Decomposition of the Time Reversal Operator for Target Detection

A thorough theory of detection problem using active time reversal has been investigated in several recent papers. Although active time reversal method is theoretically superior to the others, its practical implementation for target detection is far more difficult. This paper investigates the detection problem using passive decomposition of the time reversal operator DORT method. Provided that the signal components can be modeled as a linear combination of basis vectors with an unknown signal subspace, the generalized likelihood ratio test GLRT is derived based on Neyman-Person lemma with the unknown signal subspace replaced by its maximum likelihood estimation. The test statistics is one of the dominant eigenvalues of the time reversal operator for a point-like scatterer. Finally, the performance of the DORT detector is investigated with acoustic data collected from a waveguide tank. The experimental results show that the DORT detector can provide, respectively, 1.4 dB, 1.1 dB, and 0.8 dB performance gains over the energy detector given false alarms rate of 0.0001, 0.001, and 0.01.


Introduction
It is acoustically difficult to detect a target in shallow water environment, for instance, the performances of the traditional detectors such as the matched filter are severely degraded because the echo of active sonar undergoes distortion during two-way propagation and reflection from the target.One of the approaches to improve the detection performance is the model-based matched filter that takes advantage of the physical model to enhance signal processing 1 .Another compromising solution is the one to model the channel distortion, such as the segmented replica correlation detector and the replica correlation integration detector 2, 3 .
Time-reversal approaches provide an alternative approach to this kind of problem 4-7 .The unique feature of time reversal is that it provides a robust set of waves without any signal analysis, which compensates for the distortion induced by spatial inhomogeneities in the propagation medium 4 .Two time-reversal approaches, the iterative time-reversal, and the decomposition of the time reversal operator DORT can be employed for detection and localization of target 6-9 .
Time reversal has been intensively studied in ultrasound acoustics and electromagnetics.There have been several recent studies devoted to the detection problem from the statistic signal processing perspective 10-15 .Reference 12 investigates the detection problem using the active time reversal of DORT method and the iterative time reversal method in the context of acoustic wave processing in ocean, while 10, 11 studies the active iterative time reversal method in the context of electromagnetic wave processing in air.
Although active time reversal method is theoretically superior to the others, its practical implementation for target detection is far more difficult.As an alternative, this paper investigates the detection problem using passive DORT method in acoustic wave processing in ocean.Compared to electromagnetic wave propagation in air, sound propagation in ocean is more environmentally involved.The terms "active" and "passive" refer to the sourcereceiver arrays SRA : "active" array works when a SRA attempts to focus on a target by retransmitting an environmentally dependent excitation vector, while "passive" array functions when a SRA simply transmits signals from individual source elements followed by processing of the returned time series 16 .Moreover, the performances of the derived detectors in 12 are only evaluated by numerical simulations.This paper tests the detector with real acoustic data from waveguide experiments.
The rest of the paper is organized as follows.The mathematical model of binary hypothesis is derived, and the generalized likelihood ratio test GLRT of the DORT approach is developed in Section 2. The proposed detectors are validated by laboratory waveguide experiments in Section 3. The conclusions and summary are drawn in Section 4.

Notations
We denote scalars by lowercase letters, vector quantities by boldface lowercase letters, and matrices by uppercase boldface.• H stands for conjugate transpose; R A and I A are the real and the imaginary parts of matrix A, respectively; tr A is the sum of the diagonal elements of A; I N is the identity matrix of size N.

DORT Measurement
According to the theory of DORT, a SRA of N sensors collects data and then generates a response matrix with N × N dimension.The matrix is formed by sequentially transmitting signals from individual source elements of the SRA or orthogonal beams and recording the backscattered echo on each receiver element 8, 9, 17 .
Provided that a normalized narrow-band signal is transmitted the angular frequency ω is thus omitted in the formulas below , the received signals in time are first transformed into frequency domain.They can be modeled as a sum of interferences and possible signals whose presence we are trying to detect as follows: where s j is the signal term and v j is the interference-plus-noise term for jth transmission.The signal vector s j is assumed to be deterministic but unknown.We model them as s j Hθ j , which are a linear combination of basis vectors with an unknown signal subspace H with known rank-p corresponding to p propagation modes of a scatterer .Considering that the target is not definitely a point-like scatterer for the application of DORT method, the signal subspace has dimension p.In practical implementation, H is a N × p matrix, and each column h i is the Green's function of the channel between the SRA and one aspect of the scatterer.
Although the interference v j is the sum of the receiver noise and the reverberation signal, the DORT method has the ability to separate the echo of a target from reverberation 18 .We assume that both the receiver noise and the residual reverberation signal can be modeled as zero-mean white Gaussian random processes 2 , that is, The real and imaginary components of v j are R v j ∼ N 0, σ 2 v /2 I N and I v j ∼ N 0, σ 2 v /2 I N , respectively.Although the assumption of Gaussian reverberation may not always be true, it is often made to facilitate mathematical analysis.
After all the transmission are finished, the total received data can be arranged as a matrix by 19 Similarly, the signal matrix can be written as S s 1 . . .
The detection problem then becomes to choose one of the following two hypotheses:

The DORT Detector
As shown in 10, 12 , the suboptimal detector using GLRT in the Neyman-Pearson sense is derived as an energy detector in conventional approaches when the Green's function of unknown channel is replaced by its maximum likelihood estimation.In this paper a suboptimal detector using the GLRT approach is derived for the time reversal approach, which is named the DORT detector.
The probability density functions PDF p Y | H 1 and p Y | H 0 under condition of H 1 and H 0 are, respectively,

2.5
Recall that the generalized likelihood ratio is defined by the ratio of the likelihood functions under each hypothesis, but it is maximized over the unknown nuisance parameter space 20 , that is, where Θ denotes the nuisance parameter set, for example, the unknown signal subspace H.
It is assumed that the statistical characterization of the noise is completely known a priori in this paper.Then the detection problem becomes finding a maximum likelihood estimate of the signal subspace H, or equivalently, its orthogonal subspace A maximum likelihood principle to identify the orthogonal subspace A can form a Lagrangian for minimizing −2σ 2 v L in 2.7 under the constraints where Ξ is an N − p × N matrix, it contains the Lagrangians ξ ij .Firstly, we can estimate the signal matrix S assuming that the orthogonal subspace A is known, and then we further maximize likelihood with respect to A cf. 21 for detailed derivations .The estimate of signal matrix is where P A is the projection Substitute 2.9 into 2.7 , we obtain In practice, Y is the measurement of the transfer matrix, and R YY H is the time reversal operator.As the subspace A is unknown, we must further maximize likelihood with respect to it.Let the time reversal operator have the orthogonal decomposition below

2.11
where the eigenvalue matrix Λ 2 is sorted in descending order, and the matrix U is composed of the corresponding eigenvectors.It is then straightforward proved that L is bounded as follows: for any rank-N − p projector the bound is achieved for a projector P A onto the subspace A : Clearly, the dominant p eigenvectors form a rank-p signal subspace, and the rest of the eigenvectors builds a rank-N − p orthogonal subspace A. The estimate of the signal subspace is 2.14 Substituting 2.5 , 2.9 , and 2.13 into 2.6 , and taking the logarithm operation with constants neglected yields

2.15
Recall that only a single target is considered, that is, p 1.That means

2.16
Therefore the test statistic is one of the dominant eigenvalues of the time reversal operator if the signal components are assumed deterministic unknown and modeled as Mathematical Problems in Engineering a linear combination of basis vectors with an unknown signal subspace.The test statistic with minimal signal analysis is similar as that obtained in 19 , which is 1 λ 2 1 .We call it DORT detector.Due to its simple procedure of signal processing and no knowledge about the environment and the array configuration required, the DORT detector can be used to coarsely search the suspicious objects as a preliminary detector.In next section, we compare the detection performance of DORT detector with energy detector that is based on the same detection theory.

Experimental Results of Detector Performance
Since it is difficult to theoretically analyze the receiver-operating characteristics ROC of the DORT detector, we study the ROC by Monte Carlo simulation.This section presents the study on the detection performance of the DORT detector and the performance gain of DORT detector over the energy detector with a mix of real acoustic data from waveguide experiments and simulated noise.

Experimental Setup
The experimental setup is shown in Figure 1.The experiment was performed in a waveguide tank using a vertical SRA of 32 elements equally spaced at 0.04 m apart.The waver tank can simulate a stationary shallow water environment with multipath propagations, mainly including direct path, sea surface-reflected path, and sea bottom-reflected path.Each of the 32 elements is individually controlled and amplified during transmission and reception.The waveguide tank is 14 m in length, 1.2 m in width, and 1.4 m in height.Its three of four vertical walls are covered with anechoic tiles, the other one is a steel sheet located 12 m away from the SRA.The bottom is a sandstone basement covered with sand of 0.22 m thickness.
The transmitted signal is a 0.5 ms PCW signal centered at 18 kHz.The wavelength is about 0.08 m.The first source transducer is excited by the transmitted signal.The backscattered waves received on the N channels of the SRA are stored.This operation is repeated for all the transducers.Finally a data matrix Y is generated.The target was an airfilled steel cylinder of 0.21 m 2.6 λ in diameter and 0.51 m 6.2 λ in length suspended at a range of 8.2 m and a depth of 0.84 m.This means that the range of target to SRA is about 100 λ and suspended at depth of 10.5 λ.
The water column has a sound of equal speed profile.The bottom is modeled as a 0.22 m sand sediment layer above a sandstone basement.The environmental parameters are following: in the water column, wave speed c 1 1480 m/s; in the bottom layer, sand density ρ 2 1800 kg/m 3 , sand speed c 2 1650 m/s, and attenuation α 2 0.67 dB per wavelength; in the basement layer, density ρ 3 1800 kg/m 3 , speed c 3 1580 m/s, and attenuation α 3 0.8 dB per wavelength.

Signal Processing
In order to obtain high SNR, the time reversal operator is built from short time windows.That is, the matrix Y ω, r is the Fourier transform Y t the measurement of transfer matrix K t between time t and t Δt, where t is related to the distance r through the equation t 2r/c; c is the sound speed; Δt is the window length 18 .Especially, r 0 corresponds to the distance  of the target.In order to obtain the test statistic in 2.16 using DORT observation matrix Y H Y, the range of the target is firstly estimated.The matrix containing the target echo is then transformed into frequency domain.Finally the test statistic is got by the SVD decomposition of the matrix.A detailed illustration is shown in this section.
To estimate the distance of the target, a Δt 0.5 ms time windows is shifted by steps of 0.25 ms.The selection of the windows length will be illustrated later on.For each distancer, the matrices Y r are calculated at center frequency 18 kHz.After SVD decomposition, the first eigenvalues are normalized.In the approach of the energy detector, the average energy of the array at center frequency for each transmission is firstly calculated and then is accumulated to obtain the total average energy for the all transmissions.There is another simple method to calculate the energy is to summarize all eigenvalues after SVD decomposition.
Figure 2 depicts the first normalized eigenvalues and the normalized energies as a function of distance from 0 to 20 m.It can be seen that a peak value appears, respectively, at the distance of 8.3 m for DORT detector and 8.4 m for the energy detector, which tells that both detectors can correctly detect the target the other peak at the distance of 12 m is caused by the steel sheet .
It is worth noting that the target cannot be simply treated as a point-like scatterer compared to the wavelength.It has been shown that the diagonalization of the time-reversal operator permits the various elastic components of the scattered field to be extracted, and more than one eigenvalue is associated to the scatterer 7 .However, it has been also shown the strong specular echo can be separated from the Lamb wave by selecting temporally with a proper time window in the same reference.
Since the distance of the target has been determined, the corresponding time window Y ω, r 0 which contains the target echo can also be determined.This time-windowed segment of echo data is used for the following signal analysis and detection performance investigation.Figure 3 shows eigenvalues of the time-reversal operator at the central frequency for the windows of increasing duration of Δt 0.5, 1, 1.5, and 2 ms.With the window duration increasing, all eigenvalues except the first one which is normalized are becoming larger and larger.It is possible that other propagation modes from the target or reverberation from other objects contribute to the signals in the time windows for long durations.Hence by choosing Δt 0.5 ms as the window duration and selecting the time window associated to the strong specular echo, we still use the first eigenvalue as the test statistic for the DORT detector.
The backpropagation of the first eigenvector is calculated for the center frequency using KRAKEN 22 .The code KRAKEN is a normal mode propagation model developed   at SACLANT Undersea Research Centre now Nato Undersea Research Centre .It has been widely used for modeling ocean environments that are range independent, range dependent, or fully 3-dimensional.The image is displayed in the range from 1 m to 12 m on the whole height of the waveguide.As shown in Figure 4, the peak position in the ambiguity surface is at a range of 8.47 m and a depth of 0.85 m.The image provides information about the location

Detection Performance
As mentioned in 23, 24 , no analytical expression is available for the probability distribution of the first eigenvalue.Therefore, it is difficult to analytically express detection probability of DORT detector.The detection performance of DORT detector is numerically evaluated using Monte Carlo method.The initial SNR is calculated by the ratio of the signal energy of the target cell to the energy of the time window near the target cell, which is about 8.2 dB.Note that the initial SNR is high, we considered that the corresponding time window Y ω, r 0 only contains the target echo; that is, the signal matrix is replaced by Y ω, r 0 containing noise in practice.To obtain noisy backscatter at different SNR, we added numerically generated zero-mean white Gaussian noise to the real data of acoustic backscatter 10 .The SNR is defined as 3.1 For the practical signal processing, we set σ v 1 and scale the total signal energy to meet different SNR level.We then further modified the SNR levels in 3.1 according to the initial SNR to obtain values close to the actual ones.
To study the performance, we evaluated the detection probability P D as a function of the SNR for a fixed probability of false alarm P FA .In order to obtain the detection threshold, we, respectively, generated 10000 independent noise realizations and computed the test statistics given by 2.16 for DORT detector; the resulting 10000 test statistics are, respectively, sorted in ascending order; the thresholds are then selected so that P FA 0.0001, P FA 0.001, or P FA 0.01.To compute P D , a new set of 5000 independent noise is generated and added to the actual acoustic data.The test statistics are, respectively, computed and compared to the corresponding thresholds.The percentage of the number of times that the test statistic exceeds the threshold is used as an estimate of the detection probability.
Figure 5 shows the results of simulation versus the ones of the energy detector for P FA 0.01 developed in 12 .Figure 5 indicates that the simulation result of the energy detector is in excellent agreement with the theoretical analysis, thus it validated the proposed design of simulation experiment with a mix of real acoustic data from laboratory waveguide experiment and simulated noise.Figure 6 shows the detection performance of DORT detector and that of the energy detector.It can be seen that DORT detector provides detection gain with respect to the energy detector, except that the energy detector performs better in the case that the detection probabilities for both detectors is lower than 0.2 for extremely low SNR.For instance, we measured the performance gains by the relative SNR required to achieve detection probabilities of 0.5 given false alarm rates of 0.0001, 0.001, and 0.01.The performance gains of the DORT detector with respect to the energy detector are respectively about, 1.4 dB, 1.1 dB, and 0.8 dB.It should be pointed out that no knowledge and modeling of the propagation medium are needed for the design of DORT detector.
The reason that the DORT detector has performance gain over energy detector can be explained as follows.In fact, the test statistic of the energy detector can also be expressed as the summarization of all eigenvalues λ 2 1 for the DORT detector.The increment of the first eigenvalue is relatively larger than that of the others with SNR increasing, which can be implied in Figure 3.With the window duration increasing, the others are being more and more small compared to the first eigenvalue due to more noises are contained in the observation matrix.

Conclusion
In this paper, the target detection using passive DORT method is presented and discussed.The mathematical models of detection problem are developed using a sequence of collected data from typical DORT transmission.It is proved that the DORT detector can be derived with the unknown signal subspace replaced by its maximum likelihood estimate, if the signal components are assumed deterministic unknown and modeled as a linear combination of basis vectors with an unknown signal subspace; in addition, one of the dominant eigenvalue of the time reversal operator is the test statistic for point-like scatterer.Finally, the DORT detector and the energy detector are tested with the real acoustic data collected in the laboratory waveguide experiment.
The detection experiments using DORT method are conducted with the data measured from a 32 elements vertical SRA.A target has been detected and correctly located within the water depth.The experimental results show that the DORT detector can provide, respectively, 1.4 dB, 1.1 dB, and 0.8 dB performance gains over the energy detector given false alarms rate of 0.0001, 0.001, and 0.01.It should be pointed out that this paper only focuses on the passive time reversal method in which the SRA simply transmits conventional signals followed by processing of the returned time data series.Therefore, the detection gain is mainly provided by the decomposition that can separate the target echo from other contributions, but not by the adaption, say, the transmitted wavefront matched to the channel as the active time reversal methods.For future work, the detection problems for p independent point-like scatterers or in unstationary ocean environments using DORT method should be further investigated.

Figure 1 :
Figure 1: Experimental setup and model for the acoustic environment.

Figure 2 :
Figure 2: Target at 8.2 m: the first normalized eigenvalues solid line and the normalized energies dashed line as a function of distance from 0 m to 20 m.

Figure 3 :
Figure 3: The experimental eigenvalues of of the time-reversal operator at the central frequency and the window duration is Δt 0.5 ms circle , Δt 1 ms square , Δt 1.5 ms star , and Δt 2 ms triangle .

Figure 4 :
Figure 4: Numerical backpropagation of the first eigenvector using KRAKEN.

Figure 5 :
Figure 5: Detection probability versus SNR for P FA 0.01 using the energy detector with a mix of real acoustic data from laboratory waveguide experiment and simulated noise; analysis dashed line versus simulation solid line .

Figure 6 :
Figure 6: Detection probability versus SNR using a mix of real acoustic data from laboratory waveguide experiment and simulated noise; DORT detector solid line versus energy detector dashed line .