Cross ⌿B-energy operator-based signal detectiona) Abdel-Ouahab Boudraab兲 IRENav, Ecole Navale, Lanvéoc Poulmic BP600 29200 Brest–Armées, France, and E3I2 (EA3876), ENSIETA, 2 rue Francois Verny, 29806 Brest Cedex 9, France
Jean-Christophe Cexus E3I2 (EA3876), ENSIETA, 2 rue Francois Verny, 29806 Brest Cedex 9, France
Karim Abed-Meraim TSI department, ENST-Paris, 46 rue Barrault 75634, Paris Cedex 13, France
共Received 6 April 2007; revised 7 March 2008; accepted 4 April 2008兲 In this paper, two methods for signal detection and time-delay estimation based on the cross ⌿B-energy operator are proposed. These methods are well suited for mono-component AM-FM signals. The ⌿B energy operator measures how much one signal is present in another one. The peak of the ⌿B operator corresponds to the maximum of interaction between the two signals. Compared to the cross-correlation function, the ⌿B operator includes temporal information and relative changes of the signal which are reflected in its first and second derivatives. The discrete version of the continuous-time form of the ⌿B operator, which is used in its implementation, is presented. The methods are illustrated on synthetic and real signals and the results compared to those of the matched filter and the cross correlation. The real signals correspond to impulse responses of buried objects obtained by active sonar in iso-speed single path environments. © 2008 Acoustical Society of America. 关DOI: 10.1121/1.2916583兴 PACS number共s兲: 43.60.Bf, 43.60.Kx 关WMCs兴
I. INTRODUCTION
Signal detection and time-delay 共TD兲 estimation between signals are important problems in communications and signal processing. Typical applications involve radar, sonar, machine fault diagnostics, biomedicine, and geophysics.1 For example, in sonar signal processing the difference in arrival time of a signal at two or more spatially separated receivers is used to estimate range and bearing of the source. A common method of signal detection based on TD involves cross correlation of the receiver output, whereby an estimate of the TD is given by the argument that maximizes the crosscorrelation 共CC兲 function. Although the CC method is the simplest similarity measure used in signal processing, it is not sensitive to nonlinear dependence of the signals. In practice, the interaction between signals may be nonlinear. Furthermore, sensor characteristics may also cause nonlinear distortions. Consequently, the maximum CC may not correspond to the maximum signal interaction. In this case, the resulting TD may be erroneous. To tackle this problem a new similarity function that measures the nonlinear interaction between signals is proposed.2–4 The method is based on the cross ⌿B-energy operator, recently introduced by the authors.2 The ⌿B-energy operator is derived from a second energy-like function, called the Cross Teager-Kaiser Energy Operator 共CTKEO兲5,6 which measures the interaction between two real time functions. Compared to the CC, ⌿B a兲
Preliminary results of this work were presented at IEEE ISSPA, Sydney, Australia, 2005. b兲 Author to whom correspondence should be addressed. Electronic mail:
[email protected]; URL: http://www.ecole-navale.fr/fr/irenav/cv/ boudra/index.html J. Acoust. Soc. Am. 123 共6兲, June 2008
Pages: 4283–4289
includes temporal information and accounts for relative changes of the signal by using the first and the second derivatives of the signal. Since it is based on a nonlinear operator, ⌿B, the proposed method can be viewed as a nonlinear matched filter and we note it MBF 共for Matched ⌿B Filter兲.7 ⌿B can also be used to “detect” the presence of known signals as components of more complicated signals by measuring how much one signal is present in another one.3 Thus, the ⌿B operator can be used as a strategy for signal detection. As shown in Ref. 3, ⌿B or methods based on ⌿B are well suited for nonstationary signals such as mono-component AM-FM signals. Based on the CTKEO, both the ⌿B operator and the associated methods for TD estimation or signal detection are limited to narrowband signals. The paper is organized as follows. In the following section the ⌿B operator is presented. Section III deals with the discrete form of ⌿B which is the basis of its numerical implementation. In Sec. IV a ⌿B-based signal detection framework is introduced followed by the ⌿B-based estimation approach in Sec. V. A pseudo code of the MBF is presented in Sec. VI. Simulation and experimental results are presented in Sec. VII and concluding remarks are stated in Sec. VIII. We illustrate the method with an underwater acoustics application, where each signal is the impulse response of a buried object obtained by active sonar in a single path environment. II. THE CROSS ⌿B-ENERGY OPERATOR
Recently the CTKEO has been extended to complexvalued signals and an operator called ⌿B introduced.2 Given two complex signals x共t兲 and y共t兲, ⌿B is denned as follows:2
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⌿B共x,y兲 = 0.5关⌿C共x,y兲 + ⌿C共y,x兲兴
共1兲
=0.5关x˙*y˙ + x˙y˙ *兴 − 0.25关xy¨ * + x*y¨ + yx¨* + y *x¨兴,
共2兲
where ⌿C共x , y兲 = 0.5关x˙*y˙ + x˙y˙ *兴 − 0.5关xy¨ * + x*y¨ 兴. It has been shown that the cross ⌿B-energy operator of x共t兲 and y共t兲 is equal to the cross-Teager energies of their real and imaginary parts,2 ⌿B共x,y兲 = ⌿B共xr,y r兲 + ⌿B共xi,y i兲 ⌿B共xl,y l兲 = x˙ly˙ l − 0.5关xly¨ l + x¨ly l兴,
共3兲 l 苸 兵r,i其,
共4兲
where x共t兲 = xr共t兲 + jxi共t兲 and y共t兲 = y r共t兲 + jy i共t兲. The complex form of the signals is obtained using the Hilbert transform. For ⌿B implementation, different derivative approximations can be used.
Discretized derivatives of the continuous ⌿B operator are combined to obtain an expression closely related to the discrete form of the operator ⌿Bd and operating on discretetime signals x共n兲 and y共n兲. Different sample differences can be used, but only the two-sample backward difference is detailed herein. For simplicity, we replace t by nTs where Ts is the sampling period, and x共t兲 with x共nTs兲 or simply x共n兲. x˙共t兲 → 关xk共n兲 − xk共n − 1兲兴/Ts x¨共t兲 → 关xk共n兲 − 2xk共n − 1兲 + xk共n − 2兲兴/Ts2
+ y k共n兲xk共n − 2兲兴/Ts2 ⌿B共xk共t兲,y k共t兲兲 → ⌿Bd共xk共n − 1兲,y k共n − 1兲兲/Ts2 共5兲
The discrete form of ⌿B共x共t兲 , y共t兲兲 is given by ⌿B共x共t兲,y共t兲兲 → 关⌿Bd共xr共n − 1兲,y r共n − 1兲兲 共6兲
where 哫 denotes the mapping from continuous to discrete. Thus, from ⌿B we obtain ⌿Bd shifted by one sample to the left and scaled by Ts−2. It is easy to show that using the two-sample forward difference from ⌿B, we obtain ⌿Bd shifted by one sample to the right and scaled by Ts−2. For both asymmetric two-sample differences, ⌿B is shifted by one sample and scaled by Ts−2. If we ignore the one-sample shift and the scaling parameter, we transform ⌿B共x共t兲 , y共t兲兲 into ⌿Bd共x共n兲 , y共n兲兲 as follows: ⌿B共x共t兲,y共t兲兲 → ⌿Bd共xr共n兲,y r共n兲兲 + ⌿Bd共xi共n兲,y i共n兲兲, 共7兲 4284
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IV. ⌿B-BASED SIGNAL DETECTION
We motivate ⌿B-based detection by considering the classical binary hypothesis testing problem encountered in radar or in sonar.8 Let s共t兲 denote a baseband transmitted signal. The received signal R共t兲 is processed over an interval 关Ti , T f 兴 to detect the presence of a target. The hypotheses on R共t兲 are H0:R0共t兲 = n共t兲,t 苸 关Ti,T f 兴, H1:R1共t兲 = ␣s共t − t0兲 + n共t兲,t 苸 关Ti,T f 兴.
Under the null hypothesis, H0, the received signal contains only an additive noise, n共t兲. Under the alternative hypothesis, H1, a received time-shifted version of the transmitted signal ␣s共t − t0兲 is received in the presence of noise where ␣ denotes an unknown gain parameter. The unknown time, t0, represents the delay of the received signal and corresponds to the unknown distance of the target. Let T = 关Tmin , Tmax兴 denote the possible range of values for t0. The required observation interval is 关Ti , T f 兴 = 关Tmin , t0 + Tmax兴 in this case. For any given value of t0 苸 T, the decision whether to reject H0 is given by computing
t苸T
− 0.5关xk共n兲y k共n − 2兲
k 苸 兵i,r其. 共8兲
The three-sample symmetric difference can also be used but it leads to a more complicated expression compared to asymmetric two-sample differences. Indeed, the asymmetric approximation is less complicated for implementation and faster than the symmetric one because it requires fewer operations.
TB = arg max
⌿B共xk共t兲,y k共t兲兲 → xk共n − 1兲y k共n − 1兲/Ts2
+ ⌿Bd共xi共n − 1兲,y i共n − 1兲兲兴/Ts2 ,
+ y k共n + 1兲xk共n − 1兲兴
再
III. DISCRETIZING THE CONTINUOUS-TIME ⌿B OPERATOR
k 苸 兵i,r其.
⌿Bd共xk共n兲,y k共n兲兲 = xk共n兲y k共n兲 − 0.5关xk共n + 1兲y k共n − 1兲
冋冕
册
⌿B共R0共t兲,R1共t兲兲dt ,
T
共9兲
where TB corresponds to the time of maximum of interaction between R0共t兲 and R1共t兲.3 TB is then compared to a threshold to determine the presence 共H1兲 or absence 共H0兲 of a target. Thus, the best detector calculates the interaction between the received signal and all possible time-shifted versions of the transmitted signal and picks the largest energy interaction as the basis for the detection decision. The location of the peak is the estimate of the unknown parameter t0 = TB. V. ⌿B-BASED TIME-DELAY ESTIMATION
We have shown that the TD estimation problem measures the interaction between two FM signals by using the ⌿B operator.3 Consider a signal from a remote source being received in the presence of noise at two spatially separated receivers. The time histories of the receiver outputs, denoted by rm共t兲 and rk共t兲, are given by
再
rm共t兲 = s共t兲 + nm共t兲, rk共t兲 = s共t − 共k − m兲r兲 + nk共t兲,
where s共t兲 is the signal waveform, nm共t兲 and nk共t兲 are the noise waveforms at the respective receivers,  is an attenuation factor, and is the difference in wavefront arrival times at two consecutive receivers 共k = 2 , m = 1兲. We assume that Boudraa et al.: Cross ⌿B-energy operator
(k-m).d
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ΨB(X,X)
ΨB(Y,Y)
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FIG. 1. 共Color online兲 Geometry used to estimate the time delay associated with plane waves.
nm共t兲, nk共t兲 and s共t兲 are mutually uncorrelated.3 Propagation TD between receivers k and m is given by ⌬d / c = 共k − m兲, where ⌬d = dk − dm is the path length difference, and c is the sound speed in the medium. When the target is sufficiently distant from the receivers the wavefronts can be approximated by plane waves and the theoretical TD, TTheor, is given by TTheor = 共k − m兲d sin /c,
Amplitude
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Intersection frequency 1 0.8 0.6 0.4 ΨB(X,Y)
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0
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FIG. 3. 共Color online兲 Interaction measure between x共t兲 and y共t兲.
共10兲
where d is the distance between two consecutive receivers and is bearing angle 共See Fig. 1兲.
Pseudo Code: M =N−w+1 For z = 1 to M gz共n − z + 1兲 ← r共n兲 , n 苸 F = 兵z , . . . , w + z − 1其
VI. PSEUDO CODE OF MBF
The complex form of the signal is obtained using the Hilbert transform. The MBF implementation involves the following steps: Inputs: Emitted signal: s共p兲 = sa共p兲 + jsb共p兲, p 苸 兵1 , 2 , . . . , w其,
Compute ⌿la = ⌿Bd共sa共l兲 , gza共l兲兲 and ⌿lb = ⌿Bd共sb共l兲 , gzb共l兲兲 using Eq. 共8兲 ⌿Bd共s共l兲 , gz共l兲兲 ← ⌿la + ⌿lb , l 苸 兵1 , . . . , w其 Compute the sum I共z兲 of ⌿Bd values over F: w+z−1
Received signal: r共n兲 = ra共n兲 + jrb共n兲, n 苸 兵1 , 2 , . . . , N其, where a and b indicate the real and imaginary parts of the complex signal respectively. w and N are time durations of s共t兲 and r共t兲, respectively, and z = 1 to M denotes the index of the sliding window.
Signal X
0.5 0 −0.5
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Compute TB = arg max关I共z兲兴
Signal X
0.2 0.15 0.1 0.05
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0
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1 0.5 Signal Y
EndFor
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0 −0.5 −1
l=z
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Outputs: TB.
−1
I共z兲 = 兺 ⌿Bd共s共l兲 , gz共l兲兲
0
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100 150 200 Time Intersection frequency
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FIG. 2. 共Color online兲 Linear chirp test signals 共left兲 and their respective IFs 共right兲. J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008
FIG. 4. 共Color online兲 Interaction measure between x共t兲 and y共t兲 using CC. Boudraa et al.: Cross ⌿B-energy operator
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1
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s1(t)
0.5 0.5
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I (t) 1
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FIG. 5. Signal detection results with a = 0.95: 共a兲 First emitted signal. 共b兲 Received echo of the first signal, 共c兲 Integral, I, of ⌿Bd calculated between the emitted signal and the echo of the first signal, 共d兲 Second emitted signal. 共e兲 Received echo of the second signal, and 共f兲 Integral, I, of ⌿Bd calculated between the emitted signal and the echo of the second signal.
VII. RESULTS
The results presented below highlight the contributions of the paper. The proposed method is illustrated on synthetic and real signals and the results are compared to those of the CC method and the Matched Filter 共MF兲. The real signals correspond to impulse responses of buried objects. The present study is limited to signals obtained by active sonar in iso-speed single path environments. Both ⌿B and MBF are implemented using asymmetric two-sample differences 关Eqs. 共7兲 and 共8兲兴. A. Synthetic signals
We first show an example of interaction between two nonstationary signals measured by ⌿B operator and compare the result to the CC approach. Figure 2 shows an example of two linear FM signals x共t兲 and y共t兲 with the corresponding Instantaneous Frequencies 共IFs兲. The IF of x共t兲 increases linearly with time while that of y共t兲 decreases with time. The inter-action between x共t兲 and y共t兲 is calculated using Eq. 共2兲. Figure 3 shows the energy of each signal and the energy of
their interaction. The maximum of interaction corresponds to the instant when the two IFs intersect 共Figs. 2 and 3兲 and also where the energy of x共t兲共⌿B共x , x兲兲 and that of y共t兲共⌿B共y , y兲兲 are equal 共Fig. 3兲. The point where the IFs intersect is located at t = 125. Maximums of interaction between x共t兲 and y共t兲 occur at t = 125 and t = 240 for ⌿B and CC, respectively, as shown in Figs. 3 and 4. The CC fails to point out, as expected, the point of maximum of interaction. This result shows that the CC measure is insensitive to nonlinear dependency between x共t兲 and y共t兲. Away from the point where the IFs cross, the amplitude of the interaction decreases because there is less similarity between the two signals. As the IFs converge from the time origin to their intersection, the interaction intensity of the signals increases and the maximum of interaction is achieved at the intersection. This example shows that ⌿B is more effective with nonstationary signals than the CC. This is due to the fact that ⌿B is a nonlinear operator while CC is linear. In this simulation an example of delay estimation performed between two signals is presented. Two synthetic ref-
TABLE I. Estimated TB versus SNR signals s1共t兲 and s2共t兲 using MF and MBF methods. SNR= −6 dB Signals s1共t兲 s2共t兲
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MF 300⫾ 1 300⫾ 2
SNR= −2 dB MBF 300⫾ 1 300⫾ 1
MF 300 300⫾ 1
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SNR= 1 dB MBF 300 300⫾ 1
MF 300 300⫾ 1
SNR= 3 dB MBF 300 300⫾ 1
SNR= 5 dB
MF 300 300
MBF 300 300⫾ 1
MF 300 300
SNR= 9 dB MBF 300 300
MF 300 300
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Sensors
(a)
Samples
FIG. 7. Example of one buried target. The selected reference signal of Data2.
reference sensor, and each sensor of the array. The MBF delay estimation, of the received signal on two sensors, versus the sensor indexes along the linear array is shown in Fig. 6共b兲. Estimated delays, for all array sensors, are plotted as a function of the position indexes of the sensors along the array. Figure 6共b兲 shows a perfect agreement between theoretical estimation using Eq. 共10兲 and that of the MBF. B. Real signals
(b) FIG. 6. TD estimation results 共solid line: MBF method, dotted line: Theoretical method兲. 共a兲 Time-sensors representation in synthetic case. 共b兲 Estimated TD versus the sensor position indexes along the array in synthetic case.
erence signals, s1共t兲 and s2共t兲, with size window 共in samples兲 w set to 65 and 81, respectively, are shown in Figs. 5共a兲 and 5共d兲. The received signals, r1共t兲 and r2共t兲 shown in Figs. 5共b兲 and 5共e兲, are obtained by time-shifting 300 samples, adding noise, and attenuating the reference signals. In this example, ␣ is set to 0.95. Outputs of the MBF are shown in Figs. 5共c兲 and 5共f兲 indicating a net maximum at t = TB. As expected, the peak of the function I共t兲 is located at TB = 300. Table I lists the TB values calculated for s1共t兲 and s2共t兲 with different signal-to-noise-ratios 共SNRs兲 ranging from −6 to 9 dB, with ␣ set to 0.7. Each value of Table I corresponds to the average of an ensemble of 25 trial TD estimates. These results show that the performance of the MBF is very close to that of the MF. Both methods point to the same theoretical value for TB. In this third example we consider the TD estimation in the case of a linear array composed of 20 uniformly spaced sensors. Each received signal sensor corresponds to the backscattered echo of a punctual target 共acoustic source兲. Observed sensor output signals are shown in Fig. 6共a兲 as twodimensional plots 共time sensors兲. The delay estimation is performed between the first sensor of the array, taken as the J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008
We demonstrated the performance of the ⌿B operator on real data. These data are acoustic measurements conducted in a tank with a linear array of 20 sensors 共n = 20兲 where airfilled cylindrical objects are slightly buried under the sand bottom. Two cylinders were used for Data set 1 and one cylinder was used for Data set 2. The remaining parameter settings were c = 1485 ms−1, d = 2 mm, Data1 = 22°, Data2 = 64.15°, frequency band 关150 kHz, 250 kHz兴 and sampling rate= 2 MHz. Sensor outputs of signals 共backscattered echoes兲 arriving from one 共Data2兲 and two 共Data1兲 cylinders are shown in Figs. 7 and 8, respectively. These signals correspond to the output of the first sensor of the array and are selected as reference signals. Both the MBF and the MF
FIG. 8. Example of two buried targets. The selected reference signal of Datal. Boudraa et al.: Cross ⌿B-energy operator
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TABLE II. RMSE between MBF, MF, and theoretical TD values for Datal and Data2. Datal
Data2 RMSE共MF−Theor兲 0.250
RMSE共MBF−Theor兲 0.217
methods, applied to the filtered signals, are implemented in the time domain. Signals are smoothed using a third-order Savitzky–Golay filter over a moving window of width set to 51.9 TDs are estimated using the Theoretical 关Eq. 共10兲兴, MBF, and the MF methods. In each case delay estimation is performed between the first sensor of the array and the remaining ones. Root mean square error 共RMSE兲 between pair of sensors for Data1 and Data2 is reported in Table II:
RMSE共A−Theor兲 =
冑
n
共TDA共i兲 − TTheor共i兲兲2 兺 i=1 n−1
共11兲
RMSE共MBF−Theor兲 0.0526
RMSE共MF−Theor兲 0.180
forms the MF. The processed signals are either noiseless or moderately noisy. For very noisy signals, the robustness of the MBF must be studied. As future work, we plan to use smooth splines to give more robustness to the MBF. To confirm the effectiveness of the MBF, the method must be evaluated with a large class of signals and in different experimental conditions including high noise levels and sampling rates and sample sizes. As future work, we plan to estimate TD values in the case of a nonuniform 共distorted兲 array. We also plan to modify the proposed scheme to analyze situations in which signals and noises are mutually correlated.
C. Analysis
As shown in Fig. 9共b兲, for Data set 2 there is a perfect agreement 共except for sensor 2 where the error is of one sample兲 between the MBF and the Theoretical method 关Eq. 共10兲兴. This is confirmed by the RMSE value 共5.26%兲. The RMSE of the MF is 3.42 times higher than that of the MBF. Figure 9共a兲 shows that, for Datal, the TD estimated by the MBF and the MF deviate moderately from TD values given by Eq. 共10兲. Note that the MBF performs slightly better than the MF with a RMSE of 21.7%. For both Data1 and Data2, globally, the MBF performs better than the MF. This may be due, even partially, to the nonlinear relationship between the signals that linear method such as the MF cannot account for. The mismatch between expected TD and MBF TD values may be due to the estimation error of the bearing angles Data1 and Data2 and to c, the sound speed in the water, which depends on the temperature. It is important to keep in mind that there is no odd way for estimating the TD value. The method based on Eq. 共10兲 can be used, in the far field case, as a reference method if we have good measures of both and d.
(a)
VIII. CONCLUSION
In this paper, two methods, based on the ⌿B operator, for signal detection and time-delay estimation 关called Matched ⌿B Filter 共MBF兲兴 are introduced. MBF measures how much one signal is present in another one. The discrete version of the continuous-time operator ⌿B, which is used in its implementation, is presented. Preliminary results of signal detection and TD estimation and their comparison with matched filter and a reference method are presented. These signals show that ⌿B is sensitive to nonlinear dependencies between signals compared to the classical CC method. For signal detection the MBF gives the same results as the MF for different SNRs. For TD estimation, the MBF results are very close to those of the reference method. The result for real acoustic signals shows that the MBF globally outper4288
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(b) FIG. 9. TD estimation by theoretical, MBF, and MF methods. Estimated TD versus the sensors position indexes along the array in Datal 共a兲 and Data2 共b兲, respectively 共*: MF, 䊐: MBF, –: Theoretical methods兲. Boudraa et al.: Cross ⌿B-energy operator
ACKNOWLEDGMENTS
Authors would like to thank the Associate Editor and the anonymous reviewers for their helpful valuable comments and suggestions that helped to improve the paper. We would also like to thank Dr. J. P. Sessarego from CNRS-LMA, Marseille 共France兲 and Dr. Z. Saidi from Ecole Navale, Brest 共France兲 for making data available. Further, they express their gratitude to Dr. L. Guillon from Ecole Navale for helpful discussions about buried objects and multipath environments. S. Haykin, Adaptive Filter Theory 共Prentice–Hall, Englewood Cliffs, NJ兲 共1996兲. 2 J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy,” IEE Electronics Lett. 40, 778–780 共2004兲. 1
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A. Boudraa, J. Cexus, K. Abed-Meraim, and Z. Saidi, “Interaction measure of AM-FM signals by cross-b-operator,” in Proc. IEEE ISSPA, 775– 778 共2005兲. 4 A. Boudraa, J. Cexus, and H. Zaidi, “Functional segmentation of dynamic nuclear images by cross-b-operator,” Comput. Methods Programs Biomed. 84, 146–152 共2006兲. 5 J. Kaiser, “Some useful properties of Teager’s energy operators,” in Proc. ICASSP, 149–152 共1993兲. 6 P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Signal Process. Lett. 2, 152–154 共1995兲. 7 However, we do not claim that MBF is an optimum detector of signal in white noise as the Matched Filter 共MF兲. The main difference between the MBF and the MF is that the MF is linear measure while MBF is a nonlinear one. 8 R. McDonough and A. Whalen, Detection of Signals in Noise 共Academic, New york, 1995兲. 9 A. Savitzky and M. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 共1964兲.
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