A Joint Estimation Method of the Channel Phase Error and Motion Error for Distributed SAR on a Single Airborne Platform Based on a Time-Domain Correlation Method

Abstract

Distributed synthetic aperture radar (SAR) is a system in which transmitting or receiving arrays are distributed on multiple platforms or at different locations on one platform. Distributed SAR can be used for high-resolution wide-swath (HRWS) imaging. The typical platform used for distributed SAR is a satellite constellation, which has long baselines and an ideal trajectory. Instead of satellite constellations, this paper focuses on distributed SAR on a single airborne platform, for which the channel error and motion error are coupled. Furthermore, the traditional channel error estimation methods are invalid. Thus, based on the time-domain correlation method (TDCM), this article proposes a joint estimation method of the channel phase error and motion error for the distributed SAR on a single airborne platform. Firstly, a channel error and motion error coupled phase error model of the distributed SAR is constructed. A joint estimation method of the channel phase error and motion error is then proposed. Finally, a simulation and real data processing are provided to demonstrate the effectiveness of the proposed method.

1. Introduction

The multi-channel synthetic aperture radar (MC-SAR) system is a SAR system with multiple transmit channels or receive channels [1], which has higher dimensional degrees of freedom compared to single-channel SAR. It can effectively improve system performance and extend the scope of applications of SAR systems, such as ground moving target indication (GMTI) [2,3], high-resolution wide-swath (HRWS) imaging [4], and topographic surveys [5]. A traditional MC-SAR is achieved by dividing the whole antenna array into multiple subarrays, i.e., centralized MC-SAR. With the increasing demand for SAR applications, the centralized MC-SAR system is becoming increasingly complex and its antenna aperture is lengthening. This imposes higher requirements on the antenna design and carrier platform. In order to avoid the problems that centralized MC-SAR systems encounter, the concept of a distributed SAR is proposed.

Distributed SAR is a system in which transmitting or receiving arrays are distributed on multiple platforms or at different locations on one platform. Through the distributed placement of SAR antennas, a single long aperture antenna is decomposed into multiple small aperture antennas. Thereby, system performance, design flexibility, achievability, and system stability are improved [6,7]. The typical distributed SAR system is a satellite constellation [8,9,10], such as TerraSAR-X, Tandem-X [10,11], or TH-2 [12]. In contrast, this paper focuses on distributed SAR on a single airborne platform, which is mainly applied to platforms with high requirements for stealth and aerodynamic performance, such as supersonic fighters or reconnaissance aircrafts. Compared with the spaceborne distributed SAR, the connection between the channels of the distributed SAR on a single airborne platform is rigid, and the motion error is not negligible.

The HRWS imaging mode is a popular research area of distributed SAR [13,14,15,16,17]. For single-channel SAR systems, regardless of the operating mode, a high azimuth resolution and a wide swath contradict each other in system design due to the minimum antenna area limitation [18,19]. Distributed SAR can overcome the minimum antenna area limitation and obtain HRWS imaging by signal reconstruction methods [18,20,21,22,23]. In addition, compressed sampling has also recently been used to address undersampling SAR [24,25,26]. In [11,17,27,28], distributed SAR imaging with TerraSAR-X and Tandem-X was demonstrated. Before signal reconstruction, spaceborne distributed SAR first needs to compensate for channel errors, including channel amplitude phase errors, range sampling delays, and channel errors caused by the cross-track baseline (CTB). The above channel errors lead to amplitude errors, envelope migration, and phase errors between channels. The envelope migration is estimated and compensated by channel coregistration methods in the range time domain [29,30,31] or range frequency domain [32]. The amplitude and phase errors can be estimated and compensated by the orthogonal subspace method (OSM) [13], the adaptive weighted least square method (AWLSM) [33], the signal subspace comparison method (SSCM) [14,34], or the time-domain correlation method (TDCM) [35,36], among which the first three are more computationally intensive with the TDCM more widely used in real-time processing systems. However, the distributed SAR on a single airborne platform leads to the failure of traditional channel error estimation methods due to the non-negligible coupling of motion error and channel error (collectively referred to as “signal error” in this paper). Therefore, the joint estimation method of the channel phase and motion error needs to be studied.

For the distributed SAR motion error and channel error coupling problem, this paper firstly establishes a signal model of the coupled motion error and channel error of single-platform distributed SAR. Based on this signal model, the phase sequence form of TDCM estimation is then analyzed. A joint estimation method of the channel phase error and motion error for the distributed SAR on a single airborne platform based on the TDCM is also proposed to obtain the acceleration motion error of the platform and the phase error between the distributed channels. The effectiveness of the proposed method is verified through simulation and real data processing.

2. Signal Model

2.1. The Geometry of Distributed SAR

The geometry of the distributed SAR is shown in Figure 1. O-xyz is the Cartesian coordinate system. The y axis represents the direction of the track, the x axis represents the direction perpendicular to the track in the horizontal plane, and the z axis is determined by the right-hand rule. The antennas $a_m,m=0,1,\dots M-1$ are distributed on the platform, where the reference channel $a_0$ is the transmitting channel, and all channels receive the signal simultaneously. The height and velocity of the flight platform are $H$ and $v_{\mathrm{p}}$, respectively. The red dashed line indicates the ideal trajectory of the reference channel and the blue curve indicates the actual motion trajectory of the reference channel. The distributed SAR system operates in the side-looking mode. $\phi$ denotes the incidence angle. When the slow time $\eta =0$, the equivalent phase center (EPC) of $a_0$ is denoted as $(0,0,H)$ and the EPC of $a_m$ is $(x_m,y_m,H+z_m)$. $a_m^{\prime }$ is the projection of $a_m$ to the ideal trajectory of $a_0$ and the coordinate is $(0,y_m,H)$. The line between $a_m^{\prime }$ and $a_m$ is the CTB between the mth receiver channel and the reference channel. The solid red line in Figure 1 is the difference in slant range from the target $P$ to $a_m$ and $a_m^{\prime }$. Before signal reconstruction, the signal error between $a_m^{\prime }$ and $a_m$ needs to be compensated.

2.2. Mathematical Model

For the distributed SAR, range compression and motion compensation based on the inertial navigation system (INS) are performed first. If there is only an azimuth delay between the signal echo of $a_m^{\prime }$, i.e., ${s^{\prime }}_m\left(\tau ,\eta \right)$, and the echo signal of $a_0$, then the signal at $a_m$ can be expressed as

$$\begin{array}{l}{s}_m\left(\tau ,\eta \right)={\Gamma }_m\exp \left(j{\varphi }_m\right)\\ \hspace{1em}\hspace{1em}\hspace{1em} \exp \left(-j\frac{4\pi \Delta R_{m,me}\left(\tau \right)}{\lambda }\right)\exp \left(-j\frac{4\pi \Delta R_{m,er}\left(\tau \right)}{\lambda }\right)\exp \left(-j\frac{4\pi \Delta R_{mo}\left(\eta \right)}{\lambda }\right)\\ \hspace{1em}\hspace{1em}\hspace{1em} s_m^{\prime }\left(\left(\tau -\Delta {\tau }_m-\frac{2\Delta R_{m,me}\left({\tau }_0\right)}{c}-\frac{2\Delta R_{m,er}\left({\tau }_0\right)}{c}\right),\eta \right)\end{array}$$

(1)

where $\tau$ represents fast time, ${\tau }_0$ is the fast time corresponding to the scene center, $\lambda$ is the wavelength, $c$ is light speed, ${\Gamma }_m$ represents the channel amplitude error, and ${\varphi }_m$ represents the channel inherent phase error. $x_{m,me}$ and $z_{m,me}$ denote the measured values of the vertical CTB and the horizontal CTB of $a_m$, respectively. $x_{m,er}$ and $z_{m,er}$ denote the measured errors of the vertical CTB and the horizontal CTB. $v_c$ is the platform’s radial velocity error and $a_c$ represents the platform’s radial acceleration error. The slant range difference between the channels corresponding to the CTB $\Delta R_{m,me}\left(\tau \right)$, the slant range difference corresponding to the measurement error of the CTB $\Delta R_{m,er}$, and the slant range difference corresponding to the motion error are expressed as follows:

$$\Delta R_{m,me}\left(\tau \right)=z_{m,me}\cos \phi \left(\tau \right)-x_{m,me}\sin \phi \left(\tau \right)$$

(2)

$$\Delta R_{m,er}=z_{m,er}\cos \phi \left({\tau }_0\right)-x_{m,er}\sin \phi \left({\tau }_0\right)$$

(3)

$$\Delta R_{mo}\left(\eta \right)=v_c\eta +\frac{1}{2}{a}_c{\eta }^2$$

(4)

where

$$\phi \left(\tau \right)=\arccos \left(\frac{2H}{c\tau }\right)$$

(5)

The following assumptions are made:

Firstly, the platform flight attitude maintains the remaining constant radial velocity and the radial acceleration within the sub-aperture.

The channel envelope migration and phase error due to the CTB of the channel are azimuth time invariant. The second-order phase error due to motion error is time variant, but the coefficients are constant.

Secondly, the envelope migration corresponding to the CTB is constant within the scene.

The slant range difference corresponding to the CTB varies with fast time $\tau$, leading to a range-variant phase error. However, the range variance of envelope migration caused by the CTB is much less than half of the range cell in the imaging scene and can be considered constant.

3. Processing Method

3.1. Processing Flow

For the above signal model, the amplitude errors, channel phase errors, and motion errors within the echo data need to be compensated before signal reconstruction and SAR imaging. The processing flow chart of the distributed SAR is shown in Figure 2.

Firstly, range focusing is performed on the multi-channel echoes using range compression and then the range coregistration method proposed in [32] to eliminate envelope migration caused by the CTB and range sampling time delay. According to the measured value of the CTB, the slant range difference between channels can be obtained using Equation (2), and the phase error corresponding to the measured value of the CTB in Equation (1) is compensated. After the above processing, Equation (1) becomes

$$s_m\left(\tau ,\eta \right)={\Gamma }_m\exp \left(j{\varphi }_m^{\prime }\left(\tau \right)\right)\exp \left(-j\frac{4\pi \Delta R_{mo}\left(\eta \right)}{\lambda }\right)s_m^{\prime }\left(\tau ,\eta \right)$$

(6)

where ${\varphi }_m^{\prime }\left(\tau \right)$ includes the inherent phase error of the channel ${\varphi }_m$ and the phase error corresponding to the measurement of the CTB:

$${\varphi }_m^{\prime }\left(\tau \right)={\varphi }_m-\frac{4\pi \Delta R_{m,er}\left(\tau \right)}{\lambda }$$

(7)

After compensating for the error terms in Equation (6) using the proposed method marked in the red rectangle, non-ambiguous HRWS SAR images can be obtained by the signal reconstruction method and SAR imaging processing. However, the residual errors in Equation (6) include channel errors and motion errors, which prevents the traditional channel error estimation method from being applicable. Therefore, a joint estimation method of the channel phase error and motion error for the distributed SAR on a single airborne platform based on the TDCM is proposed in Section 3.2.

3.2. The Proposed Method

According to the analysis in Section 3.1, the residual errors include the channel amplitude phase error, the phase error caused by the CTB measurement error, and the motion error. The channel amplitude error can be estimated and compensated by a simple channel equalization. According to the analysis in [23], its corresponding phase error caused by the CTB measurement error can be considered constant in the local area, although it is range variant over the entire scene. Therefore, after compensating for the channel amplitude error, the signal of the mth channel in the local region can be expressed as

$$s_m\left(\eta \right)=\exp \left(j{\varphi }_m^{\prime }\right)\exp \left(-j\frac{4\pi \Delta R_{mo}\left(\eta \right)}{\lambda }\right)s_m^{\prime }\left(\eta \right)$$

(8)

The traditional TDCM can be used to estimate the channel phase error, but unlike the spaceborne multi-platform distributed SAR, the channels of the distributed SAR on a single airborne platform have consistent residual motion error among channels due to the steel connection. The schematic diagram of the coupling of the motion error and the CTB measurement error is shown in Figure 3. Figure 3a indicates the CTB measurement error between channels, where the red dashed line indicates the ideal trajectory and different shapes indicate different channels. The presence of the CTB measurement error shifts non-reference channels from the ideal trajectory. Figure 3b indicates the motion error of a single channel with the solid blue line indicating the actual trajectory and different colors representing different slow time samplings. Figure 3c shows the actual arrangement of the sampling points in space after coupling the two types of errors.

The proposed method is based on the TDCM, considering both channel phase error and motion error, and the main steps for error estimation are as follows.

3.2.1. Data Rearrangement

In the traditional TDCM [35], the cross-correlation operations are performed between adjacent channels at the same slow time sample point or between a certain moment of the last channel and the next moment of the first channel. The data sorting is shown in the top row of Figure 4. The TDCM operation is performed between adjacent data. However, as Figure 4 shows, the azimuth interval between adjacent data for the data sorting shown in the top row of Figure 4 is large, and the correlation between the data is poor, thus the correlation coefficient cannot be estimated. Therefore, before the phase error estimation, the data are rearranged based on the spatial distribution, as shown in the bottom figure of Figure 4, to fit the Nyquist sampling theorem.

3.2.2. TDCM Processing

After data sorting, the rearranged data are processed using the TDCM. Using reference channel data at $\eta$ as the starting point, a group of M + 1 adjacent data is formed for the time-domain cross-correlation operation. Since the adjacent data after rearrangement are not sampled at the same slow time, the phase difference obtained by the TDCM includes both the channel phase error and the motion error. From Equation (8), the phase difference between $M$ adjacent data of a set of data can be expressed as

$${\zeta }_m\left(\eta \right)=\left\{\begin{array}{l}={\varphi }_m^{\prime }-{\varphi }_{m-1}^{\prime }\\ -4\pi \frac{\Delta R_{mo}\left(\eta -m{T}_{prt}\right)-\Delta R_{mo}\left(\eta -\left(m-1\right)T_{prt}\right)}{\lambda }\\ -2\pi f_{dc}\left(\frac{d_m-d_{m-1}}{v_{\mathrm{p}}}-T_{prt}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},m=1,2,\dots ,M-1\\ =-{\varphi }_{M-1}^{\prime }\\ -4\pi \frac{\Delta R_{mo}\left(\eta +T_{prt}\right)-\Delta R_{mo}\left(\eta -\left(M-1\right)T_{prt}\right)}{\lambda }\\ -2\pi f_{dc}\left(M{T}_{prt}-\frac{d_{M-1}}{v_{\mathrm{p}}}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},m=M\end{array}\right.$$

(9)

where $T_{prt}$ represents pulse repetition time, $f_{dc}$ represents the Doppler center of the signal, and ${\zeta }_m\left(\eta \right)$ includes channel error and motion error. The sequence of phase errors by sampling point is shown in Figure 5. The phase error is a quadratic curve with slow time on single-channel data, while there are also phase jumps between channels generated by channel errors.

Through the TDCM between adjacent data, the estimation of Equation (9) is

$${\widehat{\zeta }}_m\left(\eta \right)=\left\{\begin{array}{l}\angle \left(s_m(\eta -m{T}_{prt})s_{m-1}{}^{\ast }(\eta -\left(m-1\right)T_{prt})\right),m=1,2,\dots ,M-1\\ \angle \left(s_0(\eta +T_{prt})s_{M-1}{}^{\ast }(\eta -\left(M-1\right)T_{prt})\right),m=M\end{array}\right.$$

(10)

where $\angle ( )$ represents the phase and $\ast$ is the complex conjugate sign.

Summing ${\zeta }_m\left(\eta \right)$ cumulatively, the phase difference between the channel $m$ and the reference channel is obtained:

$$\begin{array}{l}{\omega }_m\left(\eta \right)={\displaystyle \sum _{k=1}^m{\zeta }_k\left(\eta \right)}={{\varphi }^{\prime }}_m+2\pi \left(\frac{d_m}{v_a}-m{T}_{prt}\right)f_{dc}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{4\pi }{\lambda }m{T}_{prt}{v}_c+\frac{4\pi }{\lambda }\left(m\eta T_{prt}-\frac{1}{2}{m}^2{T}_{prt}{}^2\right)a_c\hspace{1em},m=1,2,\dots ,M-1\end{array}$$

(11)

The estimation of ${\omega }_m\left(\eta \right)$ is

$${\widehat{\omega }}_m\left(\eta \right)={\displaystyle \sum _{k=1}^m{\widehat{\zeta }}_k\left(\eta \right)}\hspace{1em}\hspace{1em}\hspace{1em},m=1,2,\dots ,M-1$$

(12)

Except for the phase term associated with $f_{dc}$, the remaining terms are the signal errors to be compensated before signal reconstruction.

The phase difference between the adjacent slow times of the reference channels is

$${\omega }_0\left(\eta \right)={\displaystyle \sum _{k=1}^M{\zeta }_k\left(\eta \right)}=2\pi T_{prt}{f}_{dc}-\frac{4\pi }{\lambda }{T}_{prt}{v}_c-\frac{4\pi }{\lambda }\left(\eta T_{prt}+\frac{1}{2}{T}_{prt}{}^2\right)a_c$$

(13)

The estimation of ${\omega }_0\left(\eta \right)$ is

$${\widehat{\omega }}_0\left(\eta \right)={\displaystyle \sum _{k=1}^M{\widehat{\zeta }}_k\left(\eta \right)}$$

(14)

Based on Equations (11) and (13),

$$\begin{array}{l}{\phi }_m\left(\eta \right)={\omega }_m\left(\eta \right)-\frac{{\omega }_0\left(\eta \right)}{T_{prt}}\left(\frac{d_m}{v_a}-m{T}_{prt}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em} ={\varphi }_m^{\prime }+\frac{4\pi }{\lambda }\frac{d_m}{v_a}{v}_c\\ \hspace{1em}\hspace{1em}\hspace{1em} -\frac{4\pi }{\lambda }\left(\frac{1}{2}m\left(m+1\right)T_{prt}{}^2-\frac{1}{2}\frac{d_m}{v_a}{T}_{prt}\right)a_c+\frac{4\pi }{\lambda }\frac{d_m}{v_a}{a}_c\eta \hspace{1em}\hspace{1em},m=1,2,\dots ,M-1\end{array}$$

(15)

The estimation of ${\phi }_m\left(\eta \right)$ is

$${\widehat{\phi }}_m\left(\eta \right)={\widehat{\omega }}_m\left(\eta \right)-\frac{{\widehat{\omega }}_0\left(\eta \right)}{T_{prt}}\left(\frac{d_m}{v_a}-m{T}_{prt}\right)\hspace{1em}\hspace{1em},m=1,2,\dots ,M-1$$

(16)

When there is no motion error, the channel phase error ${\widehat{\varphi }}_m^{\prime }=E\left({\widehat{\phi }}_m\left(\eta \right)\right)$ can be estimated by averaging along slow time for Equation (16). However, when the motion error is not negligible, the estimation results are biased by a constant term and a linear time-variant term compared to the channel phase error.

3.2.3. Radial Acceleration Estimation

As shown in Equation (15), the linear term of ${\phi }_m\left(\eta \right)$ is determined by the radial acceleration. By performing a linear fit to ${\widehat{\phi }}_m\left(\eta \right)$, the results of its fit are obtained:

$${\widehat{\phi }}_m^{\prime }\left(\eta \right)=p_{m,0}+p_{m,1}\eta \hspace{1em}\hspace{1em}\hspace{1em},m=1,2,\dots ,M-1$$

(17)

where $p_{m,0}$ is the estimate of the constant term in ${\phi }_m\left(\eta \right)$ and the estimation of radial acceleration is denoted as

$${\widehat{a}}_c=\frac{\lambda v_a{p}_{m,1}}{4\pi d_m}$$

(18)

3.2.4. Signal Phase Error Compensation

After compensating ${\widehat{\phi }}_m^{\prime }\left(\eta \right)$ for (8), it follows from Equations (11) and (13) that the remaining phase error between adjacent data within a set of data at the moment $\eta$ is

$$\begin{array}{l}{\phi }_m^{\prime }\left(\eta \right)=\frac{4\pi }{\lambda }\left(m{T}_{prt}-\frac{d_m}{v_a}\right)v_c\\ \hspace{1em}\hspace{1em}\hspace{1em} -\frac{2\pi }{\lambda }\left(m{T}_{prt}-\frac{d_m}{v_a}\right)T_{prt}{a}_c+\frac{4\pi }{\lambda }\left(\left(m{T}_{prt}-\frac{d_m}{v_a}\right)\right)a_c\eta \hspace{1em}\hspace{1em},m=1,2,\dots ,M-1\end{array}$$

(19)

By arranging Equation (19) in the order of the spatial azimuth of the sampling points, the phase error sequence is obtained as follows:

$$\begin{array}{l}\Delta \phi \left(k\right)=-\frac{4\pi }{\lambda }{T}_{\left(\mathrm{mod}(k,M)-1\right)}^{\prime }{v}_c-\frac{2\pi }{\lambda }{T}_{\left(\mathrm{mod}(k,M)-1\right)}^{\prime }{T}_{prt}{a}_c-\frac{4\pi }{\lambda }{T}_{\left(\mathrm{mod}(k,M)-1\right)}^{\prime }⌊\frac{k-1}{M}⌋T_{prt}{a}_c\\ \hspace{1em}\hspace{1em}\hspace{1em} -\frac{4\pi }{\lambda }⌊\frac{k-1}{M}⌋T_{prt}{v}_c-\frac{2\pi }{\lambda }{\left(⌊\frac{k-1}{M}⌋T_{prt}\right)}^2{a}_c\end{array}$$

(20)

where $T_m^{\prime }=d_m/v_a-m{T}_{prt}$ denotes the azimuth interval between the mth channel of the rearranged data and the reference channel. $\lfloor \rfloor$ denotes rounding down and $\mathrm{mod}( )$ denotes the remainder. After compensating for the radial acceleration ${\widehat{a}}_c$ estimated using Equation (18), Equation (20) is expressed as

$$\Delta \phi \left(k\right)=-\frac{4\pi }{\lambda }{T}_{\left(\mathrm{mod}(k,M)-1\right)}^{\prime }{v}_c-\frac{4\pi }{\lambda }⌊\frac{k-1}{M}⌋T_{prt}{v}_c$$

(21)

where $\Delta \phi \left(k\right)$ is the remaining phase error caused by the platform radial velocity error after compensation by the proposed method. However, the remaining phase error is in a linear form along with the azimuth sampling points, which can be equated to making the spatial sampling point arrangement deviate from the ideal track and arranging it along the new track co-linearly. A schematic diagram of the track offset after signal error compensation is shown in Figure 6.

After the track is offset, both the motion velocity of the platform and the sampling point along the track (compensated track) interval change due to the radial error, but the ratio remains unchanged. From Equation (14) in [20], it is clear that the signal reconstruction effect is not affected when the ratio of the motion velocity and the sampling point interval along the track are unchanged. After signal reconstruction, the radial velocity error basically does not affect image focus.

The above method is performed on the premise that the phase error is range invariant in the local area. The phase error of the entire scene is obtained by fitting the estimation results of each chunk by the method proposed in [23]. After the global phase error is obtained and compensated, the HRWS image can be obtained by signal reconstruction and SAR imaging processing.

4. Result of the Experiment and Discussion

In this section, the estimated accuracy of the proposed method for channel error and radial acceleration is verified by the simulation of signal error estimation and compared to the traditional TDCM and OSM; the effectiveness of the proposed method is verified by simulation of HRWS scene face targets and the processing of real data.

4.1. Signal Error Estimation Simulation

The distributed SAR system simulated in this section has four uniformly and co-linearly arranged channels with mixed baselines between channels, and the system parameters are shown in

Table 1. Distributed SAR simulation system parameters.

Parameter	Value
Platform velocity	1700 m/s
Wavelength	0.03 m
Channel spacing (x, y, z)	(10 m, 4.8 m, 4 m)
Antenna azimuthal size	1.5 m
Transmit signal bandwidth	200 MHz
Pulse repetition frequency	900 Hz
Platform height	20 km
Radial velocity	2 m/s
Radial acceleration	5 m/s2

for a typical hypersonic platform.

In this section, the estimation result curves of the TDCM and OSM in the presence of motion errors are obtained by simulation, as shown in Figure 7. The SNR of the simulation scene is 30 dB, containing 41 range samples and 512 azimuth samples. Channel 1 is the reference channel, Channel 2 has a phase error of 90°, and Channels 3 and 4 have no phase error. To illustrate the effect of motion error, the radial velocity of the platform is set at 2 m/s and the acceleration at 5 m/s². The azimuth axis of the OSM represents the azimuth Doppler frequency; the azimuth axis of the TDCM represents the slow time. When there is no motion error, the two methods should be estimated as straight lines. However, due to the existence of motion errors, the OSM estimation fails. The TDCM estimation results have a certain slope and deviate from the true value. From Equation (15), the slope is caused by the $4\pi d_m{a}_c\eta /(\lambda v_a)$ term; the shift of the mean value is caused by both velocity and acceleration.

By averaging the sampling points without estimating the motion error and with deviations from the true channel error, the OSM and the traditional TDCM obtained channel error estimates of 42.6° and 50.7°, respectively. The radial acceleration of the platform is not estimated by the TDCM or OSM. There are still large residual signal errors that remain after only compensating for the channel error estimated by the TDCM or OSM. Comparing the TDCM to the OSM, radial acceleration can be obtained by fitting the result of the TDCM but not the OSM. Therefore, the proposed method is based on the TDCM.

The proposed method obtains a constant term phase error of 50.7° and a radial acceleration estimate of 5.1 m/s² by linearly fitting the TDCM estimation results. The proposed method has the same bias related to the channel phase error as the traditional TDCM. However, from the analysis in Section 3.2, after compensating for the phase error obtained from the fit by considering the radial acceleration, the remaining phase error is equivalent to making the sampling point arrangement offset from the ideal trajectory, which does not affect the subsequent imaging. In contrast, the traditional TDCM does not estimate the motion error, and the phase error caused by the residual acceleration will affect the imaging quality. The effect on the imaging results is shown in the scene simulation in Section 4.2 and the processing of the real data in Section 4.3.

4.2. Scene Imaging Simulation

In the scene face target simulation, the real SAR image is used as the image source to simulate the echo, and the simulation parameters are consistent with Table 1 except for the motion error. Figure 8 shows the imaging results for the range of 130–131.4 km. Figure 8a–c shows the imaging results of the traditional TDCM with different motion errors, and Figure 8d shows the imaging result after signal error estimation and compensation by the method proposed in the paper. If there is only channel error, the traditional TDCM and OSM can meet the accuracy requirement of channel error estimation. However, the traditional TDCM and OSM ignore the motion error, and the deterioration of the imaging result is similar. Therefore, the traditional TDCM is taken as an example for comparison.

Defocusing is severe in Figure 8a,b due to the uncompensated motion error. Figure 8c is partially focused but has obvious azimuth ambiguity. By using the proposed method, the imaging is well focused and the ambiguity is well suppressed, as shown in Figure 8d.

The entropy of the imaging is used to evaluate image quality, as shown in

Table 2. The entropy of Figure 8.

Image	Entropy
Figure 8a	15.9
Figure 8b	15.8
Figure 8c	15.6
Figure 8d	15.4

. When the motion error is large, the imaging is defocused and the entropy is large. The entropy decreases when the motion error is smaller. The motion error is compensated by the proposed method. Therefore, the entropy is the smallest.

4.3. Real Data Processing

To further validate the effectiveness of the proposed method, an equivalent validation experiment was carried out by imaging a farmland scene using airborne dual-channel SAR at a location in Shaanxi. Figure 9 shows the optical map of the experimental scene. In addition to the farmland in the scene, there are some private houses with strong scattering.

The system used is a dual-channel radar system with a hybrid baseline, using a single-output multiple-input mode, and the specific operating parameters are shown in

Table 3. Airborne dual-channel SAR experiment system parameters.

Parameter		Value
Platform velocity		About 100 m/s
Platform height		About 6000 m
Frequency band		X
Signal bandwidth		200 MHz
Azimuth resolution		About 0.5 m
Pulse repetition frequency of the original data		1250 Hz
Pulse repetition frequency of the extraction data of a single channel		125 Hz
Channel spacing	x	3.70 m
	y	0.20 m
	z	0.08 m

.

According to the parameters shown in Table 3, the equivalent phase centers between channels are 1.85 m, 0.10 m, and 0.04 m in x, y, and z. First, the azimuth window is added to the echoes to equivalently reduce the azimuth beamwidth and azimuth resolution of the system; therefore, since the initial pulse repetition frequency of the system is larger than the azimuth bandwidth, there is no azimuth ambiguity. As a result, the data of both channels are downsampled by a factor of 10 through azimuth sampling point extraction for the equivalent azimuth downsampling case. Channel 1 extracts the 1st, 11th, 21st, etc., slow time samples and Channel 2 extracts the 5th, 15th, 25th, etc., slow time samples. After extraction, the equivalent single-channel pulse repetition frequency becomes 125 Hz, which is smaller than the signal azimuth Doppler bandwidth. The combination of the two channels’ data can constitute a sampling sequence of a distributed SAR hybrid baseline. This data processing can be used to verify the performance of the distributed SAR signal error estimation and compensation method. Figure 10 shows the schematic diagram of two-channel data extraction, where the circle indicates the actual sampling position of Channel 1, the triangle indicates the actual sampling position of Channel 2, and the solid shape indicates the actual extraction, which is used for signal error estimation and compensation.

After combining the channel data, the process is carried out according to the procedure described in Section 3.1. The TDCM is used to estimate the channel phase error, and the results of the TDCM channel error estimation are shown in Figure 11.

Figure 11 shows that the channel error has a certain slope which is consistent with the analysis of Equation (15); the slope is due to the radial acceleration of the platform. After compensating for the channel error by linear fitting, the slope and Equation (18) are used to obtain the radial acceleration estimate of 0.2736 m/s² to compensate for the effect caused by the acceleration. The spectrum is then reconstructed by using the transfer function method, and SAR imaging is performed by applying the chirp scaling (CS) method.

As a comparison, the results of single-channel undersampling imaging are shown in Figure 12, where the yellow boxes are for Target A and B and the red boxes are the positions of the ambiguous components. The ambiguity suppression effect of HRWS imaging is subsequently evaluated with Target A and B. The ambiguity in Figure 12 is severe, and the ambiguous component is clearly visible both above $f_{\mathrm{prf}}/f_{\mathrm{dr}}$ and below $-f_{\mathrm{prf}}/f_{\mathrm{dr}}$ of the real target, which substantially affects image interpretation.

The distributed SAR HRWS imaging results are shown in Figure 13, where the yellow boxes mark the positions of Target A and B and the red boxes are the positions of their ambiguous components.

Figure 13a shows the imaging results by using the traditional TDCM, and Figure 13b shows the imaging results using the proposed method. The traditional TDCM does not consider the motion error of the platform, while the proposed method does consider the error. Furthermore, though the OSM is another traditional method, the number of channels needs to be greater than the number of ambiguous components. Thus, the OSM does not fit this system. Figure 13c–f shows the local zoomed images of Target A and B. Figure 13a,c,e show that, after compensation using the traditional method, although the ambiguity is improved compared to the undersampling case, there is still a large residual. The azimuth ambiguity causes the noise floor of the imaging to increase, and the ambiguous component of the strongly scattered target is still more obvious. After adopting the two-step signal error estimation and compensation method proposed in this paper, considering the range variance of envelope migration and the coupling of motion error and channel error, the image quality is significantly improved after estimating and compensating for the signal error. The motion error is also not considered in the OSM. Therefore, the signal error is not correctly estimated and the ambiguity still exists, as in the traditional TDCM.

To further illustrate the effectiveness of the proposed method, Figure 14 shows the azimuth cuts of Target A and B in the scene. Figure 14a,c,e are the azimuth cuts of the imaging results of Target A for the single-channel downsampling, the traditional method, and the proposed method, respectively. Figure 14b,d,f are azimuth sections of the Target B imaging results for the single-channel downsampling, the traditional method, and the proposed method, respectively.

The ambiguity suppression capability of the method was measured using the Azimuth Ambiguity-to-Signal Ratio (AASR) [37]. The AASR of Figure 14 is provided in

Table 4. The AASR of the targets in Figure 14.

Targets	Methods	AASR (dB)
		Left	Right
Target A	Single-channel	−10.4	−6.1
	Traditional method	−15.9	−11.7
	Proposed method	Lower than Clutter	Lower than Clutter
Target B	Single-channel	−6.8	−9.2
	Traditional method	−12.5	−15.7
	Proposed method	Lower than Clutter	Lower than Clutter

. Figure 14a,b show that the ambiguous component is severe, and the noise floor is high in the single-channel undersampling case. As shown in Figure 14c,d, after processing by the traditional method, the ambiguity suppression is improved by about 5.6 dB, but ambiguous components are still significant. As shown in Figure 14e,f, the noise floor is reduced by processing with the proposed method, the ambiguity energy is suppressed to below the clutter level, and the imaging quality is significantly improved.

The reconstructed spectrograms obtained from the single-channel undersampling, processed by the traditional and the proposed methods, and processed by the spectrum reconstruction process are compared in Figure 15. The single-channel method generates spectral ambiguity, which leads to a significant elevation in the spectrum edges and severe spectral distortion. When spectrum reconstruction is performed after estimating and compensating the channel error by the traditional method, the spectrum diffusion phenomenon is improved and the energy within the main lobe is enhanced. However, the sidelobes of the spectrum are still high. After using the proposed method to estimate and compensate for both channel and motion errors, the reconstruction spectrum is significantly improved, and the out-of-band residual energy is small.

In summary, the effectiveness of the proposed method is verified by processing real data. The proposed method is a significant improvement over the traditional method in terms of spectrum improvement and ambiguity suppression.

5. Conclusions

In contrast to spaceborne SAR, the motion error of the airborne platform is not negligible. The traditional channel error estimation method, which neglects the motion error, would miscalculate the error. After compositing the error with the traditional method, the azimuth ambiguity component in distributed SAR HRWS imaging is still apparent.

In this paper, a signal model for the distributed SAR on a single airborne platform with channel error and motion error is constructed. A joint estimation method of the channel phase error and motion error for this distributed SAR based on the TDCM is then proposed. Finally, the effectiveness of the proposed method is verified by simulation and real data processing. The proposed method effectively solves the problem of joint estimation of channel error and motion error. Moreover, the azimuth ambiguity component in HRWS imaging is effectively suppressed after compensating for the estimated signal error. The proposed method provides a solution for correctly estimating the signal error for HRWS imaging. The estimation of signal error for distributed SAR on several airborne platforms with different error motions needs to be studied.