Imaging Method Based on Time Reversal Channel Compensation

The conventional time reversal imaging (TRI) method builds imaging function by using the maximal value of signal amplitude. In this circumstance, some remote targets are missed (near-far problem) or low resolution is obtained in lossy and/or dispersive media, and too many transceivers are employed to locate targets, which increases the complexity and cost of system. To solve these problems, a novel TRI algorithm is presented in this paper. In order to achieve a high resolution, the signal amplitude corresponding to focal time observed at target position is used to reconstruct the target image. For disposing near-far problem and suppressing spurious images, combining with cross-correlation property and amplitude compensation, channel compensation function (CCF) is introduced. Moreover, the complexity and cost of system are reduced by employing only five transceivers to detect four targets whose number is close to that of transceivers. For the sake of demonstrating the practicability of the proposed analytical framework, the numerical experiments are actualized in both nondispersive-lossless (NDL) media and dispersive-conductive (DPC) media. Results show that the performance of the proposed method is superior to that of conventional TRI algorithm even under few echo signals.


Introduction
In recent years, time reversal imaging (TRI) has attracted more attention because of its potential for target detection [1], medical imaging [2,3], nondestructive detection [4], and communications [5,6].The basic procedure of TRI contains two stages: the scatterers generate/reflect a signal and a set of receivers record the echo signals in TR forward propagation (TRFP) stage.In TR backpropagation (TRBP) stage, these recorded signals are time reversed (e.g., using analog-to-digital (A/D) converters, digital-to-analog (D/A) converters, and computational TRI [7]).After that, the TR signals are propagated from receivers to scatterer(s) and then from scatterer(s) back to transmitters.When spatial reciprocity principle holds, rebroadcast wavefields will focus at the location(s) of the original scatterer(s).
These studies assume propagation in lossless media.However, in many applications, such as optical transmission systems [8], ultrawideband communications [9], and imaging systems [10][11][12], losses and/or dispersion are existent in the intervening media.As a result, TR invariance is broken and the conventional TRI operation cannot be directly applied.Therefore, compensation techniques are proposed in these applications.In [10,11], a frequency dispersion compensation method based on conventional TRI algorithm is proposed, which consider cumulative dispersion effects stronger at later times and time-window the received signals for convenient.Combining with time reversal multiple signal classification (TR-MUSIC) algorithm, this method has also been adopted in ultrasound areas [12].However, adopting time-windowing may introduce some contaminated information, such as interference from artifacts, and overlapping factor has to be considered, which would affect imaging results seriously.Besides, conventional TRI method generates imaging map by using the maximal value of signal amplitude.Consider the propagation loss of electromagnetic (EM) wave and targets near transceivers get more power than that far from transceivers, which leads to remote targets undistinguished, that is, near-far problem [13], while TR-MUSIC needs to measure  ×  echo signals for constructing the multistatic data matrix (MDM) whose size is  × , where  is the total number of transceivers [3,13,14].Thus, the use of 2 International Journal of Antennas and Propagation conventional TRI and TR-MUSIC algorithms will increase the complexity and cost of system.Moreover, only low frequency (MHz) is considered in these researches.
For solving these problems discussed above, a novel TRI method is proposed in this paper.The signal amplitude corresponding to focal time ( focus ) observed at target position is used to reconstruct the target image.To our knowledge, this proposed imaging function is novel comparing with existing researches.By using this novel imaging function, we can not only obtain high resolution, but also avoid considering overlapping factor.As to the near-far problem and interference from artifacts, channel compensation function (CCF) is introduced to suppress them.CCF contains amplitude compensation (solving near-far problem) and cross-correlation property (alleviating interference from artifacts).Furthermore, in order to reduce the complexity and cost of system, only five transceivers are utilized to locate four targets with high resolution.Generally, for locating targets accurately, the number of transceivers is much larger than that of targets [13,15], and locating targets whose number is close to that of transceivers with high resolution appears in very few literatures.To prove the practicability of proposed method, two numerical experiments are executed in nondispersive-lossless (NDL) media and dispersiveconductive (DPC) media on the condition that the center frequency of incident pulse is 2.45 GHz (WLAN frequency band).

Description of the Proposed Method
An imaging system comprised of a linear array of  transmitters and  receivers is considered in this section and the th transmitter and th receiver are located at r  and r  , respectively.It is further assumed that each transmitter sequentially transmits probing pulse () and each receiver collects the echo signals.The frequency-domain representation of incident signal at target (located at r  ) can be illustrated as where () is the Fourier transform of () and (r  , r  , ) is the Green function which satisfies the reduced wave equation [16] (representing the "propagator" from location r  to location r  at frequency ), where ∇ is Laplace operator,  = 2/ is the wavenumber, and (r − r  ) is the Dirac delta function.
Using (r  , r  , ) to represent the propagation loss characteristic of EM wave in transmission space/medium, we can express the echo signal received by the th receiver as where (r  , ) is the object profile.To implement the numerical studies, suppose that the imaging region is discrete into  grids and each grid has a reflection coefficient   ,  = 1, . . ., .Then, we can get the (r, ) as [17] Since time reversing a signal in time domain is equivalent to taking complex conjugate in frequency domain, the TR version of (3) can be represented as Considering all receivers resubmitting corresponding TR signals to an arbitrary point r  in TRBP stage, the signal received by the th transmitter can be derived as According to spatial reciprocity principle, when r  = r  , the signal received from target position is It can be found that the TR signal at target position is a focused signal.However, this focusing phenomenon does not happen away from the target position.With enough transceivers, the signal amplitude at target position is much larger than that of off-target position, so that the target can be located accurately by using conventional TRI method.Whereas this is complex and uneconomic, thus the use of a small quantity of transceivers is an inevitable development trend.However, directly decreasing the amount of transceivers will result in obtaining fewer echo signals, by which the targets cannot be distinguished with conventional TRI algorithm.In order to solve this problem, a novel imaging map is built by using signal amplitude at  focus .According to TR concept, the TR signal at target positions will focus synchronously, and, thus,  focus is the same at all target positions, but the time corresponding to the maximal value of signal amplitude observed at off-target positions is different and not equal to  focus .In other words, as to target position, the maximal value of signal amplitude appears at  focus ; however, as to off-target position, the signal amplitude is smaller than the maximal value at  focus .Therefore, compared with conventional TRI algorithm, this new imaging map (corresponding to (11)) can enhance target resolution even under few echo signal conditions.
Further, in order to solve the near-far problem and the interference from artifacts, a CCF (  (r  , r, r  , )) is introduced.Actually, the performance of the compensation depends on several factors, such as the amount of losses (e.g., the moisture level in the soil), the target locations (e.g., depth), and the frequency of probe impulse.
In lossy and/or dispersive media, the signal   () received by each receiver is only changed with a phase shift  −e d and an amplitude attenuation  −e d , which can be expressed as where e d is the propagation direction of EM wave,  is the transmission distance of EM wave,  is the attenuation factor, and  is the phase factor.The phase aberration correction at target position is automatically compensated by the TR operation via phase conjugation.Taking into account the amplitude aberration, we can obtain the channel compensation function as where   (r  , r, r  , ),   (), and  1 () are the inverse Fourier transform of   (r  , r, r  , ),   (), and  * (), respectively.corr(  (),  1 ()) represents the cross-correlation function between   () and  1 ().Comparing ( 6) with ( 7), it can be found that the received TR signal from target position is similar to  * (), but this does not happen at off-target positions.Therefore, corr(  (),  1 ()) at target position is larger than that at off-target positions; namely, compared with off-target positions, the effect of CCF is more obvious at target position, which is able to be used to suppress ghost images effectively.By multiplying amplitude compensation factor  (|r−r  |+|r−r  |) , the CCF can solve near-far problem well.Then, based on the above analysis, we can rewrite ( 6) and (7) as follows:  ⋅   (r  , r  , r  , ) . ( The imaging formula can be formed as

Numerical Experiments and Discussions
In this section, NDL media and DPC media are studied in numerical experiments A and B, respectively.In both experiments, the size of observation area is 0.5 × 0.5 m 2 .The Gaussian modulated pulse of 0.5 ns duration centered at 2.45 GHz is used as imaging probe signal.Omnidirectional antennas operated at WLAN frequency band (2.4-2.485GHz) were used as transceivers, and these numerical experiments were simulated by MATLAB software.
For proving this method can work under few echo signal conditions, only five transceivers are employed to detect four targets.The positions of targets are  1 (0.1 m, 0.06 m),  2 (0.2 m, 0.32 m),  3 (0.36 m, 0.14 m), and  4 (0.4 m, 0.4 m), respectively, and transceivers are situated at (0.05 m, 0 m), (0.15 m, 0 m), (0.25 m, 0 m), (0.35 m, 0 m), and (0.45 m, 0 m), respectively, as shown in Figure 1.In order to facilitate comparison, all echo signals will be processed by using the conventional TRI algorithm and the proposed algorithm.
Lorentz model for the complex permittivity function [18,19] can be presented as with  where   is the number of species of media, () is the medium susceptibility,   is the resonant frequency for the  species, and   is the corresponding damping factor and   and  ∞ are the static and infinite frequency permittivities, respectively.

Numerical Experiment A:
In NDL Media.As to NDL media, () and  are equal to zero ( =  0  ∞ ).
From the imaging map obtained by conventional TRI method, as illustrated in Figure 2(a), it can be seen that targets cannot be distinguished at all in the condition of few echo signals, and near-far problem is serious.We can also find that, by using the proposed imaging map function, the resolution can be enhanced obviously, but near-far problem still exists as shown in Figure 2(b).Further, combining with CCF, targets can be distinguished and we can obtain cleaner, higher-resolution image results even under few echo signal conditions as portrayed in Figure 3.
Moreover, the received echo signal strength from each target from being large to small is  1 >  3 >  2 >  4 , while the value of the CCF corresponding to each target is opposite ( 1 <  3 <  2 <  4 ).It can be observed that, compared with  2 and  4 ,  3 is located closer to transceivers, which leads to a small compensation, and, compared with  1 , the echo signal from  3 is smaller; thus,  3 is darker than others in the imaging result.

Numerical Experiment B:
In DPC Media.The realistic soil parameters given in Table 1 are obtained by curve fitting of two-species Lorentz model as reported in [18,19] and the references therein.
Take the clay loam with moisture level of 5%, for example.Due to larger loss and more complex dispersion in DPC media, the near-far problem is more serious, imaging result obtained by using conventional TRI algorithm is worse than that in NDL media, as shown in Figure 4(a), three remote  targets disappear, and one near target is not able to be identified.Figure 4(b) gives the imaging result obtained by using the proposed imaging map function, although resolution is enhanced, targets cannot be distinguished clearly, and nearfar problem still exists as discussed in numerical experiment A; moreover, artifacts appear near the transceivers.Therefore, in order to solve problem above, CCF is introduced and Figure 5 illustrates the corresponding imaging result.It can be seen that four targets can be located accurately with high-resolution compared with Figure 4, and artifacts disappear.The similar results can also be obtained by taking the clay loam with moisture level of 2.5% and 10%, for example.

Conclusions
In this work, according to propagation loss and/or dispersion of EM wave in transmission space/medium, we introduced CCF to compensate channel attenuation and suppress spurious images; it consists of amplitude compensation and crosscorrelation property between the conjugate of incident pulse and received TR signal.The use of amplitude compensation can solve near-far problem well, and spurious images can be suppressed effectively by utilizing cross-correlation function.
In addition, the resolution can be enhanced by using new imaging map, which is constructed by the signal amplitude corresponding to focal time observed at target position.Two numerical experiments are carried out to prove the practicability of the proposed algorithm.It is worth mentioning that targets whose number is close to that of transceivers can be located accurately even under few echo signal conditions.

Figure 2 :
Figure 2: Imaging result (a) obtained by using the conventional TRI and (b) obtained by using signal amplitudes at  focus .

Figure 3 :
Figure 3: Imaging result obtained by using the proposed algorithm.

Figure 4 :Figure 5 :
Figure 4: Imaging result (a) obtained by using the conventional TRI and (b) obtained by using signal amplitudes at  focus .

Table 1 :
Lorentz model parameters of the Puerto Rico type clay loams (analysis from 30 MHz to 3840 MHz).