Range cell migration correction analysis of one-step and two-step motion compensation for millimeter-wave airborne SAR imaging

Abstract

Conventional two-step motion compensation (MOCO) method is widely adapted for airborne synthetic aperture radar (SAR) imaging due to its conciseness combining with the SAR focusing procedure. For two-step MOCO, range-independent compensation is processed before range cell migration correction (RCMC), and the range-dependent phase correction is implemented after RCMC. However, the accuracy of RCMC would be seriously decreased by the residual range-dependent phase, which is a fatal problem for high-resolution millimeter-wave (MMW) SAR imaging. In this paper, an extensive investigation on the RCMC accuracy is provided by establishing an accurate formula expression between the range cell migration error and the residual range-dependent phase error. One-step MOCO-based SAR imaging algorithm is investigated by compensating the range-dependent motion error before RCMC, so the presence of range cell migration error would be significantly suppressed. What is more, a modified azimuth match filtering (AMF) function is given by precise topography and aperture-dependent motion compensation (PTA) method to overcome the residual azimuth-dependent phase error in the azimuth compression stage. Both simulated and real-measured MMW SAR data sets are used to validate the analysis for high-resolution airborne SAR imaging.

1 Introduction

Motion compensation (MOCO) [1–4] is a crucial operating step for airborne synthetic aperture radar (SAR) [5–7] imaging because the non-ideal movement deviates the radar platform from the predetermined flight trajectory. More importantly, for high-resolution millimeter-wave (MMW) [8] SAR systems, imaging performance is more sensitive to the envelope and phase of motion errors, so a precise MOCO is essential with the availability of high-precision inertial navigation system (INS) measurement. An efficient two-step MOCO algorithm [4] is proposed by Moreira and Huang, combining with chirp scaling algorithm (CSA) [9, 10] for airborne SAR imaging. This method is divided into range-independent compensation step and range-dependent compensation step, while the first step is processed to the range compressed data and the second step is processed after range cell migration correction (RCMC). The problem of conventional two-step MOCO processing is also obvious. The residual range-dependent motion error remained after the first step seriously decreases the accuracy of RCMC in two-dimensional wavenumber domain, which presents as a curving range cell migration (RCM) range profile in two-dimensional time domain, and destroys the performance of azimuth pulse compression. Reference [11] describes the problem above for Omega-k algorithm [12, 13]. In their work, a one-step MOCO method is proposed, but the detailed analysis of the RCMC error is not given. Besides, the original MOCO methods only take range-dependent motion error into account, and the residual azimuth-dependent motion error should also be considered, which is non-ignorable for high-resolution airborne MMW SAR imaging with wide swath. The existed azimuth-dependent MOCO algorithms [14–18] could precisely compensate the azimuth-dependent motion error and modify the azimuth matched filtering function in order to eliminate the influence of azimuth-variant motion error.

Based on the signal model in [3], we investigate the cause of RCMC error as well as its definite expression deduction for MMW SAR system in this paper. A background assumption is confirmed that the trajectory information is accurately recorded by the INS and the whole motion compensation procedure is processed without autofocus step. The one-step MOCO-based imaging algorithm is investigated, which compensates range-dependent motion error before RCMC in order to suppress the residual envelope and phase error of RCM range profile. Moreover, according to the analytical expression of the residual spatial variant error, an accurate azimuth match filtering (AMF) function is modified by precise topography- and aperture-dependent motion compensation (PTA) [17], which compensates the residual azimuth-dependent motion errors remained by RCMC. In this paper, the conventional two-step MOCO-based imaging algorithm is introduced for comparison, theoretical analysis to the superiority of one-step MOCO would be adequately verified by simulated and real-measured data experiments.

The whole paper is organized as follows: Section 2 gives the signal and geometry model of the SAR imaging, RCMC accuracy respect to the residual range-dependent error is analyzed as well, and flowcharts of one-step and two-step MOCO-based SAR imaging algorithms are then given for comparison. In Section 3, RCMC error comparison between one-step and two-step MOCO is discussed, and computational burden of both methods is also analyzed in detail. In Section 4, extensive experimental results are given with both simulated and real measured MMW airborne SAR data. Conclusions are given in the last section.

2 One-step MOCO-based SAR imaging algorithm

2.1 SAR imaging and RCMC error analysis

The squinted SAR geometry is given in Fig. 1. In the model above, the SAR sensor travels along a straight-line flight path with a constant velocity v in ideal conditions, and the synthetic aperture length is L. Symbol O stands for the original point of SAR sensor. The ideal linear trajectory is defined as X-axis corresponding to the azimuth direction, point A denotes the ideal position of platform. During the data acquisition, the radar beam illuminates at squint angle φ, symbol C denotes the scene center, and r represents the range from C to radar at squint angle φ. Symbol P stands for the target located on the scene center line O ^′ C, which is parallel to the trajectory. The distance between target P and scene center C is given by x, and the distance between O ^′ and C is given by x ₀. The instantaneous range from target P to the radar in the conical coordinate system [19] is given by:

$${} {R_{n}}\left(X \right) = \sqrt {{{\left({r\cos \varphi} \right)}^{2}} + {{\left({X - x - r\sin \varphi} \right)}^{2}}},\;\; - \frac{L}{2} \le X \le \frac{L}{2}\ $$

(1)

Fig. 1

Geometric model of squint SAR imaging

where X=v t _m denotes the instantaneous azimuth coordinate of antenna phase center at slow time t _m . The expression of echo signal from target P is given by:

$${} \begin{aligned} S\left({\tau,X} \right) &= {\varepsilon_{p}} \cdot \text{rect}\left({\frac{{\tau - \Delta t}}{{{T_{p}}}}} \right) \cdot \text{rect}\left({\frac{{X - {x_{0}} - x}}{L}} \right) \\ &\quad\times\exp \left[ {j2\pi \left({ - {f_{c}}\Delta t + \frac{{\alpha {{\left({\tau - \Delta t} \right)}^{2}}}}{2}} \right)} \right] \end{aligned} $$

(2)

where α is the chirp rate, T _p is the pulse duration width, f _c is the center frequency, τ denotes the range fast-time, Δ t=2R _n (X)/c stands for the round-way propagation time of electromagnetic wave between target P and radar, c is the speed of light. ε _p corresponds to the complex-valued scattering amplitude of the point target, symbol rect $\left (\cdot \right)$ denotes the rectangular window function. According to [7], the expression of range compressed signal is given by:

$${} {\begin{aligned} S\left({\tau,X} \right) &= {\varepsilon_{p}} \cdot {\text{sinc}} \left[ {{T_{p}}\alpha \left({\tau - \frac{{2{R_{n}}\left(X \right)}}{c}} \right)} \right] \cdot {\text{rect}} \left({\frac{{X - {x_{0}} - x}}{L}} \right) \\ &\quad\times \exp \left[ { - j\frac{{4\pi }}{\lambda }{R_{n}}\left(X \right)} \right] \end{aligned}} $$

(3)

where symbol sinc $\left (\cdot \right)$ is expressed as sinc $\left (a \right) = {{\sin \left ({\pi a} \right)} / {\pi a}}$. However, the actual motion of platform is not ideal, so the range compressed signal in (3) is blurred with motion error Δ R=A ^′ P−A P. Motion compensation is then processed to the blurred range compressed signal in order to remove the envelope and phase of motion errors, so that most of the errors would be compensated. Due to our early works in [3], the signal model as well as the residual phase after deramp processing are calculated for a refined phase gradient autofocus. Based on the former accumulation, in this paper, we try to investigate the mathematical explanation of RMC error with respect to the residual motion errors remained by the range-independent MOCO. Order Δ R _E to represent the residual motion error remained by the range-independent MOCO before RCMC, which would induce serious envelope and phase error to RCMC, and decrease the azimuth pulse compression accuracy. The envelope part of Δ R _E is usually limited within a range bin in actual processing, so we ignore the small envelope of Δ R _E . Applying the range Fourier transform (FT) with respect to τ, the signal is transferred to the range wavenumber domain as shown in (4). The constant and envelope terms of signal expression could be omitted for analysis without adverse effect.

$$\begin{array}{@{}rcl@{}}{} S\left({\Delta {K_{r}},X} \right) &=& \int {S\left({\tau,X} \right)\exp \left({ - j\Delta {K_{r}}\frac{{\tau c}}{2}} \right)} \cdot d\frac{{\tau c}}{2} \notag\\ &=& \exp \left\{{\vphantom{\left[{\sqrt {{{\left({r\cos \varphi} \right)}^{2}} + {{\left({X - x - r\sin \varphi} \right)}^{2}}} + \Delta {R_{E}}\left(X \right)} \right]}}- j{K_{r}} \cdot \right.\notag\\ &&\left.\left[\!{\sqrt {{{\left({r\cos \varphi} \right)}^{2}} \,+\, {{\left({X \,-\, x - r\sin \varphi} \right)}^{2}}} \,+\, \Delta {R_{E}}\left(X \right)} \!\right] \right\} \end{array} $$

(4)

where K _r represents the range wavenumber spectrum with K _r =K _rc +Δ K _r and K _rc =4π f _c /c is the range wavenumber center with Δ K _r ∈[−2π α T _p /c,2π α T _p /c]. Then, applying the azimuth FT to (4) with respect to X, the impact of Δ R _E to the stationary phase point could be ignored for simplifying the analysis formula. The expression of signal in two-dimensional wavenumber domain is given by [3]:

$$\begin{array}{@{}rcl@{}} S\left({\Delta {K_{r}},{K_{x}}} \right) \!\!\!\!&&= \exp \left\{ { - j\left[ \!{\left(\! {\sqrt {{K_{r}^{2}} - {K_{x}}^{2}} \cos \varphi + {K_{x}}\sin \varphi}\! \right) \!\cdot r + {K_{x}} \cdot x}\! \right]}\! \right\}\notag\\ &&\quad\times \exp \left[ { - {K_{r}} \cdot \Delta {R_{E}}\left({{X^{*}}} \right)} \right] \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} &&\qquad{X^{*}} = - \frac{{r\cos \varphi \cdot {K_{x}}}}{{\sqrt {{K_{r}^{2}} - {K_{x}}^{2}} }} + r\sin \varphi + x \end{array} $$

(6)

where K _x represents the azimuth wavenumber spectrum and X ^∗ denotes the ideal stationary phase point. The first phase term of (5) is expressed by a Taylor expansion with respect to Δ K _r =K _r −K _rc around Δ K _r =0, so (5) is expressed as:

$${} \begin{array}{l} S\left({\Delta {K_{r}},{K_{x}}} \right) \approx \exp \left[ { - j\left({\sqrt {K_{rc}^{2} - {K_{x}}^{2}} r\cos \varphi + {K_{x}}\left({r\sin \varphi + x} \right)} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left({ - j\frac{{{K_{rc}}r\cos \varphi }}{{\sqrt {K_{rc}^{2} - {K_{x}^{2}}} }}\Delta {K_{r}}} \right).\exp\left[ {j\frac{{{K_{x}^{2}}r\cos \varphi }}{{2{{\left({K_{rc}^{2} - {K_{x}^{2}}} \right)}^{\frac{3}{2}}}}}\Delta {K_{r}^{2}}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[{ - {K_{r}} \cdot \Delta {R_{E}}\left({{X^{*}}} \right)}\right] \end{array} $$

(7)

where the second term of (7) represents the range cell migration and the third term represents the second-order coupling phase. One may multiply both sides of Eq. (7) by H _RCMC(K _x ,K _r ) and H _SORC(K _x ,K _r ) to perform the RCMC and second-order range compression (SORC).

$$ {H_{\text{RCMC}}}\left({{K_{x}},{K_{r}}} \right) = \exp \left[{j\Delta {K_{r}}\left({\frac{{{K_{rc}}r\cos \varphi }}{{\sqrt {K_{rc}^{2} - {K_{x}^{2}}} }} - r} \right)} \right] $$

(8)

$$ {H_{\text{SORC}}}\left({{K_{x}},{K_{r}}} \right) = \exp \left[ { - j\frac{{{K_{x}^{2}}r\cos \varphi }}{{2{{\left({K_{rc}^{2} - {K_{x}^{2}}} \right)}^{\frac{3}{2}}}}}\Delta {K_{r}^{2}}} \right] $$

(9)

The expression of signal in two-dimensional wavenumber domain processed by RCMC and SORC is given as follows:

$${} \begin{aligned} S\left({\Delta {K_{r}},{K_{x}}} \right) &= \exp \left\{{\vphantom{r\cos \varphi \sqrt {K_{rc}^{2} - {K_{x}^{2}}}}}- j\left[{\vphantom{r\cos \varphi \sqrt {K_{rc}^{2} - {K_{x}^{2}}}}} {K_{x}}\left({x + r\sin \varphi} \right) + \Delta {K_{r}}r\right.\right.\\ &\left.\left.~\qquad\quad\quad+ r\cos \varphi \sqrt {K_{rc}^{2} - {K_{x}^{2}}} \right] \right\}\\ &\quad\times \exp \left[ { - j{K_{r}} \cdot \Delta {R_{E}}\left({{X^{*}}} \right)} \right] \end{aligned} $$

(10)

where the second term of (10) is shown in (11), which is considered the residual motion error phase in two-dimension wavenumber domain, decreasing the accuracy of RCMC seriously.

$$ \Phi \left({\Delta {K_{r}},{K_{x}}} \right) = \exp \left[ { - j{K_{r}}\Delta {R_{E}}\left({{X^{*}}} \right)} \right] $$

(11)

Then, the relationship between Φ(Δ K _r ,K _x ) and RCMC error would be theoretically analyzed. It could be observed in (6) that the residual motion error Δ R _E (X ^∗) is a function of range wavenumber spectrum Δ K _r , so we may expand the second phase term in (10) by a Taylor expansion with respect to Δ K _r around Δ K _r =0, which is expressed as follows, remaining the constant term and first-order term:

$$ \Delta {R_{E}}\left({{X^{*}}} \right) \approx {\left. {\Delta {R_{E}}\left({{X^{*}}} \right)} \right|_{\Delta {K_{r}} = 0}} + {\left. {\frac{{\partial \Delta {R_{E}}\left({{X^{*}}} \right)}}{{\partial \Delta {K_{r}}}}} \right|_{\Delta {K_{r}} = 0}}\Delta {K_{r}} $$

(12)

where

$$ \begin{array}{l} \frac{{\partial \Delta {R_{E}}\left({{X^{*}}} \right)}}{{\partial \Delta {K_{r}}}} = \frac{{\partial {X^{*}}}}{{\partial \Delta {K_{r}}}} \cdot \frac{{\partial \Delta {R_{E}}\left({{X^{*}}} \right)}}{{\partial {X^{*}}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\, = \frac{{{K_{x}} \cdot \left({{K_{rc}} + \Delta {K_{r}}} \right)r\cos \varphi }}{{{{\left[ {{{\left({{K_{rc}} + \Delta {K_{r}}} \right)}^{2}} - {K_{x}^{2}}} \right]}^{\frac{3}{2}}}}} \cdot \frac{{\partial \Delta {R_{E}}\left({{X^{*}}} \right)}}{{\partial {X^{*}}}} \end{array} $$

(13)

Ignoring the impact to the stationary phase point made by Δ R _E (X ^∗), we apply azimuth inverse Fourier transform (IFT) to (10), and we have:

$${} {\begin{aligned} S\left({\Delta {K_{r}},X} \right) &= \int {S\left({\Delta {K_{r}},{K_{x}}} \right) \cdot \exp \left[ {j{K_{x}}X} \right] \cdot d{K_{x}}} \\ &= \exp \left[ { - j{K_{rc}}\left({\sqrt {{{\left({r\cos \varphi} \right)}^{2}} + {{\left({X - x - r\sin \varphi} \right)}^{2}}}} \right)} \right] \\ &\quad\times\exp \left[ {\Delta {R_{E}}\left({K_{x}^{*}} \right) \cdot \left({{K_{rc}} + \Delta {K_{r}}} \right)} \right] \end{aligned}} $$

(14)

where the stationary phase point $K_{x}^{*}$ is given by:

$$ K_{x}^{*} \approx - {K_{rc}}\frac{{X - r\sin \varphi - x}}{{\sqrt {{{\left({\cos \varphi \cdot r} \right)}^{2}} + {{\left({X - x - r\sin \varphi} \right)}^{2}}} }} $$

(15)

Substituting (15) into (12) with replacement of K _x by $K_{x}^{*}$ and ordering X−x=0 for simply analyzing, the residual motion error in azimuth time and range wavenumber domain is given by:

$$ \Delta {R_{E}}\left(X \right) = \Delta {R_{E}} + \frac{{r\sin \varphi }}{{{K_{rc}}{{\cos }^{2}}\varphi }} \cdot \frac{{\partial \Delta {R_{E}}}}{{\partial X}} \cdot \Delta {K_{r}} $$

(16)

Substituting (16) into (14), the expression of signal is translated to the following equation:

$$\begin{array}{@{}rcl@{}} S\left({\Delta {K_{r}},X} \right) &=& \exp \left[ { - j{K_{rc}}\left({\sqrt {{{\left({r\cos \varphi} \right)}^{2}} + {{\left({X - x - r\sin \varphi} \right)}^{2}}}} \right)} \right] \\ &&\times\exp \left[ { - j\left({{\vartheta_{0}} + {\vartheta_{1}}\Delta {K_{r}} + {\vartheta_{2}}\Delta {K_{r}^{2}}} \right)} \right] \end{array} $$

(17)

where the second term of (17) is the RCMC error with respect to the residual motion error Δ R _E . This phase term is expressed as a series of Δ K _r , so the phase and envelope component would be distinguished where

$$ {\vartheta_{0}} = {K_{rc}}\Delta {R_{E}}\ $$

(18)

$$ {\vartheta_{1}} = \Delta {R_{E}} + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }} \cdot \frac{{\partial \Delta {R_{E}}}}{{\partial X}} $$

(19)

$$ {\vartheta_{2}} = \frac{{r\sin \varphi }}{{{K_{rc}}{{\cos }^{2}}\varphi }} \cdot \frac{{\partial \Delta {R_{E}}}}{{\partial X}} $$

(20)

It is worthy to note that 𝜗 ₀ represents the phase component of RCMC error and 𝜗 ₁ denotes the envelope error of RCMC, which seriously destroys the imaging performance. 𝜗 ₂ is the second-order term, which slightly reflects the focusing performance in range, and the effect could be ignored in most cases.

2.2 Flowcharts of one-step and two-step MOCO-based imaging algorithms

As shown in Fig. 2, a flowchart of the one-step MOCO-based imaging algorithm is given in this subsection, while the conventional two-step MOCO-based imaging algorithm is also given for comparison. It is here needed to explain that the flowchart is displayed in such form for the purpose of distinguishing the procedure of the one-step MOCO and two-step MOCO more legible. We introduce PTA for azimuth-variant MOCO in order to eliminate the influence caused by other residual special-dependent errors. In a one-step procedure, bulk envelope compensation is processed in range wavenumber domain by multiplying H _MOCO(Δ K _r ,X) to the range compressed data. The envelope compensation function is expressed as:

$$ {H_{\text{MOCO}}}\left({\Delta {K_{r}},X} \right) = \exp \left[ { - j\Delta {R_{r}}\left({{r_{s}}} \right)\Delta {K_{r}}} \right] $$

(21)

Fig. 2

Flowchart of one-step MOCO-based imaging algorithm compared with conventional two-step MOCO-based imaging algorithm

where Δ R _r denotes the slant range projection of motion error, which is with respect to slant range r and would be expressed in (26). Then, the range-dependent motion error is compensated in range time domain by multiplying H _MOCO(r,X), where

$$ {H_{\text{MOCO}}}\left({r,X} \right) = \exp \left[ { - {jK}_{rc}\Delta {R_{r}}\left({r,X} \right)} \right] $$

(22)

The compensated data is then transformed into two-dimensional wavenumber domain, and SORC and RCMC are processed by multiplying H _SORC(K _x ,K _r ) and H _RCMC(K _x ,K _r ), shown in (8) and (9), respectively. One-step MOCO has compensated most of the range-dependent motion errors, but the residual azimuth-dependent motion error is remained after RCMC, which decreases the accuracy of azimuth pulse compression. Due to the residual azimuth-dependent motion error, 𝜗 ₀ is calculated in (18), and a method of PTA is introduced to compensate 𝜗 ₀ by modifying the AMF function. According to [17], the modified AMF function H _AMF(K _x ,r) is expressed as:

$${} \begin{aligned} {H_{\text{AMF}}}\left({{K_{x}},r} \right) &= \exp \left[ {j{K_{rc}}\Delta {R_{E}}\left({X_{e}^{*}} \right)} \right] \cdot \exp \left[ j{K_{rc}}{R_{n}}\left({r,X_{e}^{*}} \right)\right.\\ &\left.\quad+ j{K_{x}}\left({X_{e}^{*} - x} \right) \right] \end{aligned} $$

(23)

where $X_{e}^{*}$ is calculated by

$$ \frac{{\partial {R_{n}}\left({{X_{e}}^{*}} \right)}}{{\partial {X_{e}}^{*}}} + \frac{{\partial \Delta {R_{E}}\left({{X_{e}}^{*}} \right)}}{{\partial {X_{e}}^{*}}} + \frac{{{K_{x}}}}{{{K_{rc}}}} = 0 $$

(24)

As the one-step MOCO-based imaging algorithm is described in detail, we briefly describe the procedure of the two-step MOCO-based imaging algorithm for comparison. For two-step MOCO, the envelope compensation and the first-phase compensation are processed with respect to the reference slant range r _s before RCMC, and the second range-dependent MOCO step is processed after RCMC, but serious RCM envelope error is remained at this step. In order to make the comparison more equitable, azimuth-dependent motion error is then compensated by method of PTA.

3 Comparative analysis of one-step and two-step MOCO

3.1 RCMC error comparison between one-step and two-step MOCO

In the previous section, we analyze the RCMC error with respect to the residual motion error Δ R _E , but Δ R _E is different for the one-step and two-step MOCO. In this subsection, we focus on calculating Δ R _E and comparing the RCMC error between the one-step and two-step MOCO.

The actual range history at slant range r is expressed as:

$${} \tilde R\left(r,X \right) = \sqrt {{{\left({X - x - r\sin \varphi - \Delta x} \right)}^{2}} + {{\left({{Y_{t}} - \Delta y} \right)}^{2}} + {{\left({H - \Delta z} \right)}^{2}}} $$

(25)

where Δ x, Δ y, and Δ z represent the along track error, cross track error, and height error, respectively. X _t =r sinφ denotes the projection of r on x direction, ${Y_{t}} = \sqrt {{r^{2}}{{\cos }^{2}}\varphi - {H^{2}}}$ denotes the projection of r on y direction, and H is the height of the platform. The azimuth-independent motion error respect to range r is expressed as:

$$ \Delta R_{r}\left(r \right) = \frac{{{X_{t}}}}{r}\Delta x + \frac{{{Y_{t}}}}{r}\Delta y + \frac{H}{r}\Delta z $$

(26)

For the two-step MOCO algorithm, the range-independent component of motion error is compensated in the first step. The residual motion error $\Delta {\tilde R_{E}}$ contains range-dependent component and azimuth-dependent component, which is given by:

$$ \begin{array}{l} \Delta {{\tilde R}_{E}}\left({r,X} \right) = \tilde R\left(r,X \right) - R_{n}\left({{r,X}} \right) - \Delta R_{r}\left(r_{s} \right)\\ = \frac{{{r_{s}} - r}}{{{r_{s}^{2}}}} \cdot \left({ - \frac{{{H^{2}}}}{{\sqrt {{r_{s}^{2}}{{\cos }^{2}}\varphi - {H^{2}}} }}\Delta y + H\Delta z} \right) + \delta {R_{a}}\left(X \right) \end{array} $$

(27)

where r denotes the range bin of target, r _s denotes the slant range from the radar to the beam center, δ R _a is the azimuth-dependent motion error. It is obvious in (27) that the residual motion error is in proportion to the range between target and scene center, so the error is diffused along range direction.

For one-step MOCO, range-dependent motion error is compensated before RCMC. However, a new error δ R _o is introduced for squint SAR, the residual motion error before RCMC is given by:

$${} {\begin{aligned} \Delta {R_{E}}\left(X \right) &= \delta {R_{o}}\left(X \right) + \tilde R\left({r,X} \right) - {R_{n}}\left({r,X} \right) - \Delta {R_{r}}\left(r \right)\\ &= \delta {R_{o}}\left(X \right) + \delta {R_{a}}\left(X \right)\\ &\approx - \frac{{\left({X - x} \right)\sin \varphi }}{{{r_{s}^{2}}}} \cdot \left({ - \frac{{{H^{2}}}}{{\sqrt {{r_{s}^{2}}{{\cos }^{2}}\varphi - {H^{2}}} }}\Delta y + H\Delta z} \right)\\ &\quad+ \delta {R_{a}}\left(X \right) \end{aligned}} $$

(28)

where x denotes the azimuth position of the target. In this section, we mainly focus on providing the comparison of RCMC envelope error between the two-step and one-step MOCO, which is expressed as 𝜗 ₁ in (19). Order

$$ u\left(X \right) = - \frac{{{H^{2}}}}{{\sqrt {{r_{s}^{2}}{{\cos }^{2}}\varphi - {H^{2}}} }}\Delta y + H\Delta z $$

(29)

Substituting (29) into (19), the RCMC envelope error for the two-step MOCO is given by:

$$ \Delta {T_{1}} \approx \frac{{{r_{s}} - r}}{{{r^{2}}}}\left[ {u\left(X \right) + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }} \cdot u'\left(X \right)} \right]\ $$

(30)

Similarly, substituting (30) into (19), the RCMC envelope error for the one-step MOCO is given by:

$$ \Delta {T_{2}} \approx \frac{{\left({X - x} \right)\sin \varphi }}{{{r^{2}}}}\left[ {u\left(X \right) + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }} \cdot u'\left(X \right)} \right]\ $$

(31)

where the azimuth-dependent motion error δ R _a has little impact on the RCMC envelope, so it is not considered in this paper. The ratio between Δ T ₁ and Δ T ₂ is expressed as:

$$ \frac{{\Delta {T_{1}}}}{{\Delta {T_{2}}}} \approx \left| {\frac{{{r_{s}} - r}}{{\left({X - x} \right)\sin \varphi }}} \right| $$

(32)

where the numerator of (32) represents the range difference between target and scene center and the denominator represents the length of range walk. It’s clear that the RCMC envelope error is in proportion to the range difference for two-step MOCO, so the imaging performance would be seriously destroyed at the points far from scene center line. However, in some cases, the effect of residual range-dependent motion error could be neglected if the maximum of RCMC envelope is within 1/4 rang bin, which is shown as:

$$ \max \left| {\Delta {R_{E}} + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }} \cdot \frac{{\partial \Delta {R_{E}}}}{{\partial X}}} \right| \le \frac{1}{4}\Delta r\ $$

(33)

where max|·| denotes the maximum of the absolute value. For the two-step MOCO, the inequality becomes:

$$ \max \left| {\frac{{{r_{s}} - r}}{{{r^{2}}}}\left[ {u\left(X \right) + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }} \cdot u'\left(X \right)} \right]} \right| \le \frac{1}{4}\Delta r $$

(34)

And for the one-step MOCO, the inequality becomes:

$$ \max \left| {\frac{{\left({X - x} \right)\sin \varphi }}{{{r^{2}}}}\left[ {u\left(X \right) + \frac{{r\sin \varphi }}{{{{\cos }^{2}}\varphi }}u'\left(X \right)} \right]} \right| \le \frac{1}{4}\Delta r $$

(35)

It could be found that, for SAR imaging with wide swath, the requirement of inequation (34) is hard to satisfy at points far from the scene center line, while inequation (35) is easier to meet. A simulation result of the RCM error comparison with real-measured trajectory deviations between the two-step and one-step MOCO is shown in Fig. 3. The trajectory deviations are extracted from a real position and orientation system shown in Fig. 3 a. The residual RCMC envelope error is calculated with a squint angle of 5^∘ and a pitch angle of 45^∘, the center closest slant range is 4000 m, and the range difference between target and scene center is 100 m. The corresponding residual RCMC envelope errors are shown in Fig. 3 b. It is clear that the envelope error of the two-step MOCO is up to several meters, while the one-step MOCO significantly overcomes this problem.

Fig. 3

Real measured motion error and corresponding residual RCMC envelope error. a Real-measured motion error. b Corresponding residual RCMC envelope error

3.2 Computational burden analysis

In this subsection, the computational burden of the one-step and two-step MOCO-based SAR imaging algorithms is respectively measured by operating number of fast Fourier transform (FFT), inverse fast Fourier transform (IFFT), and complex multiplication for comparison. As shown in Fig. 2, suppose the azimuth and range point numbers are denoted by N _a and N _r . It needs to be noticed that we analyze the operand by merging adjoining phase terms and without regard of the calculation of PTA operation. For the one-step MOCO-based imaging algorithm, there are 4N _a times N _r -point FFT/IFFT operators, 2N _r times N _a -point FFT/IFFT operators, and 5 times N _r ×N _a -point complex multiplications to obtain a focused imaging. Comparing with the conventional two-step based imaging algorithm, there are 2N _a times N _r -point FFT/IFFT operators, 2N _r times N _a -point FFT/IFFT operators, and 5 times N _r ×N _a -point complex multiplications. It could be found that the one-step MOCO-based imaging algorithm adds 2N _r more times N _a -point FFT/IFFT operators, which slightly increases the computational burden and exchanges for a better focused imagery.

4 Simulated and real data experiments

4.1 Experiments with simulated data

In this subsection, a group of point target simulation experiments are set to demonstrate the effectiveness of the one-step MOCO-based imaging algorithm compared with the two-step MOCO-based imaging algorithm. Simulation parameters and three-point target coordinates are shown in Table 1. It needs to be emphasized that points A and C are far from the scene center line, while point B lies on the scene center line. In this simulation experiment, the performance of the one-step MOCO imaging algorithm is validated by simulated data mixed with motion errors which are extracted from airborne INS, which are shown in Fig. 3 a.

Table 1 Simulation parameters

At first, we present the comparison of RCMC error with respect to the residual range-dependent motion error between the two-step MOCO and one-step MOCO. In order to illustrate the RCMC error explicitly, the range profile of three points are shown in two-dimensional time domain. It could be easily found in Fig. 4 a that the residual RCMs of points A and C are obvious, except for point B because it is lying on the scene center line without range-dependent motion error before RCMC. These RCMC errors are deemed to coincide well with the expression of 𝜗 ₁ in (19). It is also shown in Fig. 4 a that the residual RCMC errors of points A and C processed by two-step MOCO are spanning across several range cells, so the focusing performance would be seriously damaged. The RCM corrected range profiles of points A, B, and C processed by the one-step MOCO are shown in Fig. 4 b for comparison, while all the RCMC errors of points A, B, and C are well removed.

Fig. 4

Comparison of the two-step MOCO and one-step MOCO in RCMC results of points A, B and C. a Two-step MOCO results. b One-step MOCO results

Then, azimuth compression performance of the one-step MOCO-based imaging algorithm is investigated. The two-dimensional images of points A, B, and C processed by the conventional two-step MOCO-based imaging algorithm and one-step MOCO-based imaging algorithm are shown in Fig. 5. Because of the existence of residual RCMC errors, the imaging algorithm based on the conventional two-step MOCO fails to focus in azimuth for points A and C, as shown in Fig. 5 a. While the well-focused imaging results of points A, B, and C processed by the one-step MOCO-based imaging algorithm are shown in Fig. 5 b. Figure 6 shows the comparison of azimuth point spreading response of points A, B, and C. In order to evaluate the focused improvement of the proposed algorithm comparing with the conventional one, three quantitative metrics are introduced to measure the point impulse responses of points A, B, and C, which are shown in Table 2. The quantitative metrics are peak side-lobe ratio, (PSLR), integrated side-lobe ratio (ISLR), and impulse response width (IRW). It is obvious that the focusing degradation of the conventional two-step MOCO-based imaging algorithm is overcame by the one-step MOCO-based imaging algorithm, especially for points A and C, which are far from the scene center line.

Fig. 5

Comparison of the two-step MOCO and one-step MOCO in imaging results of points A, B, and C. a Two-step MOCO results. b One-step MOCO results

Fig. 6

Comparison of the two-step MOCO and one-step MOCO in azimuth pulse response curves of points A, B, and C

Table 2 Focusing performance comparison between two focusing algorithms

4.2 Experiments with real-measured data

In this subsection, a set of comparison experiment is provided based on the processing of measured data recorded by an experimental airborne MMW SAR system. The instantaneous position and motion parameters of platform are measured by a high-accuracy INS equipped on the platform. Detailed radar parameters are shown in Table 3. An imaging result processed by the one-step MOCO-based imaging algorithm is shown in Fig. 7. In order to compare the imaging performance between the conventional two-step MOCO-based imaging algorithm and one-step MOCO-based imaging algorithm more vividly, two typical areas named scene 1 and scene 2 with obvious point-like targets are highlighted by the yellow rectangles in the scene. The picked two scenes are respectively magnified in Fig. 8 a, b for imaging performance comparison, where the two-step MOCO imaging results are lied on the left, while the one-step MOCO imaging results are lied on the right. It is shown that defocusing of targets in the images is distinct for the two-step MOCO-based imaging algorithm, and the one-step MOCO significantly removes the RCMC error, so the targets are well focused. Moreover, in order to check the azimuth point spreading response improvement of the one-step MOCO-based imaging algorithm, two isolated point-like targets named point A and point B are extracted from Fig. 7 by yellow circle for azimuth point spreading response function comparison. The comparison results of point A and point B are shown in Fig. 9 a, b, respectively. A better azimuth pulse response function could be obtained by the one-step MOCO-based imaging algorithm, compared with the two-step. The quantitative analysis results of the azimuth point spreading response functions of Fig. 9 are listed in Table 4. It could be found in the experiments above that, focusing performance is sensitive to the residual RCMC error especially for high-resolution MMW SAR imaging, so conventional two-step MOCO-based imaging algorithm is not suitable for this case. The one-step MOCO removes the range-dependent motion error before RCMC, so that the range profile after RCMC is accurate.

Fig. 7

Imaging result processed by the one-step MOCO-based imaging algorithm

Fig. 8

Comparison of the two-step MOCO-based imaging algorithm and one-step MOCO-based imaging algorithm in local amplification results of scene 1 and scene 2. a Comparison of scene 1 (the two-step MOCO-based imaging result is on the left, the one-step MOCO-based imaging result is on the right). b Comparison of scene 2 (two-step MOCO-based imaging result is on the left, the one-step MOCO-based imaging result is on the right)

Fig. 9

Azimuth point spreading response function comparison of A and B. a Point A. b Point B

Table 3 Experiment parameters

Table 4 Focusing performance comparison of points A and B between two focusing algorithms

In order to verify the calculation analysis in Section 2, we record the calculation time for both two-step and one-step MOCO-based SAR imaging algorithms. The computer platform is installed with Windows10 64-bit operating system, Core i7-4720HQ@2.6GHz CPU, 16-GB memory and Matlab with version of R2015a. A block of 16,384×8192 (range ×azimuth) points SAR data is used for test, the whole data is divided into four sub-blocks in azimuth, and the calculation time of the two-step and one-step MOCO-based imaging algorithms are 855.13s and 909.49s, respectively. With the nearly equal computation complexity compared with the two-step MOCO, the one-step MOCO is applicative for practical MMW SAR imaging application.

5 Conclusions

The conventional two-step MOCO algorithm remains the range-dependent motion error before RCMC, which decreases the accuracy of RCMC in two-dimensional wavenumber domain, inducing serious envelope and phase error to the range profile. In this paper, analytical expressions of these errors are deduced in detail. The one-step MOCO-based imaging algorithm is also investigated to compare with the conventional two-step MOCO-based imaging algorithm, which removes the range-dependent motion error before RCMC, so the RCMC error is significantly suppressed. Simulations and measured MMW data experiments illustrate the outperforms of the one-step MOCO-based SAR imaging algorithm, which verify the analysis in this paper.