**Jun Wan 1, Yu Zhou 1,\*, Linrang Zhang 1, Zhanye Chen <sup>2</sup> and Hengli Yu <sup>1</sup>**


Received: 16 August 2019; Accepted: 19 September 2019; Published: 22 September 2019

**Abstract:** The synthetic aperture radar (SAR) image of moving targets will defocus due to the unknown motion parameters. For fast-maneuvering targets, the range cell migration (RCM), Doppler frequency migration and Doppler ambiguity are complex problems. As a result, focusing of fast-maneuvering targets is difficult. In this work, an efficient SAR refocusing algorithm is proposed for fast-maneuvering targets. The proposed algorithm mainly contains three steps. Firstly, the RCM is corrected using sequence reversing, matrix complex multiplication and an improved second-order RCM correction function. Secondly, a 1D scaled Fourier transform is introduced to estimate the remaining chirp rate. Thirdly, a matched filter based on the estimated chirp rate is proposed to focus the maneuvering target in the range–azimuth time domain. The proposed method is computationally efficient because it can be implemented by the fast Fourier transform (FFT), inverse FFT and non-uniform FFT. A new deramp function is proposed to further address the serious problem of Doppler ambiguity. A spurious peak recognition procedure is proposed on the basis of the cross-term analysis. Simulated and real data processing results demonstrate the validity of the proposed target focusing algorithm and spurious peak recognition procedure.

**Keywords:** complex Doppler ambiguity; fast-maneuvering target refocusing; non-uniform FFT (NUFFT); 1D scaled Fourier transform (1D SCFT); synthetic aperture radar (SAR)

### **1. Introduction**

Synthetic aperture radar (SAR) can image the scenes of interest during the day and night regardless of weather conditions, which attracts considerable attention worldwide [1–10]. SAR has been widely used in numerous remote sensing applications, such as marine observation, traffic monitoring and antiterrorism. The growing demand for surveillance of moving targets has made imaging these targets a major task for modern SAR systems [11–15]. Nevertheless, the unknown motion parameters between the SAR platform and the moving target result in range cell migration (RCM) and Doppler frequency migration (DFM) [16,17]. These factors lead to the defocused image of moving targets. Thus, the defocusing effects induced by the RCM and DFM should be effectively removed.

Several methods have been presented to remove the RCM with the SAR system. The Hough/Radon transforms [18,19] were utilised to search the trajectory and correct the RCM. However, they suffer from high computational complexity due to the searching of the trajectory. On this basis, the first-order keystone transform (FOKT) [20,21], second-order keystone transform (SOKT) [22,23] and Doppler keystone transform [24] were proposed to remove the corresponding RCM without knowing a priori knowledge of the moving target. Although these transforms avoid searching of the target trajectory, their performance is limited by the effects of DFM and Doppler ambiguity. Methods, such as FOKT-based methods [25,26], stationary phase-based methods [27,28], joint time–frequency analysis-based methods [17,29] and modified SOKT-based methods [30], were presented to deal with these issues. However, these methods ignore the acceleration motion and only consider the moving target with a low-order (i.e., second-order) phase model. The acceleration motion and third-order phase should be further considered for the fast-maneuvering target [16,31,32]. Thus, the aforementioned methods may be inappropriate.

The components of RCM and DFM become increasingly complex due to the acceleration motion and third-order phase. Different from the RCM and DFM for moving targets with a low-phase model, which only includes first-order RCM (FRCM), second-order RCM (SRCM) and linear DFM (LDFM), the third-order RCM (TRCM) and quadratic DFM (QDFM) should be included for fast-maneuvering targets. The first-order phase of the target signal induces the Doppler centre shift; then, the Doppler centre ambiguity emerges due to the limitation of pulse repetition frequency (PRF) for the SAR system [25,27]. The complex azimuth Doppler spectrum induced by Doppler centre shift and DFM may distribute into one, two or multiple PRF bands. When the azimuth Doppler spectrum occupies two or multiple PRF bands, the target spectrum split occurs, which induces the Doppler spectrum ambiguity. The TRCM, QDFM, Doppler centre blur and spectrum ambiguity lead to the difficulty in the focusing of fast-maneuvering targets.

The axis mapping-based coherently integrated cubic phase function (CICPF) method [31] was introduced in consideration of the acceleration motion and third-order phase. However, this method directly applies SOKT to remove the SRCM and ignores the complex Doppler spectrum ambiguity (i.e., Doppler spectrum spanning over two or multiple PRF bands). If the Doppler spectrum is not located entirely on one PRF band, then the target trajectory will split into multiple parts after performing this method. A SOKT-based generalised Hough-high-order ambiguity function (SOKT-GHHAF) method [32] was proposed to focus the maneuvering targets for dealing with the aforementioned issue. This method uses the operation of Doppler centre shifting by PRF/2 to eliminate the effect of Doppler spectrum split. However, if the target spectrum bandwidth is larger than PRF/2, then the operation of Doppler centre shifting by PRF/2 will be invalid, and the effect of Doppler spectrum split will still persist [25]. The parameter searching-based methods [16,33] were introduced without the effect of Doppler spectrum ambiguity. Although these algorithms are effective, they suffer from large computational complexity induced by a brute-force parameter searching procedure.

We present a new computationally efficient algorithm for refocusing of ground fast-maneuvering targets on the basis of the previous works. In this algorithm, the RCM is corrected using sequence reversing, matrix complex multiplication and an improved SRCM correction function in the range frequency and azimuth slow time domain. The Doppler centre shift is removed simultaneously. Then, a 1D scaled Fourier transform (SCFT) with the constant factor ε is used to estimate the chirp rate of the target signal. Thereafter, a matched filter based on the estimated chirp rate is presented to focus the moving target in the range–azimuth time domain. In addition, a new deramp function with the constant factor ϕ is proposed to further deal with the Doppler spectrum ambiguity. Then, the operation of combining the new deramp function and SOKT is introduced to address the mismatch of the improved SRCM correction function. The cross-term interference for multiple targets is analysed, and a spurious peak recognition procedure is proposed. The simulated and real data processing results verify the proposed target focusing algorithm and spurious peak recognition procedure.

The main contributions of this work are listed as follows: (1) the proposed algorithm can achieve a well-focused result in the range–azimuth time domain because the acceleration motion and third-order phase of the fast-maneuvering target are considered; (2) the presented algorithm has low computational complexity given that it can be implemented by the fast Fourier transform (FFT), inverse FFT (IFFT) and non-uniform FFT (NUFFT); (3) two constant factors, namely, ε and ϕ, are introduced to expand the applicability of the proposed algorithm; (4) a new deramp function is introduced to further deal with the complex Doppler ambiguity; and (5) a spurious peak recognition procedure is presented to address the cross-term interference.

The rest of the paper is organised as follows: Section 2 provides the signal model and characteristics. Section 3 describes the proposed algorithm. Section 4 gives specific analysis related to the proposed algorithm. Section 5 presents the simulated and real data processing results. Section 6 gives the discussion of the proposed algorithm. Section 7 provides the final conclusions.

#### **2. Signal Model and Characteristics**

### *2.1. Signal Model*

The motion geometry between the SAR platform under the side-looking strip-map mode and the ground maneuvering target on a slant-rang plane is shown in Figure 1. During the synthetic time *Ta*, the SAR platform flies with constant velocity *v*. The maneuvering target with cross-track velocity *vc*, cross-track acceleration *ac*, along-track velocity *va* and along-track acceleration *aa* moves from point A to point B. *R*<sup>0</sup> and *Rs*(*tn*) denotes the nearest and instantaneous slant ranges between the SAR platform and the maneuvering target. *tn* represents the azimuth slow time.

**Figure 1.** Motion geometry between the synthetic aperture radar platform and the ground maneuvering target on a slant-rang plane.

In accordance with the motion geometry described in Figure 1, *Rs*(*tn*) is expressed as

$$R\_{\delta}(t\_n) = \sqrt{\left(\upsilon t\_n - \upsilon\_a t\_n - \frac{1}{2} a\_a t\_n^2\right)^2 + \left(R\_0 - \upsilon\_c t\_n - \frac{1}{2} a\_c t\_n^2\right)^2}.\tag{1}$$

The instantaneous slant range *Rs*(*tn*) can be expanded on the basis of the Taylor series expansion. Considering the accuracy of the range model, a third-order range model is used as follows [16,31–33]:

$$R\_s(t\_n) \approx R\_0 - v\_t t\_n + \frac{\left(\upsilon - \upsilon\_a\right)^2 - R\_0 a\_c}{2R\_0} t\_n^2 + \frac{v\_c \left(\upsilon - \upsilon\_a\right)^2 + R\_0 a\_a \left(\upsilon\_a - \upsilon\right)}{2R\_0^2} t\_n^3. \tag{2}$$

We suppose that the radar transmits the linear frequency modulation (LFM) signal with the form as follows [20]:

$$s\_t(t) = \text{rect}(\frac{t}{T\_p}) \exp(j2\pi f\_c t + j\pi \gamma t^2),\tag{3}$$

where *t* indicates the range time, rect(·) is the rectangle window function, *Tp* denotes the pulse length, γ is the chirp rate of the transmitted signal and *fc* represents the carrier frequency. The received baseband signal is written as

$$\mathbf{s}\_{\text{base}}(t, t\_n) = \text{correct}\left[\frac{t - 2R\_s(t\_n)/c}{T\_p}\right] \mathbf{w}\_a(t\_n) \exp\left\{j\pi\gamma \left[t - \frac{2R\_s(t\_n)}{c}\right]^2\right\} \exp\left[-j\frac{4\pi}{\lambda}R\_s(t\_n)\right],\tag{4}$$

where σ is the backscattering coefficient of the moving target, *c* denotes the speed of electromagnetic wave, *wa*(·) is the azimuth window function and λ is the wavelength of the transmitted signal.

By substituting Equations (2) into (4) and performing the range compression [20,22], the received signal omitting the envelope in the range frequency and azimuth time domain yields

$$\begin{split} s\_1(f, t\_n) &= \text{rect}\left(\frac{f}{\mathcal{B}}\right) w\_d(t\_n) \exp\left[-\frac{j4\pi}{\varepsilon}(f + f\_\varepsilon)\right. \\ &\cdot \left(R\_0 - v\_\varepsilon t\_n + \frac{\left(v - v\_\varepsilon\right)^2 - R\_0 a\_\varepsilon}{2R\_0} t\_n^2 + \frac{v\_\varepsilon \left(v - v\_\varepsilon\right)^2 + R\_0 a\_\varepsilon \left(v\_\varepsilon - v\right)}{2R\_0^2} t\_n^3\right) \end{split} \tag{5}$$

where *f* denotes the range frequency variable and *B* = γ*Tp* is the bandwidth of the transmitted signal.

After the range IFFT is applied to Equation (5), the received signal in the range and azimuth slow time domain is expressed as

$$\begin{cases} s\_1(t, t\_n) &= \text{sinc}\left\{ B \left[ t - 2 \left( R\_0 - v\_c t\_n + \frac{(v - v\_v)^2 - R\_0 a\_c}{2R\_0} t\_n^2 + \frac{v\_c (v - v\_u)^2 + R\_0 a\_d (v\_b - v)}{2R\_0^2} t\_n^3 \right) \right] \right\} \\\ \cdot w\_a(t\_n) \exp\left[ -\frac{j4\pi}{\lambda} \left( R\_0 - v\_c t\_n + \frac{(v - v\_a)^2 - R\_0 a\_c}{2R\_0} t\_n^2 + \frac{v\_c (v - v\_a)^2 + R\_0 a\_s (v\_b - v)}{2R\_0^2} t\_n^3 \right) \right] \end{cases} \tag{6}$$

where sinc(*x*) = sin(π*x*)/(π*x*) denotes the sinc function.

#### *2.2. Signal Characteristics*

In accordance with Equation (5), the range frequency variable *f* is coupled with the azimuth slow time variable *tn*. Not only the coupling effects caused by the low-order terms, namely, the *tn*- and *t* 2 *<sup>n</sup>*-term, but also those induced by the high-order one, namely, the *t* 3 *<sup>n</sup>*-term, exist. Therefore, the range position and Doppler frequency of the moving target change with the azimuth slow time.

As described in the sinc function term of Equation (6), the *tn*-term induces FRCM, the *t* 2 *<sup>n</sup>*-term causes SRCM and the *t* 3 *<sup>n</sup>*-term leads to TRCM in the range dimension. According to the last exponential term of Equation (6), the *tn*-term, *t* 2 *<sup>n</sup>*-term and *t* 3 *<sup>n</sup>*-term result in Doppler centre shift, LDFM and QDFM, respectively, in the Doppler frequency dimension. The RCM and DFM are severe for fast-maneuvering targets. This condition makes the trajectory span over multiple ranges and Doppler frequency cells. Thus, the complex RCM and DFM should be effectively removed to focus the moving target.

The target azimuth Doppler spectrum distribution must be further studied to obtain a well-focused result [34]. The Doppler centre shift of the fast-maneuvering target is larger than PRF/2, and the target shows Doppler centre blur. In the 2D spectrum dimension, the potential azimuth spectrum distributions caused by Doppler centre shift and DFM consist of the following cases. When *fB* < *PRF*/2, where *fB* denotes the azimuth spectrum bandwidth of the target signal, two azimuth spectrum distributions are introduced: case I: the azimuth spectrum is located entirely on one PRF band, as shown in Figure 2a, where *fdc* represents the Doppler centre shift in the figure; case II: the azimuth spectrum spans over two PRF bands, as shown in Figure 2b. When *PRF*/2 < *fB* < *PRF*, two other azimuth spectrum distributions are obtained: case III: the azimuth spectrum still occupies one PRF band, as displayed in Figure 2c; case IV: the azimuth spectrum also distributes into two PRF bands, as shown in Figure 2d. When *fB* > *PRF*, as shown in Figure 2e, the azimuth spectrum distributes into several PRF bands. When the azimuth spectrum does not entirely occupy one PRF band, namely, cases II, IV and V, the target spectrum split will occur; this phenomenon induces the azimuth Doppler spectrum ambiguity [34,35].

**Figure 2.** Potential azimuth Doppler spectrum distributions: (**a**) case I: *fB* < *PRF*/2, spectrum entirely in one pulse repetition frequency (PRF) band; (**b**) case II: *fB* < *PRF*/2, spectrum spanning two PRF bands; (**c**) case III: *PRF*/2 < *fB* < *PRF*, spectrum entirely in one PRF band; (**d**) case IV: *PRF*/2 < *fB* < *PRF*, spectrum spanning two PRF bands; (**e**) case V: *fB* > *PRF*, spectrum spanning several PRF bands.

In summary, the complex RCM and DFM lead to severe integration loss and defocusing of the target image. In the existing methods, the Doppler ambiguity number searching [25,34] and Doppler centre shifting by PRF/2 operations [26,27,32] were usually used to deal with the Doppler centre blur and spectrum ambiguity, respectively. However, the Doppler ambiguity number searching operation increases the computational complexity. If the spectrum distribution belongs to case IV or V, then the Doppler centre shifting by PRF/2 operation will be invalid. Accordingly, the moving targets become difficult to be focused in the presence of complex Doppler ambiguity by applying transitional FOKT or SOKT-based [20–26,31,32] and stationary phase-based [27,28] methods. We present a new algorithm for refocusing of fast-maneuvering targets to deal with aforementioned problems.

#### **3. Proposed Algorithm Description**

The RCM, DFM, Doppler centre shift and Doppler ambiguity are the key issues for refocusing of fast-maneuvering targets according to the target signal properties described in the previous section. Therefore, a new fast algorithm is presented in this section.
