A Simple Algorithm for Compensation for Range Cell Migration in a Stripmap SAR

Introduction.  Range Cell Migration (RCM) is a source of image blurring in synthetic aperture radars (SAR). There are two groups of signal processing algorithms used to compensate for migration effects. The first group includes algorithms that recalculate the SAR signal from the "along–track range – slant range" coordinate system into the "along-track range  –  cross-track range"  coordinates using the method of interpolation. The disadvantage of these algorithms is their considerable computational cost. Algorithms of the second group do not rely on interpolation thus being more attractive in terms of practical application.Aim. To synthesize a simple algorithm for compensating for RCM without using interpolation.Materials and methods. The synthesis was performed using a simplified version of the Chirp Scaling algorithm.Results.  A simple algorithm, which presents a modification of the Keystone Transform algorithm, was synthesized. The synthesized algorithm based on Fast Fourier Transforms and the Hadamard matrix products does not require interpolation.Conclusion. A verification of the algorithm quality via mathematical simulation confirmed its high efficiency. Implementation of the algorithm permits the number of computational operations to be reduced. The final radar image  produced using the proposed algorithm is built in the true Cartesian coordinates. The algorithm can be applied for SAR imaging of moving targets. The conducted analysis showed that the algorithm yields  the  image of a moving target provided that the coherent processing interval is sufficiently large. The image lies along a line, which angle of inclination is proportional to the projection of the target relative velocity on the line-of-sight. Estimation of the image parameters permits the target movement parameters to be determined.

Introduction. The Range Cell Migration (RMC) takes place in synthetic aperture radar (SAR) of high range resolution due to the range walk of the compressed signal through range resolution cells. The RCM causes significant image blurring [1,2]. Nowadays there are a number of signal processing algorithms to compensate for its negative influence on the SAR image quality. These algorithms can be subdi-vided into two groups. Algorithms of the first group use the function interpolation. The SAR 2D signal matrix undergoes the transformation from the "along-track range -slant range" coordinate system to the "along-track range -cross-track range" system. The Range -Doppler Algorithm [3][4][5][6], the Wave Number Algorithm [7][8][9][10] and the Keystone Transform (KT) [11,12] belong to this group. When the received signal matrix is large, the nonlinear transformation from the first coordinate system to the second one is a burdensome computational problem that requires significant computing power. Besides, the interpolation can cause the appearance of spurious targets in SAR images. Algorithms of the second group do not implement the interpolation and, in this sense, they are more attractive for implementation. The Chirp Scaling Algorithm [13][14][15][16][17], the Extended Chirp Scaling Algorithm [18] and the Frequency Scaling Algorithm [19,20] are in this group. Quality of algorithms of the second group is not worse, but they are faster due to the Fast Fourier Transforms (FFT) used for their realization.
The KT, which was firstly introduced in [11], occupies a special position among algorithms of the first group because it can be adopted for SAR imaging of moving targets [12]. Extension of the KT for the moving target imaging is based on the assumption that the range curvature is not large during the coherent processing interval and the range walk can be considered only velocity-based (linear). The success of the algorithm gave rise to additional search of efficient signal processing algorithms to compensate for the RMC. In [21] the second order KT algorithm was suggested especially for the moving target imaging. This algorithm permits to compensate the range walk in case of large range curvature. In [22] the second order double-Keystone Transform is presented. This algorithm can correct for both velocity-based and acceleration-based (quadrature) range walk.
Later the KT without interpolation, which is named the ZLZ-algorithm in the paper using the first letters of the authors' names, was proposed in [23]. The Chirp Scaling (CS) developed in [24] is used in this algorithm to avoid the interpolation. The algorithm combines the KT simplicity with a high efficiency of the CS. Although this algorithm was synthesized to compensate for the linear component of the range walk for the moving target, it can be used for SAR imaging of stationary targets too.
In this paper the synthesis and the analysis of a new algorithm to compensate for the RCM in a stripmap SAR are presented. The algorithm is based on a reduced version of the CS to get rid of the interpolation. The algorithm is computationally simpler than the ZLZ-algorithm and has the same image quality.
In case of stationary target imaging the final radar image is built in the "along-track range -cross-track range" coordinate system, thus avoiding geometrical distortions that appear in the image constructed in the "along-track range -slant range" coordinates by the ZLZ-algorithm. Analysis of moving target imaging of the algorithms is performed in the paper using the approach firstly implemented in [25] and based on the theory of distributions (generalized functions). It is proved that the proposed algorithm and the ZLZ-algorithm yield different radar images of a moving target. Particularly the image produced by the proposed algorithm has a noticeable inclination in the "along-track range -cross-range range" coordinate plane and the inclination angle depends to the projection of the target relative velocity on the target line-of-sight.
is the along-track range; is the range wavenumber corresponding to the spectral frequency is the spectrum of the signal complex envelop   at written as a function of k.
Let us suppose that the cross-track range of the point is much larger than the length of the CPI, i. e. vT Then the following equation is true for the slant range: are the values of   Rx and its first two derivatives in the point 0. x  Equation (2) can be rewritten as, Presence of k in the argument of the second exponent in (3) is a result of the RCM. Transition to a new along-track range permits to factorize phasors of   , sS x k depending on the range wavenumber k and the along-track range x.
In the KT-algorithm transition (4) is performed via the function interpolation, and in this way the RCM compensation is realized. Translation (4) is a scale transform. To perform it without interpolation let us use the following identity: 22 2 exp exp exp , where 1 C and 2 C are constants independent on the range and azimuth wavenumbers k and K.
It is easy to prove that the product Introduction of C, which is considered constant in further calculation, is very important because it "consumes" all other constants and slowly varying terms that will appear.
The last phasor in equation (6) After this multiplication   Finally, it is necessary to calculate the Fourier transforms over x and k that yield For a stripmap SAR 2 0 00 0 Substitution of (9) into (8) yields   0 00 2 0 , ,.
x K k y y Ss K y C R

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A flowchart of the synthesized algorithm is shown in Fig. 1, a. In Fig. 1, b a flowchart of the ZLZ-algorithm is depicted and its notation is adapted to the present article. Comparing the algorithms, it is easy to realize that they are almost similar. The difference is the additional multiplications of the 2D spectra  Hence, the SAR image is constructed by this algorithm in the coordinates "along-track range  slant range". This is the difference between the algorithms, since the proposed one constructs the radar image in the "along-track range  cross-track range", which is a true Cartesian coordinate system. The synthesized algorithm has one more advantage: it requires two Hadamard matrix products less. If the received signal is a -matrix, NM  where M and N are the numbers of the range and azimuth resolution cells correspondingly, implementation of the proposed algorithm allows to save 2MN complex multiplications. SAR imaging of moving targets. Let us consider the case when the point is uniformly moving. Suppose that its velocity is ˆx   (3) can be rewritten as, (11) Let us apply the designed algorithm starting from the step, which corresponds to the calculation of the spectrum   ,. SS K k In further computations it is supposed that the stationary phase method (see, e. g. [26]) can be applied the perform necessary integrations.
The Fourier transformation over the variable x in (11) Hx of (7) and performing some simplifications, the following equation can be received: To perform the calculation, it necessary to do integration in (12)   Therefore, in case of a moving target its image is shifted from the true position. The shift depends on the relative projection of the target velocity on the line-of-sight.
Analysis in this section shows that the proposed algorithm produces the well-focused image of a moving target if the CPI is infinitely large. In case of a finite CPI, as it is shown in the next section, the image looks like a crossing of two ridges. One ridge is parallel to the y axis, the other one is inclined to the x axis. The focused image lays along the second ridge. It is possible to show that this ridge approximately lies along the line (14) The line passes the point xy   and the tangent of its inclination angle is tan 0.  Estimation of the moving parameters in this case is more difficult.
Results of the computer simulation. Computer simulation of the synthesized algorithm was executed with the following scenario parameters:  [1], which includes sequential execution of the matched filtering of the received signal at each repetition period, quadratic term compensation and harmonic signal analysis using the Fast Fourier Transform over the "slow" time t. From the figure it follows that signal compression is not effective. The radar image of the source is "smeared" in the along-track and the cross-track ranges. xy as it was indicated above. Thus, the simulation proves the efficiency of the synthesized algorithm in case of stationary targets.
In Fig. 3  Quality of the radar image of the standard SAR algorithm (Fig. 3, a) deteriorates significantly in comparison with the image in Fig. 2, a: it is much more blurred and shifted from the true target position. Qualities of the images of the proposed algorithm (Fig. 3, b) and the ZLZ-algorithm (Fig. 3, c) are much better. They are slightly "smeared" and shifted, as it was predicted in the previous section. It is worth to note that the shape of the target image of the ZLZ-algorithm lays along the dashed magenta line that corresponds to the parabola (15) and the envelope of the image is parallel to the axis x. At the same time the image of the proposed algorithm changes its shape: the ridge that was parallel to the axis x in Fig. 2, b has changed its inclination and lays along the dashed magenta line, which equation is (14). This inclination can be used to estimate the target radial speed although such estimation can be realized via implementation of complicate algorithms using the Wigner-Ville Distribution [28,29], the Fractional Fourier Transform [30,31] or the Radon Transform [32,33]. These algorithm permit to estimate the transverse target speed, as well. Estimating of the image position 00, xy   and parameters  and  it is possible to determine the initial coordinates and the velocity of the target.
Conclusion. The range cell migrations (RCM) is one of the main factors that deteriorates the quality of SAR images. The article synthesizes an RCM compensation algorithm without interpolation. The synthe- In this case the algorithm would produce a focused image of the moving target that is shifted from the target true position and slightly rotated. The shift and the inclination angle of the image depend on the projection of the target relative velocity on the line-of-sight, i. e. the radial speed of the target. Estimation of the image parameters permits to determine the target movement parameters -its true position and velocity.