Long-Time Coherent Integration for Maneuvering Target Based on Second-Order Keystone Transform and Lv’s Distribution

Abstract

Maneuvering target detection is a challenging task for radar due to its maneuverability and weak energy. Long-term coherent integration can remarkably increase signal energy to improve the target detection ability. Unfortunately, high velocity and acceleration of the target will produce linear range migration (LRM), quadratic range migration (QRM), and Doppler frequency migration (DFM), which seriously degrades the coherent integration gain and further deteriorates target detection performance. To solve this problem, a method based on second-order Keystone transform (SKT) and Lv’s distribution (LVD), also combined with Radon Fourier transform (RFT), i.e., SKTLVD, is proposed in this paper. The LRM is firstly corrected by using RFT. Then, the SKT is employed to remove QRM. Finally, LVD is utilized to eliminate the DFM and achieve coherent integration. Compared with several representative methods, the SKTLVD consumes low computation and obtains good target detection performance, striking a balance between computational cost and target detection ability. Numerical simulations and real measured radar data demonstrate that the proposed method can obtain considerable coherent integral gain under acceptable computational complexity.

1. Introduction

Over the past few years, radar has been widely applied in civil and military due to its all-day and all-weather target detection capability. However, the gradual development of stealth technology makes the signal-to-noise ratio (SNR) of radar echoes extremely low, posing a significant challenge to the traditional radar detection technology [1,2,3,4,5]. It is known that integrating the energy of multiple echoes can effectively improve the ability of weak target detection without changing the radar hardware. Especially for long-time coherent integration, it can almost achieve linear integration gain with the number of pulses via compensating for the phase difference between different echoes [6,7,8,9,10].

Unfortunately, the emergence of highly maneuvering targets causes the envelope and phase of the echoes signal to migrate over a long integration time, resulting in range migration (RM) and Doppler frequency migration (DFM). These detrimental effects seriously deteriorate the detection performance and invalidate conventional integration methods. Hence, it is necessary to compensate the RM and DFM to concentrate the signal energy and improve the gain.

According to the different ways of compensation, the existing methods can be roughly divided into envelope-phase independent compensation methods and envelope-phase joint compensation methods, as shown in

Table 1. Classification and characteristics of long-time coherent integration methods.

Algorithm Type	Processing	Representative Methods	Advantage	Inferiority
Envelope-phase independent compensation methods	Cascade	ARMTD ACCFLVD KTLVD SKTRFT	Fast implementation	Model error
			Intensively real time	Integration loss
Envelope-phase joint compensation methods	Parallel	RFT GRFT RFRFT RLVD	Accurate model	Search complex
			High gain	Vast computing

.

The independent compensation methods adopt cascade processing, which regards the envelope term and phase term as independent parts, compensating the RM and DFM generated by them. Specifically, RM generated by the envelope term includes: (1) the first-order term, namely linear RM (LRM); (2) the second-order term, namely quadratic RM (QRM), also known as range curvature; (3) the higher-order term. Keystone transform (KT) [11,12], axis rotation (AR) [13], and scaled inverse Fourier transform (SCIFT) [14] are proposed to solve LRM, in which KT eliminates the coupling between fast time and slow time via variable scale transformation, and it can realize correction without a priori information. Adjacent cross-correlation function (ACCF) [15], second-order KT (SKT) [16,17], and other algorithms are utilized to deal with QRM. On the other hand, the phase term migration includes the first-order term, i.e., Doppler frequency shift (DFS), and the second-order term, i.e., DFM. For DFS, the moving target detection algorithm (MTD) [18] based on fast Fourier transform (FFT) is simple and effective in engineering. While for DFM, the performance of FFT, which can only handle fixed frequency signals, becomes worse. Thus, a series of time-frequency analysis tools that can conduct time-varying frequency signals are widely used, e.g., Wigner Ville distribution (WVD) [19], fractional Fourier transform (FRFT) [20], and LV distribution (LVD) [21]. LVD is a novel method proposed in recent years, and it can gather the energy-dispersed linear frequency-modulated (LFM) signal into an impulse in the centroid frequency and chirp rate (CFCR) domain without any searching process. The above envelope and phase compensation methods are organically cascaded to form complete envelope-phase independent compensation methods, i.e., ARMTD [22] and KTFFT [23] for uniform moving targets; and ACCFLVD [24], KTLVD [25], and SKTRFT [26] for uniformly accelerated targets. In the case of signal envelope and phase decoupling, these methods can exploit their characteristics to select the corresponding transformation and fast algorithm for compensation flexibly. Therefore, the amount of calculation is usually small, and the real-time performance is strong. However, when dealing with targets with complex motion states, these methods adopt approximation to simplify the motion model and make the correction algorithms realize smoothly. Moreover, the approximation will inevitably cause errors and reduce the coherent integration gain.

The joint compensation methods wield parallel processing, which realizes the joint compensation of the signal envelope and phase term via searching the unknown motion parameters of the target. Radon Fourier transform (RFT) [27,28,29] is one of the classic algorithms, attracting extensive attention since it was proposed. It realizes the accumulation of uniform velocity targets through the two-dimensional joint search of distance and velocity. On this basis, generalized RFT (GRFT) [30] is used to complete the coherent accumulation of complex moving targets. In addition, [31] combines RFT and fractional Fourier transform (FRFT) to propose RFRFT, and [32] combines RFT and Lv’s distribution (LVD) to propose RLVD. The advantages of these methods are accurate compensation and high coherent accumulation gain. Ideally, when the search parameter value exactly matches the real parameter value of the target, the RM and DFM can be fully compensated, and the best coherent integration gain can be obtained. However, implementing these methods is not straightforward because the scope and interval of parameter search are challenging to determine without prior knowledge. If we pursue high gain, the search range needs to be expended, and the search interval needs to be refined, increasing search volume. For maneuvering targets with acceleration, the amount of computation increases geometrically due to the further improvement of the search dimension, which seriously affects the real-time performance of these methods.

The above analysis shows that the existing two kinds of algorithms have the problems of unsatisfactory integration gain and poor real-time performance, which are challenging to meet the increasing detection needs of high maneuvering weak targets on the whole. Therefore, this paper proposes a method based on RFT, SKT, and LVD, i.e., SKTLVD, to solve the problem of high-speed maneuvering target coherent integration and detection. In this method, RFT is firstly employed to search the target velocity and jointly eliminate the LRM in envelop and DFS in the phase; then, SKT is performed to compensate QRM in the envelope independently. Finally, LVD is utilized to correct the DFM in the phase and realize the coherent integration. It can be seen that SKTLVD jointly compensates the first-order term migration in the envelope and phase via a one-dimensional search and independently compensates the second-order term migration in the envelope and phase via SKT and LVD. Thus, it combines the advantages of independent and joint compensation methods, reconciling the coherent integration gain with real-time requirements.

The rest of this paper is organized as follows: In Section 2, the signal model is presented. Section 3 analyzes the traditional methods and explains the problems. In Section 4, the coherent integration method based on SKTLVD is proposed, and the main procedure and computational complexity analysis of the proposed algorithm are also drawn. In Section 5, the numerical experiments are carried out to verify the effectiveness of SKTLVD. Finally, conclusions and future works are given in Section 6.

2. Signal Model

Suppose that the radar transmits an LFM signal, which can be written as:

$$s_t(t_k,t_m)=\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{t_k}{t_p}\right)\exp \left(j\pi K{t}_k{}^2\right)\exp \left[j2\pi f_c\left(t_k+t_m\right)\right]$$

(1)

where

$$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}(\frac{t_k}{t_p})=\left\{\begin{array}{c}1,\left|t_k\right|<t_p/2\\ 0,\left|t_k\right|>t_p/2\end{array}\right.$$

(2)

is the rectangular window function. $t_k$ represents fast time, and $t_p$ denotes the pulse duration. $t_m=m\mathrm{P}\mathrm{R}\mathrm{T}(m=1,2,...,M)$ denotes slow time, in which PRT is the pulse repetition time, and M is the integration pulse number. K denotes the frequency modulated rate, and $f_c$ is the carrier frequency. Therefore, the $mth$ pulse-echo signal scattered from the target received by the radar after down-conversion can be stated as:

$$s_r\left(t_k,t_m\right)=A_0\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left[\frac{t_k-2R\left(t_m\right)/c}{t_p}\right]\mathrm{e}\mathrm{x}\mathrm{p}\left\{j\pi K{\left[t_k-\frac{2R\left(t_m\right)}{c}\right]}^2\right\}\mathrm{e}\mathrm{x}\mathrm{p}\left[-j\frac{4\pi f_{c}R\left(t_m\right)}{c}\right]$$

(3)

where $A_0$ denotes the target reflectivity, and $c$ denotes the speed of light. $R\left(t_m\right)$ is the instantaneous slant range between the radar and target. Neglecting the high-order motion components, the $R\left(t_m\right)$ of the maneuvering target with a constant acceleration satisfies

$$R\left(t_m\right)=r_0+v_0{t}_m+\frac{1}{2}{a}_0{t}_m^2$$

(4)

where $r_0$, $v_0$, and $a_0$ denote the initial slant range, the radial velocity, and the radial acceleration between the radar and the target, respectively. Use match filter $h(t_k)=\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(t_k/t_p\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(j\pi K{t}_k^2\right)$, and according to the stationary phase principle, the signal after pulse compression in $f_k-t_m$ domain can be expressed as:

$$s_{pc}\left(f_k,t_m\right)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-j\frac{4\pi \left(f_c+f_k\right)}{c}\left(r_0+v_0{t}_m+\frac{1}{2}{a}_0{t}_m^2\right)\right]$$

(5)

where $A_1$ is the signal amplitude after pulse compression, and B is the signal bandwidth, then using inverse FFT (IFFT) to transform the signal to $t_k-t_m$ domain:

$$s_{pc}\left(t_k,t_m\right)=A_2\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[B\left(t_k-\frac{2\left(r_0+v_0{t}_m+\frac{1}{2}{a}_0{t}_m^2\right)}{c}\right)\right]\mathrm{e}\mathrm{x}\mathrm{p}\left[-j\frac{4\pi \left(r_0+v_0{t}_m+\frac{1}{2}{a}_0{t}_m^2\right)}{\lambda }\right]$$

(6)

where $A_2$ is signal amplitude after IFFT, $\lambda$ is signal wavelength, and $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(x\right)$ is the sinc function. From the $s_{pc}\left(t_k,t_m\right)$, it can be found that:

• The peak position of the signal envelope $t_k=2\left(r_0+v_0{t}_m+0.5a_0{t}_m^2\right)/c$, which moves with the slow time. Moreover, when the migration exceeds one range resolution unit, the RM is generated. The first-order term, i.e., $2v_0{t}_m/c$ leads to LRM and the second-order term, i.e., $a_0{t}_m^2/c$ leads to QRM.
• The signal phase term is an LFM signal with respect to the $t_m$, its centroid frequency is $2v_0/\lambda$, and chirp rate is $a_0/\lambda$. As we know, the frequency of the LFM signal changes with time, and when the variation exceeds one frequency resolution unit, the DFM is generated.

Consequently, the LRM, QRM in the signal envelop, and DFM in the signal phase must be eliminated to improve the gain of coherent integration.

3. Conventional Algorithm Analysis

This section selects several representative methods from the envelope-phase independent compensation algorithm and envelope-phase joint compensation algorithm, mainly analyzes their principles, and highlights their problems.

3.1. Envelope-Phase Independent Compensation Methods

We focus on the KTLVD [25] and SKTRFT [26] in envelope-phase independent compensation methods.

KTLVD first applies KT to compensate RM and then uses LVD to compensate DFM, while it does not compensate QRM. Actually, the signal after KT can be written as:

$$\mathrm{K}\mathrm{T}\left(f_k,t_n\right)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\exp \left\{-\frac{j4\pi }{c}\left[\left(f_k+f_c\right)r_0\right.\right.\left.\left.+f_c{v}_0{t}_n+\frac{f_c^2}{f_k+f_c}\frac{a_0}{2}{t}_n^2\right]\right\}$$

(7)

It can be seen that the first-order term of the coupling between $f_k$ and $t_n$, which generates the LRM, has been eliminated, but the second-order term coupling still exists. The algorithm adopts approximate simplification, i.e., $f_c^2{t}_n^2/\left(f_k+f_c\right)\approx f_c{t}_n^2$ to neglect QRM. Unfortunately, the error caused by approximation will increase with the increase in target acceleration, which seriously deteriorates the performance of coherent integration.

On the other hand, the procedure of SKTLVD is as follows: SKT is utilized to correct the QRM. Then, fractional Fourier transform is employed to estimate the targets’ acceleration and compensate for the DFM. Finally, RFT is used to correct the RM for target coherent detection. Bring SKT into (5) to obtain:

$$\mathrm{S}\mathrm{K}\mathrm{T}(f_k,t_n)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\exp \left\{-\frac{j4\pi }{c}\left[\left(f_k+f_c\right)r_0\right.\right.\left.\left.+\sqrt{(f_k+f_c)f_c}{v}_0{t}_n+f_c\frac{a_0}{2}{t}_n^2\right]\right\}$$

(8)

Although SKT eliminates the second-order coupling, it transforms the remaining first-order coupling $\left(f_k+f_c\right)v_0{t}_n$ to complex high-order coupling $\sqrt{(f_k+f_c)f_c}{v}_0{t}_n$. This method addresses approximation and only retains first-order Taylor series expansion over $\sqrt{(f_k+f_c)f_c}$, obtaining $\sqrt{(f_k+f_c)f_c}\approx \left(f_c+f_k/2\right)$, thus simplifying first-order coupling. However, the approximation inevitably has errors and reduces the coherent integration gain.

3.2. Envelope-Phase Joint Compensation Methods

RFRFT [31] and RLVD [32] in the envelope-phase joint compensation methods are mainly analyzed.

RFRFT is a combination of RFT and FRFT, which first employs second-order RFT to extract the signal from the data and then uses FRFT to compensate DFM. The processing can be written as:

$$\begin{gathered} \mathrm{RFRFT}=\underset{r,v,a}{\max }\left\{\mathrm{F}\mathrm{R}\mathrm{F}\mathrm{T}\left\{s_{pc}\left[\frac{2\left(r+v{t}_m+\frac{1}{2}a{t}_m^2\right)}{c},t_m\right]\right\}\right\} \\ r\in \left[r_{\min },r_{\max }\right],v\in \left[v_{\min },v_{\max }\right],a\in \left[a_{\min },a_{\max }\right] \end{gathered}$$

(9)

where $r$, $v$, and $a$ represent the search values of $r_0$, $v_0$, and $a_0$. Meanwhile, $\left[r_{\min },r_{\max }\right]$,$\left[v_{\min },v_{\max }\right]$, and $\left[a_{\min },a_{\max }\right]$ denote the search scope. The operations of RFRFT include three-dimension (3D) search and FRFT, which are incredibly complex and time-consuming.

The principle of RLVD is similar to RFRFT, except that LVD is used instead of FRFT, as shown in Equation (10):

$$\begin{gathered} \mathrm{RLVD}=\underset{r,v,a}{\max }\left\{\mathrm{L}\mathrm{V}\mathrm{D}\left\{s_{pc}\left[\frac{2\left(r+v{t}_m+\frac{1}{2}a{t}_m^2\right)}{c},t_m\right]\right\}\right\} \\ r\in \left[r_{\min },r_{\max }\right],v\in \left[v_{\min },v_{\max }\right],a\in \left[a_{\min },a_{\max }\right] \end{gathered}$$

(10)

Like RFRFT, RLVD has a tremendous amount of computation, which affects the practical effect.

4. Coherent Integration Based on SKTLVD

4.1. The Principle of the Proposed Method

Aiming at the deficiencies of existing algorithms, a novel long-time coherent integration method for maneuvering targets is proposed. Its principle and main steps are as follows:

Step 1. Perform pulse compression on the received signal and apply FFT along the $t_k$ dimension.

We can obtain the signal $s_{pc}\left(f_k,t_m\right)$ in the range frequency-slow time domain, as shown in Equation (6). There are LRM, QRM and DFM in the envelope and phase of the signal at present, which need to be eliminated.

Step 2. Linear range walk correction via RFT.

In order to tackle LRM, the first-order coupling between $f_k$ and $t_n$ in Equation (5) must be eliminated. KT can effectively realize blind correction, but it will bring more trouble to QRM correction, as shown in Equation (7). It is difficult for KT-based methods to compensate LRM and QRM simultaneously. Therefore, the proposed method first borrows the idea of RFT and directly searches the target velocity to jointly compensate for first-order migration in the envelope and phase, which can avoid the Doppler ambiguity of the KT-based method at the same time. In addition, the search for the slant range will be completed later.

Then, we determine the searching scope of target velocity $\left[v_{\min },v_{\max }\right]$ based on related prior information, and the search interval is set to $\mathrm{\Delta }v=\lambda /2T$, where T denotes the total coherent integration time. Thus, the velocity search value can be expressed as:

$$v_s=v_{\min }+n_v\mathrm{\Delta }v n_v=1,2,...,N_v$$

(11)

where $N_v=\left(v_{\max }-v_{\min }\right)/\mathrm{\Delta }v$ denotes the number of searches. Furthermore, the compensation function $H_{v_s}\left(f_k,t_m\right)=\exp \left[j4\pi \left(f_k+f_c\right)v_s{t}_m/c\right]$ is constructed accordingly and multiplied by Equation (5) to obtain:

$$s_{v_s}\left(f_k,t_m\right)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\exp \left\{-\frac{j4\pi }{c}\left[\left(f_k+f_c\right)R_0\right.\right.\left.\left.+\left(f_k+f_c\right)\left(v_0-v_s\right)t_m+\frac{1}{2}\left(f_k+f_c\right)a_0{t}_m^2\right]\right\}$$

(12)

When $v_s=v_0$, the LRM in the envelope and DFS in the phase are completely compensated, and the highest coherent integration gain is obtained. Hence, $v_s$ is determined via maximizing the final coherent integration result.

Step 3. Range curvature correction using SKT.

For the convenience of analysis, suppose that $v_s=v_0$; then, Equation (12) can be written as:

$$s_{v_s}(f_k,t_m)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\exp \left\{-\frac{j4\pi }{c}\left[\left(f_k+f_c\right)R_0+\frac{1}{2}\left(f_k+f_c\right)a_0{t}_m^2\right]\right\}$$

(13)

Now, there is no first-order coupling term interference. Then, SKT is directly adopted to eliminate QRM, and the scaling expression of the SKT is defined as:

$$t_n=\sqrt{\frac{f_c}{f_k+f_c}}$$

(14)

Substituting (14) into (13) yields:

$$\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s}(f_k,t_n)=A_1\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\left(\frac{f_k}{B}\right)\exp \left\{-\frac{j4\pi }{c}\left[\left(f_k+f_c\right)R_0+f_c\frac{a_0}{2}{t}_n^2\right]\right\}$$

(15)

It can be seen that all coupling in the signal envelope has been eliminated after SKT. Hence, LRM and QRM are completely compensated. Applying IFFT on Equation (15) with respect to $f_k$, we can obtain:

$$\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s}\left(t_k,t_n\right)=A_3\sin c\left[B\left(t_k-\frac{2R_0}{c}\right)\right]\exp \left[-\frac{j4\pi }{\lambda }\left(R_0+\frac{a_0}{2}{t}_n^2\right)\right]$$

(16)

where $A_3$ is the signal amplitude. We can observe that the target energy has been concentrated in the same range cell. To further improve the coherent integration gain, it is necessary to continue the slant range search in RFT. With the given search interval $\mathrm{\Delta }r=c/2B$ and search scope $\left[r_{\min },r_{\max }\right]$, the extracted signal $\mathrm{S}\mathrm{K}\mathrm{T}\left(t_n\right)$ can be expressed as:

$$\begin{gathered} \mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n\right)=\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s}\left(\frac{2r_s}{c},t_n\right)=A_3\sin c\left[B\left(\frac{2r_s}{c}-\frac{2R_0}{c}\right)\right]\exp \left[-\frac{j4\pi f_c}{c}\left(R_0+\frac{a_0}{2}{t}_n^2\right)\right] \\ {r}_s=r_{\min }+n_r\mathrm{\Delta }r, n_r=1,2,\dots ,N_r, N_r=\left(r_{\max }-r_{\min }\right)/\mathrm{\Delta }r \end{gathered}$$

(17)

Step 4. Doppler walk correction employing LVD.

The signal after range search can be written as:

$$\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n\right)=A_3\exp \left(-\frac{j4\pi R_0}{\lambda }\right)\exp \left(-\frac{j2\pi a_0{t}_n^2}{\lambda }\right)$$

(18)

We can see that $\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n\right)$ becomes a chirp signal in the slow time dimension, where the centroid frequency is 0, and the chirp rate is $-2a_0/\lambda$. Thus, the LVD is conducted to compensate DFM, and the specific procedures are as follows:

• Calculate the parametric symmetric instantaneous autocorrelation function $R^C(t_n,\tau )$ of Equation (18):

  $$R^C(t_n,\tau )=\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n+\frac{\tau +b}{2}\right)\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}^{*}\left(t_n-\frac{\tau +b}{2}\right)=A_3^2\exp \left[-\frac{j4\pi a_0}{\lambda }\left(\tau +b\right)t_n\right]$$

  (19)

  where b denotes a constant time-delay related to a scaling operator, and τ denotes lag variable.
• Use variable substitution $t_n=\frac{t_b}{h(\tau +b)}$ to remove the coupling between $t_n$ and $\tau$ in the exponential phase term:

  $$R^C(t_b,\tau )=A_3^2\exp \left[-\frac{j4\pi a_0}{h\lambda }{t}_b\right]$$

  (20)
• Perform two-dimension (2D) FT on Equation (20) with respect to $t_b$ and $\tau$, the LVD is obtained:

  $$\mathrm{S}\mathrm{K}\mathrm{T}\mathrm{L}\mathrm{V}{\mathrm{D}}_{v_s,r_s}\left(f,\gamma \right)=A_4\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(f\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(\gamma +\frac{2a}{\lambda }\right)$$

  (21)

  where $A_4$ is the transformed amplitude after LVD. So far, all three migrations, i.e., LRM, QRM, and DFM, have been eliminated, and the spreading energy has been concentrated in the same cell. Hence, the target’s energy coherently integrates as a peak in the CFCR domain, and the peak value is located in $f=0,\gamma =-2a_0/\lambda$, which is consistent with the actual value of target motion parameters.

Step 5. Estimate unknown motion parameters include ${\widehat{v}}_0,{\widehat{r}}_0,{\widehat{a}}_0$ via maximizing the $\mathrm{S}\mathrm{K}\mathrm{T}\mathrm{L}\mathrm{V}{\mathrm{D}}_{v_s,r_s}\left(f,\gamma \right)$, as shown in Equation (22):

$$\left({\widehat{v}}_0,{\widehat{r}}_0,{\widehat{a}}_0\right)=\underset{v_s,r_s,f,\gamma }{\arg \max }\left|\mathrm{S}\mathrm{K}\mathrm{T}\mathrm{L}\mathrm{V}{D}_{v_s,r_s}\left(f,\gamma \right)\right|$$

(22)

When the searching range, velocity, and acceleration, respectively, match the actual value of the target, the LVD output can reach its maximum value.

Step 6. Carry out the constant false alarm ratio (CFAR) detector and take the maximum with the adaptive threshold to confirm a target:

$$\max \left|\mathrm{S}\mathrm{K}\mathrm{T}\mathrm{L}\mathrm{V}{D}_{v_s,r_s}\left(f,\gamma \right)\right|\underset{H_0}{\overset{H_1}{\gtrless }}\eta$$

(23)

where $\eta$ is the detection threshold given by the false alarm probability; if the maximum is smaller than the threshold, the moving target detection is announced. Otherwise, there will be no target.

In conclusion, the flow of the proposed SKTLVD can be summarized as follows:

In Figure 1, the green steps correspond to using RFT to compensate LRM, the red steps represent adopting SKT to eliminate QRM, and the yellow steps mean applying LVD to remove DFM.

4.2. Computational Complexity Analysis

In this part, the computational computation of the proposed method SKTLVD is analyzed in terms of the number of complex multiplications and compared with KTLVD, SKTRFT, RFRFT, and RLVD methods. Assume that the number of echo pulses, searching range, searching velocity, searching acceleration, and searching Doppler ambiguity are denoted as $M,N_r,N_v,N_a,N_d$. The number of FRFT rotation orders and LVD time lag are expressed as $N_p,N_{\tau }$. The amount of computation required for the main steps of each method is described below.

SKTRFT eliminates QRM via SKT ($O(N_{r}M{\log }_{2}M)$), estimates acceleration via FRFT ($O(N_r{N}_{p}M{\log }_{2}M)$), and eliminates the LRM via RFT ($O(N_r{N}_{v}M)$).

For KTLVD, its main procedures include include KT ($O(N_{r}M{\log }_{2}M)$), range search ($O(N_r)$), LVD operation ($O(N_{\tau }M{\log }_{2}M)$), and Doppler ambiguity search ($O(N_d)$).

The implementation of SKTLVD needs velocity search ($O(N_v)$), QRM correction via SKT ($O(N_{r}M{\log }_{2}M)$), range search ($O(N_r)$), and DFM correction via LVD ($O(N_{\tau }M{\log }_{2}M)$).

The main procedures of RFRFT contain $r-v-a$ 3-D search ($O(N_r{N}_v{N}_a)$) and FRFT operation ($O\left(N_{p}M{\log }_{2}M\right)$).

RLVD consists of a $r-v-a$ 3-D search ($O(N_r{N}_v{N}_a)$) and LVD operation ($O(N_{\tau }M{\log }_{2}M)$).

By overlaying the computations of each step, the computational complexity of the above methods is listed in

Table 2. Comparison of computational complexities.

Methods	Computation Complexity
SKTRFT	$O\left(N_{r}M{\log }_{2}M+N_r{N}_{a}M{\log }_{2}M+N_r{N}_{v}M\right)$
KTLVD	$O\left(N_d\left(N_{r}M{\log }_{2}M+N_r{N}_{\tau }M{\log }_{2}M\right)\right)$
SKTLVD	$O\left(N_v\left(N_{r}M{\log }_{2}M+N_r{N}_{\tau }M{\log }_{2}M\right)\right)$
RFRFT	$O\left(N_r{N}_v{N}_a{N}_{p}M{\log }_{2}M\right)$
RLVD	$O\left(N_r{N}_v{N}_a{N}_{\tau }M{\log }_{2}M\right)$

.

Under the assumption of N_r = N_v = N_a = N_d = N_p = N_τ = M, Figure 2 intuitively depicts the computational curves according to Table 2.

Consequently, the computational complexity of each method is ranked from low to high as follows: SKTRFT < KTLVD ≈ SKTLVD < RFRFT ≈ RLVD. It can be seen from Figure 2 that due to 3D search, RFRFT and RLVD consume the most considerable computation, which is not suitable for real-time processing. Compared with the above two joint compensation methods, KTLVD and SKTLVD avoid the search of acceleration and reduce the computational complexity by two orders of magnitude. Meanwhile, SKTRFT has the lowest amount of computation because it sacrifices the integration performance and adopts a more straightforward RFT operation.

5. Simulation Results and Analysis

In this section, we carry out several numerical experiments to verify the performance of the proposed method under the complex zero-mean white Gaussian noise environment, where the simulated parameters are given in

Table 3. Simulation parameters.

Parameters	Value
Carrier frequency	3 GHz
Bandwidth	10 MHz
Sample frequency	20 MHz
Pulse repetition frequency	1000 Hz
Pulse duration	50 μs
Number of pulses	1024
Initial slant range	100 km
Radial velocity	200 m/s
Radial acceleration	100 m/s2

. Furthermore, other popular coherent integration methods, i.e., MTD, RFT, KTLVD, SKTRFT, RFRFT, and RLVD, are also given for comparative analysis.

5.1. Migration Correction Ability

First, the effect of migration correction includes LRM correction, QRM correction, and DFM correction, which are depicted in Figure 3, and the noise is ignored temporarily for the convenience of observation.

Figure 3a shows the target trajectory after pulse compression, which indicates severe LRM and QRM due to the high speed and maneuverability of the target. Figure 3b shows the result after velocity search, first-order phase compensation, and SKT, where the target energy could be seen in the same range cell after LRM and QRM correction. Figure 3c is the frequency spectrum of $\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n\right)$, which is a typical LFM signal. It can be seen that there is severe DFM, and the target energy is dispersed in different frequency cells. Fortunately, from Figure 3d, we can find that a significant peak is formed in the CFCR domain after LVD operation on the $\mathrm{S}\mathrm{K}{\mathrm{T}}_{v_s,r_s}\left(t_n\right)$, confirming the effectiveness of DFM correction.

5.2. Coherent Integration for a Weak Target

In this subsection, the long-time coherent integration performance of the proposed method for a single weak maneuvering target is evaluated, and the SNR after pulse compression is −16 dB.

Figure 4a shows the result of MTD, in which the target is totally buried in the noise because all migrations are not handled. It is evident from Figure 4b that the RFT cannot integrate the target energy without eliminating QRM and DFM. In addition, because the error caused by neglecting QRM sacrifices some energy, KTLVD is also invalid, which is shown in Figure 4c. Affected by the approximation error that arose out of simplifying the first-order coupling, although SKTRFT integrates energy, it is still not enough to detect the weak maneuvering target in Figure 4d. On the other hand, Figure 4e presents the result of the proposed SKTLVD. We can clearly see that the target energy is focused as one prominent peak via long-time coherent integration, making the target magnify out from the noise and contributing to target detection and parameters estimation. The above simulation results indicate the integration performance of SKTLVD for a weak maneuvering target.

5.3. Coherent Integration for Multi-Targets

Furthermore, we analyze the coherent integration performance of the proposed method for multiple targets in Figure 5, where the motion parameters of two maneuvering targets are listed in

Table 4. Motion parameters of targets.

Motion Parameters	Target A	Target B
Initial slant range	100 km	105 km
Radial velocity	200 m/s	150 m/s
Radial acceleration	100 m/s2	80 m/s2
SNR (after pulse compression)	10 dB	5 dB

.

Figure 5a shows the result after pulse compression, which indicates that serious RM occurs on the trajectories of both targets A and B. Meanwhile, we can find that the energy of targets A and B are concentrated in their initial range cell from Figure 5b. Figure 5c,d display the proposed method integration results of targets A and B. The initially dispersed target energy has integrated into two significant peaks, which demonstrates the ability of SKTLVD for multi-target integration.

5.4. Target Detection Performance

The detection performances of the proposed method, MTD, RFT, RFRFT, RLVD, KTLVD, and SKTLVD are investigated via Monte Carlo trials. SNRs after pulse compression vary from −25 to 10 dB with the step of 1 dB, and for each SNR value, 500 independent simulations are performed. In addition, the constant false alarm probability is set as $10^{-2}$. Figure 6 shows the detection probability curves of different methods versus SNR.

It is apparent that the MTD has the poorest detection probability without any compensation, while RFT is only slightly better than MTD due to ignoring the QRM and DFM caused by acceleration. It is noteworthy that the proposed method SKTLVD outperforms the RFRFT thanks to the higher gain of LVD operation. Moreover, since there is no approximation error in the motion model, the detection performance of the proposed method is superior to KTLVD and SKTRFT. Specifically, when SNR = −15 dB, the detection probabilities of SKTRFT, KTLVD, and SKTLVD are 0%, 17%, and 96%, respectively, which proves that the proposed method has more vital detection ability. Furthermore, RLVD has the optimal detection capability, but the computational complexity is enormous, limiting its practical application. Compared with the RLVD, the proposed method has close detection capability and much lower computational complexity.

5.5. Real Data Processing

The real measured radar data are used to further illustrate the practicability of the proposed method. The data were collected by VHF-band radar, which has remarkable detection performance for stealth targets, where the system parameters of the radar are given in

Table 5. Radar system parameters for real data.

Parameters	Value
Carrier frequency	317 MHz
Bandwidth	2 MHz
Sample frequency	5 MHz
Pulse repetition frequency	1500 Hz
Integration time	1 s

.

The real data processing results are illustrated in Figure 7. Figure 7a shows the result of MTD, in which the target energy is dispersed over multiple range and velocity cells, which creates difficulties for target detection. Furthermore, Figure 7b gives the result after velocity search and compensation. Since the LRM has been eliminated, the RM is alleviated, and the integration performance improves. However, the energy is not fully concentrated because QRM and DFM have not been corrected. Moreover, Figure 7c shows the result after SKT and LVD; it can be seen that the target energy is well-focused in the same centroid frequency-chirp rate cell, which significantly benefits target detection. The centroid frequency is transformed into the corresponding target velocity for visual display, and the chirp rate is transformed into the corresponding target acceleration. Finally, Figure 7d shows the coherent integration results in the velocity-acceleration domain. We can see a significant peak formed via coherent integration at the target, and from the peak location, we could obtain $a_0=10.1 \mathrm{m}/{\mathrm{s}}^2$. Consequently, we can conclude that the target coherent integration and detection of the proposed method is valid.

6. Conclusions

A long-time coherent integration method is proposed in this paper to address the problem of detecting maneuvering targets. Drawing on the superiority of the existing independent and joint compensation methods, the proposed method SKTLVD adopts RFT to compensate the LRM jointly and performs SKT and LVD to compensate QRM and DFM independently. The simulation results show that compared with the conventional independent compensation methods, i.e., SKTRFT, and KTLVD, the detection probability of SKTLVD is increased by 96% and 79%, respectively, when SNR = −15 dB. On the other hand, compared with the traditional joint compensation methods, i.e., RFRFT, and RLVD, the computational efficiency of SKTLVD is improved by two orders of magnitude when the integration performance is close. Moreover, the measured data processing results demonstrate that the proposed method possesses satisfactory integration and detection performance. Since the target motion components above the second-order are ignored, the possible future work might concern the long-time coherent integration for the target with jerk motion and deal with cubic-range curvature quadratic Doppler frequency migration.