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PAPER

Signal Detection in Underwater Sound Using the Empirical Mode Decomposition Fu-Tai WANG†,†† , Nonmember, Shun-Hsyung CHANG†,†††a) , Member, and Jenny Chih-Yu LEE††† , Nonmember

SUMMARY In this article, the empirical mode decomposition (EMD) is introduced to the problem of signal detection in underwater sound. EMD is a new method pioneered by Huang et al. for non-linear and nonstationary signal analysis. Based on the EMD, any input data can be decomposed into a small number of intrinsic mode functions (IMFs) which can serve as the basis of non-stationary data for they are complete, almost orthogonal, local and adaptive. Another useful tool for processing transient signals is discrete wavelet transform (DWT). In this paper, these IMFs are applied to determine when the particular signals appear. From the computer simulation, based on the receiver operating characteristics (ROC), a performance comparison shows that this proposed EMD-based detector is better than the DWT-based method. key words: empirical mode decomposition, wavelet, underwater sound, signal detection

1.

Introduction

For processing non-stationary signals, the most basic method is the spectrogram. The Wigner-Ville distribution is another one that is the Fourier transform of the central covariance function. The wavelet transform is an adjustable window Fourier spectral analysis. Modifying the global representation of the Fourier analysis enables these methods to analyze the non-stationary data. Based on a limited time window-width Fourier spectral analysis, they get a time-frequency distribution [1]. Various time-frequency approaches have been developed to detect signal underwater [2]–[7]. The necessary conditions for the basis to represent a non-stationary signal are: (a) complete; (b) orthogonal; (c) local; and (d) adaptive [8]. The first requirement guarantees the degree of accuracy of the expansion; the second requirement guarantees positive value of energy. The locality and adaptive condition are the most decisive criteria for non-stationary data. Only by adapting to the local alteration of the processes can the decomposition fully explain the hidden physics of the signals. The authors of [8] and [9] developed a new data analytical method, based on Manuscript received February 10, 2006. Manuscript revised April 20, 2006. Final manuscript received May 16, 2006. † The authors are with the Department of Electrical Engineering, National Taiwan Ocean University, Keelung, Taiwan, Republic of China. †† The author is with the Department of Electrical Engineering, Hwa Hsia Institute of Technology, Chung Ho, Taipei, Taiwan, Republic of China. ††† The authors are with the Department of Microelectronic Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan, Republic of China. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.9.2415

the empirical mode decomposition (EMD), that a collection of intrinsic mode functions (IMFs) can be generated. Because these IMFs have well-behaved Hilbert transforms, the instantaneous frequencies can be produced from these IMFs and any event can be localized on the time and the frequency axes. These IMFs can serve as the basis of non-stationary data and this basis is complete, almost orthogonal, local and adaptive. An application of the EMD to identify the components of the wave spectra was investigated by Veltcheva [10]. In a broadband stochastic situation such as fractional Gaussian noise, the built-in adaptivity of EMD acting as a “wavelet-like” dyadic filter bank was examined in [11]. Some approaches have been used for signal detection by involving predefined models that are designed for particular situations. The authors of [4] used an empirical model of the noise for signal detection. They found that summary features of wavelet decompositions of underwater sounds appeared to exhibit distinctive multivariate behavior when signals occurred. By noting that IMFs can serve as the basis of non-stationary data and this basis is complete, almost orthogonal, local and adaptive, in this article, instead of imposing wavelet on underwater sounds, these IMFs of them that are decomposed by the EMD are applied to this method to identify transient signals underwater. Using density estimation of the joint distribution of the multivariate vectors of these IMFs, a period of noise-only data is used to build an adaptive noise model of the background continuum. Observations considered to be outliers from this noise model at any time are then flagged as potential signals. The computer simulation, based on the receiver operating characteristics (ROC), shows that the performance of this proposed EMDbased detector is better than that of the DWT-based method. 2.

Empirical Mode Decomposition and Signal Detection

Hilbert-Huang transform is a new method for analyzing non-stationary signal [8]. Any input data can be decomposed into a small number of intrinsic mode functions by using the empirical mode decomposition (EMD). From the Hilbert transform of the IMF, the instantaneous frequencies of the signal can be given. Finally, the input data is presented by the Hilbert spectrum in an energy-frequency-time distribution. In order to obtain a meaningful instantaneous frequency, some limitations should be adopted on the data. Functions that satisfy those restrictions are called intrinsic mode functions [8]: (1) the number of zero crossings and

c 2006 The Institute of Electronics, Information and Communication Engineers Copyright


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the number of extrema must either equal or differ by one in the whole data set at most; and (2) the mean value of the envelope defined by the local minima and the local maxima is zero at any point. To show these restrictions physically, let us examine the function x(t) = k + cos 2t,

(1)

where k is the mean of x(t). When k = 0, x(t) = cos 2t with frequency f = 1/π ≈ 0.318 Hz and zero mean and it is symmetric with respect to the zero mean. Its Hilbert transform is sin(2t) as in Fig. 1. The phase function is a straight dash line as illustrated in Fig. 2 and its instantaneous frequency f ≈ 0.318 Hz is a dash line with k = 0 shown in Fig. 3. If k < 1, x(t) becomes an asymmetric wave form with non-zero mean. Under this condition, the phase function and the instantaneous frequency are very different but still meaningful as shown in Figs. 2 and 3 with k < 1. If k > 1, x(t) is a riding wave and both of the phase function and the instantaneous frequency have negative values which are meaningless as shown in Figs. 2 and 3 with k > 1.

Fig. 1

Fig. 2 form.

The Hilbert transform of cos 2t is sin 2t.

This example using the local restrictions to define a meaningful instantaneous frequency physically indicates that the function should be symmetric with respect to the local zero mean. The local mean of the envelopes defined by the local minima and the local maxima is used to force the local symmetry instead. With the definition of the zero crossings, the IMF involves only one oscillation mode, excluding complex riding waves. At any given time, the input data X(t) are not IMFs. More than one oscillatory mode is involved in most of them. The process to reduce the data into IMF components is designated as the empirical mode decomposition (EMD) [8]. The decomposition process starts with the envelopes constructed by the local minima and maxima separately. Once the extrema are found, all the local minima are linked by a cubic spline as the lower envelope. For the local maxima, the procedure is repeated to produce the upper envelope. Then, all the data are covered by the lower and upper envelopes. The mean of these two envelopes is denoted as m1 . The first component h1 is the difference between the data and m1 . This sifting process is illustrated in Fig. 4. Two

Fig. 3 The instantaneous frequency of the model functions via the Hilbert transform.

The unwrapped phase of the model functions via the Hilbert transFig. 4

Illustration of the sifting process.

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purposes of the sifting process are: (1) to eliminate riding waves; and (2) to make the wave-profiles more symmetrical [8]. Ideally, h1 should be an intrinsic mode function. Although all the local minima are negative and all the local maxima are positive as shown in Fig. 4, wave h1 is still asymmetric. Now h1 is treated as the data and then take the the second sifting process: the sifting procedure has to be repeated l times, until h1l is IMF. This IMF h1l is then designated as the first IMF component of the data. The stop of the sifting process is determined by the size of the standard deviation SD: ⎡ ⎤ I ⎢
⎢⎢ |h1(l−1) (i) − h1l (i)|2 ⎥⎥⎥⎥ ⎢ SD = (2) ⎢⎣ ⎥⎦ , h2 (i) i=0

1(l−1)

Fig. 5

Reconstructing details of dolphin signals using DWT.

which computed from the two successive sifting results. Other stopping criteria are provided in [9], [10]. The shortest period content of the data should be contained in c1 . Separating c1 from the rest of the data, we have the residue r1 . The sifting process then can be repeated on r1 and all the following rm s and get X(t) =

M


cm + r M .

(3)

m=1

Then a decomposition of the input data into M IMFs and one residue is achieved. The detail of the decomposition process is presented in [8]. The EMD is adaptive and highly efficient, because it extracts IMF directly from the signal associated with intrinsic time scales. Based on the local properties of the data, the EMD decomposition is applicable to non-stationary processes. The authors of [4] used an empirical model of the noise for signal detection and found that summary features of discrete wavelet transform (DWT) decompositions of underwater sounds appeared to exhibit distinctive multivariate behavior when signals occurred. According to [4], underwater sound recordings are divided into appropriate time “window,” subdividing any given input data into periods of 0.743 second (with each period containing 215 = 32768 data points). Each of these 0.743 second segments split into 256 time windows with a length of 128 samples is subjected to the DWT analysis. Figure 5 shows the wavelet reconstructing details D1,...,D8 of dolphin signals S [12] based on the DWT. The spectral analysis of these reconstructing details D1,...,D8 is shown in Fig. 6. The reconstructing details D1, ..., D5 of DWT in level 1 to level 5 covering the frequency ranges that are of interest in this case (i.e. 1–20 kHz) are summarized in terms of five mean sums of squares, (vi,1 , · · · , vi,5 ) for each time window i, so that 128 2 D jk , j = 1 , · · · , 5, vi, j = k=1 128 where D2 jk are taken from time window i. From the results of the spectral analysis in Fig. 6, the 3rd through the 5th components of DWT (i.e. D3, D4 and D5) are selected but the 1st and the 2nd components are rejected. In the comparison of EMD with DWT, we focus on these three mean sums

Fig. 6 Spectral analysis of the reconstructing details of dolphin signals using DWT.

of squares for our particular application and form a vector of multivariate observation vi = (vi,3 , vi,4 , vi,5 ) in time window i. The behavior of these observations vi , i = 1, ···, 256 covers the range 1–5 kHz. In other application, the definition of vi will be different. The behavior of the multivariate observation by DWT vi = (vi,3 , vi,4 , vi,5 ) of typical dolphin signals that cover the range 1–5 kHz is shown in Fig. 7. These dolphin signals are selected extracts of a recording [12]. During a period of noise when no “signals” are present, the behavior of vi = (vi,3 , vi,4 , vi,5 ) by DWT is depicted in Fig. 8. The joint behavior of these mean sums of squares of noise is significantly different from that of signals. This result suggests the usage of density estimates to establish the initial density estimate of noise and then to detect the signal. Inspired by this application and by noting that IMFs can serve as the basis of non-stationary data and this basis is complete, almost orthogonal, local and adaptive, in this article, these IMFs of underwater sounds [12] that are decomposed by the EMD are applied to this method. IMFs c1 , c2 and c3 of underwater sounds are summarized in terms of three mean sums of squares, (vi,1 , vi,2 , vi,3 ) for each time window i, so that

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Fig. 7 Behavior of vi for typical dolphin signals by DWT. – – –, vi,3 ; —, vi,4 ; · · ·, vi,5 .

Fig. 9 Behavior of vi for typical dolphin signals by EMD. – – –, vi,3 ; —, vi,4 ; · · ·, vi,5 .

Fig. 8 Behavior of vi for typical background noise by DWT. – – –, vi,3 ; —, vi,4 ; · · ·, vi,5 .

Fig. 10 Behavior of vi for typical background noise by EMD. – – –, vi,3 ; —, vi,4 ; · · ·, vi,5 .

128 vi, j =

k=1

c2j,k

128

,

j = 1 , 2 , 3,

c2j,k

where are taken from time window i, focusing on these mean sums of squares and forming a vector of multivariate observation vi = (vi,1 , vi,2 , vi,3 ) in time window i. The behavior of the multivariate observation vi = (vi,1 , vi,2 , vi,3 ) of typical dolphin signals by EMD is shown in Fig. 9. During a period of noise when no “signals” are present, the behavior of vi = (vi,1 , vi,2 , vi,3 ) by EMD is depicted in Fig. 10. The joint behavior of these mean sums of squares of noise is significantly different from that of signals. Based on the receiver operating characteristics (ROC), the following section will show that these vectors of multivariate observations vi produced by EMD can establish the initial density estimate of noise and get a better detection performance then that proposed by DWT. 3.

Empirical Model and Computer Analysis

The authors of [4] suggested the usage of density estimates

to establish the initial density estimate of noise and then to detect the signal. With given multiple-component data vi , ..., vI , density estimates can be obtained by using a multivariate kernel density estimator with a multivariate normal kernel [13],

I (v − vi ) Q−1 (v − vi ) (detQ)1/2
exp − yˆ (v) = , (4) (2π)3/2 Ih3 i=1 2h2 where Q is a robust estimate of the covariance matrix [14], [15], and h is a suitable global window width. The optimal choice of h can be determined by cross-validation, by I maximizing the approximate log-likelihood for the data yi (vi )), where yˆ i refers to the kernel density estii=1 log(ˆ mate based on all given data v1 , ..., vI , except vi itself. The duration I = 256 is selected to ensure the stability of kernel estimates [13]. In this multiple-component data situation, resistant linear regressions are needed to estimate both of variances and ˆ and cˆ ob recovariances in Q, denoting both of them by bar spectively. We define them for the specially chosen com-

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ponents v1 , v2 − bv1 , v3 − cv2 − dv1 , taking zero as the cˆ ob between each pair of them, and taking the square of a resisˆ for each of them. The results tant estimate of scale as the bar are: ˆ ˆ bar(v 2 )
bar((v2 − bv1 ) + bv1 ) 2ˆ ˆ = bar(v 2 − bv1 ) + b bar(v1 ), cˆ ob(v1 , v2 )
cˆ ob(v1 , (v2 − bv1 ) + bv1 ) ˆ = bbar(v 1 ), 2ˆ ˆ ˆ ) = bar(v − cv bar(v 3 3 2 − dv1 ) + c bar(v2 − bv1 ) ˆ + (d + bc)2 bar(v 1 ), ˆ cˆ ob(v1 , v3 ) = (cb + d)bar(v1 ), ˆ ˆ cˆ ob(v2 , v3 ) = cbar(v 2 − bv1 ) + b(cb + d)bar(v1 ), ˆ and cˆ ob are defined for all linear combinations of so that bar v1 , v2 , v3 . The constants b, c, d can be chosen by biweight fit [14]. The arithmetic mean is a nonresistant summary, on the other hand, biweight is an abbreviation for bisquare weight and is resistant since a change of a small part of the data fails to change the summary substantially [14]. Weighting observations according to  (1 − u2 )2 , |u| ≤ 1, w(u) = 0, elsewhere, with ui =

vi,1 − vˆ1 , pF

choosing p = 9 and F = 12 (H) where the interquartile range H (distance between the hinges or 25% points) is a resistant estimate of scale. The biweight estimate is defined as I i=1 w(ui )vi,1 . vˆ1 =  I i=1 w(ui ) According to [16], the estimated variance of the biweight estimate vˆ1 is expressed by  n  (v1 − vˆ1 )2 (1 − u2 )4 ns2bi =  , (5)  [ (1−u2 )(1−5u2 )][−1+  (1−u2 )(1−5u2 )]  where  indicates summation for u2 ≤ 1 only and n is the number of ui that satisfies u2i ≤ 1. (1 − u2 )4 = w2 (u) implies that weights are defined by the u’s. For a small u2 , the weights tend to be equal and Eq. (5) reduces to  (v1 − vˆ1 )2 . (n − 1) It seems to be a reasonable variance estimate and the ns2bi is named the biweight variances. ˆ Taking the biweight variances in (5), we define bar(v 1 ), ˆ ˆ bar(v 2 − bv1 ), and bar(v3 − cv2 − dv1 ) as: 2 ˆ (for the value of v1 ), bar(v 1 ) = nsbi ˆbar(v2 − bv1 ) = ns2 (for the value of v2 − bv1 ), bi

2 ˆ bar(v 3 − cv2 − dv1 ) = nsbi

(for that of v3 − cv2 − dv1 ).

After that, Q can be defined as ⎡ ˆ ⎢⎢⎢ bar(v1 ) cˆ ob(v1 , v2 ) cˆ ob(v1 , v3 ) ⎢ ˆ Q = ⎢⎢⎢⎢ cˆ ob(v2 , v3 ) bar(v 2) ⎣ ˆ bar(v 3)

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎦ ,

where Q is a robust estimate of the covariance matrix used in the estimator (4). This kernel estimator yˆ (v) is first applied to an initial sample, v1 , ..., vI , taken from the sound recording, which consists of pure background noise without signals of interest. Once this initial kernel estimate is established, it is treated as an empirical model of the noise, and the next observed vector of the selected mean sums of squares v is considered. Using a detection criterion to test if v is a signal or not H #{ˆy(v) ≥ yˆ i (vi )} + 1 1 ≶ η, I+1 H0

(6)

where “#{·}” means “the number of” for i = 1, ..., I; yˆ denotes the kernel density estimate based on v1 , ..., vI ; yˆ i denotes the kernel density estimate based on v1 , ..., vI but excluding vi . If this test produces a value smaller than a threshold value η, v is identified as an outlier from the current kernel density estimate and it is flagged as a potential signal and then subsequently ignored. Otherwise, this v replaces vI , vi is replaced by vi+1 , i = 1, ..., I − 1, and the optimal window width and the kernel density are updated before the next v. If this model is allowed to adapt with time and to be a baseline at any time, the signal can be identified. More detail of this detection criterion was presented in [4]. The performance of the proposed EMD-based detector is compared to the DWT-based method by using the ROC curves in the following experiment. The empirical mode decomposition (EMD) is first applied to an initial sample (32, 768 data points) taken from a sound recording consisting of pure background noise without signals of interest. The intrinsic mode functions (IMFs) c1 , c2 , c3 decomposed by the EMD are summarized in terms of three mean sums of squares. Each one has 256 time windows. Density estimation of the joint distribution of these summaries is used to obtain an adaptive noise model of the background continuum and to choose a threshold value η, as mentioned before. Once this EMD-based initial kernel estimate is established, an observed vector of the mean sums of squares of noise-free signal v is considered. The detection criterion in Eq. (6) is used to test if v is a signal or not. Observations considered to be outliers from this noise model at any time are then flagged as a potential signal. If this test produced a value smaller than the threshold η, this observation v is identified as an outlier from the current kernel density estimate and flagged as a potential signal, and the detection counter is increased. The same is done for an observed vector of the mean sums of squares of noise-only data v. If this test is smaller than the threshold, a false alarm is declared and its corresponding counter is increased. For every threshold value η, with the experiment repeated 256 times, the PD and

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Fig. 11 ROCs for the EMD-based detector ‘o’ and that of the DWTbased detector ‘x.’

Fig. 13 The DWT reconstructing details (Top) and the DWT-based detection result (Bottom) of a dolphin sound.

Fig. 12 Performance of the EMD-based detector ‘o’ and that of the DWT-based detector ‘x.’

Fig. 14 The EMD-based decompositions (Top) and the EMD-based detection result (Bottom) of a dolphin sound.

the PF for that particular threshold value can be evaluated. This experiment is run for several of the threshold values η, and the ROC of the EMD-based detector by representing the PD as a function of the PF can be obtained and is plotted in Fig. 11. The number of detection index or SNR is defined as 10 log(E/No), where E stands for the energy of signals and No stands for the energy of noise. The value of SNR is set at 3.11 dB for the curves in Fig. 11. A comparison of the ROC between the EMD-based detector and the DWT-based method in Fig. 11 shows that the proposed method is better than the DWT-based detector if the PF is lower than 0.83. Figure 12 indicates the performance of the EMD-based detector and that of the DWT-based method under the same value of PF = 0.02. The DWT reconstructing detail and the DWT- based detection result of a dolphin sound [12] are depicted in Fig. 13. The three lines of top panel of Fig. 13 are components of D3, D4 and D5. In contrast, the IMFs and the EMD- based detection result of a dolphin sound are plotted in Fig. 14. The three lines of top panel of Fig. 14 are components of c1 , c2 and c3 . The vertical axis of the graph in the

upper side of each figure is biased for top and center curves to avoid the overlap. In the bottom panel of each figure, the white region is the part that means no “signal” under a detection criterion. These results show that the performance of the proposed EMD-based detector is better than that of the DWT-based method under the same PF , because IMFs can serve as the basis of non-stationary data and this basis is complete, almost orthogonal, local and adaptive. 4.

Conclusion

The empirical mode decomposition (EMD) is introduced to the problem of signal detection in underwater sound. Based on the EMD, any input data can be decomposed into a small number of intrinsic mode functions (IMFs) which can serve as the basis of non-stationary data for they are complete, almost orthogonal, local and adaptive. These IMFs are applied to determine when the particular signals appear. Using density estimation of the joint distribution of the multivariate vectors of these IMFs, a period of noise-only data is used

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to build an adaptive noise model of the background continuum. It is treated as an empirical model of the noise. Observations considered to be outliers from this noise model at any time are then flagged as potential signals. From the computer simulation, based on the receiver operating characteristics (ROC), a performance comparison shows that this proposed EMD-based detector is better than the DWT-based method. Acknowledgments This work was supported by the National Science Council of the R.O.C. under contract NSC 93-2611-E-019-024. References [1] A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice-Hall, NJ, 1989. [2] P. Abry and P. Flandrin, “Multiresolution transient detection,” Proc. IEEE-SP Int. Symp. Time-Freq. Time-Scale Anal., pp.225–228, Oct. 1994. [3] S.A. Benno and J.M.F. Moura, “On translation invariant subspaces and critically sampled wavelet transforms,” Multidimensional Systems and Signal Process., vol.8, pp.89–110, Jan. 1997. [4] T.C. Bailey, T. Sapatinas, K.J. Powell, and W.J. Krzanowski, “Signal detection in underwater sound using wavelets,” J. Amer. Statist. Assoc., vol.93, no.441, pp.73–83, March 1998. [5] S.R. Massel, “Wavelet analysis for processing of ocean surface wave records,” Ocean Engineering, no.28, pp.957–987, 2001. [6] S.H. Chang and F.T. Wang, “Application of the robust discrete wavelet transform to signal detection in underwater sound,” Int. J. Electron., vol.90, no.6, pp.361–371, 2003. [7] F.T. Wang and S.H. Chang, “Signal detection in underwater sound by dual-tree discrete wavelet transform,” submitted for publication. [8] N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung, and H.H. Liu, “The empirical decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A., vol.454, pp.903–995, 1998. [9] N.E. Huang, M.L. Wu, S.R. Long, S.P. Shen, W.Q. Per, P. Gloersen, and K.L. Fan, “A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis,” Proc. R. Soc. Lond., no.459, pp.2317–2345, 2003. [10] A.D. Veltcheva and C.G. Soares, “Identification of the components of wave spectra by the Hilbert Huang transform method,” Applied Ocean Research, no.26, pp.1–12, 2004. [11] P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filter bank,” IEEE Signal Process. Lett., vol.11, no.2, pp.112–114, Feb. 2004. [12] M. Goodall, The Way of the Ocean, New World Music, 1998. [13] B.W. Silverman, Density Estimation for Statistics and Data Analysis, p.78, Chapman and Hall, London, 1986. [14] F. Mosteller and J.W. Tukey, Data Analysis and Regression, pp.203– 219, Addison-Wesley, 1977. [15] P.A. Tukey and J.W. Tukey, Graphical Display of Data sets in 3 or more Dimensions, pp.189–275, Wesley, Chichester, 1981. [16] A.M. Gross, “Confidence-interval robustness with long-tailed symmetric distribution,” J. Amer. Statist. Assoc., vol.71, pp.409–416, 1976.

Fu-Tai Wang received his B.S. degree in electronic engineering from the Chinese Culture University, Taipei, Taiwan, in 1996 and his M.S. degree in electronic engineering from National Taiwan Ocean University, Keelung, Taiwan, in 1999. Since 1999, he has been an instructor at the Electronic Engineering Department at Hwa Hsia Institute of Technology, Chung Ho, Taipei, Taiwan. His research interests include wavelet transform, digital signal processing, signal detection, classification, and communication techniques. He is currently working toward to his Ph.D. at the Electronic Engineering Department at National Taiwan Ocean University, Keelung, Taiwan.

Shun-Hsyung Chang received the Ph.D. degree in electrical engineering from the National Sun Yat Sen University, Taiwan, in 1990. Afterwards, he served as an associate professor and has been a professor since 1997 in the Department of Electrical Engineering at National Taiwan Ocean University where he was the chairman 1998–2003. Since August 2003, he also has been serving as the Vice President of the National Kaohsiung Marine University, Taiwan, and a professor in the Department of Microelectronic Engineering at this university. He has published over 120 technical papers. His research interests include underwater signal processing, electrical engineering, and communication engineering. His outstanding performance in research brought him great honor. In 1984, he was granted “The Creative Youth” prize by the Ministry of Education. He received the Long-Term Paper Award, organized by Acer Incorporated, in 1994, 1996 and 1998. He jointly with his graduate students won the Third Prize of Graduate Team in TI-Taiwan 1994 DSP Design Championship. He contributes greatly in editorship. He was the editor-in-chief of the Journal of Marine Science and Technology 1998–2001, of which he is an editor at present. He is currently an editor of the Journal of Ocean and Undersea Technology.

Jenny Chih-Yu Lee received her B.S. degree in mathematics from the University of California at Davis in 1988 and her M.S. degree in mathematics from the California State University at Hayward in 1991. Since 1996, she has been an instructor of mathematics at National Kaohsiung Marine University, Taiwan. Her research interests cover numerical analysis, mathematical physics and signal processing.