Three-Dimensional Motion Compensation Method Based on Sparse Array Designed for Time-Division Multiplexing Multiple-Input-Multiple-Output Through-Wall Radar

Abstract

A large-aperture radar composed of a multiple-input-multiple-output (MIMO) planar array can complete 3D through-wall imaging (TWI), but the simultaneous work of the multiple transceiver channels leads to difficulties in designing the hardware. In engineering, multiple transceiver channels are usually realized by time-division multiplexing (TDM) in MIMO radar, which is called TDM MIMO radar. A time delay will be introduced when the channels are switched, which will cause high sidelobes and position deviation in the 3D imaging of moving targets, also known as range migration. This paper proposes a motion compensation algorithm based on sparse array, designed to eliminate range migration in moving targets in 3D TWI scenes. In the proposed algorithm, the coincident array elements of the equivalent array are used as the compensation channels to calculate the position difference of the target, which can correct the remaining MIMO channels. The proposed algorithm is compared with no compensation, and the reference-channel-based motion compensation algorithm (RCMCA). According to the simulation and experimental results, the proposed motion compensation algorithm can effectively eliminate sidelobes, and keep the position deviation within 0.30 m in the 3D TWI of moving targets under the TDM MIMO radar, without increasing the system complexity.

1. Introduction

Multiple channels in the multiple-input-multiple-output (MIMO) through-wall radar (TWR) can work simultaneously to collect data quickly, which can realize real-time through-wall imaging (TWI) [1,2,3,4]. In addition, sparsely distributed transmitting and receiving antennas can form a large aperture, to obtain a high resolution. Therefore, MIMO TWR is an essential technology in the detection of through-wall targets [5,6,7].

The larger the aperture of the array in the MIMO radar, the higher the azimuth resolution [8,9], but more transceiver channels are required to form the large aperture, and the corresponding number of transmitters and receivers will lead to great difficulties in designing the hardware. In order to reduce the complexity of the system, and maintain the consistency of the channels, multiple transmitting and receiving channels in MIMO radar are usually implemented by time-division multiplexing (TDM) through microwave switches in engineering. This structure requires only one radar transmitter and receiver to implement a MIMO system, also known as TDM MIMO radar [10,11]. At the same time, there are also shortcomings to TDM MIMO radar. The delay caused by the switching process of TDM MIMO radar cannot be ignored in the TWI of moving targets. In the signal acquisition of the through-wall moving target, the position of the moving target has changed within the acquisition time of the single-frame echo [10]. There will be high sidelobes and position deviation when the echo of the moving target is used for imaging directly, which is called range migration here [1].

Compensating for range migration in TDM MIMO radar is essential before imaging. Achieving through-wall, real-time, high-precision motion compensation is a major challenge. Hu et al. proposed a random switching strategy, using sparse reconstruction technology to reconstruct the doppler spectrum of the target, to solve the range migration [12]. However, the timing of random switching needs to be realized using a convex optimization algorithm for specific scenarios. In [13], the phase error is reduced by optimizing the switching scheme of the transmitting antenna. In [14], a signal modulation scheme with an alternating emission is designed to compensate the coupling-phase change produced by the velocity. Both methods can eliminate range migration, but the scan rate of the system is reduced. A motion compensation algorithm based on multi-frame data fusion (MFDF) is proposed in [3] for through-wall motion compensation. It constructs a geometric model based on a uniform array, to obtain the position difference of the target. The geometric solution is then used for echo correction. However, this method is only applicable to 2D TWI, and cannot be extended to 3D TWI.

For the motion compensation of 3D TWI, we have proposed the single-channel motion compensation algorithm (SCMCA) and the reference channel-based motion compensation algorithm (RCMCA) in our previous work [10]. Two adjacent frame echoes of a single channel in the SCMCA are used to correct the echoes of moving targets, then its accuracy is limited by the switching rate of the switch. A reference channel is installed in the center of the array to obtain the position difference of the target in the RCMCA, which can compensate the MIMO channels accurately. But the extra reference channel will increase the complexity of the hardware design. In this paper, a motion compensation algorithm based on a sparse array design is proposed, to solve the range migration of 3D TWI in TDM MIMO radar. An improved sparse array is designed in the proposed method; the coincident array elements in the equivalent array are used as compensation channels to calculate the motion information of the target, which is used to correct the remaining MIMO channels. The system complexity is not increased at any point during the compensation process.

The remainder of this paper is as follows. The model of 3D TWI for moving targets is established in Section 2. Section 3 presents the principle of the RCMCA, and the derivation of the proposed algorithm. Section 4 analyzes the simulation and experimental results in detail. Finally, Section 5 presents the conclusion.

2. Signal Model

The 3D TWI scene of the moving target is shown in Figure 1. The planar array of the radar includes $M$ transmitting antennas and $N$ receiving antennas, with a total of $M\times N$ channels. Both the transmitting and receiving channels are realized via TDM through microwave switches, which is called TDM MIMO 3D TWR. There will be a delay between the channels, due to switching. During the signal acquisition process of the moving target, the position of the target has changed from $P_1$ to $P_{M\times N}$ within the single-frame (containing $M\times N$ channels) time, as is shown in Figure 1. The echo of the moving target can be described as:

$$S_j(t)=S_0(t-{\tau }_j)$$

(1)

where $S_0(t)$ is the transmitting signal, and ${\tau }_j={\tau }_{S\_j}+{\tau }_{M\_j}$ is the echo delay of the target calculated by channel $j$, $j=1,\dots ,M\times N$. ${\tau }_{S\_j}$ is the time delay of the starting position $P_1$ measured by channel $j$. ${\tau }_{M\_j}$ is the time delay introduced by the target movement within the time when the switch is switched to channel $j$. ${\tau }_{S\_j}$ can be expressed as:

$$\begin{array}{l}{\tau }_{S\_j}=(\sqrt{{(x_{T\_j}-P_x)}^2+{(y_{T\_j}-P_y)}^2+{(z_{T\_j}-P_z)}^2}\\ +\sqrt{{(x_{R\_j}-P_x)}^2+{(y_{R\_j}-P_y)}^2+{(z_{R\_j}-P_z)}^2})/c\end{array}$$

(2)

where $(x_{T\_j},y_{T\_j},z_{T\_j})$ and $(x_{R\_j},y_{R\_j},z_{R\_j})$ are the positions of the transceiver antennas of the $j$th channel, respectively. $(P_x,P_y,P_z)$ are the coordinates of the initial position of the target. $c$ is the propagation speed of the electromagnetic waves.

The ${\tau }_{M\_j}$ will introduce different effects to different channels. The echoes of the through-wall stationary and moving targets are given in Figure 2. It can be seen that the echo-delay trend measured using different channels has changed significantly, which is known as range migration [1]. There will be high sidelobes and position deviation when the echo $S_j(t)$ is used for imaging directly. Therefore, the echo of the moving target needs to be compensated before the imaging.

3. Motion Compensation Method

3.1. Reference-Channel Motion Compensation Algorithm

In our previous work, a reference-channel motion compensation algorithm (RCMCA) was proposed for echo correction in moving targets in 3D TWI [10]. The RCMCA is implemented on an improved hardware system. In order to form a large aperture, while reducing the number of array elements, four transmitting and eight receiving antennas are used in the designed TDM MIMO radar. In addition, a reference channel is added to the system, as is shown in Figure 3a. The designed sparse MIMO array is based on the MIMO array optimization method of concentric circle rotation. The four transmitting antennas are installed at the endpoints of the square, and the eight receiving antennas are equally divided into two groups, and placed on two concentric circles. The projections of the receiving antennas are the same in all dimensions, as shown in Figure 3b. Adjusting the rotation angles and radii of the two concentric circles can reduce the sidelobes in all dimensions simultaneously. The array can provide optimal 3D imaging.

In the RCMCA-based system, the switching process of the MIMO channel and reference channel is given in Figure 4. $T_M/R_N$ represents the channel formed by the $M$th transmitting antenna and the $N$th receiving antenna, and $Ref\_C$ is the reference channel. The reference channel and the MIMO channel work interleaved. The reference channel will work once after each MIMO channel completes data acquisition. The position of the reference channel remains unchanged, it can obtain the position difference of the moving target and use it in the compensation of the MIMO channel.

The RCMCA can improve the quality of the 3D TWI of moving targets significantly [10]. However, the additional reference channel will lead to more difficulties in the hardware design. In this paper, the range migration of the 3D TWI of moving targets will be solved, without increasing the complexity of the system.

3.2. Proposed Motion Compensation Algorithm

A motion compensation algorithm based on an improved MIMO sparse array, in which the reference channel is removed, is proposed in this paper. The improved MIMO array is shown in Figure 5a; in addition, the switching sequence of the transmitting and receiving antennas is marked. Figure 5b gives the equivalent array of the improved MIMO array, in which four equivalent array elements overlap (channels 1, 10, 19 and 28), and are located in the center of the array. These overlapping equivalent array elements can be used in motion compensation.

The simulation results of the resolution before and after the array improvement are shown in Figure 6. It can be seen that the resolution of the improved array is basically the same as that of the original array. The height sidelobe of the array after improvement deteriorated a little, but remained below −14 dB, which would not affect the subsequent 3D TWI.

The single-frame data of the MIMO array contain 32 channels; the switching sequence of the channels is shown in Figure 7, where the blue triangles represent equivalent channels with overlapping positions. The echo time delay of moving targets measured by all channels is ${\tau }_j=[{\tau }_{C1} {\tau }_{C2} \dots {\tau }_{C32}]$, which can be expressed as:

$$\begin{array}{l}{\tau }_j=[{\tau }_{C1} {\tau }_{C1}+\Delta {\tau }_1 {\tau }_{C1}+2\Delta {\tau }_1 \dots {\tau }_{C1}+8\Delta {\tau }_1\\ {\tau }_{C10} {\tau }_{C10}+\Delta {\tau }_2 {\tau }_{C10}+2\Delta {\tau }_2 \dots {\tau }_{C10}+8\Delta {\tau }_2\\ {\tau }_{C19} {\tau }_{C19}+\Delta {\tau }_3 {\tau }_{C19}+2\Delta {\tau }_3 \dots {\tau }_{C19}+8\Delta {\tau }_3\\ {\tau }_{C28} {\tau }_{C28}+\Delta {\tau }_4 {\tau }_{C28}+2\Delta {\tau }_4 \dots {\tau }_{C28}+4\Delta {\tau }_4]\end{array}$$

(3)

where ${\tau }_j$ represents the echo delay of the target when the $j$th channel is working. ${\tau }_{C1}$ is the echo delay of the target when channel 1 receives data. The target can be regarded as moving at a constant speed during the time when channel 1 is switched to channel 10. Then, the echo delay of the target increases uniformly, and the increase rate of the delay is $\Delta {\tau }_1$. The change in the delay within a single frame time can be divided into four sections, according to the overlapping channels of the equivalent array, channel 1–channel 10, channel 10–channel 19, channel 19–channel 28, and channel 28–channel 32. The increase rates of the delays are $\Delta {\tau }_1$, $\Delta {\tau }_2$, $\Delta {\tau }_3$, and $\Delta {\tau }_4$, respectively. It is worth noting that $\Delta {\tau }_4$ is calculated by combining channel 28 of the current frame and channel 1 of the next frame (denoted as channel 33).

Channel 1 and channel 10 are in the same position in the first section of the delay change, and the time delay change of the target during this period can be measured. Then, the increase rate of the delay $\Delta {\tau }_1$ can be calculated as:

$$\Delta {\tau }_1=[S_1(t)\otimes S_{10}(t)]/9$$

(4)

where “$\otimes$” represents the convolution operation, which can be used to calculate the time difference of the two signals [10]. Similarly, if the positions of channel 10 and channel 19 are the same, the positions of channel 19 and channel 28 are the same, and the positions of channel 28 and channel 1 in the next frame (represented as channel 33) are the same, then the increase rates of the delays ($\Delta {\tau }_2$, $\Delta {\tau }_3$, and $\Delta {\tau }_4$) of the remaining three sections can be calculated as:

$$\Delta {\tau }_2=[S_{10}(t)\otimes S_{19}(t)]/9$$

(5)

$$\Delta {\tau }_3=[S_{19}(t)\otimes S_{28}(t)]/9$$

(6)

$$\Delta {\tau }_4=[S_{28}(t)\otimes S_{33}(t)]/5$$

(7)

The increase rate of the echo delay can be used to compensate for the delay in the echo signal measured by each channel in each section. The compensated signal ${\tilde{S}}_j(t)$ can be described as:

$${\tilde{S}}_j(t)=\left\{\begin{array}{c}slide[S_j(t),(\mathrm{n}1-1)\Delta {\tau }_1], \mathrm{n}1=1,\dots ,9\\ slide[S_j(t),(\mathrm{n}2-10)\Delta {\tau }_2+T_{1-10}], \mathrm{n}2=10,\dots ,18\\ slide[S_j(t),(\mathrm{n}3-19)\Delta {\tau }_3+T_{1-19}], \mathrm{n}3=19,\dots ,27\\ slide[S_j(t),(\mathrm{n}4-28)\Delta {\tau }_4+T_{1-28}], \mathrm{n}4=28,\dots ,32\end{array}\right.$$

(8)

where “$slide[ ]$” is a shift function which can shift the original signal according to the corresponding time delay [1]. $T_{1-10}$, $T_{1-19}$, and $T_{1-28}$ are the time difference between the overlapping channels 10, 19, and 28, and channel 1, respectively, which can be obtained using the convolution formula.

$$\begin{array}{l}slide[S_j(t),\Delta \tau ]=\\ \left\{\begin{array}{c}〚S_j(\Delta \tau :\mathrm{end}),S_j(0:\Delta \tau )〛,\mathrm{when} \Delta \tau \ge 0\\ 〚S_j(\mathrm{end}+\Delta \tau :\mathrm{end}),S_j(0:\mathrm{end}+\Delta \tau )〛,\mathrm{when} \Delta \tau <0\end{array}\right.\end{array}$$

(9)

The flowchart of the proposed compensation algorithm is given in Figure 8. The compensated signal ${\tilde{S}}_j(t)$ is used for 3D TWI. The modified Kirchhoff 3D TWI algorithm is employed in the system, which has been described in detail in [10].

The algorithm is implemented on Matlab 2020a, and the pseudocode of the proposed motion compensation method is shown in Figure 9.

4. Simulation and Experimental Data

In this section, the proposed motion compensation method is compared with the results before compensation and after RCMCA compensation. The results of the simulation and experiment prove the superiority of the proposed method.

4.1. Simulation Data

We use gprMax software (v3) to build the model of the 3D TWI for a moving target [15]. The model is given in Figure 10a, which is constructed strictly according to the real scene in Figure 10b. The thickness of the wall is 24 cm, and the relative permittivity ${\epsilon }_1=6.0$. The radar parameters designed in the simulation are given in

Table 1. The simulation parameters of the radar.

Parameters	Value
Signal	Pulse
Center frequency	1 GHz
Bandwidth	1 GHz
Scanning rate	3 frame/s

. The array structure of the MIMO radar is given in Figure 5b. In the simulation, the human target is equivalent to a cylinder. The target moves away from the radar at a speed of 0.8 m/s from (0, 0, 4.0 m). Based on the normal direction of the wall, the moving direction of the target is 30°.

The 3D imaging, azimuth-range 2D projection, and azimuth-height 2D projection results without compensation are given in Figure 11a–c. The sidelobes in the imaging are high, and the position shows obvious deviation according to the 2D projections of the azimuth range and azimuth height. Figure 11e shows the azimuth-range 2D projection compensated by the RCMCA; the sidelobes of the target are low, and the imaging position is basically close to the starting point (0, 0, 4.0 m). Figure 11g–I are the imaging results after compensation using the algorithm proposed in this paper. The problem of the sidelobes and the position deviation of the target is also well resolved, and the effect has reached the same level as the RCMCA.

Figure 12 is the projection comparison of the 3D imaging for the different algorithms in the azimuth and height directions. The sidelobe levels in the azimuth and height directions before compensation are −1.3 dB and −6.0 dB, respectively, and the position deviation levels in the azimuth and height directions are 0.32 m and 0.30 m, respectively. The RCMCA eliminates the sidelobes in the azimuth and height directions, and the position deviations in the azimuth and height directions are 0.06 m and 0.05 m, respectively. The sidelobes in the azimuth and height directions are also basically eliminated after compensation using the proposed method, and the position deviations in the azimuth and height directions are 0.03 m and 0.08 m, respectively, which are basically consistent with the RCMCA.

In addition, the simulations are carried out separately for the targets whose velocity is between 0 and 1.8 m/s, and whose distance is between 2 and 9 m. Setting the starting point of the target as (0, 0, 4.0 m), Figure 13a shows the positioning deviation (3D positioning deviation) of the targets at different speeds. The position deviation before the motion compensation increases with the moving speed. After the RCMCA or the proposed algorithm, the position deviation is kept within 0.30 m at different speeds. Figure 13b gives the positioning deviation when the target passes through different positions at a speed of 0.8 m/s. The positioning deviation increases with distance without compensation. After being compensated using the RCMCA or the proposed algorithm, the position deviation Is kept within 0.30 m.

Table 2. Performance comparison of the different compensation algorithms.

Performance	Sidelobes	Position Deviation	Imaging Time	System Complexity
Method
Before compensation	−1.3 dB @azimuth −6.0 dB @height	<1.40 m	0.25 s	low
RCMCA	no sidelobes	<0.30 m	0.26 s	high
The proposed	no sidelobes	<0.30 m	0.26 s	low

shows the performance comparison between the different motion compensation algorithms. The proposed algorithm can eliminate the imaging sidelobes of moving targets, and control the positioning deviation within 0.30 m, without increasing the complexity of the system. In addition, in terms of the calculation time, compared with no compensation, the proposed algorithm only takes 0.10 s, which is the same as the RCMCA.

4.2. Experimental Data

The 3D TWI experiment of a single moving target is shown in Figure 10b. The 3D through-wall imaging radar was developed by the Aerospace Information Research Institute, Chinese Academy of Sciences. The MIMO radar includes four transmitting antennas and eight receiving antennas, and the planar array is consistent with that in Figure 5. The Archimedes spiral antenna (the horizontal and vertical radiation angles are both −60° to +60°) is adopted in the system. The specific parameters of the radar used in the experiment are given in

Table 3. The parameters of the radar in the experiment.

Parameters	Value
Signal	Pseudo-random encoded signal
Antenna	The Archimedes spiral antenna
Radiation angle	−60°–+60°
Center frequency	1 GHz
Bandwidth	1 GHz
Scanning rate	3 frame/s

. A pseudo-random coded signal is used in the radar to improve the penetration performance. The radar center frequency and bandwidth are both 1 GHz. The scanning frequency of the system is three frame/s. The obstacle in the experimental scene is a red brick wall with a thickness of 37 cm. Assuming that the wall is of a homogeneous medium, the relative permittivity of the wall is approximately 5.6, measured by auxiliary equipment such as the vector network analyzer, and the conductivity $\sigma =0.05{ \mathrm{Sm}}^{-1}$. In the experiment, the starting position of the target is (−1.5, 6.0 m). Based on the normal direction of the wall, the moving direction of the target is 30°, and the speed is 0.8 m/s.

Figure 14a–i show the 3D imaging and 2D projection with no compensation, the RCMCA, and the proposed algorithm, respectively. It is obvious that the sidelobes of the imaging are high without compensation, and the position shows deviation. After compensation using the RCMCA, the sidelobes of the target are eliminated, and the position deviation is also corrected. Figure 14g–i are the imaging results after compensation using the proposed algorithm. The problems of the target sidelobes and deviation have also been well resolved, and the imaging effect is basically the same as that of the RCMCA. The target positions before motion compensation, after the RCMCA, and after the proposed algorithm are (−1.95, 6.31 m), (−1.53, 6.16 m), and (−1.52, 6.18 m), respectively. The target position compensated using the proposed method is basically consistent with the RCMCA, which is close to the initial point (−1.5, 6.0 m).

In order to verify the stability of the proposed method in practical experiments, experiments are performed on targets in different states (including different speeds and different distances). Setting the starting point of the target as (0, 6.0 m),

Table 4. The position deviation at different speeds.

Speed	0.3 m/s	0.7 m/s	1.1 m/s	1.5 m/s
Method
Before compensation	0.02 m	0.54 m	1.04 m	1.20 m
RCMCA	0.00 m	0.15 m	0.18 m	0.28 m
The proposed	0.00 m	0.16 m	0.20 m	0.28 m

gives the position deviations when the target moves at a speed of 0.3 m/s, 0.7 m/s, 1.1 m/s, and 1.5 m/s, respectively. For scenarios with different speeds, with the RCMCA or the proposed algorithm, the position deviation can be controlled in 0.30 m.

Table 5. The position deviation at different distances.

Range	2 m	4 m	6 m	8 m
Method
Before compensation	0.25 m	0.42 m	0.60 m	0.98 m
RCMCA	0.05 m	0.12 m	0.17 m	0.27 m
The proposed	0.04 m	0.12 m	0.19 m	0.26 m

gives the position deviation of the target passing the distances of 2 m, 4 m, 6 m, and 8 m, at a speed of 0.8 m/s, respectively. At different distances, the position deviation using the RCMCA and the proposed method remains within 0.30 m.

5. Conclusions

This paper proposes a motion compensation algorithm based on a sparse planar array, designed to eliminate the range migration in the 3D TWI of moving targets in TDM MIMO radar. The coincident array elements of the equivalent array are used as the compensation channel, to obtain the motion state of the target, which can correct the remaining MIMO channels. The proposed motion compensation algorithm is implemented under the assumption that the wall is a homogeneous medium. The results of the gprMax simulation and through-wall experiments show that the proposed algorithm can effectively eliminate the sidelobes, and contain the position deviation within 0.30 m, without increasing the complexity of the system, which achieves the same effect as the RCMCA. The proposed method can also be extended to motion compensation in radars at different frequency bands.