0 Introduction

Conventional inverse synthetic aperture radar(ISAR) imaging uses range-Doppler (RD) method, in which a simple fast Fourier transform (FFT) is applied in the cross-range dimension to achieve the cross-range resolution. The RD method works under the assumptions of plane wave-front and small rotation-angle in the coherent processing interval (CPI)^[1]. Motion compensation including range alignment and phase compensation is necessary before the FFT in cross-range to remove the translational motion component between the target and the radar. This FFT-based imaging method becomes invalid if the target undergoes a larger rotation relative to the radar accompanied by migration through resolution cell (MTRC)^[1-3].

Many literatures for ISAR imaging of space targets have appeared^[4-9]. In space target imaging, the trajectory information of space targets is always available. The targets mostly undergo stationary motion. This in fact enables the applications of the polar format algorithm (PFA)^{[10, 11]} and back-projection (BP)^[12-14] imaging method to integrate longer data than conventional RD method, and hence obtain higher image resolution. As compared to RD and PFA, the BP imaging method has several advantages: (1) it does not need motion compensation required by RD algorithm; (2) it does not need the assumption of plane wave-front; (3) it is applicable to arbitrary trajectories and imaging geometries; (4) it can be implemented efficiently exploiting parallel processing and high speed hardware, e.g. graphic processing unit (GPU). In this paper, we are interested in performing the image formation of space targets using BP method with the known trajectory information.

Due to the limited accuracy of the trajectory data, the errors in the trajectory may lead to mispositioning and smearing of the space target in the reconstructed image. We provide an analysis of the effect of trajectory errors on the image reconstruction and use the microlocal analysis to get explicit expressions of the positioning errors. The results show the relationship between the positioning errors and the trajectory errors, and the dependency of the positioning errors on the space target trajectory, the flying velocity of the space target, etc. We specify the positioning errors due to trajectory errors for some typical types of trajectory errors. Numerical simulations are performed to demonstrate our analysis. Our results can be easily extended to the bistatic ISAR imaging case. The approach for analyzing the positioning errors due to space target trajectory errors is initially presented in our previous conference paper^[15].

1 ISAR Imaging Principles

The ISAR imaging from the perspective of BP theory is briefly described in this section.

The received signal of ISAR is given as

$ d\left( {s,t} \right) = \int {{{\rm{e}}^{ - {\rm{j}}\omega \left( {t - \frac{{R\left( {\mathit{\boldsymbol{X}}\left( s \right)} \right)}}{{{c_0}}}} \right)}}A\left( {\mathit{\boldsymbol{X}}\left( s \right),\omega } \right)T\left( {\mathit{\boldsymbol{X}}\left( s \right)} \right){\rm{d}}\omega {\rm{d}}\mathit{\boldsymbol{X}}\left( s \right)} $

(1)

where t denotes the fast time, s the slow-time, c₀ the electromagnetic wave speed, ω the angular frequency of the transmitted waveform, X(s) the coordinates of the scatterers on the target, and T(X(s)) the target reflectivity. The s dependency of the scatterer position accounts for the position change as the space target flies along the trajectory. Note that the stop-go assumption is used in Eq. (1), which is valid for the cases of pulsed transmitted waveform and little radial motion existing between the space target and the radar. A(X(s), ω) is a complex amplitude function including the transmitted waveform and the antenna beam patterns.R(X(s)) denotes the total range of the two-way propagation of the electromagnetic wave. For monostatic ISAR, R(X(s)) is given by

$ R\left( {\mathit{\boldsymbol{X}}\left( s \right)} \right) = 2\left| {\mathit{\boldsymbol{X}}\left( s \right) - \mathit{\boldsymbol{p}}} \right| $

(2)

where p denotes the position of radar. Without the loss of generality, we focus on monostatic ISAR imaging in the rest of the discussion.

$ {\rm{Let}}\;\;\;\;\;\;\mathit{\boldsymbol{X}}\left( s \right) = \mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{X'}} $

(3)

where γ(s) denotes the trajectory of space target and X′ the coordinates of the target scatterers in the local coordinate system embedded on the target, as shown in Fig. 1.

Fig. 1 presents the imaging geometry for space targets. Ox₁x₂x₃ is the absolute or fixed reference coordinate system. O′x₁′x₂′x₃′ is the local coordinate system embedded on the space target where the origin O′ is usually chosen to be at the mass center of the target. The coordinate system O′x₁′x₂′x₃′ is translated from the origin of Ox₁x₂x₃ by the space target trajectory γ(s). Note that the rotation of the space target is not considered during the synthetic aperture time.

Substituting Eqs. (2), (3) into Eq. (1), and making the change of variables X→X′, we have

$ d\left( {s,t} \right) = \int {{{\rm{e}}^{ - {\rm{j}}\omega \left( {t - \frac{{2\left| {\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{X'}} - \mathit{\boldsymbol{p}}} \right|}}{{{c_0}}}} \right)}}A\left( {s,\mathit{\boldsymbol{X'}},\omega } \right)T\left( {\mathit{\boldsymbol{X'}}} \right){\rm{d}}\omega {\rm{d}}\mathit{\boldsymbol{X'}}} $

(4)

In the rest of our discussion, X′ will be replaced with X for notational simplicity.

A three-dimensional (3D) image can be reconstructed as long as the data is collected in 3D space, i.e., γ(s) undergoes sufficient variation in R³. We focus on the two-dimensional (2D) image formation. Let z∈ R² denote the position in the reconstructed 2D image. By using the filtered BP (FBP) method^[12], the image can be formed as

$ \hat T\left( \mathit{\boldsymbol{z}} \right) = \int {{{\rm{e}}^{{\rm{j}}\omega \left( {t - \frac{{R\left( {s,\mathit{\boldsymbol{Z}}} \right)}}{{{c_0}}}} \right)}}Q\left( {\mathit{\boldsymbol{z}},s,\omega } \right)d\left( {s,t} \right){\rm{d}}\omega {\rm{d}}s{\rm{d}}t} $

(5)

Here

$ R\left( {s,{\bf{Z}}} \right) = 2\left| {\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}} \right| $

(6)

where Z∈ R³, Z=(z, 0). The image is assumed to be reconstructed on a zero-height plane, i.e., z₃=0.

In Eq. (5), the phase term accounts for the matched filtering. The filter Q can be chosen according to a variety of criteria^{[16, 17]} to compensate the amplitude term in Eq. (5) or to enhance the edges in the images, etc.

Substituting Eq. (4) into Eq. (5), we have

$ \hat T\left( \mathit{\boldsymbol{z}} \right) = \int {{{\rm{e}}^{{\rm{j}}\mathit{\Phi }\left( {\mathit{\boldsymbol{\omega }},\mathit{\boldsymbol{Z}},\mathit{\boldsymbol{X}},s} \right)}}Q\left( {\mathit{\boldsymbol{z}},s,\mathit{\boldsymbol{\omega }}} \right)A\left( {\mathit{\boldsymbol{X}},s,\mathit{\boldsymbol{\omega }}} \right){\rm{d}}\mathit{\boldsymbol{\omega }}{\rm{d}}s} $

(7)

where Φ(ω, Z, X, s)=$\frac{\mathit{\boldsymbol{\omega }}}{{{c_0}}}\left[{R\left( {s, \mathit{\boldsymbol{X}}} \right)-R\left( {s, \mathit{\boldsymbol{Z}}} \right)} \right]$. Note that X=(x, x₃) and Z=(z, 0).

The Hörmander-Sato lemma^[18-22] tells us that the imaging operator in fact reconstructs a scatterer at location x on the target at position z in the image satisfying the following conditions^{[12, 19, 20]}.

$ {\partial _\omega }\mathit{\Phi }\left( {\mathit{\boldsymbol{Z}},\mathit{\boldsymbol{X}},s,\mathit{\boldsymbol{\omega }}} \right) = 0 \Rightarrow R\left( {s,\mathit{\boldsymbol{Z}}} \right) = R\left( {s,\mathit{\boldsymbol{X}}} \right) $

(8)

$ {\partial _s}\mathit{\Phi }\left( {\mathit{\boldsymbol{Z}},\mathit{\boldsymbol{X}},s,\mathit{\boldsymbol{\omega }}} \right) = 0 \Rightarrow {f_{\rm{d}}}\left( {s,\mathit{\boldsymbol{Z}}} \right) = {f_{\rm{d}}}\left( {s,\mathit{\boldsymbol{X}}} \right) $

(9)

Here f_d denotes the Doppler frequency which is given by

$ {{f}_{\text{d}}}\left( s,\mathit{\boldsymbol{X}} \right)=\frac{\mathit{\boldsymbol{\omega }}}{{{c}_{0}}}\mathit{\boldsymbol{\dot \gamma }}\left( s \right)\cdot \overset\frown{\mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{X}}-\mathit{\boldsymbol{p}}} $

(10)

where ${\mathit{\boldsymbol{\hat u}}}$ denotes the unit vector of the vector u, $\overset\frown{\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}}$ the radar look-direction^{[12, 23]}. $\mathit{\boldsymbol{\dot \gamma }}\left( s \right)$ is the first derivative of γ(s) with respect to s denoting the flying velocity of the space target. Eqs. (8) and (9) show that the reconstructed scatterer at z has the same range and Doppler frequency with the scatterer located at X^{[12, 19, 20]}.

2 Analysis of Positioning Errors Due to Trajectory Errors

In this section, we analyze reconstruction errors in ISAR images due to errors in the trajectory of the space target using microlocal analysis^{[18, 24, 25]}.

Let γ(s) denote the ideal or assumed trajectory and γ^ε(s)=γ(s)+εΔγ(s) the true trajectory of the space target where ε is a small constant. In an ideal case where the target is tracked with sufficient accuracy, i.e.Δγ(s)≈0, the scatterer X on the target is mapped to z₀ in the image. We refer to z₀ as the correct position of the scatterer in the reconstructed image. Note that the mapping from X to z₀ is associated with the projection of 3D coordinates to 2D coordinates.

Let z=z₀+Δz be the erroneous reconstructed position due to trajectory error Δγ(s). Using Eqs. (8) and (9), we have^[23]

$ \begin{array}{l} R\left( {{\mathit{\boldsymbol{\gamma }}^\varepsilon }\left( s \right),\mathit{\boldsymbol{z}}} \right) = R\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),{\mathit{\boldsymbol{z}}_0}} \right) \Rightarrow \\ R\left( {\mathit{\boldsymbol{\gamma }}\left( s \right) + \varepsilon \Delta \mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right) = R\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}} - \Delta \mathit{\boldsymbol{z}}} \right) \end{array} $

(11)

$ \begin{array}{l} {f_{\rm{d}}}\left( {{\mathit{\boldsymbol{\gamma }}^\varepsilon }\left( s \right),\mathit{\boldsymbol{z}}} \right) = {f_{\rm{d}}}\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),{\mathit{\boldsymbol{z}}_0}} \right) \Rightarrow \\ {f_{\rm{d}}}\left( {\mathit{\boldsymbol{\gamma }}\left( s \right) + \varepsilon \Delta \mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right) = {f_{\rm{d}}}\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}} - \Delta \mathit{\boldsymbol{z}}} \right) \end{array} $

(12)

By expanding the terms on the left side of Eqs. (11) and (12) in Taylor series around ε=0 and keeping the first-order terms, Eqs. (11) and (12) will respectively become

$ \begin{array}{l} {\nabla _z}R\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right) \cdot \Delta \mathit{\boldsymbol{z}} = \\ - \varepsilon {\partial _\varepsilon }R\left( {\mathit{\boldsymbol{\gamma }}\left( s \right) + \varepsilon \Delta \mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right)\left| {_{\varepsilon = 0}} \right. \end{array} $

(13)

$ \begin{array}{l} {\nabla _z}{f_{\rm{d}}}\left( {\mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right) \cdot \Delta \mathit{\boldsymbol{z}} = \\ - \varepsilon {\partial _\varepsilon }{f_{\rm{d}}}\left( {\mathit{\boldsymbol{\gamma }}\left( s \right) + \varepsilon \Delta \mathit{\boldsymbol{\gamma }}\left( s \right),\mathit{\boldsymbol{z}}} \right)\left| {_{\varepsilon = 0}} \right. \end{array} $

(14)

By simplifying Eqs. (13) and (14), we will obtain

$ \Delta \mathit{\boldsymbol{z}}\cdot \overset\frown{\mathit{\Xi }\left( s,\mathit{\boldsymbol{z}} \right)}=\frac{-\varepsilon \Delta \mathit{\boldsymbol{\gamma }}\left( s \right)\cdot \overset\frown{\left( \mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}} \right)}}{\left| \mathit{\Xi }\left( s,\mathit{\boldsymbol{z}} \right) \right|} $

(15)

$ \begin{align} &\Delta \mathit{\boldsymbol{z}}\cdot \overset\frown{\mathit{\dot{\Xi }}\left( s,\mathit{\boldsymbol{z}} \right)}=\frac{1}{\left| \mathit{\dot{\Xi }}\left( s,\mathit{\boldsymbol{z}} \right) \right|}\left[ -\varepsilon \Delta \dot{\gamma }\left( s \right)\cdot \right. \\ &\left. \overset\frown{\left( \mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}} \right)}-\varepsilon {\mathit{\boldsymbol{\dot \gamma }}}\left( s \right)\cdot \frac{\Delta {{\mathit{\boldsymbol{\gamma }}}^{\bot }}\left( \mathit{\boldsymbol{Z}},s \right)}{\left| \mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}} \right|} \right] \\ \end{align} $

(16)

where

$ \mathit{\Xi }\left( s,\mathit{\boldsymbol{z}} \right)={{\mathit{\boldsymbol{D}}}_{z}}\overset\frown{\mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}}} $

(17)

$ \mathit{\dot \Xi }\left( {s,\mathit{\boldsymbol{z}}} \right) = {\mathit{\boldsymbol{D}}_z}\frac{{{{\mathit{\boldsymbol{\dot \gamma }}}^ \bot }\left( {\mathit{\boldsymbol{Z}},s} \right)}}{{\left| {\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}} \right|}} $

(18)

and

$ {\mathit{\boldsymbol{D}}_z} = \left( {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0 \end{array}} \right) $

(19)

where D_z denotes the projection matrix that projects a 3D vector onto the image plane. Thus, Ξ denotes the projection of radar look-direction onto the image plane. We refer to $\Delta \mathit{\boldsymbol{z}} \cdot \overset\frown{\mathit{\Xi }\left( {s, \mathit{\boldsymbol{z}}} \right)}$ as the radial positioning error and denote it by Δz^r, z₁Oz₂.

In Eq.(18), ${{\mathit{\boldsymbol{\dot \gamma }}}^ \bot }\left( {\mathit{\boldsymbol{Z}}, s} \right)$ denotes the component of the velocity perpendicular to the radar look-direction, which is given by

$ \begin{align} &{{{{\mathit{\boldsymbol{\dot \gamma }}}}}^{\bot }}\left( \mathit{\boldsymbol{Z}},s \right)={\mathit{\boldsymbol{\dot \gamma }}}\left( s \right)-\left[ {\mathit{\boldsymbol{\dot \gamma }}}\left( s \right)\cdot \overset\frown{\left( \mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}} \right)} \right]\cdot \\ &\ \ \ \ \ \ \overset\frown{\left( \mathit{\boldsymbol{\gamma }}\left( s \right)+\mathit{\boldsymbol{Z}}-\mathit{\boldsymbol{p}} \right)} \\ \end{align} $

(20)

We refer to the direction of ${{\mathit{\boldsymbol{\dot \gamma }}}^ \bot }\left( s \right)$ as the transverse radar look-direction^[23]. ${\mathit{\dot \Xi }}$ defined by Eq. (18) denotes the projection of the transverse radar look-direction onto the image plane. Thus, $\Delta \mathit{\boldsymbol{z}} \cdot \overset\frown{\mathit{\dot \Xi }\left( {s, \mathit{\boldsymbol{z}}} \right)}$ is refered to as the transverse positioning error and can be denoted by Δz^t. Accordingly, Δγ^⊥(s) in Eq. (16) denotes the component of the trajectory error perpendicular to the radar look-direction $\overset\frown{\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}}.\Delta \mathit{\boldsymbol{\dot \gamma }}\left( s \right)$ is the derivative of Δγ(s) with respect to s, denoting the velocity error of the space target. The details of the derivations of Eqs. (15) and (16) are presented in Ref.[26].

From Eqs. (15) and (16), we have the following findings:

(1) The radial positioning error Δz^r is determined by the radial component of the trajectory errors. The transverse positioning error Δz^t depends on the radial component of the velocity error, the transverse component of the trajectory error, the velocity of the space target and the range between the radar and target. Note that we refer to the radial component as the projection of a vector onto the radar look-direction and the transverse component as the projection of a vector onto the plane perpendicular to the radar look-direction.

(2) The first term in the square bracket of Eq. (16) makes more contribution to the transverse positioning error than the second term due to the large range term in the denominator of the second term. This implies that the value of Δz^t is more affected and determined by the radial velocity error of the space target.

(3) The dependency of Δz^r and Δz^t on s gives the explanation of smearing in the reconstructed image. In the case of wide-aperture imaging where the change of the radar look-direction becomes significant during the CPI and the case where the trajectory errors are time-varying, the positioning errors accordingly change with time leading to the defocusing of the targets.

The positioning errors induced by typical trajectory errors including constant, linear, quadratic and sinusoidal trajectory errors are analyzed. The results are listed in Table 1. The angles are defined as follows.

ψ:Grazing angle of radar look direction (the angle between $\overset\frown{ {\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}} }$ and its projection on the image plane).

ψ_⊥:Grazing angle of radar transverse look direction (the angel between ${{\mathit{\boldsymbol{\dot \gamma }}}^ \bot }\left( {\mathit{\boldsymbol{Z}}, s} \right)$ and its projection on the image plane).

θ:The angel between $\mathit{\boldsymbol{\dot \gamma }}\left( s \right)$ and Δγ^⊥(s).

φ:Viewing angle (the angle between $\mathit{\boldsymbol{\dot \gamma }}\left( s \right)$ and $\overset\frown{\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}}$).

Fig. 2 illustrates the related vectors and angles. The gray plane denotes the image plane denoted by z₁Oz₂. The coordinates Oz₁z₂z₃ coincides with the local coordinates O′x₁′x₂′x₃′ at the starting time of the synthetic aperture. The dark red and green arrows denote the directions of Ξ and ${\mathit{\dot \Xi }}$, respectively.

Note that in Table 1, the superscript r and t denote the radial and transverse components of a vector, respectively. For example, Δγ₀^r=Δγ₀·$\overset\frown{\left( {\mathit{\boldsymbol{\gamma }}\left( s \right) + \mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{p}}} \right)}$ and Δγ₀^t = |Δγ₀^⊥|. A_e sinωs in the last row denotes a 3D sinusoidal function, i.e., A_esinωs=[A_esin¹ω₁s A_esin²ω₂s A_esin³ω₃s].

Nextly the positioning errors for each case in Table 1 are analyzed in detail.

(1) Constant trajectory error Δγ₀ The second row of Table 1 gives positioning errors induced by the constant trajectory error. The radial error Δz^r is dependent on the radial component of Δγ₀ and the transverse error Δz^t is dependent on the transverse component of Δγ₀. For a relatively short synthetic aperture, the radar look-direction does not change much. Δz^r and Δz^t are approximately constant with only position shift in the reconstructed image rather than defocusing. However, for a large synthetic aperture, the radar look-direction undergoes relatively large change leading to large changes of angles ψ, ψ_⊥ and θ. Thus, Δz^r and Δz^t may vary significantly with slow time, which leads to the smearing of the reconstructed target even if the trajectory error is constant.

(2) Linear trajectory error v_es The third row of Table 1 shows the positioning errors due to the linear trajectory error v_es where v_e denotes the velocity error. For a short synthetic aperture where the radar look-direction and related angles are almost constant, Δz^r and Δz^t vary linearly with slow time with the radial velocity error v_e^r and transverse velocity error v_e^t being the slopes, respectively. As shown by the first term in the third row and the third column, Δz^t is also dependent on v_e^r. This v_e^r dependent term is much larger than the second linear term in Δz^t and also larger than the radial positioning error Δz^r due to the multiplication of the range term |γ(s)+Z-p|.

(3) Quadratic trajectory error a_es²/2 Ignoring the changes of the radar look-direction and the related angles in a relatively short imaging time, Δz^r and Δz^t are both quadratically dependent on the radial and transverse acceleration error, a_e^r and a_e^t, respectively. The transverse positioning error Δz^t is much larger than the radial component Δz^r due to the range dependent term, as listed in the fourth row and the third column.

(4) Sinusoidal trajectory error A_esinωs As show in the fifth row of Table 1, Δz^r and Δz^t oscillate with time in the case of sinusoidal trajectory errors. Similar to the cases (2) and (3), Δz^t is much larger than radial positioning error and mainly depends on the radial component of the sinusoidal trajectory error. Thus, the target gets smeared in the transverse radar look-direction if there is a radial component of the sinusoidal trajectory error (A_eωcosωs)^r.

3 Demonstrations 3.1 Configuration and parameters

In this section, simulations are conducted to demonstrate our analysis. The satellite tool kit (STK) is used to obtain the real trajectory data of in-orbit space targets, which is required by the raw data generation. The transmitted waveform is a linear frequency modulated signal. The parameters of the orbit, waveform and system configuration are listed in Table 2.

The total passing time of the orbit used in the simulations is 860.43 s. We select a part of the trajectory in the simulations, which corresponds a CPI of 3.75 s having 1 500 pulses. The range between the target and the radar at the starting time is 680.85 km.

In the current configuration, the synthetic aperture length is much smaller than the radar-to-target range. Thus, the radar look-direction and transverse look-direction do not change significantly during the coherent integration time.

The image plane is selected as the x₁′O₁x₂′ plane at the starting time of the synthetic aperture. The size of the scene considered in the image reconstruction is 20 m²×20 m² and is discretized into 300 pixels×300 pixels with a pixel size of 0.067 m×0.067 m approximately. A point target is assumed in the simulations. The target is located at [0 0 0] in the local coordinate system O₁x₁′x₂′x₃′ with unit reflectivity, corresponding to the [150 150] th pixel in the image.

The position errors in the reconstructed images are measured and compared with the analytic ones calculated using Eqs. (15), (16). The sub-aperture processing is used in measuring the position errors. The data is divided into several patches with or without overlapping. The images are reconstructed using each patch of data, which refer to as the sub-images. The position errors in each sub-image are measured and then interpolated to obtain the positioning errors over the aperture.

3.2 Results

Fig. 3 shows the reconstructed image of the point target with correct trajectory information. The target is well-focused at [-0.085 m, -0.085 m], which basically coincides with the true position of the point [0 m, 0 m]. The trails intersecting at the target position indicate the directions of Ξ and ${\mathit{\dot \Xi }}$, i.e. the projections of the radar look-direction and transverse look-direction on the image plane, as indicated by the white arrows in Fig. 3.

3.2.1 Constant trajectory errors

We assumed a constant trajectory error, Δγ₀=[2 m, 2 m, 0 m] in the image formation. Fig. 4 shows the reconstructed result using the erroneous trajectory. The yellow circle denotes the true position of the target.

It can be seen that the target is well reconstructed, however at an erroneous position of [-2.01 m, -2.01 m]. The positioning error is a constant position shift, which is constant with our analysis in Section 2.

Fig. 5 presents the analytic and measured radial and transverse positioning error. It can be seen that the measured and analytic values match very well. Note that the discretization of the image and the imaging resolution determined by the configuration and waveform parameters affect the accuracy of the measured position errors, which leads to a discrepancy between the measured and analytic values.

By comparing the radial and transverse components of the positioning errors, we can see that Δz^t is smaller than Δz^r. This is also indicated in the image shown in Figure 4, in which the position shift in the radial direction is dominant. The curves in Fig. 5 show that Δz^r is nearly constant during the integration time, while Δz^t undergoes a larger change with the slow time than Δz^r. This may be due to the larger change of angles θ and φ, as shown by the term in the second row and the third column. However, since Δz^t varies in a small range, there is no visible smearing in the reconstructed image, as shown in Fig. 4.

3.2.2 Liner trajectory error

We considered a linear trajectory error, Δγ(s)=v_es, where v_e=[0.001 m/s, 0.001 m/s, 0 m/s] in the image formation. Fig. 6 shows the reconstructed image under the assumption of linear trajectory error. It can be seen that the target is reconstructed at a position shifting mainly in the transverse look-direction as compared to the target true position. This is consistent with our analysis in Section 2. The radial component of the velocity error induces a much larger transverse positioning error than the radial positioning error.

Fig. 7 shows the analytic and measured radial and transverse positioning error. It can be seen that the measured values match with the analytic ones well. Both the radial and transverse error change linearly with time. Δz^t is larger than Δz^r, as indicated by the red and black lines in Fig. 7. This is consistent with the image shown in Fig. 6 and the theoretical result in Section 2.

3.2.3 Quadratic trajectory errors

We assume Δγ(s)= $\frac{1}{2}$a_es² where a_e=[0.000 2 m/s², 0.000 2 m/s², 0 m/s²]. Fig. 8 shows the reconstructed image. Note that in this case the size of the scene is increased to be 40 m²×40 m² to show the complete signature of the target in the reconstructed image.

It is seen that the target is defocused with a streaking artifact. As compared to the correct reconstruction in Fig. 3, the smearing occurs mainly in the transverse look-direction.

Fig. 9 presents the analytic and measured radial and transverse positioning errors in the case of a quadratic trajectory error. The measured positioning errors indicated by the dashed lines match with the analytic ones indicated by the solid lines. The radial positioning error approaches zero, while the transverse positioning error is much larger than the radial component and undergoes a large change during the integration time. This in fact leads to the smeared target signature, as shown in Fig. 8. It can be also seen from Fig. 8 that the transverse positioning error changes quadratically with the time. This is consistent with our theoretical analysis in Section 2.

3.2.4 Sinusoidal trajectory errors

We assume a sinusoidal error, Δγ(s)=[0.01sin(πs/16) m, 0.01cos(πs/16) m, 0 m] in the trajectory of the target. Fig. 10 shows the reconstructed image under the assumption of this sinusoidal trajectory error.

From Fig. 10, it is seen that the target is not focused at its true position. The target energy spreads severely in the transverse look-direction. Comparing the target signature with that of in Fig. 8, i.e. the image reconstructed in the case of a quadratic trajectory error, we see the smearing occurs mainly in the transverse look-direction similarly, but in the case of a sinusoidal trajectory error, the smearing is centered around the true position of the target rather than the one-side smearing in Fig. 8 in the case of a quadratic trajectory error.

Fig. 11 shows the analytic and measured radial and transverse components of the positioning error. The consistency of the measured and analytic values verifies the correctness of our formulas of the positioning errors.

It can be seen from Fig. 11 that the radial positioning error is much close to zero, but the transverse positioning error is large and varies in a sinusoidal law. This explains the smeared target signature in Fig. 10, which is due to the oscillation of the transverse positioning error around zero.

3.3 Remarks

From the demonstration above, we see that our analysis provides a qualitative relationship between the smeared target signatures in the reconstructed images and the trajectory errors. The deterministic trajectory errors in the demonstration including constant, linear, quadratic and sinusoidal errors are mainly considered. The random trajectory errors are wideband noise-like errors. Given a specified data set, the realization of a random trajectory error can be regarded as a combination of a number of sinusoidal signals, which leads to a superposition of the defocusing effects as shown in Fig. 10.

Our results given by Eqs. (15), (16) indicate that a straightforward inversion from 2D positioning errors (a common 2D imaging is assumed.) to six independent unknowns (three components of the transmitter trajectory errors and three for the receiver) is not solvable. However, under certain scenarios, the relationship between the positioning errors and the trajectory errors may be simplified, e.g., in the monostatic configuration. In the monostatic case, the trajectory error can be expressed in terms of a radial component and a transverse component. Thus, the inversion may be solvable. Since the true position of the target is not known a prior, the position errors cannot be determined with only the measurements of the erroneous positions. We may divide the data into several sub-apertures and measure the change of the erroneous positions through the sub-apertures. The differential position errors in horizontal and vertical directions are then used to retrieve the radial and transverse trajectory errors. We are developing autofocusing methods on this track.

4 Conclusions

We presented an approach to analyze the reconstruction errors in ISAR imaging of space targets due to errors in the trajectory information. We derived the explicit formulas of the positioning errors in the radar look-direction and transverse look-direction using microlocal analysis and obtained a quantitative relationship between the trajectory errors of the space targets and positioning errors in the reconstructed images. The constant trajectory error leads to constant position shifts in the reconstructed images for a relatively short imaging time. The linear trajectory error leads to a much large position shift in the transverse look-direction for a relatively short imaging time. For wide-aperture imaging or long imaging time, the dependency of the positioning errors on slow time cannot be ignored and may lead to defocusing of the target rather than only a constant position shift. The quadratic trajectory error leads to obvious smearing in radar traverse look-direction and small radial positioning error. The sinusoidal trajectory error leads to smearing of the target in radar transverse look-direction centered at the target true position. Numerical simulation results demonstrated our theoretical analysis. The autofocusing research may benefit from our results. To develop the autofocusing method exploiting the obtained quantitative relationship between the trajectory error and reconstructed position error is one of our undergoing works.

Acknowledgements

This work is supported by the National Natural Science Foundation of China(No.61871217) and the Foundation of Graduate Innovation Center in Nanjing University of Aeronautics and Astronautics (No. kfjj20170404), China.