research article evaluation of seven time-frequency

Research ArticleEvaluation of Seven Time-Frequency Representation AlgorithmsApplied to Broadband Echolocation Signals

Josefin Starkhammar1 and Maria Hansson-Sandsten2

1Department of Biomedical Engineering, Lund University, P.O. Box 118, 221 00 Lund, Sweden2Centre for Mathematical Sciences, Mathematical Statistics, Lund University, P.O. Box 118, 221 00 Lund, Sweden

Correspondence should be addressed to Josefin Starkhammar; [email protected]

Received 22 April 2015; Accepted 23 June 2015

Academic Editor: Marc Thomas

Copyright © 2015 J. Starkhammar and M. Hansson-Sandsten. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

Time-frequency representation algorithms such as spectrograms have proven to be useful tools in marine biosonar signal analysis.Although there are several different time-frequency representation algorithms designed for different types of signals with variouscharacteristics, it is unclear which algorithms that are best suited for transient signals, like the echolocation signals of echolocatingwhales. This paper describes a comparison of seven different time-frequency representation algorithms with respect to theirusefulness when it comes to marine biosonar signals. It also provides the answer to how close in time and frequency two transientscan be while remaining distinguishable as two separate signals in time-frequency representations. This is, for instance, relevant instudies where echolocation signal component azimuths are compared in the search for the exact location of their acoustic sources.The smallest time difference was found to be 20 𝜇s and the smallest frequency difference 49 kHz of signals with a −3 dB bandwidthof 40 kHz. Among the tested methods, the Reassigned Smoothed Pseudo Wigner-Ville distribution technique was found to be themost capable of localizing closely spaced signal components.

1. Introduction

The present paper discusses the advantages and disadvan-tages of seven time-frequency representation algorithms ofshort transient acoustic signals such as the echolocationsignals fromdolphins. Previous studies have shown that time-frequency representations of transient biosonar signals arevaluable tools when analyzing details in the signals generatedby for instance bottlenose dolphins Tursiops truncatus [1,2]. However, it is unclear from the literature which time-frequency representation method is best suited for echoloca-tion signals with respect to their inherent properties (signallength, signal to noise levels, frequency component composi-tions, etc.) or which limitations the different methods have.

This review of methods aims to serve twomain purposes.First and foremost it is outlined as a guide to help researcherschoose the time-frequency distribution algorithm that isbest suited for their specific application in biosonar studies.Secondly, it aims to investigate how close together in time

and frequency two transients, with only slightly differenttiming and/or frequency content, can be from each otherwhile remaining distinguishable by currently existing time-frequency representation algorithms [3, 4].

This smallest distinguishable time difference is relevant infor example marine biosonar studies when comparing com-ponent azimuths of echolocation signals (“clicks”) in differentpositions within the sonar beam. Such studies would, forinstance, be useful when trying to determine the propagationpath of echolocation clicks inside a dolphin’s head andwhen trying to establish whether certain echolocation signalsoriginate from one or two sound sources. Reports by Dormer[5], Amundin and Andersen [6], Madsen et al. [7], and Auet al., [8] suggest that the echolocation beam is generatedby only one source, the right set of phonic lips. Lammersand Castellote [9] suggest that beluga whales (Delphinapterusleucas) may generate their echolocation clicks by simultane-ous activation of both the right and left pairs of phonic lips.Cranford et al. [10] indicate that either the left or the right

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2015, Article ID 342503, 13 pageshttp://dx.doi.org/10.1155/2015/342503

2 Advances in Acoustics and Vibration

set of phonic lips or simultaneous activation of both sets ofphonic lips present in the animals may produce echolocationsignals.

The application of adequate time-frequency representa-tion methods will also be useful in studies of the distributionof the frequency components within the beam, as wellas in investigations of the internal beam steering abilityof echolocating animals [11–13]. Future studies comparingazimuths and frequency contents of different parts of theecholocation beam are likely to be useful in these areasof biosonar research. However, the smallest distinguish-able time difference of transients must first be establishedtheoretically. These results are provided in the presentpaper.

The signal processing methods have been evaluated afterapplication to echolocation-inspired synthetic test signalswhere the parameters are based on previously reported mea-surements and theories regarding the generation of dolphinecholocation [8–11, 14]. As a reference, the methods havealso been applied to measurement data recorded from abottlenose dolphin (Tursiops truncatus) performing a targetdetection task [15].

2. Method

2.1. Chosen Time-Frequency Representation Methods. Thetime-frequency representation methods included in thisstudy are Spectrogram (Sp), Wigner-Ville distribution (WV),Choi-Williams distribution (CW), Reassigned MultitaperSpectrogram (RMSp), Locally Stationary Process MultitaperSpectrogram (LMSp), Reassigned Smoothed PseudoWigner-Ville distribution (RSmWV), and SpectrogramMultiplication(SpM). All implementations of the algorithms are made bythe authors.

The evaluation is based on holistic visual judgments ofthe ability of the time-frequency representations to localizethe signal components rather than on individual quantifiedparameters. This strategy was chosen after careful considera-tion and was based on the fact that an analysis that is strictlybased on quantified parameters tends to narrow down theperspective of the algorithms’ abilities to visualize the signalsin an intuitively interpretable way.This limitation was largelyavoided by the chosen method.

The Fractional Fourier Transform has been applied toecholocation signals before [1]. However, we chose herenot to include the Fractional Fourier Transform in thealgorithm comparison study since others have shown thatits implementation is problematic. For instance, Bultheel andMart́ınez Sulbaran [16] demonstrated that the different waysof implementing the Fractional Fourier Transform did notgive coherent results.

2.2. Spectrogram (Sp). Traditional spectrum analysis tech-niques characterize the frequency content of signals withoutproviding any information of when in time the differentfrequency components of the signal appear. However, thespectrogram divides the signal into several shorter sections,where a regular spectrum is estimated from each section,

revealing inwhich shorter time interval the different frequen-cies occur. The spectrogram of a signal 𝑥(𝑡) is defined as

𝑆 (𝑡, 𝑓) = [∫

∞

−∞

𝑥 (𝑡1) ℎ (𝑡1 − 𝑡) 𝑒−𝑗2𝜋𝑓𝑡1𝑑𝑡1]

2, (1)

where the window function ℎ(𝑡), centered at time 𝑡, ismultiplied with the signal 𝑥(𝑡) before the Fourier transformand squared absolute value (see [3]). There are severaladvantages of using the spectrogram, for example, a fastimplementation thanks to the fast Fourier transform (FFT)and a straightforward interpretation. In the ideal case, thewindow and thereby the time slot should be very short inorder to clearly reveal the exact time when the frequenciesstart and stop. However, a short time slot gives a poorfrequency resolution, with the resultant problem that wecannot differ between closely spaced frequencies. Therefore,each application requires a reasonable compromise betweenhigh time resolution and high frequency resolution. Thelength (and shape) of the window function ℎ(𝑡) determinesthe resolution in time and frequency. Usually, a propertrade-off between time and frequency resolution is foundby choosing a window length of roughly the same size asthe length of the signal. The 3 dB bandwidth will then beapproximately (2/window length) ∗ 𝐹

𝑠, which is also the

best achievable frequency resolution (with 𝐹𝑠as the sampling

frequency).

2.3. Wigner-Ville Distribution (WV). The Wigner-Ville dis-tribution is based on the Fourier transform of the so-called instantaneous autocorrelation, which provides the bestpossible concentration in time as well as frequency (see [3]).Usually, the expression for the Wigner-Ville distribution isreferred to as

𝑊(𝑡, 𝑓) = ∫

∞

−∞

𝑥(𝑡 +𝜏

2)𝑥∗

(𝑡 −𝜏

2) 𝑒−𝑗2𝜋𝑓𝜏

𝑑𝜏, (2)

where 𝑥(𝑡) is the measured signal or the corresponding ana-lytical signal. However, this representation can only be usedfor simple signals, consisting of a single component in thetime-frequency plane.Withmore than one component, crossterms appear. These are located in between the componentsin the time-frequency plane and can be twice as large as thetrue signal components.The cross terms show up between allseparate components and if the signal is disturbed by highlevels of noise, which usually generates many time-frequencycomponents and therefore also many cross terms, it becomesdifficult to interpret the Wigner-Ville distribution.

The implementation we have made for this paper isbased on the discrete-time, discrete-frequency representationdescribed in Chapter 6 of Boashash [4].

2.4. Choi-Williams Distribution (CW). Various kernels forsmoothing of the Wigner-Ville distribution have been pro-posed to reduce cross terms. The smoothing is formulated as

𝑊CW (𝑡, 𝑓) = 𝑊(𝑡, 𝑓) ∗ ∗𝜙 (𝑡, 𝑓) , (3)

where 𝜙(𝑡, 𝑓) is the smoothing kernel and ∗∗ denotesdouble convolution (smoothing), in time as well as frequency.

Advances in Acoustics and Vibration 3

Possibly the most applied method is the Choi-Williamsdistribution [17], also called the exponential distributionwhere

𝜙 (𝑡, 𝑓) = ∬

∞

−∞

𝑒−]2𝜏2/𝛼

𝑒𝑗2𝜋(]𝑡−𝜏𝑓)

𝑑𝜏 𝑑] (4)

and 𝛼 is a smoothing parameter to be chosen. The Choi-Williams distribution gives a reduction of the cross terms butkeeps most of the concentration of the auto-terms, when thesignal components are located at different frequencies as wellas different times. However, for two components located atthe same frequency, or at the same time, the cross terms willstill be more or less present and the remaining amplitudeof the cross terms could cause a misinterpretation of thedistributions.

The Choi-Williams kernel is implemented using the sam-pled discrete-time and discrete-frequency form presented inChapter 6 of Boashash [4].

2.5. Reassigned Multitaper Spectrogram Using Hermite Func-tions (RMSp). The concept of multitapers replaces the cal-culation of the two-dimensional convolution between thesmoothing kernel and the Wigner-Ville distribution with so-called multitaper spectrograms, defined as

𝑆MT (𝑡, 𝑓) =1𝐼

𝐼

∑

𝑖=1𝑆𝑖(𝑡, 𝑓) , (5)

where 𝑆𝑖(𝑡, 𝑓) is the spectrogram using the window ℎ

𝑖(𝑡)

and 𝐼 is the number of multitapers. Using Hermite func-tions as windows has been shown to give the best time-frequency localization and orthonormality in the time-frequency domain. Orthonormality leads to optimal reduc-tion of variance if the disturbances can be defined as whiteGaussian noise, which results in a better estimate when thesignals are heavily disturbed by noise. Another advantage ofthe multitaper technique is its easy and fast implementation.The multitaper spectrogram smoothes the time-frequencyrepresentation resulting in a lower time-frequency resolutionand therefore the reassignment technique is applied as pro-posed in Xiao and Flandrin [18].

2.6. Locally Stationary Process Multitaper Spectrogram(LMSp). A locally stationary process is here designed tohave a covariance function that is a multiplication of acovariance function of a stationary process

𝑟 (𝜏) = 𝑒−𝛽(𝜏

2/8) (6)

and a time-variable function

𝑞 (𝑡) = 𝑒−𝑡

2/2 (7)

as

𝑟LSP

(𝑡, 𝜏) = 𝑞 (𝑡) ⋅ 𝑟 (𝜏) . (8)

The parameter 𝛽 is a design parameter. The process is non-stationary with properties suitable for modeling measured

signals that, for example, have a transient amplitude behaviorlike the echolocation signals of odontocetes. For such signals,themean square error optimal smoothing kernel and the cor-respondingmultitaper spectrogram representation is definedaccording to

𝑆LSP (𝑡, 𝑓) =𝐼

∑

𝑖=1𝜌𝑖𝑆𝑖(𝑡, 𝑓) , (9)

where 𝑆𝑖(𝑡, 𝑓) is the spectrogram using the 𝑖th Hermite

function. The parameters 𝜌𝑖, 𝑖 = 1 ⋅ ⋅ ⋅ 𝐼, correspond to a set

of weighting factors that are defined depending on the valueof the model-parameter 𝛽 [19].

2.7. Reassigned Smoothed Pseudo Wigner-Ville Distribution(RSmWV). The reassignment principle has existed for sometime but was not applied to any large extent until Auger andFlandrin reintroduced the method [20]. The idea behind thereassignment is to first reduce the cross terms by smoothingof theWigner distribution and then recapture the localizationof a single component by reassigning its mass to the center ofgravity. In RSmWV, the smoothing is defined as

𝑊SP (𝑡, 𝑓) = 𝑔1 (𝑡) ∗ 𝑊 (𝑡, 𝑓) ∗ 𝐺2 (𝑓) , (10)

where 𝑔1(𝑡) and 𝐺2(𝑓) are separate smoothing windowfunctions for time and frequency, respectively. This so-calledseparable kernel replaces the 2D convolution of the quadratictime-frequency representation with two 1D convolutions,which might simplify the choice of suitable parameters inthe design of the windows. In this form, the spectrogramis easily seen as a weighted sum of all the Wigner-Villedistribution values at the neighboring points. The “mass”at these points should be assigned to the centers of gravity,which are calculated as

𝑡SP (𝑡, 𝑓) = 𝑡 −(𝑡 ⋅ 𝑔1 (𝑡)) ∗ 𝑊 (𝑡, 𝑓) ∗ 𝐺2 (𝑓)

𝑊SP (𝑡, 𝑓),

𝑓SP (𝑡, 𝑓) = 𝑓−𝑔1 (𝑡) ∗ 𝑊 (𝑡, 𝑓) ∗ (𝑓 ⋅ 𝐺2 (𝑓))

𝑊SP (𝑡, 𝑓).

(11)

The reassignment is then made according to

RSSP (𝑡, 𝑓) = ∬

∞

−∞

𝑊SP (𝑢, V) 𝛿 (𝑡 − 𝑡SP (𝑢, V))

⋅ 𝛿 (𝑓 −𝑓SP (𝑢, V)) 𝑑𝑢 𝑑V,(12)

where 𝛿 is the Dirac impulse which reassigns (moves)the “mass” of 𝑊SP(𝑢, V) to the time and frequency valuesspecified by 𝑡SP(𝑢, V) and 𝑓SP(𝑢, V). This leads to perfectlylocalized signals in the case of single component chirp signals,frequency tones, and impulse signals.

We have implemented the discrete time-frequency algo-rithm called MSPWVD in Auger and Flandrin [20] usingthe smoothing parameters 𝑀 and 𝑁. The time-smoothingwindow 𝑔

1(𝑡) has the length 2𝑀 − 1 samples (approximately

a time-smoothing width of (2𝑀 − 1)/𝐹𝑠sec). The frequency

smoothing window 𝐺2(𝑓) has a 3 dB frequency bandwidth


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(2/(2𝑁 − 1)) ∗ 𝐹𝑠Hz, determining the possible frequency

resolution.The reassignment procedure can be made for all time and

frequency values. However, there are several advantages bysetting the spectrum values to a lower limit, below whichno reassignment is performed. This limit is mostly relatedto the “noise floor,” the disturbances showing up randomlyeverywhere in the time-frequency plane, caused by the noisedisturbance of the signal. The other advantage is the fastercomputation as the reassignment made at every time andfrequency value is a heavy calculation procedure.

2.8. Multiplication of Short Time Fourier Transforms (SpM).The spectrogram multiplication method is based on anoptimal combination of spectrograms from a minimummean cross-entropy criterion, [21] and the solution becomesa multiplication of a long-window and a short-windowspectrogram (geometric mean),

𝑆SpM (𝑡, 𝑓) = √

2∏

𝑖=1𝑆𝑖(𝑡, 𝑓). (13)

The idea of this method is to overcome some of the disadvan-tages of having a trade-off between a high time resolution anda high frequency resolution. The signal components that arepresent in both spectrograms (both the long-window and theshort-window spectrograms) are enhancedwith thismethod.

2.9. Description of the Test Signals. The frequency contentsof three different synthetic signals over time are representedwith all sevenmethods.The noise sensitivities of themethodsare compared by adding white Gaussian noise to the thirdtest signal.The fourth test signal is a measured click recordedfrom a bottlenose dolphin (Tursiops truncatus).

The first test signal (S1) is a synthetic two-componentsignal composed of two transient chirps overlapping in timebut separate in frequency content; see Figure 1. The linearchirp rate (𝑎) is 640 ∗ 10

6Hz/s, the start frequencies (𝑓𝑖)

of each signal component are 40 kHz and 100 kHz, and thetotal signal is approximately 70𝜇s long. It is generated by thesuperposition of two chirp components (𝑐

𝑖, 𝑖 = 1, 2) defined

by

𝑐𝑖= cos(2𝜋(𝑎

2𝑡2+𝑓𝑖⋅ 𝑡)) , (14)


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where 𝑡 = time. Each component was sampled with 𝐹𝑠=

1MHz and windowed with a Gaussian window (GW) beforesummation. The window can be described by

GW = 𝑒−(𝑡−3𝜎)2/2𝜎2

, (15)

where

𝜎 = 10−5 s. (16)

The windows smoothed the frequency spectra so that thetrue start frequencies were in fact lower than 𝑓

𝑖. The start

frequency of the 100 kHz signal component was stepwisedecreased until all of the time-frequency representationmethods displayed S1 as one single signal and the twocomponents merged into one single click structure. This wasdone in order to establish the smallest frequency differencethat the two components of S1 could have while still beingrepresented as two components by each of the studied time-frequency representations.

The second test signal (S2) (see Figure 2) was a synthetictwo-component signal comprising two successive transient

chirps with overlapping frequency content; see Figure 2. Bothchirps were created in the same way as those in S1. The startfrequency of both components was set to 50 kHz and thestart time difference was 35𝜇s. The start time difference waslater stepwise decreased until the two signal componentswererepresented as a single signal by all methods.This was done inorder to establish the smallest distinguishable time differencebetween two signal components.

The third tested signal (S3) comprised one transientup-sweep chirp and one shorter transient chirp partiallyoverlapping in both time and frequency content; see Figure 3.The shorter of the two components was windowed with aGaussian window described by (15) where 𝜎 = 10

−5

/3 s. Theamplitude of that component was doubled compared to theother one beforewindowing to compensate for signal loss dueto the narrower window.

Generally, the further apart the components are in eitherfrequency or time, the easier they are to accurately representin both time and frequency. Therefore, the frequency and/ortime differences between the signal components have beenchosen in order for some of the studied methods to collapseand not be able to separate the two components. This was


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done to enlighten both the advantages and limitations of thedifferent methods.

The capacity of the methods to cope with noise was testedby addingwhite Gaussian noise with amean value of zero anda variance of 0.1 to S3.

The fourth tested signal (S4) was an echolocation clickfrom a male bottlenose dolphin (Tursiops truncatus). Thesignal was recorded 45 degrees off the main acoustic axis of

the animal and contained two transients, separated in time(see Figure 4). A complete description of how the signal wasmeasured can be found by Branstetter et al. [15]. In summary,the animal was performing a strictly echoic target detectiontask while stationed on a bite plate. The signal was recordedwith 300 kHz and band pass filtered from 100Hz to 200 kHz.

All the time-frequency representation figures (Figures 5–10) consist of grids of 300 time points and 819 frequency


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M: 50N: 10

Window length 1: 25Window length 2: 75

Window length: 150𝜇s𝛽 = 1.2

𝛼 = 0.8

Figure 5: Time-frequency representations of S1.

points (𝐹𝑠

= 1MHz, FFT-length 4096). The gray scalein the figures ranges linearly from the minimum value tothe maximum value of each time-frequency representation.Three of the methods, that is, WV, CW and LMSp, providednegative values in the plots. Hence, the zero level has not

been represented as black in these plots, as it has in the othermethods. This is also the reason why we chose a linear scalein all figures instead of a logarithmic one.

The applied window functions used in the Sp, RSmWV,and SpMmethods are Hanning windows.


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Number of windows: 4

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𝛼 = 1.2



3. Results and Discussion

Figure 5 shows the resulting time-frequency representationsof S1, plotted with manually optimized parameters for thisspecific signal.The two signal components that overlapped intime but were separate in frequency content were represented

as two partially merged areas by the Sp, the LMSp, and theSpM method. Cross-terms between the signal componentsdominated theWV and CW time-frequency representations.The RMSp, reassignment based on a multitaper spectrogram,requires rather long time windows to achieve the requiredfrequency resolution of the two time-frequency components.


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The smoothing in time, combined with the reassignmenttechnique, produces picture where there actually seems to befour components. RSmWV, however,managed to separate thetwo signal components and demonstrated that there was achirp-like tendency in them.

The minimum frequency difference between the signalcomponents in S1 was 49 kHz. The components were not

distinguishable when the frequency difference was below thisvalue, except with RSmWV, which continued to display twosignals (see Table 1).

The visualization usingWV and CW are also here heavilydisturbed by the cross terms between the two components(see Figure 6). For the Sp and LMSp the leakage causesartifacts and when the time distance of the two components


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Figure 8: Test signal 3 after adding white Gaussian noise with a mean value of zero and a variance of 0.1 to S3.

Table 1: Summary of results.

Are the two clickcomponents clearly

localized in both time andfrequency for dfmin =

49 kHz?

Are the two clickcomponents clearly

localized in both time andfrequency for dtmin =

20𝜇s?Sp No NoWV No NoCW No NoRMSp No YesLMSp No NoRSmWV Yes YesSpM No Yes

is narrowed, the resulting picture will visualize just onecomponent. When the time difference between the signalcomponents was decreased to 20𝜇s, only RMSp, RSmWV,and SpM displayed them as separate signals. Signals closer intime than 20𝜇s were visualized as a single signal. It is alsonotable that the shape of the slope of the chirp in S2 wasdiscerniblewithRMSp, in this case (compared to the previousexample with S1), and the slope could not be clearly visualizedby any of the other methods.

S3 is a good example of a signal that is difficult todisplay accurately in the time-frequency plane (see Figure 7).RMSp, RSmWV, and LMSp displayed it as containing twocomponents separated in time and partially overlapping infrequency content, as the frequency resolution was too lowusing the best possible time-resolution. The RSmWV andthe LMSp representations factually indicated that the signalcontained one short broadband transient in addition to onelonger and more narrow band transient. All other methodsrepresented S3 as a triangular structure, caused by the crossterms.

Adding white Gaussian noise of 0.29Vrms to S3 (seeFigure 8) gave the result shown in Figure 9. With the excep-tion of RSmWV, all methods demonstrated a collapse inthe presence of intense noise. The Sp, LMSp, and Spmall become difficult to interpret, due to the smoothing ofsignal components and additional interfering with noisecomponents. The cross terms with noise components in WV

and CW make these time-frequency spectra impossible tointerpret. In the RMSp reassignment of noise componentsto single time-frequency points also makes the visualizationunclear.

According to all methods, S4 contained two click com-ponents and one low-frequency click approximately 63 𝜇sbefore a more high-frequency click (Figure 10). The centerfrequency of the two clicks was 25 kHz and 103 kHz. Thetime-frequency visualizations aremost similar to the examplesignal S3, as there are both a time and a frequency differencebetween components. The typical view with cross termsbetween components is seen for the WV and CW andthe smoothing, especially in frequency, is clear for the Sp,LMSp, and SpM. However, RMSp and RSmWV showedmorelocalized signal energies as compared to the other methods.

Table 1 summarizes the results in terms of whichmethodsthat were able to localize the signal components in bothtime and frequency for the smallest possible frequency andtime differences. The RSmWVmethod was the only one thatmanaged to localize both signal components accurately forextremely small time and frequency differences between thesignal components.

The two-component Gaussian windowed chirp signalused in this study consists of components that are approx-imately 70 𝜇s long. For components overlapping only intime, a frequency difference no smaller than 49 kHz wasrequired for signals with a −3 dB bandwidth of 40 kHz inorder to be able to clearly localize both signal components.For components overlapping only in frequency, the timedifference had to be at least 20 𝜇s for any of the investigatedmethods to visually resolve them as two components. For sig-nals of more complex structure, for example, overlapping infrequency as well as in time, and where the two componentswere of different length, special care had to be taken wheninterpreting the results. For such signals, it is advisable to basethe interpretation on the results frommore than onemethod.

Furthermore, the results indicate that the RSmWVmethod was the least sensitive to noise disturbances. It wasalso the most robust when taking into account that differentnoise realizations result in rather different appearances of thetime-frequency representations. RSmWVandRMSp are bothbased on smoothing or multitapering, which reduces noisedisturbances in the time-frequency plane. The reason for theRMSp and RSmWV to perform different and better than


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Figure 9: Time-frequency representations of test signal 3 (S3) with noise added.

the other methods is however the reassignment technique,which moves mass of the spectrum to the mass center ofthe corresponding spectrum. Therefore, the reassignmentbased methods are not useful for finding, for example,the energy or amplitude of components but is a techniquefor visualization to resolve time-frequency components. Ageneral reflection is that these results also indicate that the

reassignment technique is very useful when it comes tonoisy transient signals and would thus benefit scientists inmarine biosonar research. Nevertheless, there is room forfurther developments of for instance design parameters andsmoothing/multitaper techniques of time-frequency repre-sentations to further optimize the reassignment methods toshort transient biosonar signals in future studies.


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Figure 10: Time-frequency representations of a bottlenose dolphin (Tursiops truncatus) echolocation click recorded 45 degrees off axis, tothe right of the animal.

4. Conclusions

The smallest frequency difference between signal compo-nents overlapping in time while remaining distinguishablewith currently existing time-frequency representation algo-rithms was found to be 49 kHz. The corresponding smallest

time difference between signal components overlapping infrequency content was found to be 20 𝜇s.

Among the tested methods, RSmWV was the one mostcapable of localizing closely spaced signal components. Theresults also suggested that it was themost robust technique innoisy environments. However, a disadvantage with RSmWV


is the need to find suitable 𝑀 and 𝑁 parameters, in combi-nation with deciding a threshold value for the reassignmentprocedure. The method hence requires some degree ofparameter “tweaking.” Although the SpM method requiresthe setting of two parameters, these are easily interpretableas time window sizes. This straightforward interpretation isnot possible for the𝑀 and𝑁 values of the RSmWVmethod.Therefore, the SpM method might be preferred in studieswhere there are no extreme demands when it comes to signalcomponent localization precision.

The reassignment technique is effective also for noisytransient signals and is thus well suited for use in marinebiosonar research.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors thank Brian Branstetter for letting us test themethods on one of his highly interesting recordings.

References

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[8] W. W. L. Au, B. K. Branstetter, P. W. Moore, and J. J. Finneran,“Dolphin biosonar signals measured at extreme off-axis angles:insights to sound propagation in the head,” The Journal of theAcoustical Society of America, vol. 132, no. 2, pp. 1199–1206, 2012.

[9] M. O. Lammers andM. Castellote, “The beluga whale producestwo pulses to form its sonar signal,” Biology Letters, vol. 5, no. 3,pp. 297–301, 2009.

[10] T. W. Cranford, W. R. Elsberry, W. G. Van Bonn et al.,“Observation and analysis of sonar signal generation in thebottlenose dolphin (Tursiops truncatus): evidence for two sonarsources,” Journal of Experimental Marine Biology and Ecology,vol. 407, no. 1, pp. 81–96, 2011.

[11] J. Starkhammar, P. W. Moore, L. Talmadge, and D. S. Houser,“Frequency-dependent variation in the two-dimensional beampattern of an echolocating dolphin,” Biology Letters, vol. 7, no. 6,pp. 836–839, 2011.

[12] L. N. Kloepper, P. E. Nachtigall, C. Quintos, and S. A. Vlachos,“Single-lobed frequency-dependent beam shape in an echolo-cating false killer whale (Pseudorca crassidens),” Journal of theAcoustical Society of America, vol. 131, no. 1, pp. 577–581, 2012.

[13] P. W. Moore, L. A. Dankiewicz, and D. S. Houser, “Beamwidthcontrol and angular target detection in an echolocating bot-tlenose dolphin (Tursiops truncatus),”The Journal of the Acous-tical Society of America, vol. 124, no. 5, pp. 3324–3332, 2008.

[14] W. W. L. Au, The Sonar of Dolphins, Springer, New York, NY,USA, 1993.

[15] B. K. Branstetter, P. W. Moore, J. J. Finneran, M. N. Tormey,and H. Aihara, “Directional properties of bottlenose dolphin(Tursiops truncatus) clicks, burst-pulse, and whistle sounds,”Journal of the Acoustical Society of America, vol. 131, no. 2, pp.1613–1621, 2012.

[16] A. Bultheel and H. E. Mart́ınez Sulbaran, “Computation ofthe fractional Fourier transform,” Applied and ComputationalHarmonic Analysis, vol. 16, no. 3, pp. 182–202, 2004.

[17] H.-I. Choi and W. J. Williams, “Improved time-frequencyrepresentation of multi-component signals using exponentialkernels,” IEEE Transactions on Acoustics, Speech, and SignalProcessing, vol. 37, no. 6, pp. 862–871, 1989.

[18] J. Xiao and P. Flandrin, “Multitaper time-frequency reas-signment for nonstationary spectrum estimation and chirpenhancement,” IEEE Transactions on Signal Processing, vol. 55,no. 6, pp. 2851–2860, 2007.

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