Fast ISAR imaging method for complex manoeuvring target based on local polynomial transform ‐ fast chirp Fourier transform

In ISAR imaging, echo signals of complex manoeuvring targets have to be modelled as multi ‐ component cubic phase signals (m ‐ CPS). In this condition, the Doppler diffusion will seriously affect the quality of ISAR imaging. Herein, a complex manoeuvring target imaging method based on local polynomial transform ‐ fast chirp Fourier transform (LPT ‐ FCFT) is proposed. In this method, the instantaneous frequency and the quadratic chirp rate of the echo signal are estimated by local polynomial transformation, then the chirp rate and the amplitude are estimated by fast chirp Fourier transform, and finally the high ‐ resolution ISAR image of the complex manoeuvring target is obtained by CLEAN technology. Compared with the traditional method, the proposed method has the advantages of less computational complexity, clearer imaging results and better anti ‐ noise performance. Simulation results indicate the effectiveness and superiority of the proposed method.


| INTRODUCTION
Manoeuvring target imaging has been the focus of ISAR imaging. In the early stage of the manoeuvring target ISAR imaging study, the echo signals are generally modelled as LFM signals in the slow time domain. For such a model, imaging methods are divided into two types, one is using the timefrequency distributions to realise the range instantaneous Doppler (RID) imaging, such as Wigner-Ville distribution (WVD) and its various modified methods [1][2][3], the other is through the parameter estimation of the chirp rate and the instantaneous frequency of the echo to imaging, such as Lv's distribution (LVD) methods [4][5][6] and matching Fourier transform (MFT) methods [7][8][9].
With the increasing mobility of air targets, the LFM signal model is no longer suitable to describe the complex manoeuvring targets, so the echo should be further modelled as a multi-component cubic phase signal (m-CPS) [10,11]. At present, most of the imaging methods of m-CPS adopt parameter estimation. As many parameters need to be estimated in m-CPS, it is often necessary to estimate each parameter in turn, such as high-order ambiguity function-integrated cubic phase function (HAF-ICPF) [12,13] and product high-order ambiguity function-modified integrated cubic phase function (PHAF-MICPF) [14] methods. However, error transfer may occur in this process, which affects the accuracy of parameter estimation. Therefore, it is useful to estimate multiple parameters simultaneously, such as integrated cubic phase bilinear autocorrelation function (ICPBAF) [15][16][17], generalised scaled Fourier transform (GSCFT) [18][19][20] and chirp rate-quadratic chirp rate distribution (CRQCRD) [21,22] methods. On the other hand, the CLEAN technique is often used for parameter estimation-based ISAR imaging methods, which increase the computational complexity greatly. Therefore, some computation-heavy operations, such as non-uniform Fourier transform and integration with multiple variables should be avoided when designing imaging algorithms.
In conclusion, a complex manoeuvring target ISAR imaging method based on local polynomial transform-fast chirp Fourier transform (LPT-FCFT), is proposed here. This method refers to the local polynomial Wigner distribution (LPWD) [23,24] to propose LPTwith lower computational complexity, and uses it to estimate the instantaneous frequency and the quadratic chirp rate simultaneously. Then, fast chirp Fourier transform (FCFT) is used to estimate the chirp rate and amplitude. Finally, the reconstructed signal obtained by CLEAN technology is used for high-resolution ISAR imaging. The simulation results show that this method has advantages in less computational complexity, accuracy of parameter estimation, and anti-noise performance.

| SIGNAL MODEL
The rotation model of the target is shown in Figure 1, where p is a scattering centre, O is the centre of the target rotation, L is the unit vector of the radar's line of sight (LOS), r p is the position vector of p, Ω is the rotation vector, which can be decomposed into vertical to the L and parallel to the L of the two parts.
When the scattering centre p rotates, its radial velocity can be written as where Ω E , L and r p are, respectively, expressed as 8 < : Therefore the radial velocity of p is For a complex manoeuvring target, the elements of the Ω E have the following form If the radar transmits the LFM signal, the echo signal of the scattering centre p after Dechirp and translational compensation can be written as where B is the bandwidth of transmitting signal, λ is the wavelength, σ p is the scattering coefficient of p, À ω x;2 ω y;2 ω z;2 � Φ p ¼ À l y r pz − l z r py l z r px − l x r pz l x r py − l y r px � where y p = R p (0)−R ref is the range coordinate of scattering centre p, R p (0) is the radial distance between p and radar at t m = 0. Substitute Equations (6) and (7) into Equation (5), after the migration through range cells (MTRC) correction [25] and range compression, the echo of complex manoeuvring target containing N scattering points can be written as It can be seen that the echo of complex manoeuvring target is m-CPS.

| PARAMETER ESTIMATION METHOD BASED ON LPT-FCFT
LPT is defined with reference to LPWD. LPWD is defined as where τ is the delay variable, Ω is a vector that containing n+1 variables such as ω 1 , ω 3 …ω 2nþ1 , θðΩ; τÞ is LPWD has the advantages of non-search and multiparameter simultaneous estimation, and has broad application prospects in m-CPS signal parameter estimation and ISAR imaging of manoeuvring targets. However, LPWD needs to calculate the integral separately when multiple variables take different values, which significantly increase the computational complexity. Therefore, LPT is proposed, which has much lower computational complexity.
After the MTRC correction, the echo of the k-th range bin can be written as where A p ¼ σ p sincðBðf r;k − 2y p =cÞÞ is the echo amplitude, f r,k is the fast time frequency corresponding to the k-th range bin, the kernel function of LPT can be constructed as where ζ is the quadratic chirp rate, according to the prior knowledge of target, the roughly value range of ζ can be determined.
In particular, when t m = 0, Equation (11) is changed into the following form where R cross ðτ; ζÞ is the cross-term caused by conjugate multiplication and the first term of Equation (13) is the selfterm.
The LPT of the echo is obtained by applying the fast Fourier transform of R LPT ðτ; ζÞ in the τ domain The transformation processes of the self-term and crossterm in Equation (12) are analysed, respectively. The results of self-term LPT can be calculated by the standing-phase method. Set a slow-changing phase According to reference [26], the standing phase point is τ * ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The self-term LPT can be obtained by Equation (17) Substitute Equations (15) and (16) into Equation (17) and take the absolute value as Equation (18) 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ζ þ 2ϕ 3;p p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi It can be seen that, (18) can be maximised. Therefore, the estimated values of the instantaneous frequency and the quadratic chirp rate can be obtained by peak search after LPT.
For the cross-term, any of its components has the following form where the q is a scattering centre different from the p and A q ,ϕ 1;q , ϕ 2;q ,ϕ 3;q have mathematical meaning similar to A p ,ϕ 1;p , ϕ 2;p ,ϕ 3;p . The results of cross-term LPT are as follows Obviously, if b ≠ 0, no matter what values of ζ and f τ are taken, there will be no peak in the result of cross-term LPT. If b = 0, ϕ 2;p ¼ ϕ 2;q must be present. However, according to Equation (6), the echoes of two scattering centres with different space positions must have different chirp rates. Therefore, the cross-terms cannot form the false peaks.
In conclusion, LPT can simultaneously estimate the instantaneous frequency and the quadratic chirp rate of the 668echo, and suppress the cross-term by its own properties. The specific method for parameter estimation using LPT is as follows where f τm and ζ m , respectively, represent the values of f τ and ζ at the peak in Equation (23) Using the quadratic chirp rate estimated by Equations (23) and (24), the echo is processed as follows At this point, the cubic phase term of the scattering centre p will be eliminated; while the cubic phase term of other scattering centres still exists. Therefore Equation (25) can be written as where other(t m ) represents the echo of the scattering centres except the p. Considering that the echo of scattering centre p only has second-order phase term, the chirp rate and amplitude can be simultaneously estimated by CFT [27,28].
where γ is the value range of chirp rate, which can be roughly determined by the target. When x(t) is the LFM signal, its instantaneous frequency and chirp rate can be estimated by peak search. However, CFT requires a lot of multiplication and integral. When the length of γ and f are M, the number of multiplication will be more than M 3 , so the fast algorithm, FCFT, is adopted. The specific operation of FCFT is as follows If the lengths of γ and f were M, the total multiplication number of Equations (29) and (30) are M 2 + 0.5M 2 log 2 M, which is significantly lower than CFT.
When γ ¼ ϕ 2;p in Equation (29), the second-order phase of the first term of S 0 Rk ðt m Þ will be compensated and the peak will be obtained after FFT, which can be used for amplitude estimation. While the high-order phase in the second term of Equation (27) still exists, there is no peak after FFT.
According to the above principle, the chirp rate and amplitude of the echo of scattering centre p are estimated where M is the length of slow time sampling number, A and γ m , respectively, represent the peak value of Equation (31) and the corresponding value of γat the peak.

| ANALYSIS OF COMPUTATIONAL COMPLEXITY
In order to demonstrate the superiority of LPT-FCFT in computational complexity, HAF-ICPF and LPWD are selected for comparison, where HAF-ICPF is a common m-CPS parameter estimation method and LPWD is similar to LPT-FCFT. For the convenience of analysis, the length of each variable in each method, such as the value range of slow time, delay variable and chirp rate, is set as M. The number of multiplications is taken as the standard to measure the computational complexity. For the HAF-ICPF, M multiplications are needed to reduce the order of m-CPS signal by using the delay constant. When ICPF is used, M 2 multiplications are needed to construct the kernel function, and M 3 multiplications are needed to carry out the non-uniform Fourier transform. M + 0.5Mlog 2 M multiplications are needed to compensate the high-order phase and to estimate the instantaneous frequency and amplitude. Therefore, the total number of multiplications required by HAF-ICPF is For LPWD, M multiplications are required to calculate the instantaneous autocorrelation function at t m = 0. M 3 multiplications are required to calculate LPWD. ICPF is also used in the LPWD to estimate the chirp rate, and M 2 + M 3 multiplications are required by ICPF. Therefore, the total multiplications required by the parameter estimation method based on LPWD are For the LPT-FCFT proposed in this paper, M 2 multiplications are needed to calculate the kernel function at t m = 0, 0.5M 2 log 2 M multiplication is needed to complete LPT, and M 2 + 0.5M 2 log 2 M multiplications are required by FCFT. Therefore, the total multiplications required by LPT-FCFT are LIU ET AL. -669 When the M takes different values, the computational complexity of the three methods is shown in Figure 2.
It can be seen that when the value of M exceeds 60, the computational complexity of LPT-FCFT starts to be significantly lower than that of the other two methods. Moreover, in practical applications, the value of M is usually hundreds or even thousands, at which time LPT-FCFT will have more obvious advantages in computational complexity and will greatly reduce the time required by ISAR imaging (Figure 3).

| ISAR IMAGING METHOD BASED ON LOCAL POLYNOMIAL TRANSFORM-FAST CHIRP FOURIER TRANSFORM
LPT-FCFT is combined with CLEAN technology to achieve ISAR imaging of complex manoeuvring targets. The specific steps are as follows: Step (1) Carry out translational compensation, MTRC correction and range compression for the echo signal.
Step (2) Select the echo data of the k-th range bin, namely S Rk (t m ).
Step (3) The S Rk (t m ) is processed by LPT. The estimated value of instantaneous frequency and quadratic chirp rate of scattering centre p are obtained by peak search, namelŷ ϕ 1;p andφ 3;p .
Step (4) Using f 1 (t m ) andφ 3;p to compensate the thirdorder phase of the echo of scattering centre p, and FCFT is used to obtain the estimated value of chirp rate and amplitude, namelyφ 2;p andÂ p .
Step (5) The signal is reconstructed and superimposed according to the estimated parameters. The reconstructed signal is shown in Equation (36), whereŷ p ¼ cf rk =2 is the estimated range coordinate, f rk is the fast time frequency of the k-th range bin.
Step (6) The reconstructed echo should be deleted from the total echo, and the signal used for deletion is shown in Equation (37).
Step (7) Repeat steps 3−6 until the remaining energy of the k-th range bin reaches 5-12% of the initial energy. If the SNR of the echo is lower than 0 dB, the termination criteria need to be increased appropriately. Moreover, the termination criteria can also be determined by the dynamic range as: whereÂ p1 represents the amplitude of the first scattering centre to be extracted, ΔA represent the given dynamic range.
Step (8) All range bins are processed in steps from 2 to 7 and the reconstructed signal is processed by FFT in slow time domain to obtain high-resolution ISAR images of complex manoeuvring targets as in Equation (38).
where f d is the Doppler frequency. Considering that f d and f r are usually discrete, according to the range compression theory and basic properties of FFT, the mathematical expressions of range and cross-range resolutions are ρ r ¼ c=2B, where θ is the total rotation angle.

| VERIFICATION OF LPT-FCFT PARAMETER ESTIMATION PERFORMANCE WITH SYNTHETIC DATA
In this section, a m-CPS with two components is used to verify the parameter estimation performance of LPT-FCFT. The slow time sampling length is 512 and the signal parameters are as follow: Under different SNR, the LPT results of the above m-CPS are shown in Figure 4.
When the SNR is −5-10 dB, the estimated values of the instantaneous frequency and quadratic chirp rate are: ϕ 1;1 ¼ −14:97,φ 3;1 ¼ −15:14,φ 1;2 ¼ 10:08,φ 3;2 ¼ −20:02. The relative errors of these estimated parameters were 0.15%, 0.93%, 0.8% and 0.1%, respectively. These errors are mainly caused by the interval of discretisation value. Considering that these errors are very small, their influence can be ignored in the high-order phase compensation and ISAR imaging. Next, parameter estimation performance of FCFT is verified. Under the condition that the SNR is 0, FCFT processing is carried out for the following two cases, respectively, and the results are shown in Figure 5:  In case 1, the cubic phase term of component 1 is basically compensated, so the chirp rate of component 1 will be estimated by FCFT, and the same is true in case 2. Unique peaks exist in Figures 5(a) and 5(b), and the estimated values of chirp rate areφ 2;1 ¼ 19:96 andφ 2;2 ¼ 14:95, with relative errors of 0.2% and 0.3%, respectively. The estimated values of amplitude were 1.0100 and 0.9876, respectively, with relative errors of 1% and 1.24%, which were mainly caused by noise. Therefore, it can be proved that FCFT can accurately estimate the chirp rate and amplitude of the echo.
In conclusion, LPT-FCFT is more accurate in parameter estimation and has good performance when the SNR is not too low.

| ISAR IMAGING VERIFICATION OF COMPLEX MANOEUVRING TARGETS WITH SYNTHETIC DATA
ISAR imaging verification was performed using a simulated aircraft containing 330 scattering centres, the RCS of the scattering centres to the Rayleigh distribution of σ ¼ 0:5. The radar carrier frequency was set at 10 GHz, bandwidth at 2 GHz, number of pulse at 512, and pulse repetition frequency (PRF) at 200 Hz. The origin of the target coordinate system was 20 km from the radar. The angular velocity was 0.028 rad/s, the angular acceleration 0.007 rad/s 2 , and the angular jerk 0.006 rad/s 3 . The shape of the simulated aircraft and the ideal ISAR imaging result are shown in Figure 6: When the condition of SNR is 0, the six methods range-Doppler (RD), WVD, smooth pseudo Wigner-Ville distribution (SPWVD), HAF-ICPF, LPWD, and LPT-FCFT were used to conduct ISAR imaging on the simulation target. The result of the RD method is shown in Figure 7, the results of the WVD, SPWVD, HAF-ICPF, and LPWD methods are shown in Figure 8, and the result of the LPT-FCFT method is shown in Figure 9.
It can be seen that the result of the RD method is seriously defocused, in particular the two sides of the wing are very fuzzy. The result of the WVD method is difficult to distinguish the target, due to the dense scattering centres distribution, resulting in false peaks created by the cross-terms seriously affecting the imaging quality [26]. The result of the SPWVD method is relatively clear, but the adjacent scatters are aliasing. Moreover, due to the limitation of the method itself, the crossrange boundary of the WVD and SPWVD methods reduces by half, which results in some changes in the target shape. In the result of the HAF-ICPF method, some scatters of the nose and two wings are missing or fuzzy, due to a large number of crossterms caused by twice conjugate multiplication. The result of the LPWD method shows that some scatters on the edges of the two wings are fuzzy. This is because ICPF uses incoherent accumulation, which may lead to the error of chirp rate estimation when the noise is strong [29]. In comparison, the result of the LPT-FCFT method is much clearer and basically consistent with Figure 6, which proves the superiority of this method.
Next, the performance of the LPT-FCFT method under different SNR is verified. When SNR is -5 dB, -3 dB, -1 dB, and 1 dB, the ISAR imaging results of LPT-FCFT method are shown in Figure 10.
It can be observed that when the SNR is −5, most strong scattering centres are well focused, but the weaker scattering centres are overwhelmed by noise. When the SNR is −3, the noise scatters are significantly reduced and the ISAR image is clearer. When the SNR is above −1, the noise scatters almost disappear, and the ISAR imaging results are quite clear. In conclusion, the LPT-FCFT method has good performance when the SNR is higher than −3 dB. Finally, the effectiveness of the LPT-FCFT method is verified by mig-25 simulation data. The data come from the Naval Research Laboratory and are measured by the radar in 9 GHz carrier frequency, 512 MHz bandwidth, and 15 kHz pulse repetition frequency. The results of the RD and LPT-FCFT methods at the condition of SNR are 10 dB are shown in Figures 11(a) and 11(b) respectively, and the results at 0 dB are shown in Figures 11(c) and 11 (d).
When the SNR is 10 dB, the results of the RD method are blurred due to target manoeuvre; especially, the scattered points in the nose are completely aliased. The result of the LPT-FCFT method shows that the scatters of MIG-25 are well focussed and the imaging quality is very good. When the SNR is 0 dB, only the outline of the target can be seen roughly in the result of the RD method. However, the results of the LPT-FCFT method are basically close to those at 10 dB. This further proves that the LPT-FCFT method can effectively perform high-resolution ISAR imaging on complex manoeuvring targets with good anti-noise performance and certain practical significance.

| CONCLUSION
Herein, a complex manoeuvring target ISAR imaging method based on LPT-FCFT is proposed. The method simultaneously estimates the instantaneous frequency and the quadratic chirp rate, the chirp rate, and the amplitude, respectively, and then uses CLEAN technology to reconstruct the echo and conduct ISAR imaging. The advantages of this method are: (1) the computational complexity is significantly lower than other methods; (2) it can estimate multiple parameters simultaneously and be more accurate; and (3) ISAR imaging results in complex manoeuvring targets that are clearer and have good anti-noise performance.