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

- Guanyong Wang
^{1, 2}, - Lei Zhang
^{1}Email author, - Jun Li
^{2}and - Qingrong Hu
^{2}

**2016**:115

https://doi.org/10.1186/s13634-016-0416-1

© The Author(s) 2016

**Received: **13 May 2016

**Accepted: **28 October 2016

**Published: **10 November 2016

## 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.

## Keywords

## 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

*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:

*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:

*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*=2

*R*

_{ 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:

*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.

*K*

_{ r }represents the range wavenumber spectrum with

*K*

_{ r }=

*K*

_{ rc }+

*Δ*

*K*

_{ r }and

*K*

_{ rc }=4

*π*

*f*

_{ c }/

*c*is the range wavenumber center with

*Δ*

*K*

_{ r }∈[−2

*π*

*α*

*T*

_{ p }/

*c*,2

*π*

*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]:

*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:

*H*

_{RCMC}(

*K*

_{ x },

*K*

_{ r }) and

*H*

_{SORC}(

*K*

_{ x },

*K*

_{ r }) to perform the RCMC and second-order range compression (SORC).

*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:

*Δ*

*R*

_{ E }(

*X*

^{∗}), we apply azimuth inverse Fourier transform (IFT) to (10), and we have:

*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:

*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

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

*H*

_{MOCO}(

*Δ*

*K*

_{ r },

*X*) to the range compressed data. The envelope compensation function is expressed as:

*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*

_{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:

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.

*r*is expressed as:

*Δ*

*x*,

*Δ*

*y*, and

*Δ*

*z*represent the along track error, cross track error, and height error, respectively.

*X*

_{ t }=

*r*sin

*φ*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:

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.

*δ*

*R*

_{ o }is introduced for squint SAR, the residual motion error before RCMC is given by:

*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

*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:

^{∘}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.

### 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 4*N*
_{
a
} times *N*
_{
r
}-point FFT/IFFT operators, 2*N*
_{
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 2*N*
_{
a
} times *N*
_{
r
}-point FFT/IFFT operators, 2*N*
_{
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 2*N*
_{
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

Simulation parameters

Carrier frequency | 35 GHz |

Pulse repetition frequency | 2000 Hz |

Velocity | 70 m/s |

Pulse width | 2 |

Center closest slant range | 4000 m |

Squint angle | 5 |

Grazing angle | 45 |

Range resolution | 0.15 m |

Azimuth resolution | 0.15 m |

Point A coordinate | (0, 400) |

Point B coordinate | (0, 0) |

Point C coordinate | (0,−400) |

*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.

Focusing performance comparison between two focusing algorithms

Target point | Method | PSLR (dB) | ISLR (dB) | IRW (m) |
---|---|---|---|---|

A | One-step MOCO | −13.0416 | −10.4187 | 0.1444 |

Two-step MOCO | −4.1975 | −4.4064 | 0.3631 | |

B | One-step MOCO | −13.3593 | −11.3825 | 0.1531 |

Two-step MOCO | −11.5402 | −11.1442 | 0.1575 | |

C | One-step MOCO | −13.4320 | −11.4704 | 0.1663 |

Two-step MOCO | −4.9471 | −5.0134 | 0.3894 |

### 4.2 Experiments with real-measured data

Experiment parameters

Carrier frequency | 35 GHz |

Pulse repetition frequency | 5000 Hz |

Velocity | 70 m/s |

Pulse width | 20 |

Center closest slant range | 4000 m |

Squint angle | 5 |

Grazing angle | 45 |

Range resolution | 0.15 m |

Azimuth resolution | 0.15 m |

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

Target point | Method | PSLR (dB) | ISLR (dB) | IRW (m) |
---|---|---|---|---|

A | One-step MOCO | −16.5387 | −10.7167 | 0.1575 |

Two-step MOCO | −15.5048 | −8.6454 | 0.2975 | |

B | One-step MOCO | −14.6624 | −10.6820 | 0.1702 |

Two-step MOCO | −0.6995 | 2.3087 | 0.3755 |

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.

## Declarations

### Acknowledgements

The authors thank the anonymous reviewers for their valuable comments to improve the paper quality. This work was supported by the National Natural Science Foundation of China under grant numbers 61301280 and 61301293 and the High-Resolution Earth Observation System Major Special Project Youth Innovation Foundation of China under grant number GFZX04060103.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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