for providing the humpback whalevocalizations.
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