A Frequency-Domain Adaptive Matched Filter for Active Sonar Detection

sensors

Article A Frequency-Domain Adaptive Matched Filter for Active Sonar Detection

Zhishan Zhao 1,2, Anbang Zhao 1,2,*, Juan Hui 1,2,*, Baochun Hou 3, Reza Sotudeh 3 and Fang Niu 1,2

1 Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China; [email protected] (Z.Z.); [email protected] (F.N.) 2 College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China 3 School of Engineering and Technology, University of Hertfordshire, Hatﬁeld AL10 9AB, UK; [email protected] (B.H.); [email protected] (R.S.) * Correspondence: [email protected] (A.Z.); [email protected] (J.H.); Tel.: +86-138-4511-7603 (A.Z.); +86-136-3364-1082 (J.H.)

Received: 1 May 2017; Accepted: 29 June 2017; Published: 4 July 2017

Abstract: The most classical detector of active sonar and radar is the matched ﬁlter (MF), which is the optimal processor under ideal conditions. Aiming at the problem of active sonar detection, we propose a frequency-domain adaptive matched ﬁlter (FDAMF) with the use of a frequency-domain adaptive line enhancer (ALE). The FDAMF is an improved MF. In the simulations in this paper, the signal to noise ratio (SNR) gain of the FDAMF is about 18.6 dB higher than that of the classical MF when the input SNR is −10 dB. In order to improve the performance of the FDAMF with a low input SNR, we propose a pre-processing method, which is called frequency-domain time reversal convolution and interference suppression (TRC-IS). Compared with the classical MF, the FDAMF combined with the TRC-IS method obtains higher SNR gain, a lower detection threshold, and a better receiver operating characteristic (ROC) in the simulations in this paper. The simulation results show that the FDAMF has higher processing gain and better detection performance than the classical MF under ideal conditions. The experimental results indicate that the FDAMF does improve the performance of the MF, and can adapt to actual interference in a way. In addition, the TRC-IS preprocessing method works well in an actual noisy ocean environment.

Keywords: adaptive line enhancer; frequency-domain processing; matched ﬁlter; time reversal convolution; active sonar detection

1. Introduction The concept of the matched ﬁlter (MF) was proposed as early as the period of the Second World War [1]. In electronic information systems of communication, radar, and sonar, the MF is always the most commonly used among a variety of signal detection methods, because the MF is the optimal linear ﬁlter that obtains the maximum output signal to noise ratio (SNR) to detect the known signals in white ambient noise. The MF is employed in a wide range of actual applications such as target ranging, channel estimating, and spread spectrum communication. Dwork [2] gave the expression of the MF in the frequency domain, which will be discussed in detail in Section2. The appearances of the frequency-domain MF and the fast Fourier transform simplify the implementation of the MF, and make real-time processing possible for it. The application of the MF in sonar can be traced back to [3,4], which discussed some limitations to the ambiguity function theory of a few common sonar waveforms in terms of their narrow-band approximation, and investigated the echo-to-reverberation power ratio on the output of the MF by

Sensors 2017, 17, 1565; doi:10.3390/s17071565 www.mdpi.com/journal/sensors Sensors 2017, 17, 1565 2 of 12 Sensors 2017, 17, 1565 2 of 12

Afterwards, many new techniques based on the MF for the application in sonar detection using the ambiguity function. These discussions laid the foundation for the application of the MF in emerged. By way of example, we state four such techniques. The first is the model-based MF [5–7], sonar detection systems. which is generated by correlating the received signal with a reference channel that consists of the Afterwards, many new techniques based on the MF for the application in sonar detection emerged. transmitted signal convolved with the impulse response of the medium, and, when Doppler By way of example, we state four such techniques. The first is the model-based MF [5–7], which is compensated, can improve the performance of the MF of the Doppler-shifted signals in a time generated by correlating the received signal with a reference channel that consists of the transmitted dispersive ocean environment. The second is the adaptive MF [8–10], which is designed to resolve signal convolved with the impulse response of the medium, and, when Doppler compensated, can the mismatch problem between the channel model parameters and the actual time-varying and improve the performance of the MF of the Doppler-shifted signals in a time dispersive ocean environment. space-varying ocean acoustic channel. The replica of the adaptive MF is adaptively weighted by The second is the adaptive MF [8–10], which is designed to resolve the mismatch problem between extracting the time–frequency features of the echo signal with an adaptive filter. The adaptive MF the channel model parameters and the actual time-varying and space-varying ocean acoustic channel. has better matching performance than the model-based MF in slow time-varying coherent-multipath The replica of the adaptive MF is adaptively weighted by extracting the time–frequency features of channels. The third technique is the stochastic MF [11–13], which can be applied to frequency the echo signal with an adaptive filter. The adaptive MF has better matching performance than the time-varying signals such as the wide band modulated sonar signal propagated in shallow water. model-based MF in slow time-varying coherent-multipath channels. The third technique is the stochastic The stochastic MF is able to take into account uncertainties and variations of the multipath channel, MF [11–13], which can be applied to frequency time-varying signals such as the wide band modulated thus enhancing the detection efficiency. The fourth technique is the compressive MF [14,15], which sonar signal propagated in shallow water. The stochastic MF is able to take into account uncertainties is a correlation-based technique that combines analytical techniques in the field of compressive and variations of the multipath channel, thus enhancing the detection efficiency. The fourth technique is sensing for estimating the unknown delay and amplitude of a received signal using only a small the compressive MF [14,15], which is a correlation-based technique that combines analytical techniques number of noisy, randomly chosen frequency-domain samples of the signal. in the field of compressive sensing for estimating the unknown delay and amplitude of a received signal Since the MF has been proven to be the optimal detector under ideal conditions, the study of it using only a small number of noisy, randomly chosen frequency-domain samples of the signal. is mostly under a specific interference background rather than on its own features to improve its Since the MF has been proven to be the optimal detector under ideal conditions, the study of performance. Moreover, the better the performance of the detector under ideal conditions, the it is mostly under a specific interference background rather than on its own features to improve its better the performances of the other improved methods concerning the detector for a specific performance. Moreover, the better the performance of the detector under ideal conditions, the better interference. Is it possible to find a better detector than the MF under ideal conditions? Within the the performances of the other improved methods concerning the detector for a specific interference. range of linear operation and constant filters, the answer is no. However, jumping out of this range, Is it possible to find a better detector than the MF under ideal conditions? Within the range of linear we propose a frequency-domain adaptive matched filter (FDAMF), which has better expected operation and constant filters, the answer is no. However, jumping out of this range, we propose a processing performance. frequency-domain adaptive matched filter (FDAMF), which has better expected processing performance. In this paper, we studied the FDAMF by a theoretical, a simulation, and an experimental In this paper, we studied the FDAMF by a theoretical, a simulation, and an experimental method method for active sonar detection. We can find a better algorithm than the traditional MF in the for active sonar detection. We can find a better algorithm than the traditional MF in the field of field of adaptive processing, because the FDAMF is not a constant filter. It is undoubtedly a great adaptive processing, because the FDAMF is not a constant filter. It is undoubtedly a great challenge in challenge in the field of detection to find a better detector than the optimal one under ideal the field of detection to find a better detector than the optimal one under ideal conditions, and it will conditions, and it will also improve the performances of the above methods when combined with also improve the performances of the above methods when combined with them in an actual ocean them in an actual ocean environment. The FDAMF will provide more reliable and efficient environment. The FDAMF will provide more reliable and efficient detection techniques to engineering detection techniques to engineering applications in active sonar, radar, and communications. applications in active sonar, radar, and communications.

2.2. Analysis Analysis of of Output Output Spectrum Spectrum of of the the MF MF InIn this this section, section, we we analyze analyze the the output output spectrum spectrum of of the the MF MF to to prove that the frequency-domain outputoutput containscontains sinusoidalsinusoidal components. components. In otherIn other words, words, that therethat arethere line are components line components in the spectrum in the spectrumof the output of the spectrum. output Thus, spectrum. we can Thus, transform we can the outputtransform spectrum the output of the spectrum MF into a frequency-domainof the MF into a frequency-domainline detection problem. line Thisdetection is the theoreticalproblem. basisThis ofis usingthe theoretical an adaptive basis line of enhancer using an (ALE) adaptive to improve line enhancerthe performance (ALE) to of improve the MF. the performance of the MF.

2.1.2.1. Theoretical Theoretical Basis Basis TheThe block diagram ofof aa frequency-domainfrequency-domain MF MF is is shown shown in in Figure Figure1, in1, whichin which the the FT andFT and the IFTthe IFTrespectively respectively represent represent the Fourierthe Fourier transform transform and and the inversethe inverse Fourier Fourier transform. transform.

x()t X () Y () Y ()

y() H ()

FigureFigure 1. 1. BlockBlock diagram diagram of of a a frequency- frequency-domaindomain matched matched filter ﬁlter (MF).

Assume that the time-domain input signal of the MF is:

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Assume that the time-domain input signal of the MF is:

x(t) = s(t) + n(t), (1) where s(t) is the known speciﬁc signal, except the arrival time and amplitude are unknown, and n(t) is additive zero-mean band-limited white noise. Then, the spectral function of x(t) can be expressed as:

X(ω) = S(ω) + N(ω), (2) where S(ω) and N(ω) denote the FT of s(t) and n(t), respectively, and ω = 2πf. The frequency-domain output Y(ω) of the MF is given by:

Y(ω) = H(ω)X(ω) = H(ω)S(ω) + H(ω)N(ω) , (3) = So(ω) + No(ω) where H(ω) is the transmission function of the MF, that is, the FT of the impulse response h(t). So(ω) and No(ω) denote the output spectrum of s(t) and n(t) of the MF, respectively. When the input ambient noise is white, the transmission function of the MF is proved to be a complex conjugation of the input spectrum [16]. Assuming that the output signal component so(t) obtains the maximum output instantaneous SNR at t = t0, we have:

H(ω) = kS∗(ω)e−jωt0 , (4) where k is a constant, and * means complex conjugation. From (3) and (4), it follows that:

Y(ω) = kS(ω)S∗(ω)e−jωt0 + kN(ω)S∗(ω)e−jωt0 2 . (5) = k|S(ω)| e−jωt0 + kN(ω)S∗(ω)e−jωt0

The second term in (5) is a random spectrum generated by ambient noise, which is white if the signal bandwidth is wide enough. The ﬁrst term in (5) is the signal component, which can be seen as a continuous wave (CW) with the variable of frequency and the intensity of k|S(ω)|2. From (3) and (5), it follows that:

So(ω) = A(cos ωt0 − j sin ωt0), (6) where A = k|S(ω)|2 is deﬁned as an amplitude function. To see this more clearly, the real part and the imaginary part of the output spectrum are respectively given by:

Re[S (ω)] = A cos(2πt f ) o 0 . (7) Im[So(ω)] = A sin(2πt0 f )

The above equations indicate that the frequency-domain output of a speciﬁc signal processed by the MF is a CW signal with the variable of f. As we know, the time-domain output Y(τ) of the MF is the IFT of the frequency-domain output Y(ω). Let us deﬁne the spectral spectrum Y(Ω) as the FT of Y(ω):

Z ∞ Y(Ω) = Y(ω)e−jΩωdω. (8) −∞

Then, it follows that: ∞ ( ) = R −jωt0 −jΩω So Ω −∞ Ae e dω ∞ . (9) = R −jω(Ω+t0) −∞ Ae dω Sensors 2017, 17, 1565 4 of 12

SensorsFrom 2017 the, 17, above1565 equations, we can see that the real part and the imaginary part of the spectral4 of 12 spectrum Y(Ω) each have a line component at the spectral frequency Ω = −t0. Hence, we can use the frequency-domainfrequency-domain ALE ALE to to enhance enhance the the line line component, component, and and then then improve improve the the output output SNR SNR to to make make weakweak signals signals be be found found easily. easily.

2.2.2.2. Principle Principle Simulation Simulation InIn this this subsection, subsection, we verifywe verify the above the above theoretical theoretical analysis analysis by simulation by simulation with MATLAB. with MATLAB. Assume theAssume target signalthe targets(t) issignal a linear s(t) frequencyis a linear modulationfrequency modulation (LFM) signal (LFM) with signal a starting with frequency a starting of frequency 10 kHz, aof bandwidth 10 kHz, a of bandwidth 10 kHz, and of 10 a duration kHz, and time a duration of 30 ms. time As of shown 30 ms. in As Figure shown2a, thein Figure starting 2a, time the ofstartings(t) istime−15 of ms s( andt) is the−15 ending ms and time the isending 15 ms. time Then, is the15 ms. time Th ofen, the the maximum time of the output maximum SNR should output be SNR at t0should= 15 ms be, i.e., at thet0 = end15 ms, of the i.e., signal’s the end duration, of the signal’s which canduration, be seen which in Figure can2 bec. Theseen output in Figure spectrum 2c. The andoutput the spectral spectrum spectrum and the of spectral the MF spectrum are shown of in the Figure MF are2b,d shown respectively. in Figure Itcan 2b,d be respectively. seen that there It can is a be lineseen component that there atis Ωa line= − tcomponent0 = −15 ms. at Ω = −t0 = −15 ms.

FigureFigure 2. 2.Normalized Normalized outputs outputs of of the the MF. MF. (a )(a The) The target target signal signals(t );s(t ();b )(b The) The output output spectrum spectrum|Y (ωY ())|; ; (c)(c The) The time-domain time-domain output output|y (τy)()|;(;d ()d The) The spectral spectral spectrum spectrum|Y (YΩ())|. .

3.3. Principle Principle of of the the FDAMF FDAMF

TheThe block block diagram diagram of of the the time-domain time-domain ALE ALE is is shown shown in in Figure Figure3. 3.∆ Δ= =m mτ0τ0is is a a delay delay with with a a samplingsampling period period of ofτ 0τand0 and a a delay delay unit unit of of m. m.∆ Δshould should be be larger larger than than the the correlation correlation time time of of ambient ambient noisenoise to to make make the the noise noise components components of x( kof− xm(k) and− mx) (andk) uncorrelated. x(k) uncorrelated. Meanwhile, Meanwhile,∆ should Δ be should smaller be thansmaller the correlationthan the correlation time of the time signal of the components signal components to make themto make correlated them correlated [17]. W(k )[17]. is the W weight(k) is the vectorweight calculated vector calculated by the least by mean the squareleast mean (LMS) square iterative (LMS) algorithm, iterative and εalgorithm,(ω) is the estimationand ε(ω) erroris the toestimation adjust the valueerror ofto Wadjust(k). In the this value paper, of weW( applyk). In this the time-domainpaper, we apply ALE the to the time-domain frequency-domain ALE to bythe changingfrequency-domain the time-variable by changing to a frequency-variable. the time-variable to a frequency-variable. An ALE can detect a narrow-band signal or a CW pulse signal with unknown frequency effectively in ambient noise. Because there is a line in the spectral spectrum of the MF, we can use an ALE to x()ksknk () ()   ()k process the spectrum Y(ω) in order to improve the SNR gain. We can then use the IFT to transform the spectrum to the time-domain output yALE(τ) of the FDAMF, as shown in Figure4. The simulation signal generator is shown in the left parts of Figure4, in which the BP represents a bandpass ﬁlter. The target signal s(t) is an LFM signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. The ambient noise n(t) isy zero-mean()k white noise. The input Wk() x()km

Figure 3. Block diagram of the adaptive line enhancer (ALE).

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frequency-domain ALE to enhance the line component, and then improve the output SNR to make weak signals be found easily.

2.2. Principle Simulation In this subsection, we verify the above theoretical analysis by simulation with MATLAB. Assume the target signal s(t) is a linear frequency modulation (LFM) signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. As shown in Figure 2a, the starting time of s(t) is −15 ms and the ending time is 15 ms. Then, the time of the maximum output SNR should be at t0 = 15 ms, i.e., the end of the signal’s duration, which can be seen in Figure 2c. The output spectrum and the spectral spectrum of the MF are shown in Figure 2b,d respectively. It can be seen that there is a line component at Ω = −t0 = −15 ms.

Figure 2. Normalized outputs of the MF. (a) The target signal s(t); (b) The output spectrum Y () ; Sensors 2017, 17, 1565 5 of 12 (c) The time-domain output y() ; (d) The spectral spectrum Y() .

SNR3.Sensors Principle is −201710, dB 17of, 1565 bythe taking FDAMF the noise after passing a bandpass ﬁlter (BPF) into account. In this simulation,5 of 12 the FT is carried out by the fast Fourier transform (FFT). We put the ending time of the target signal to The block diagram of the time-domain ALE is shown in Figure 3. Δ = mτ0 is a delay with a the locationAn ALE of 0can ms fordetect easier a narrow-band observation. Assignal this studyor a CW involves pulse the signal detection with ofunknown the LFM frequency signal in sampling period of τ0 and a delay unit of m. Δ should be larger than the correlation time of ambient noise,effectively we choose in ambient an effective noise. detecting Because tool,there the is a concise line in fractionalthe spectral Fourier spectrum transform of the (CFRFT)MF, we can [18 ],use as aan noise to make the noise components of x(k − m) and x(k) uncorrelated. Meanwhile, Δ should be comparisonALE to process to the the MF. spectrum The output Y(ω of) in the order CFRFT to improve is shown the in FigureSNR gain.5a. After We can processing then use the the target IFT to smaller than the correlation time of the signal components to make them correlated [17]. W(k) is the signaltransform contaminated the spectrum by the to the noise time-domain with the block output diagrams yALE(τ) of in the Figures FDAMF,1 and as4 shown respectively, in Figure we 4. can weight vector calculated by the least mean square (LMS) iterative algorithm, and ε(ω) is the obtain the outputs of the MF and the FDAMF, as shown in Figure5b,c, in which the step of the ALE is estimation error to adjust the value of W(k). In this paper, we apply the time-domain ALE to the 10−5 and the number of weights is 256 (in this paper, the number of weights and the step of the ALE frequency-domain by changingst() the time-variable to a frequency-variable. are taken to be these values unlessx otherwise()t X () stated).Y() yALE () nt() x()ksknk () ()   ()k H ()   Figure 4. Block diagram of the frequency-domain adaptive matched filter (FDAMF). Sensors 2017, 17, 1565  5 of 12 The simulation signal generator is shown in the left partsy()k of Figure 4, in which the BP An ALE can detect a narrow-band signalWk() or a CW pulse signal with unknown frequency effectivelyrepresents in aambient bandpass noise. filter. Because Thex()km target there signalis a line s( tin) isthe an spectral LFM signal spectrum with ofa startingthe MF, frequencywe can use of an 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. The ambient noise n(t) is zero-mean ALE to process the spectrum Y(ω) in order to improve the SNR gain. We can then use the IFT to white noise. The input SNR is −10 dB by taking the noise after passing a bandpass filter (BPF) into transform the spectrumFigure to the 3. time-domainBlock diagram output of the adaptiveyALE(τ) of line the enhancer FDAMF, (ALE). as shown in Figure 4. account. In this simulation,Figure 3.the Block FT diagramis carried of theou tadap by tivethe linefast enhancer Fourier (ALE).transform (FFT). We put the ending time of the target signal to the location of 0 ms for easier observation. As this study involves st() the detection of the LFM signal in noise, we choose an effective detecting tool, the concise fractional x()t X () Y() yALE () Fourier transform (CFRFT) [18], as a comparison to the MF. The output of the CFRFT is shown in Figure 5a. After processingnt() the target signal contaminated by the noise with the block diagrams in Figures 1 and 4 respectively, we can obtainH () the outputs of the MF and the FDAMF, as shown in Figure 5b,c, in which the step of the ALE is 10−5 and the number of weights is 256 (in this paper, the number ofFigure Figureweights 4. 4. BlockandBlock the diagram diagram step ofof of the the frequency-domainALE frequency-domain are taken to adaptive adaptivebe these matched matchedvalues unlessfilter ﬁlter (FDAMF). (FDAMF). otherwise stated).

The simulation signal generator is shown in the left parts of Figure 4, in which the BP represents a bandpass filter. The target signal s(t) is an LFM signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. The ambient noise n(t) is zero-mean white noise. The input SNR is −10 dB by taking the noise after passing a bandpass filter (BPF) into account. In this simulation, the FT is carried out by the fast Fourier transform (FFT). We put the ending time of the target signal to the location of 0 ms for easier observation. As this study involves the detection of the LFM signal in noise, we choose an effective detecting tool, the concise fractional Fourier transform (CFRFT) [18], as a comparison to the MF. The output of the CFRFT is shown in Figure 5a. After processing the target signal contaminated by the noise with the block diagrams in Figures 1 and 4 respectively, we can obtain the outputs of the MF and the FDAMF, as shown in Figure 5b,c, in which the step of the ALE is 10−5 and the number of weights is 256 (in this paper, the number of weights and the step of the ALE are taken to be these values unless otherwise stated).

FigureFigure 5. 5. NormalizedNormalized outputs of of the the concise concise fractional fractional Fourier Fourier transform transform (CFRFT), (CFRFT), the the MF MF and and the theFDAMF FDAMF when when input input signal signal to noise to noise ratio ratio (SNR) (SNR) = −10 = dB.−10 (a dB.) CFRFT (a) CFRFT (SNR (SNR= 20.2 = dB); 20.2 (b dB);) MF (b (SNR) MF = (SNR21.4 =dB). 21.4 (c dB).) FDAMF (c) FDAMF (SNR = (SNR 35.3 dB). = 35.3 dB).

From the comparison in Figure 5, we can see that the processing effect of the MF is slightly From the comparison in Figure5, we can see that the processing effect of the MF is slightly better better than that of the CFRFT, which is in accord with the conclusion that the MF is the optimal than that of the CFRFT, which is in accord with the conclusion that the MF is the optimal detector under detector under ideal conditions. The SNR gain of the FDAMF is significantly higher than that of the ideal conditions. The SNR gain of the FDAMF is signiﬁcantly higher than that of the MF. The output MF. The output SNR of the MF is 21.4 dB and that of the FDAMF is 35.3 dB, while the SNR gain of SNR of the MF is 21.4 dB and that of the FDAMF is 35.3 dB, while the SNR gain of the FDAMF to the the FDAMF to the conventional MF is 13.9 dB in this simulation. conventional MF is 13.9 dB in this simulation. 4. Improvement of the FDAMF under Low Input SNR Although the FDAMF has achieved more obvious SNR gain than the MF in white ambient noise, the ALE processing gain depends on the input SNR, based on the property itself. The ALE Figure 5. Normalized outputs of the concise fractional Fourier transform (CFRFT), the MF and the FDAMF when input signal to noise ratio (SNR) = −10 dB. (a) CFRFT (SNR = 20.2 dB); (b) MF (SNR = 21.4 dB). (c) FDAMF (SNR = 35.3 dB).

From the comparison in Figure 5, we can see that the processing effect of the MF is slightly better than that of the CFRFT, which is in accord with the conclusion that the MF is the optimal detector under ideal conditions. The SNR gain of the FDAMF is significantly higher than that of the MF. The output SNR of the MF is 21.4 dB and that of the FDAMF is 35.3 dB, while the SNR gain of the FDAMF to the conventional MF is 13.9 dB in this simulation.

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4. Improvement of the FDAMF under Low Input SNR Although the FDAMF has achieved more obvious SNR gain than the MF in white ambient noise, the ALE processing gain depends on the input SNR, based on the property itself. The ALE processing gain will drop sharply when the input SNR is very low [19]. In order to make the ALE work more efﬁciently, it is necessary to preprocess the input signals. As proved above, the frequency-domain output of the target signal processed by the MF is a CW signal with the variable of frequency, whereas the frequency-domain output of the band-limited white noise processed by the MF is still band-limited white. Therefore, we propose a time reversal convolution and interference suppression (TRC-IS) method in the frequency-domain to preprocess the spectrum of the MF to improve the input SNR of the ALE, based on the differences of the spectrums of the target signal and noise processed by the MF, and ﬁnally improve the SNR gain of the FDAMF. Assume that a weak CW signal which is contaminated by additive noise can be expressed as:

x(k) = A cos(ωk + ϕ) + Bn(k), (10) where k is an integer (1, 2, 3, ... , N), and ω = 2πfTs, where f denotes frequency and Ts denotes sampling period. A and B are constants which denote the intensity of the signal and noise. Expressing the sequence of the samples as a vector, we have:

T x = [x1, x2, ··· , xN] , (11)

T where xk = x(k), and the correlation matrix of x is R = xx .   x11 x21 ··· xN,1    x12 x22 ··· xN,2  R =  , (12)  . . .. .   . . . .  x1,N x2,N ··· xN,N where xi,j = x(i)x(j). The output of the time reversal convolution (TRC) is a sequence in the order of diagonal direction of matrix R, i.e., the TRC converts R into a sequence:

y(k) = y1 + y2 + ··· + yN + ··· y2N−1,  y = x N  1 1,  y = x + x  2 1,N−1 2,N  ··· . (13) where y = x + x + ··· + x  N 11 22 N,N   ···  y2N−1 = xN,1

The TRC, which contains all of the elements of the correlation matrix, calculates these elements in a simple and fast way, i.e., the TRC contains all of the information of the correlation matrix. In (13), yN is the auto-correlation term, which has no anti-interference ability. When the SNR is low, yN mainly contains the energy of the interference, and therefore it should be suppressed. The simulation below is based on Figure6, which shows the block diagram of the TRC-IS. We use low-frequency simulation signals to show the processing effects more clearly. In this simulation, the input signal s(t) is a CW signal with a frequency of 2 Hz and a bandwidth of 10 s. The noise n(t) is an additive white Gaussian noise (AWGN). The input SNR is −10 dB. Figure7a,b show the TRC results of s(t) and n(t), from which we can see that the energy of the noise is mainly concentrated at the spike of t = 10 s, whereas the energy of the signal is relatively scattered. The result of the signal x(t) contaminated by noise n(t) processed by the TRC is shown in Figure7c; suppressing the interference Sensors 2017, 17, 1565 7 of 12

of which is to set zero to the spike of the noise energy, then we obtain the output of the TRC-IS y(t). Normalizing the spectrums of x(t) and y(t), we get Figure7d, in which the solid line represents the normalized spectrum of x(t), and the dashed line represents the normalized spectrum of y(t), the output SNRs of whom are 20.5 dB and 41.1 dB respectively. From the comparison, it can be seen that Sensors 2017, 17, 1565 7 of 12 Sensorsthe TRC-IS 2017, 17 can, 1565 improve the detecting performance of single-frequency signals. 7 of 12

x()tstnt () () y()t Y () x()tstnt () () y()t Y ()

Figure 6. Block diagram of time reversal convolution and interference suppression (TRC-IS). FigureFigure 6. 6. BlockBlock diagram diagram of of time time reversal reversal convolution convolution and and interference interference suppression (TRC-IS).

Figure 7. Simulation results of the TRC-IS when the SNR = −10 dB. (a) time reversal convolution Figure 7. Simulation results of the TRC-IS when the SNR = −10 dB. (a) time reversal convolution Figure(TRC) of7. theSimulation signal s (resultst); (b) TRCof the of TRC-IS the noise whenn(t); the (c) TRCSNR of= − the10 signaldB. (a) contaminated time reversal byconvolution noise x(t); (TRC) of the signal s(t); (b) TRC of the noise n(t); (c) TRC of the signal contaminated by noise x(t); (TRC)(d) Normalized of the signal spectrum s(t); (b with) TRC and of withoutthe noise TRC-IS. n(t); (c) TRC of the signal contaminated by noise x(t); (d) Normalized spectrum with and without TRC-IS. (d) Normalized spectrum with and without TRC-IS. We verify the improving effects of the TRC-IS on the FDAMF in the following simulation. The target signal is an LFM signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. The ambient noise is a zero-mean AWGN, and the input SNR is −15 dB. The normalized outputs of the MF and the FDAMF are shown in Figure8a,b, in which the output SNRs are 13.3 dB and 19.5 dB respectively. If we preprocess the input signal of the ALE in Figure4 with the TRC-IS in Figure6, we can obtain the TRC-IS-FDAMF output, as shown in Figure8c, in which the SNR is 46.8 dB. It is seen from Figure8 that the FDAMF cannot detect useful signals effectively when the input SNR is pretty low, whereas after the TRC-IS preprocessing, improving the input SNR of the ALE, the performance of the FDAMF is improved a lot. Combining the FDAMF with the TRC-IS makes it possible to detect weak signals in strong noise more effectively.

Figure 8. Normalized outputs of the MF, the FDAMF, and the TRC-IS-FDAMF when the input SNR = Figure 8. Normalized outputs of the MF, the FDAMF, and the TRC-IS-FDAMF when the input SNR = −15 dB. (a) The MF (SNR = 13.3 dB); (b) The FDAMF (SNR = 19.5 dB); (c) The TRC-IS-FDAMF (SNR = −15 dB. (a) The MF (SNR = 13.3 dB); (b) The FDAMF (SNR = 19.5 dB); (c) The TRC-IS-FDAMF (SNR = 46.8 dB). 46.8 dB). We verify the improving effects of the TRC-IS on the FDAMF in the following simulation. The We verify the improving effects of the TRC-IS on the FDAMF in the following simulation. The target signal is an LFM signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a target signal is an LFM signal with a starting frequency of 10 kHz, a bandwidth of 10 kHz, and a duration time of 30 ms. The ambient noise is a zero-mean AWGN, and the input SNR is −15 dB. The duration time of 30 ms. The ambient noise is a zero-mean AWGN, and the input SNR is −15 dB. The

Sensors 2017, 17, 1565 7 of 12

x()tstnt () () y()t Y ()

Figure 6. Block diagram of time reversal convolution and interference suppression (TRC-IS).

Figure 7. Simulation results of the TRC-IS when the SNR = −10 dB. (a) time reversal convolution (TRC) of the signal s(t); (b) TRC of the noise n(t); (c) TRC of the signal contaminated by noise x(t); Sensors 2017, 17, 1565 8 of 12 (d) Normalized spectrum with and without TRC-IS.

Sensors 2017, 17, 1565 8 of 12

normalized outputs of the MF and the FDAMF are shown in Figure 8a,b, in which the output SNRs are 13.3 dB and 19.5 dB respectively. If we preprocess the input signal of the ALE in Figure 4 with the TRC-IS in Figure 6, we can obtain the TRC-IS-FDAMF output, as shown in Figure 8c, in which the FigureFigureSNR is 8. 8.46.8 NormalizedNormalized dB. It is seen outputs from ofFigure of the the MF, 8 MF, that the the th FDAMF,e FDAMF,FDAMF and cannot and the theTRC-IS-FDAMF detect TRC-IS-FDAMF useful signals when when effectively the input the input SNRwhen = SNR = −15 dB.(a) The MF (SNR = 13.3 dB); (b) The FDAMF (SNR = 19.5 dB); (c) The TRC-IS-FDAMF −the15 dB.input (a )SNR The MFis pretty (SNR low,= 13.3 whereas dB); (b) afterThe FDAMFthe TRC-IS (SNR preprocessing, = 19.5 dB); (c )improving The TRC-IS-FDAMF the input SNR(SNR of = (SNR = 46.8 dB). 46.8the ALE,dB). the performance of the FDAMF is improved a lot. Combining the FDAMF with the TRC-IS makes it possible to detect weak signals in strong noise more effectively. 5. PerformanceWe verify Analysis the improving effects of the TRC-IS on the FDAMF in the following simulation. The 5. Performance Analysis targetIn thissignal section, is an weLFM use signal the Monte with a Carlo starting method frequency to compare of 10 thekHz, performance a bandwidth of of the 10 MF kHz, and and the a FDAMF.duration The timeIn simulationthis of section, 30 ms. target we The use ambient signal the Monte is annoise LFMCarlo is signalamethod zero-mean with to compare a startingAWGN, the frequency andperformance the input of 10 of kHz,SNRthe MF aisbandwidth and−15 thedB. The FDAMF. The simulation target signal is an LFM signal with a starting frequency of 10 kHz, a of 10 kHz, and a duration time of 30 ms. The interference is a zero-mean AWGN. bandwidth of 10 kHz, and a duration time of 30 ms. The interference is a zero-mean AWGN. The SNRThe gain SNR curves gain curves of the of MF the and MF the and FDAMF the FDAM at differentF at different SNRs SNRs in AWGN in AWGN are shown are shown in Figure in 9, in whichFigure the number9, in which of samplesthe number at each of samples input SNRat each is 1000,input andSNR theis 1000, SNR and gain the is SNR deﬁned gain as is thedefined difference as betweenthe the difference mean of between the output the mean SNR of and the the output input SNR SNR. and The the dashedinput SN lineR. The denotes dashed the line SNR denotes gain the of the MF, andSNR the gain solid of the line MF, denotes and the that solid of line the denotes FDAMF. that It of can the be FDAMF. seen that It can the be SNR seen gainthat the of theSNR FDAMF gain of is signiﬁcantlythe FDAMF higher is thansignificantly that of higher the MF than in thethat rangeof the thatMF in Figure the range9 shows. that Figure With the9 shows. SNR increasing,With the the SNRSNR gain increasing, of the MF the gradually SNR gain tendsof the toMF about gradually 26 dB tends when to theabout input 26 dB SNR when is higher the input than SNR−10 is dB, whereashigher the SNRthan gain−10 dB, of whereas the FDAMF the SNR increases gain of tothe about FDAMF 240 increases dB. This to is about in accord 240 dB. with This the is in property accord of the ALE,with i.e., the the property higher of the the input ALE, SNRi.e., the the high betterer the the input performance. SNR the better the performance.

Figure 9. The SNR gain curves of the MF and the FDAMF at different SNRs in additive white Figure 9. The SNR gain curves of the MF and the FDAMF at different SNRs in additive white Gaussian Gaussian noise (AWGN). noise (AWGN). Figure 10 shows the detection probability curve when the false-alarm probability is 10−3. The Figuredashed 10 line shows denotes the the detection MF and probabilitythe solid line curve denotes when the theFDAMF. false-alarm When a probabilityspecific detection is 10 −3. The dashedprobability line denotesis required, the such MF as and 0.8, the the soliddetection line threshold denotes theof the FDAMF. FDAMF Whenis about a 5 speciﬁc dB lower detection than probabilitythat of is the required, MF, i.e., suchthe FDAMF as 0.8, can the work detection more efficiently threshold than of thethe FDAMFMF at a lower is about input 5 SNR. dB lower When thana specific input SNR is required, such as −15 dB, the detection probability of the FDAMF is significantly higher than that of the MF, i.e., the FDAMF is better at detecting targets than the MF in a certain ambient interference.

Sensors 2017, 17, 1565 9 of 12 that of the MF, i.e., the FDAMF can work more efficiently than the MF at a lower input SNR. When a specific input SNR is required, such as −15 dB, the detection probability of the FDAMF is significantly Sensors 2017, 17, 1565 9 of 12 higher than that of the MF, i.e., the FDAMF is better at detecting targets than the MF in a certain Sensors 2017, 17, 1565 9 of 12 ambient interference.

Figure 10. The detection probability curves of the MF and the FDAMF when false-alarm probability Figure 10.10. TheThe detectiondetection probabilityprobabilitycurves curves of of the the MF MF and and the the FDAMF FDAMF when when false-alarm false-alarm probability probability is is 10−3 in AWGN. 10is −103−in3 in AWGN. AWGN. Figure 11 shows the receiver operating characteristic (ROC) curves of the MF and the FDAMF Figurewhen 11the showsshows input thetheSNR receiverreceiver = −16 dB. operatingoperating The dashed characteristiccharacte line denotesristic (ROC) the MF curves and the of thesolid MFMF line andand denotes thethe FDAMFFDAMF the whenwhen theFDAMF.the input input It SNR isSNR seen = −= in 16− Figure16 dB. dB. The 11 The that dashed dashed the ROC line line denotesof the denotes FDAMF the MF the is and obviouslyMF the and solid betterthe line solid than denotes thatline ofdenotes the the FDAMF. MF. the ItFDAMF. is seenWhen It in is Figurea seen specific in11 Figure thatfalse-alarm the 11 ROC that probability the of the ROC FDAMF is of required, the isFDAMF obviously such isas obviously0.1, better the detection than better that thanprobability of the that MF. of of Whenthe the MF. a FDAMF can be as high as 0.98, while that of the MF is only 0.50. When a specific detection speciﬁcWhen a false-alarm specific false-alarm probability probability is required, is such required, as 0.1, thesuch detection as 0.1, probabilitythe detection of theprobability FDAMF canof the be FDAMFprobability can be asis required,high as such0.98, aswhile 0.8, thethat false-alarm of the MF probability is only of0.50. the WhenFDAMF a isspecific less than detection one as highthousandth, as 0.98, while while that that of of thethe MF reaches is only about 0.50. 0.48. When a speciﬁc detection probability is required, suchprobability as 0.8, theis required, false-alarm such probability as 0.8, ofthe the false-alarm FDAMF is lessprobability than one of thousandth, the FDAMF while is thatless ofthan the one MF reachesthousandth, about while 0.48. that of the MF reaches about 0.48.

Figure 11. The receiver operating characteristic (ROC) curves of the MF and the FDAMF when SNR = −16 dB in AWGN.

6. Experimental Results Figure 11.11. The receiver operating characteristic (ROC (ROC)) curves of the MF and the FDAMFFDAMF whenwhen SNR = The−1616 dBsea dB in intrial AWGN. AWGN. installation is shown in Figure 12. The LFM signal is transmitted from the flank array sonar of the surveying ship and received by the towed line array sensor located at a depth of 6. Experimental25 m. The distanceResults between the transmitting sensor and the receiving sensor is about 200 m. The 6. Experimental Results target is about 2 km away from the surveying ship, and is located at a depth of 35 m. The average The sea trial installation is shown in Figure 12. The LFM signal is transmitted from the flank Thedepth sea of trial the installation sea area is is55 shownm or more, in Figure and the12 .seabed The LFM consists signal of issediment. transmitted The sound from thespeed ﬂank array sonar of the surveying ship and received by the towed line array sensor located at a depth of array sonarmeasured of the in surveyingthe sea trial ship was and 1470 received m/s. We by use the the towed data received line array in sensorthis sea located trial to atverify a depth the of 25 m. The distance between the transmitting sensor and the receiving sensor is about 200 m. The 25 m. Theperformance distance and between practicability the transmitting of the FDAMF sensor for and signal the detection. receiving sensor is about 200 m. The target target is about 2 km away from the surveying ship, and is located at a depth of 35 m. The average is about 2 km away from the surveying ship, and is located at a depth of 35 m. The average depth depth of the sea area is 55 m or more, and the seabed consists of sediment. The sound speed measured in the sea trial was 1470 m/s. We use the data received in this sea trial to verify the

performance and practicability of the FDAMF for signal detection.

Sensors 2017, 17, 1565 10 of 12 of the sea area is 55 m or more, and the seabed consists of sediment. The sound speed measured in the sea trial was 1470 m/s. We use the data received in this sea trial to verify the performance and practicability of the FDAMF for signal detection. Sensors 2017, 17, 1565 10 of 12

Sensors 2017, 17, 1565 10 of 12

FigureFigure 12. 12.The The sea sea trial trial installation.installation. Figure 12. The sea trial installation. The processingThe processing results results of the of receivedthe received signals signals from from the the sea sea trial trial by by using using the the MF, MF,the the FDAMF,FDAMF, and and theThe TRC-IS-FDAMF processing results respectively of the received are shown signals in from Figure the sea13. trialWe useby using logarithmic the MF, coordinatesthe FDAMF, to the TRC-IS-FDAMF respectively are shown in Figure 13. We use logarithmic coordinates to display the displayand the resultsTRC-IS-FDAMF because ofrespectively the strong are interference shown in betweenFigure 13. 0 Wes to use1 s andlogarithmic the high coordinates fluctuations to of resultsthe because displayprocessing ofthe the resultsresults. strong because We interference can of estima the strongte between the interference arrival 0 stime to between 1 of s andthe target 0 the s to high echo1 s and ﬂuctuations to thebe aroundhigh fluctuations of2.8 the s from processing of the results.relative Wethe can processing position estimate ofresults. the transmitting arrival We can timeestima array, ofte thethe thearrival target receiving time echo of to array,the be target around and echo the 2.8 totarget. be s from around It can the 2.8 relativebe s fromseen thefrom position of theFigure transmittingrelative 13a thatposition array,the targetof thethe echotransmitting receiving is almost array, array, submerge andthe re thedceiving in target. the array,background It canand bethe interference seentarget. from It can when Figure be seen processing 13 froma that the Figure 13a that the target echo is almost submerged in the background interference when processing targetthe echo received is almost signals submerged with the MF, in the and background the processing interference results of the when FDAMF processing is also not the satisfactory received signals in the received signals with the MF, and the processing results of the FDAMF is also not satisfactory in with theFigure MF, 13b and due the to processing the low SNR, results whereas of the a significant FDAMF ispeak also of not the satisfactory target echo is in observed Figure 13 at b2.76 due s of to the Figure 13b due to the low SNR, whereas a significant peak of the target echo is observed at 2.76 s of the received signals after processing with the TRC-IS-FDAMF in Figure 13c. The SNR of the received low SNR,the whereas received a signals signiﬁcant after processing peak of the with target the TRC- echoIS-FDAMF is observed in Figure at 2.76 13c. s ofThe the SNR received of the received signals after signals can be estimated to be approximately −8 dB according to the position of the target echo. We processingsignals with can the be TRC-IS-FDAMF estimated to be approximately in Figure 13c. −8 The dB according SNR of the to the received position signals of the target can be echo. estimated We to be approximatelycalculatecalculate the the output− 8output dB SNR according SNR of ofeach each tomethod method the position in in Figure Figure of 13 13 the toto target bebe about echo. 33 dB,dB, We 77 dB, dB, calculate and and 16 16 dB. thedB. So, So, output the the SNR SNR SNR of gains of the MF, the FDAMF, and the TRC-IS-FDAMF are 11 dB, 15 dB, and 24 dB in this example. each methodgains in of Figurethe MF, 13 the to FDAMF, be about and 3 the dB, TRC-IS-FDAMF 7 dB, and 16 dB.are 11 So, dB, the 15 SNR dB, and gains 24 dB of thein this MF, example. the FDAMF, The experimental results indicate that the FDAMF does improve the performance of the MF and can and the TRC-IS-FDAMFThe experimental areresults 11 dB,indicate 15 dB,that andthe FDAMF 24 dB in does this improve example. the performance The experimental of the MF results and can indicate adaptadapt to theto the actual actual interference interference in in a away. way. In In ad addition,dition, thethe TRC-IS preprocessing method method works works well well that thein FDAMFveryin very noisy noisy does ambient ambient improve noise. noise. the performance of the MF and can adapt to the actual interference in a way. In addition, the TRC-IS preprocessing method works well in very noisy ambient noise.

(a)(a)

(b) (b)

Figure 13. The comparison of the processing results of the received signals from the sea trial: (a) Using

the MF (SNRFigure = 313. dB); The (b comparison) Using the of FDAMF the processing (SNR =results 7 dB); of ( cthe) Using received the signals TRC-IS-FDAMF from the sea (SNR trial: =(a) 16 dB). The arrivalFigureUsing time 13. the The of MF the comparison (SNR target = 3 echodB); of ( b isthe) Using 2.76 processing s. the FDAMF results (SNR of =the 7 dB);received (c) Using signals the TRC-IS-FDAMFfrom the sea trial: (SNR ( a) Using= 16 the dB). MF The (SNR arrival = 3 time dB); of(b the) Using target the echo FDAMF is 2.76 (SNRs. = 7 dB); (c) Using the TRC-IS-FDAMF (SNR = 16 dB). The arrival time of the target echo is 2.76 s.

Sensors 2017, 17, 1565 11 of 12

7. Conclusions In this paper, we proposed an improved MF in AWGN combined with the ALE by analyzing the output spectrum and the spectral spectrum of the MF, which is the FDAMF. When the input SNR is too low, we propose a preprocessing method—TRC-IS—to improve the performance of the FDAMF. We also verified that the performance of the FDAMF is significantly better than that of the MF by simulations and experiments. The methods proposed in this paper can make the detector work more efficiently than the MF at a lower input SNR. It is a great challenge in the field of detection to find a better detector than the optimal one (i.e., the MF) under ideal conditions, and it will provide more reliable and efficient detection technology for the engineering applications in active sonar, radar, and communications. Combined with the appropriate methods, the FDAMF can also be used in actual interferences such as colored noise, reverberation, multipath, and Doppler shift to improve the performance of these methods.

Acknowledgments: This work was supported in part by the National Natural Science Foundation of China under Grant 11374072 and 61371171. All authors are grateful to Junying Hui from Harbin Engineering University for insightful discussions. Author Contributions: Z.Z. proposed the novel method and wrote the paper; A.Z. conceived and designed the experiments; J.H. performed the numerical simulations; B.H. and R.S. contributed with valuable discussions and scientific advice; F.N. analyzed the data. Conflicts of Interest: The authors declare no conflict of interest.

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