Cross Ψ[sub B]-energy operator-based signal detection

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Cross �B-energy operator-based signal detectiona)

Abdel-Ouahab Boudraab�

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 CexusE3I2 (EA3876), ENSIETA, 2 rue Francois Verny, 29806 Brest Cedex 9, France

Karim Abed-MeraimTSI department, ENST-Paris, 46 rue Barrault 75634, Paris Cedex 13, France

�Received 6 April 2007; revised 7 March 2008; accepted 4 April 2008�

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-FMsignals. The �B energy operator measures how much one signal is present in another one. The peakof the �B operator corresponds to the maximum of interaction between the two signals. Comparedto the cross-correlation function, the �B operator includes temporal information and relativechanges of the signal which are reflected in its first and second derivatives. The discrete version ofthe continuous-time form of the �B operator, which is used in its implementation, is presented. Themethods are illustrated on synthetic and real signals and the results compared to those of thematched filter and the cross correlation. The real signals correspond to impulse responses of buriedobjects obtained by active sonar in iso-speed single path environments.© 2008 Acoustical Society of America. �DOI: 10.1121/1.2916583�

PACS number�s�: 43.60.Bf, 43.60.Kx �WMCs� Pages: 4283–4289

I. INTRODUCTION

Signal detection and time-delay �TD� estimation be-tween signals are important problems in communications andsignal processing. Typical applications involve radar, sonar,machine fault diagnostics, biomedicine, and geophysics.1 Forexample, in sonar signal processing the difference in arrivaltime of a signal at two or more spatially separated receiversis used to estimate range and bearing of the source. A com-mon method of signal detection based on TD involves crosscorrelation of the receiver output, whereby an estimate of theTD is given by the argument that maximizes the cross-correlation �CC� function. Although the CC method is thesimplest similarity measure used in signal processing, it isnot sensitive to nonlinear dependence of the signals. In prac-tice, the interaction between signals may be nonlinear. Fur-thermore, sensor characteristics may also cause nonlineardistortions. Consequently, the maximum CC may not corre-spond to the maximum signal interaction. In this case, theresulting TD may be erroneous. To tackle this problem a newsimilarity function that measures the nonlinear interactionbetween signals is proposed.2–4 The method is based on thecross �B-energy operator, recently introduced by theauthors.2 The �B-energy operator is derived from a secondenergy-like function, called the Cross Teager-Kaiser EnergyOperator �CTKEO�5,6 which measures the interaction be-tween two real time functions. Compared to the CC, �B

a�Preliminary results of this work were presented at IEEE ISSPA, Sydney,Australia, 2005.

b�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 �6�, June 2008 0001-4966/2008/123�6

ution subject to ASA license or copyright; see http://acousticalsociety.org/c

includes temporal information and accounts for relativechanges of the signal by using the first and the second de-rivatives of the signal. Since it is based on a nonlinear op-erator, �B, the proposed method can be viewed as a nonlin-ear matched filter and we note it MBF �for Matched �B

Filter�.7 �B can also be used to “detect” the presence ofknown signals as components of more complicated signalsby measuring how much one signal is present in anotherone.3 Thus, the �B operator can be used as a strategy forsignal detection. As shown in Ref. 3, �B or methods basedon �B are well suited for nonstationary signals such asmono-component AM-FM signals. Based on the CTKEO,both the �B operator and the associated methods for TDestimation or signal detection are limited to narrowband sig-nals.

The paper is organized as follows. In the following sec-tion the �B operator is presented. Section III deals with thediscrete form of �B which is the basis of its numericalimplementation. In Sec. IV a �B-based signal detectionframework is introduced followed by the �B-based estima-tion approach in Sec. V. A pseudo code of the MBF is pre-sented in Sec. VI. Simulation and experimental results arepresented in Sec. VII and concluding remarks are stated inSec. VIII. We illustrate the method with an underwateracoustics application, where each signal is the impulse re-sponse of a buried object obtained by active sonar in a singlepath environment.

II. THE CROSS �B-ENERGY OPERATOR

Recently the CTKEO has been extended to complex-valued signals and an operator called �B introduced.2 Given

2
two complex signals x�t� and y�t�, �B is denned as follows:
© 2008 Acoustical Society of America 4283�/4283/7/$23.00

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�B�x,y� = 0.5��C�x,y� + �C�y,x�� 1�

=0.5�x*y + xy*� − 0.25�xy* + x*y + yx* + y*x� , �2�

where �C�x ,y�=0.5�x*y+ xy*�−0.5�xy*+x*y�. It has beenshown that the cross �B-energy operator of x�t� and y�t� isequal to the cross-Teager energies of their real and imaginaryparts,2

�B�x,y� = �B�xr,yr� + �B�xi,yi� �3�

�B�xl,yl� = xlyl − 0.5�xlyl + xlyl�, l � �r,i� , �4�

where x�t�=xr�t�+ jxi�t� and y�t�=yr�t�+ jyi�t�. The complexform of the signals is obtained using the Hilbert transform.For �B implementation, different derivative approximationscan be used.

III. DISCRETIZING THE CONTINUOUS-TIME �BOPERATOR

Discretized derivatives of the continuous �B operatorare combined to obtain an expression closely related to thediscrete form of the operator �Bd

and operating on discrete-time signals x�n� and y�n�. Different sample differences canbe used, but only the two-sample backward difference is de-tailed herein. For simplicity, we replace t by nTs where Ts isthe sampling period, and x�t� with x�nTs� or simply x�n�.

x�t� → �xk�n� − xk�n − 1��/Ts

x�t� → �xk�n� − 2xk�n − 1� + xk�n − 2��/Ts2

�B�xk�t�,yk�t�� → xk�n − 1�yk�n − 1�/Ts2

− 0.5�xk�n�yk�n − 2�

+ yk�n�xk�n − 2��/Ts2

�B�xk�t�,yk�t�� → �Bd�xk�n − 1�,yk�n − 1��/Ts

2

k � �i,r� . �5�

The discrete form of �B�x�t� ,y�t�� is given by

�B�x�t�,y�t�� → ��Bd�xr�n − 1�,yr�n − 1��

+ �Bd�xi�n − 1�,yi�n − 1��/Ts

2, �6�

where � denotes the mapping from continuous to discrete.Thus, from �B we obtain �Bd

shifted by one sample to theleft and scaled by Ts

−2. It is easy to show that using thetwo-sample forward difference from �B, we obtain �Bdshifted by one sample to the right and scaled by Ts

−2. For bothasymmetric two-sample differences, �B is shifted by onesample and scaled by Ts

−2. If we ignore the one-sample shiftand the scaling parameter, we transform �B�x�t� ,y�t�� into�Bd

�x�n� ,y�n�� as follows:

�B�x�t�,y�t�� → �Bd�xr�n�,yr�n�� + �Bd

�xi�n�,yi�n�� ,

�7�

4284 J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008


�Bd�xk�n�,yk�n�� = xk�n�yk�n� − 0.5�xk�n + 1�yk�n − 1�

+ yk�n + 1�xk�n − 1�� k � �i,r� . �8�

The three-sample symmetric difference can also be used butit leads to a more complicated expression compared to asym-metric two-sample differences. Indeed, the asymmetric ap-proximation is less complicated for implementation andfaster than the symmetric one because it requires fewer op-erations.

IV. �B-BASED SIGNAL DETECTION

We motivate �B-based detection by considering theclassical binary hypothesis testing problem encountered inradar or in sonar.8 Let s�t� denote a baseband transmittedsignal. The received signal R�t� is processed over an interval�Ti ,Tf� to detect the presence of a target. The hypotheses onR�t� are

�H0:R0�t� = n�t�,t � �Ti,Tf� ,

H1:R1�t� = �s�t − t0� + n�t�,t � �Ti,Tf� .

Under the null hypothesis, H0, the received signal containsonly an additive noise, n�t�. Under the alternative hypothesis,H1, a received time-shifted version of the transmitted signal�s�t− t0� is received in the presence of noise where � denotesan unknown gain parameter. The unknown time, t0, repre-sents the delay of the received signal and corresponds to theunknown distance of the target. Let T= �Tmin,Tmax� denotethe possible range of values for t0. The required observationinterval is �Ti ,Tf�= �Tmin, t0+Tmax� in this case. For any givenvalue of t0�T, the decision whether to reject H0 is given bycomputing

TB = arg maxt�T

�T

�B�R0�t�,R1�t��dt , �9�

where TB corresponds to the time of maximum of interactionbetween R0�t� and R1�t�.3 TB is then compared to a thresholdto determine the presence �H1� or absence �H0� of a target.Thus, the best detector calculates the interaction between thereceived signal and all possible time-shifted versions of thetransmitted signal and picks the largest energy interaction asthe basis for the detection decision. The location of the peakis the estimate of the unknown parameter t0=TB.

V. �B-BASED TIME-DELAY ESTIMATION

We have shown that the TD estimation problem mea-sures the interaction between two FM signals by using the�B operator.3 Consider a signal from a remote source beingreceived in the presence of noise at two spatially separatedreceivers. The time histories of the receiver outputs, denotedby rm�t� and rk�t�, are given by

�rm�t� = s�t� + nm�t� ,

rk�t� = �s�t − �k − m�r� + nk�t� ,

where s�t� is the signal waveform, nm�t� and nk�t� are thenoise waveforms at the respective receivers, � is an attenu-ation factor, and � is the difference in wavefront arrival times
at two consecutive receivers �k=2,m=1�. We assume that
Boudraa et al.: Cross �B-energy operator


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nm�t�, nk�t� and s�t� are mutually uncorrelated.3 PropagationTD between receivers k and m is given by �d /c= �k−m��,where �d=dk−dm is the path length difference, and c is thesound speed in the medium. When the target is sufficientlydistant from the receivers the wavefronts can be approxi-mated by plane waves and the theoretical TD, TTheor, is givenby

TTheor = �k − m�d sin �/c , �10�

where d is the distance between two consecutive receiversand � is bearing angle �See Fig. 1�.

VI. PSEUDO CODE OF MBF

The complex form of the signal is obtained using theHilbert transform. The MBF implementation involves thefollowing steps:

Inputs:

Emitted signal: s�p�=sa�p�+ jsb�p�, p� �1,2 , . . . ,w�,

Received signal: r�n�=ra�n�+ jrb�n�, n� �1,2 , . . . ,N�,where a and b indicate the real and imaginary parts of thecomplex signal respectively. w and N are time durations ofs�t� and r�t�, respectively, and z=1 to M denotes the index ofthe sliding window.

Outputs: TB.

0 50 100 150 200−1

−0.5

0

0.5

1

Time

Sig

nalX

0 50 100 150 200−1

−0.5

0

0.5

1

Time

Sig

nalY

0 50 100 150 2000.05

0.1

0.15

0.2

0.25

Time

Nor

mal

ized

freq

uenc

y

0 50 100 150 2000.05

0.1

0.15

0.2

0.25

Time

Nor

mal

ized

freq

uenc

y

Signal X

Signal Y

Intersection frequency

Sensor k

Plane wave

(k-m).d

θθ

Target(source)

Sensor m

FIG. 1. �Color online� Geometry used to estimate the time delay associatedwith plane waves.

J. Acoust. Soc. Am., Vol. 123, No. 6, June 2008


Pseudo Code:M =N−w+1

For z=1 to Mgz�n−z+1�←r�n� ,n�F= �z , . . . ,w+z−1�

Compute �al =�Bd

�sa�l� ,gaz�l�� and

�bl =�Bd

�sb�l� ,gbz�l�� using Eq. �8�

�Bd�s�l� ,gz�l��←�a

l +�bl , l� �1, . . . ,w�

Compute the sum I�z� of �Bdvalues over F:

I�z�= �l=z

w+z−1

�Bd�s�l� ,gz�l��

EndFor

Compute TB=arg max1�z�M

�I�z��

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time

Am

plitu

de

ΨB

(X,X)ΨB

(Y,Y)

ΨB

(X,Y)

Intersection frequency

FIG. 3. �Color online� Interaction measure between x�t� and y�t�.

FIG. 4. �Color online� Interaction measure between x�t� and y�t� using CC.
FIG. 2. �Color online� Linear chirp test signals �left� and their respective IFs�right�.
Boudraa et al.: Cross �B-energy operator 4285


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VII. RESULTS

The results presented below highlight the contributionsof the paper. The proposed method is illustrated on syntheticand real signals and the results are compared to those of theCC method and the Matched Filter �MF�. The real signalscorrespond to impulse responses of buried objects. Thepresent study is limited to signals obtained by active sonar iniso-speed single path environments. Both �B and MBF areimplemented using asymmetric two-sample differences �Eqs.�7� and �8��.

A. Synthetic signals

We first show an example of interaction between twononstationary signals measured by �B operator and comparethe result to the CC approach. Figure 2 shows an example oftwo linear FM signals x�t� and y�t� with the correspondingInstantaneous Frequencies �IFs�. The IF of x�t� increases lin-early with time while that of y�t� decreases with time. Theinter-action between x�t� and y�t� is calculated using Eq. �2�.Figure 3 shows the energy of each signal and the energy of

TABLE I. Estimated TB versus SNR signals s1�t� and s2�t� using MF and M

SNR=−6 dB SNR=−2 dB SNR

Signals MBF MF MBF MF MBFs1�t� 300�1 300�1 300�1 300 300s2�t� 300�2 300�2 300�1 300�1 300�1

0 20 40 600

0.5

1

0 200 400 600−1

−0.5

0

0.5

1

1.5

0 200 400 600

−1

0

1

2

3

s 1(t)

r 1(t)

TimeI 1(t

)

Time

Time

(a)

(b)

(c)TB

FIG. 5. Signal detection results with a=0.95: �a� First emitted signal. �b�emitted signal and the echo of the first signal, �d� Second emitted signal. �e�the emitted signal and the echo of the second signal.



their interaction. The maximum of interaction corresponds tothe instant when the two IFs intersect �Figs. 2 and 3� and alsowhere the energy of x�t��B�x ,x�� and that of y�t��B�y ,y��are equal �Fig. 3�. The point where the IFs intersect is lo-cated at t=125. Maximums of interaction between x�t� andy�t� 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, asexpected, the point of maximum of interaction. This resultshows that the CC measure is insensitive to nonlinear depen-dency between x�t� and y�t�. Away from the point where theIFs cross, the amplitude of the interaction decreases becausethere is less similarity between the two signals. As the IFsconverge from the time origin to their intersection, the inter-action intensity of the signals increases and the maximum ofinteraction is achieved at the intersection. This exampleshows that �B is more effective with nonstationary signalsthan the CC. This is due to the fact that �B is a nonlinearoperator while CC is linear.

In this simulation an example of delay estimation per-formed between two signals is presented. Two synthetic ref-

methods.

B SNR=3 dB SNR=5 dB SNR=9 dB

MF MBF MF MBF MF MBF MF300 300 300 300 300 300 300

00�1 300�1 300 300�1 300 300 300

0 20 40 60 80−1

−0.5

0

0.5

1

0 200 400 600

−1

−0.5

0

0.5

1

0 200 400 600

−1

0

1

2

s 2(t)

r 2(t)

I 2(t)

Time

Time

Time

(d)

(e)

(f)TB

ved echo of the first signal, �c� Integral, I, of �Bdcalculated between the

ved echo of the second signal, and �f� Integral, I, of �Bdcalculated between

BF

=1 d

3

ReceiRecei



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erence signals, s1�t� and s2�t�, with size window �in samples�w set to 65 and 81, respectively, are shown in Figs. 5�a� and5�d�. The received signals, r1�t� and r2�t� shown in Figs. 5�b�and 5�e�, are obtained by time-shifting 300 samples, addingnoise, and attenuating the reference signals. In this example,� is set to 0.95. Outputs of the MBF are shown in Figs. 5�c�and 5�f� indicating a net maximum at t=TB. As expected, thepeak of the function I�t� is located at TB=300. Table I liststhe TB values calculated for s1�t� and s2�t� with differentsignal-to-noise-ratios �SNRs� ranging from −6 to 9 dB, with� set to 0.7. Each value of Table I corresponds to the averageof an ensemble of 25 trial TD estimates. These results showthat the performance of the MBF is very close to that of theMF. Both methods point to the same theoretical value for TB.

In this third example we consider the TD estimation inthe case of a linear array composed of 20 uniformly spacedsensors. Each received signal sensor corresponds to the back-scattered echo of a punctual target �acoustic source�. Ob-served sensor output signals are shown in Fig. 6�a� as two-dimensional plots �time sensors�. The delay estimation is

Sensors

Samples(a)

(b)

FIG. 6. TD estimation results �solid line: MBF method, dotted line: Theo-retical method�. �a� Time-sensors representation in synthetic case. �b� Esti-mated TD versus the sensor position indexes along the array in syntheticcase.

performed between the first sensor of the array, taken as the



reference sensor, and each sensor of the array. The MBFdelay estimation, of the received signal on two sensors, ver-sus the sensor indexes along the linear array is shown in Fig.6�b�. Estimated delays, for all array sensors, are plotted as afunction of the position indexes of the sensors along thearray. Figure 6�b� shows a perfect agreement between theo-retical estimation using Eq. �10� and that of the MBF.

B. Real signals

We demonstrated the performance of the �B operator onreal data. These data are acoustic measurements conducted ina tank with a linear array of 20 sensors �n=20� where air-filled cylindrical objects are slightly buried under the sandbottom. Two cylinders were used for Data set 1 and onecylinder was used for Data set 2. The remaining parametersettings were c=1485 ms−1, d=2 mm, �Data1=22°, �Data2

=64.15°, frequency band �150 kHz,250 kHz� and samplingrate=2 MHz. Sensor outputs of signals �backscattered ech-oes� arriving from one �Data2� and two �Data1� cylinders areshown in Figs. 7 and 8, respectively. These signals corre-spond to the output of the first sensor of the array and areselected as reference signals. Both the MBF and the MF

FIG. 7. Example of one buried target. The selected reference signal ofData2.

FIG. 8. Example of two buried targets. The selected reference signal of
Datal.


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methods, applied to the filtered signals, are implemented inthe time domain. Signals are smoothed using a third-orderSavitzky–Golay filter over a moving window of width set to51.9 TDs are estimated using the Theoretical �Eq. �10��,MBF, and the MF methods. In each case delay estimation isperformed between the first sensor of the array and the re-maining ones. Root mean square error �RMSE� between pairof sensors for Data1 and Data2 is reported in Table II:

RMSE�A−Theor� =��

i=1

n

�TDA�i� − TTheor�i��2

n − 1�11�

C. Analysis

As shown in Fig. 9�b�, for Data set 2 there is a perfectagreement �except for sensor 2 where the error is of onesample� between the MBF and the Theoretical method �Eq.�10��. This is confirmed by the RMSE value �5.26%�. TheRMSE of the MF is 3.42 times higher than that of the MBF.Figure 9�a� shows that, for Datal, the TD estimated by theMBF and the MF deviate moderately from TD values givenby Eq. �10�. Note that the MBF performs slightly better thanthe MF with a RMSE of 21.7%. For both Data1 and Data2,globally, the MBF performs better than the MF. This may bedue, even partially, to the nonlinear relationship between thesignals that linear method such as the MF cannot account for.The mismatch between expected TD and MBF TD valuesmay be due to the estimation error of the bearing angles�Data1 and �Data2 and to c, the sound speed in the water, whichdepends on the temperature. It is important to keep in mindthat there is no odd way for estimating the TD value. Themethod based on Eq. �10� can be used, in the far field case,as a reference method if we have good measures of both �and d.

VIII. CONCLUSION

In this paper, two methods, based on the �B operator,for signal detection and time-delay estimation �calledMatched �B Filter �MBF�� are introduced. MBF measureshow much one signal is present in another one. The discreteversion of the continuous-time operator �B, which is used inits implementation, is presented. Preliminary results of signaldetection and TD estimation and their comparison withmatched filter and a reference method are presented. Thesesignals show that �B is sensitive to nonlinear dependenciesbetween signals compared to the classical CC method. Forsignal detection the MBF gives the same results as the MFfor different SNRs. For TD estimation, the MBF results arevery close to those of the reference method. The result for

TABLE II. RMSE between MBF, MF, and theoretica

Datal

RMSE�MBF−Theor� RMSE�MF−Theor�

0.217 0.250

real acoustic signals shows that the MBF globally outper-



forms the MF. The processed signals are either noiseless ormoderately noisy. For very noisy signals, the robustness ofthe MBF must be studied. As future work, we plan to usesmooth splines to give more robustness to the MBF. To con-firm the effectiveness of the MBF, the method must be evalu-ated with a large class of signals and in different experimen-tal conditions including high noise levels and sampling ratesand sample sizes. As future work, we plan to estimate TDvalues in the case of a nonuniform �distorted� array. We alsoplan to modify the proposed scheme to analyze situations inwhich signals and noises are mutually correlated.

values for Datal and Data2.

Data2

RMSE�MBF−Theor� RMSE�MF−Theor�

0.0526 0.180

(a)

(b)

FIG. 9. TD estimation by theoretical, MBF, and MF methods. Estimated TDversus the sensors position indexes along the array in Datal �a� and Data2

l TD

�b�, respectively �*: MF, �: MBF, –: Theoretical methods�.



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ACKNOWLEDGMENTS

Authors would like to thank the Associate Editor and theanonymous reviewers for their helpful valuable commentsand suggestions that helped to improve the paper. We wouldalso like to thank Dr. J. P. Sessarego from CNRS-LMA,Marseille �France� and Dr. Z. Saidi from Ecole Navale, Brest�France� for making data available. Further, they expresstheir gratitude to Dr. L. Guillon from Ecole Navale for help-ful discussions about buried objects and multipath environ-ments.

1S. Haykin, Adaptive Filter Theory �Prentice–Hall, Englewood Cliffs, NJ��1996�.

2J. Cexus and A. Boudraa, “Link between cross-Wigner distribution andcross-Teager energy,” IEE Electronics Lett. 40, 778–780 �2004�.



3A. Boudraa, J. Cexus, K. Abed-Meraim, and Z. Saidi, “Interaction mea-sure of AM-FM signals by cross-b-operator,” in Proc. IEEE ISSPA, 775–778 �2005�.

4A. Boudraa, J. Cexus, and H. Zaidi, “Functional segmentation of dynamicnuclear images by cross-b-operator,” Comput. Methods ProgramsBiomed. 84, 146–152 �2006�.

5J. Kaiser, “Some useful properties of Teager’s energy operators,” in Proc.ICASSP, 149–152 �1993�.

6P. Maragos and A. Potamianos, “Higher order differential energy opera-tors,” IEEE Signal Process. Lett. 2, 152–154 �1995�.

7However, we do not claim that MBF is an optimum detector of signal inwhite noise as the Matched Filter �MF�. The main difference between theMBF and the MF is that the MF is linear measure while MBF is a non-linear one.

8R. McDonough and A. Whalen, Detection of Signals in Noise �Academic,New york, 1995�.

9A. Savitzky and M. Golay, “Smoothing and differentiation of data bysimplified least squares procedures,” Anal. Chem. 36, 1627–1639 �1964�.



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