Sea Surface Moving Target Detection Using a Modified Keystone Transform on Wideband Radar Data

Abstract

The echoes collected by wideband radar systems provide abundant information on target scatterers, which is beneficial to target detection, classification, and recognition. However, as the radar range resolution increases, range cell migration (RCM) during the coherent integration (CI) period happens much easier, which may cause a degradation of target detection probability. In addition, due to the target’s orientation and structure relative to the radar, the distribution characteristics of the target scatterers in high-resolution range profiles (HRRPs) and the detection window length may vary from pulse to pulse, which may reduce the performance of traditional energy integration (EI) detectors. To solve those problems, moving range-spread target (RST) detection combining the modified keystone transform (MKT) and improved EI (IEI) is proposed in this paper. Firstly, based on waveform entropy minimization, MKT using hunter–prey optimization (HPO) is introduced to reduce the CI gain loss. The target Doppler ambiguity factor is estimated using such an effective optimization technique. Then, the IEI detector optimized by the adaptive threshold and detection window is utilized to achieve target detection, which minimizes the sensitivity of the traditional EI detector to the detection window length. The proposed method significantly improves the performance of moving RSTs in sea clutter without prior knowledge of the target Doppler ambiguity factor. Experiments are conducted by comparing the proposed method with other competing methods on both simulation data and real sea clutter data. The results demonstrate that the proposed method can obtain the CI more efficiently and has a higher detection probability.

1. Introduction

Adaptive detection of moving range-spread targets (RSTs) in sea clutter has received considerable attention in recent decades, such as with hypersonic cruise missiles observing marine targets. The relative velocity between the radar and the target contributes phase terms to the received signals, causing range cell migration (RCM) and defocusing of the target’s high-resolution range profiles (HRRPs). Thus, it would be difficult to integrate multiple pulses coherently due to the RCM effect, which may decrease the target detection performance [1,2].

To solve this problem, some range migration effect elimination methods have been developed that can improve the CI gain [3,4], such as the Hough transform [5] and the track-before-detect technique [6]. Nevertheless, these methods cannot compensate for the phase fluctuation, so the detection performance in a low signal-to-clutter ratio (SCR) environment is poor. The narrow-band cross-ambiguity function (NBCAF) and the wide-band cross-ambiguity function (WBCAF) are widely applied for CI detection in a low SCR environment. However, they have low precision in parameter estimation in practice [7]. To resolve this problem, the Radon transform [8] is introduced, but its high complexity may make it inapplicable in real-time processing. Other effective methods are applied for moving target detection, such as adjacent cross-correlation function (ACCF) [9], scaled inverse Fourier transform (SCIFT) [10], sequence reversing transform (SRT) [11,12,13], keystone transform (KT) [14,15], etc. In particular, the correlation between two adjacent pulses at low SCR is weak, and the signal envelope cannot be accurately aligned; thus, ACCF is only suitable for target detection in a high SCR environment. SCIFT and SRT can achieve the RCM correction by fast Fourier transform (FFT) and inverse FFT, and the computational burden is low. However, SCIFT and SRT also have high requirements for SCR. KT is an effective algorithm to improve the target energy in a long-time CI period, which is helpful to the increase in SCR gain. However, KT may be invalid without a priori knowledge of the Doppler ambiguity factor when the target velocity is ambiguous. To solve this problem, in [16], a new KT algorithm by introducing the scale transformation factor and the Doppler frequency compensation function is proposed, but the simulation results show that the CI gain is heavily sensitive to the ambiguity of the inter-pulse Doppler frequency, thus the robustness is low. In [17,18], the KT algorithm based on entropy minimization is proposed to deal with defocusing due to RCM. However, the time costs may be high due to iterative searching. Recently, some other energy integration (EI) algorithms have been developed, such as sparse fractional ambiguity function (SFRAF) [19], square-law noncoherent pulse integration [20], and modified axis rotation transform (MART) [21]. SFRAF enables higher-order phase signals to be well aggregated, but it needs the prior sparsity of the returns, which is often unknown and may vary in realistic applications. The square-law noncoherent pulse integration method is proposed based on the assumption that the arbitrary partial correlation for both target radar cross section (RCS) and clutter speckle can be accommodated. However, the detection background in real civilian and military applications is rather complex (thermal noise and compound clutter), so the assumption of knowing the correlation for both target RCS and clutter speckle is often invalid. MART can reduce the RCM by searching the range search area and rotation angle, but its computational complexity is huge, which results in difficulties in adaptation to large-scale signal processing.

In real radar applications, to further improve the performance of moving RSTs, an effective detection method is another key optional way [22,23]. In [24,25], two integration detectors are presented to detect RSTs in Gaussian clutter. One is referred to as an EI detector, which is generated by comparing all target scatterer energy within the detection window with a threshold. The other is known as the scatterer density-dependent generalized likelihood ratio test (SDD-GLRT) detector. Therein, the experimental results indicate that SDD-GLRT outperforms the EI detector for a sparse RST. Nevertheless, SDD-GLRT is devised based on exponential and logarithmic operations, with higher complexity. Moreover, it is exploited under the assumption that the target scatterer density is known, so this assumption in real radar applications can hardly be satisfied. Recently, in [26], based on the radar system’s prior structure knowledge, a persymmetric adaptive detector for RSTs in compound Gaussian clutter with inverse Gaussian texture [27] is proposed, and numerical results indicate that it possesses a constant false alarm rate (CFAR) property under the assumption that the clutter covariance matrix (CCM) and the inverse Gaussian texture are known. However, CCM and the inverse Gaussian texture of real sea clutter are difficult to acquire in advance. Moreover, target detection essentially belongs to a binary hypothesis testing case. In statistical detection theory, some detection problems based on the group invariant’s decision principle have been analyzed in depth, including the adaptive radar detection problems in Gaussian disturbance plus structured interference [28,29] and Gaussian disturbance with structured covariance matrix [30], but the detection performance of RSTs needs to be further verified. Motivated by it, in [31], one elegant attempt to extend the framework of invariant testing to RST detection is exploited. However, it is derived based on the assumption that the target energy is completely contained in radar detection cells, which may not always be satisfied in practical scenarios. The frequency-diverse array multiple-input multiple-output radar has the advantage of additional target range information, which is beneficial to detection performance improvement. In this term, an adaptive detector based on a two-step GLRT criteria is derived [32]. In [33], an adaptive detector for moving RSTs in a clutter-plus-noise environment is proposed. Moreover, by resorting to the Rao and Wald tests [34], two Bayesian frameworks for moving RSTs are devised. In [35], two adaptive target detectors, referred to as the structured GLRT and the unstructured GLRT, are studied. These methods have provided various solutions for target detection problems, but there is no specific method to achieve the best solution for all target detection problems. Therefore, how to efficiently improve the performance of RSTs is still an open and significant topic.

In this paper, we propose a sea surface moving target detection method using a modified keystone transform. In detail, the proposed method is accomplished by a two-step procedure. Firstly, based on waveform entropy minimization, a modified keystone transform (MKT) using hunter–prey optimization (HPO) is introduced to reduce the CI gain loss without prior knowledge of the Doppler ambiguity factor. In detail, the HPO has been developed to efficiently estimate the Doppler ambiguity factor for moving RSTs, and consequently, the MKT in this paper has lower time costs than the conventional KT algorithm based on entropy minimization [17,18]. Furthermore, the function of waveform entropy minimization is used as the fitness function. Then, an IEI method optimized by the adaptive threshold and detection window is exploited, which effectively reduces the sensitivity of the traditional EI detector to the detection window length, and decreases the clutter energy integration, which is beneficial to the SCR improvement. Finally, the performance assessments are carried out by comparing some conventional methods using the simulated clutter and the real sea clutter. The numerical results show that the proposed method can achieve better detection performance for moving RSTs in a clutter background.

The rest of this paper is organized as follows: In Section 2, the detection problem is introduced, and the received signal model of moving RSTs and the clutter model are demonstrated. In Section 3, the proposed method combining MKT and IEI is put forward. In Section 4, the validity and efficiency of MKT are verified by several experiments, and then the detection results of the proposed method and other traditional detection methods in a clutter environment are shown. Finally, Section 5 summarizes this paper.

2. Problem Formulation

To provide theoretical and data support for moving RST detection, in this section we introduce the radar system model, mainly including the received signal model of moving RSTs and clutter mode.

2.1. Received Signal Model of Moving RSTs

Assuming that the wideband radar transmits linear frequency-modulated (LFM) pulses, which can be expressed as [36,37],

$$s(t)=\exp (j2\pi f_{0}t+j\pi \epsilon t^2),0\le t\le \tau$$

(1)

where $\tau$ is the pulse width, $t$ is the fast time in the range time, $\epsilon$ and $f_0$ denote the frequency modulation rate and carrier frequency, respectively. For the wideband radar system, the target may be considered as multiple physical scatterers distributed over a few range cells, which can be described by a scattering center model [38]. The target returned signal at the mth pulse after down-converting can be represented by

$$s_r(t)={\displaystyle \sum _{l=1}^L{A}_l}(t)\exp [-j2\pi f_0{\tau }_l+j\pi \epsilon {(t-{\tau }_l)}^2]$$

(2)

where $L$ is the number of range cells occupied by the target, $A_l(t)$ is the target scattering center amplitude of a RST in the lth radar range cell, which is related to the radar polarization way, target material, incidence angle, etc. ${\tau }_l=\frac{2R_l}{c}=\frac{2(R_{0,l}+v{t}_a)}{c}$ is the time delay of the target’s lth scattering center, $c=3\times {10}^8$ m/s is the light speed, $R_{0,l}$ is the initial radial range between the radar and the target’s lth scattering center, $v$ is the target velocity along the radar line of sight (LOS), and $t_a$ is the slow-time variable.

After matched filtering, the target returned signal in the slow time-range frequency domain can be written as follows:

$$S_r(f_r,t_a)={\displaystyle \sum _{l=1}^L{A}_l}(f_r)\exp [-j4\pi (f_r+f_0)\frac{R_{0,l}+v{t}_a}{c}]$$

(3)

From Equation (3), it can be observed that the RCM problem arises from the coupling between range frequency $f_r$ and slow time variable $t_a$ in the exponential term, which would reduce the CI gain and RST detection performance.

2.2. Clutter Model

Through the abundant analyses of clutter data, common statistical models of the clutter can be expressed by log-normal, Gamma, Weibull, K distributions, etc. Since Weibull-distributed clutter can not only describe the scattered reflections collected by wideband radars excellently but is also suitable for the different types of clutter environments at medium and low grazing angles, in this paper, the Weibull distribution of clutter is used to assess the detection performance of moving RSTs.

Generally, clutter can be characterized by its probability density function (PDF). The PDF of the Weibull-distributed clutter can be written as:

$$f(x)=(\frac{\mu }{\beta }){(\frac{x}{\beta })}^{\mu -1}\exp [-{(\frac{x}{\beta })}^{\mu }],\begin{array}{cc}& x\ge 0\end{array}$$

(4)

where $\mu >0$ and $\beta >0$ denote the shape parameter and scale parameter, respectively. More specifically, the shape parameter $\mu$ reflects the heavy tail of the clutter distribution, and the scale parameter $\beta$ represents the clutter power.

3. Proposed Approach

3.1. RCM Correction Based on Modified KT

3.1.1. Review of the Traditional KT Algorithm

KT is well known for its capability of eliminating RCM without the kinetic information of the moving targets, which can effectively improve the CI gain in a low SCR environment.

We define that as:

$$t_a=\frac{f_0}{f_0+f_r}{t}_m$$

(5)

where $f_r$ is the range-frequency variable, $t_m$ is the new slow-time variable. It can be observed that the range corresponding to $t_m$ is scaled by the factor of $f_0+f_r/f_0$ from the initial slow-time variable $t_a$. Substitute $t_a$ into Equation (3), and then we can get $S_r(f_r)$ in the range time dimension expressed as

$$\begin{array}{ll}{S}_r\left(f_r,t_m\right)& ={\displaystyle \sum _{i=1}^L{S}_r\left(f_r,\frac{f_0}{f_r+f_0}{t}_m\right)}\\ & ={\displaystyle \sum _{i=1}^L{A}_i\exp \left[-j\frac{4\pi \left(f_r+f_0\right)R_i}{c}\right]\exp \left(j\frac{4\pi f_0}{c}v{t}_m\right)}\exp (-j2\pi F\frac{f_0}{f_0+f_r})\hfill \end{array}$$

(6)

where $F=\mathrm{round}(v/v_{blind})$ is the Doppler ambiguity factor, $v_{blind}=\frac{c}{2f_0{T}_p}$, $T_p$ is the pulse repetition interval. We can find that the coupled term between the range frequency $f_r$ and slow time variable $t_a$ is removed.

The remapping process for KT is shown in Figure 1.

From Figure 1a, it is observed that the equiphase lines of the target’s returns are not parallel to each other before KT. After KT processing, the rectangular target’s return samples are obtained, as shown in Figure 1b. The equiphase lines of radar returns are parallel, and the gap between each line is not dependent on the range frequency $f_r$. Therefore, the coupling between the range frequency $f_r$ and slow time variable $t_a$ in the exponential term in Equation (3) can be removed, and then the range migration during the long-time CI period can be corrected. However, prior knowledge of the Doppler ambiguity factor $F$ in Equation (6) is unknown in practice, which greatly limits the application of KT. To solve this problem, the KT algorithm based on waveform entropy minimization is derived [17,18].

3.1.2. KT Algorithm Based on Waveform Entropy Minimization

Wave entropy is generally used to describe system uncertainty and plays a key role in the fields of cybernetics, astrophysics, life sciences, etc. In the radar community, waveform entropy could be utilized to characterize the radar signal power’s dispersion along the parameter axis.

Assume that mth radar return after pulse compression is $X=\left\{x_1,\dots ,x_i,\dots ,x_{N’}\right\}$, $N’=T_p\cdot f_s$ is the sampling number, and $f_s$ is the sampling frequency. The waveform entropy of mth radar return can be expressed as:

$$E(X)=-{\displaystyle \sum _{i=1}^{N’}{p}_i}\log p_i$$

(7)

where $p_i=\left|x_i\right|/\Vert X\Vert$ and $\Vert X\Vert ={\displaystyle \sum _{i=1}^{N’}\left|x_i\right|}$.

From Equation (2), it is observed that the target return power is uniformly distributed along the parameter axis (such as the fast time domain $t$) under a low-SCR environment. In this case, the return power’s dispersion is strong, and the return uncertainty is large, so the waveform entropy value is relatively larger. On the contrary, the return power is more concentrated in a high-SCR environment, so the waveform entropy of target returns tends to be smaller. Therefore, the waveform entropy can be used to quantitatively describe the concentration of the target return power after CI processing. In this term, the minimum waveform entropy in HRRPS after CI in Equation (7) corresponds to the estimated Doppler ambiguity factor $F$ in Equation (6) for moving targets. Afterward, the RCM can be corrected via KT during a long coherent processing interval with the estimated Doppler ambiguity factor $F$, which will be verified in Section 4. However, its computational complexity may be very high due to iterative searching. To estimate the Doppler ambiguity factor efficiently and effectively, the proposed MKT utilizes the HPO optimization technique. The estimation process of the Doppler ambiguity factor has two main blocks: fitness function calculation and optimization algorithm application.

3.1.3. MKT Algorithm

• Fitness Function

The KT algorithm based on waveform entropy minimization could be considered an optimization issue, with the solutions being the optimum values of the undetermined algorithm parameters. The fitness function could be used to determine whether or not the estimated Doppler ambiguity factor matches the real value. The fitness function can be expressed as:

$$\mathrm{G}(X)=\min (\mathrm{E}(X))$$

(8)

From Equation (8), the fitness function can be formed by calculating the Doppler ambiguity factor that results in the minimum entropy.

2.

HPO Algorithm

The HPO algorithm is a new intelligent optimization algorithm with the advantages of high optimization ability and fast convergence speed, and it has drawn important attention recently [40]. The mathematical formulation of the HPO can be expressed as:

• Step 1: Randomly initialize the population in the HPO, which can be expressed as:

$$y_a=rand(1,d).*(ub-lb)+lb$$

(9)

where $d$, $ub$, and $lb$ are the tested problem’s dimensions, solution’s upper and lower boundaries, respectively.

• Step 2: Perform the search mechanism, which consists of two random steps:

Hunter search mechanism: The mathematical formulation of the hunter search mechanism can be shown as follows:

$$y_{a,b}(t’+1)=y_{a,b}(t’)+0.5[(2CZ{P}_{pos}(b)-y_{a,b}(t’))+(2(1-C)Z{\eta }_{(b)}-y_{a,b}(t’))]$$

(10)

where $a\in [1,N_1]$ is the number of the current search agent, $N_1$ is the number of the search agents, $b\in [1,d]$ is the number of the current variable, $y_{a,b}(t’)$ and $y_{a,b}(t’+1)$ represent hunter’s current and next positions accordingly. $Z=R_2\otimes IDX+{\overrightarrow{R}}_3\otimes (~IDX)$ represents an adaptable variable, where $IDX=(P=\begin{array}{cc}=& 0\end{array})$ is the index value, and $P={\overrightarrow{R}}_1<C$, ${\overrightarrow{R}}_1$ and ${\overrightarrow{R}}_3$ are random vectors within $[0,1]$, $R_2\in [0,1]$ is a random value. $P_{pos}{}_{(b)}$ and ${\eta }_{(b)}$ are the position of the victim and the average of all possible places, respectively. $C$ can be used to achieve the balance between exploration and exploitation, which can be computed as follows:

$$C=1-it(\frac{0.98}{MaxIt})$$

(11)

where $it$ and $MaxIt$ represent the current iteration value and maximum number of iterations, respectively.

The location $P_{pos}{}_{(b)}$ can be obtained when first calculating the mean of all positions as follows:

$$\eta =\frac{1}{n}{\displaystyle {\sum }_{a=1}^n{\overrightarrow{y}}_a}$$

(12)

Then the search agent’s distance from the mean position could be calculated as follows:

$$D_{euc(a)}={({{\displaystyle {\sum }_{b=1}^d(y_{a,b}-{\eta }_b)}}^2)}^{\frac{1}{2}}$$

(13)

The search agent can be determined as follows:

$${\overrightarrow{P}}_{pos}={\overrightarrow{y}}_a|aissorted{D}_{euc}(kbest)$$

(14)

where $kbest$ is a new random variable:

$$kbest=round(C\times N_1)$$

(15)

Prey search mechanism: The prey could change positions to flee to a secure position when it is assaulted, which can be described as the following:

$$y_{a,b}(t’+1)=T_{pos(b)}+CZ\cos (2\pi R_4)\times (T_{pos(b)}-y_{a,b}(t’))$$

(16)

where $y_{a,b}(t’)$ and $y_{a,b}(t’+1)$ represent the prey’s current and next positions; consequently, $R_4\in [0,1]$ is a random value, $T_{pos(b)}$ is the optimum global position, which can be considered the finest position where the prey is safe.

• Step 3: Determine the hunter and prey, which are evaluated based on a random process written as follows:

$$y_a(t’+1)=\{\begin{array}{l}{y}_a(t’)+0.5[(2CZ{P}_{pos}-y_a(t’))+(2(1-c’)Z\eta -y_a(t’))];\begin{array}{c}\begin{array}{cc}if& R_5<\beta \end{array}\end{array}\\ T_{pos}+CZ\cos (2\pi R_4)\times (T_{pos}-y_a(t’));\begin{array}{c}else\end{array}\end{array}$$

(17)

where $R_5\in [0,1]$ is a random variable. If $R_5<\beta$, the search agent is regarded as a hunter, otherwise, the search agent is regarded as a prey. After determining the hunter and prey, the upcoming location of the search agent can be obtained from Equation (17).

3.2. Detection Method

3.2.1. Traditional EI Method

Assuming that the radar returns occupy multiple contiguous range cells, after pulse compression, the returned signal can be defined as $s(n),n=1,\dots ,N$, $N$ is the total range cell number in the detection window.

The test statistic ${\lambda }_{EI}$ of the EI method can be expressed as:

$$\{\begin{array}{l}{\lambda }_{EI}=\frac{1}{{\sigma }^2}{\displaystyle \sum _{n=1}^N{\left|s(n)\right|}^2}\\ \{\begin{array}{l}{H}_1:{\lambda }_{EI}\ge T\\ H_0:{\lambda }_{EI}<T\end{array}\end{array}$$

(18)

where ${\sigma }^2$ is the clutter power, $T$ is the detection threshold, which can be obtained by applying Equation (18) to the pure clutter data by Monte Carlo tests. $H$ denotes the binary hypothesis test, expressed as follows:

$$\{\begin{array}{l}{H}_0:s(n)=w(n)\begin{array}{cc},& (\begin{array}{cc}\mathrm{no}& \mathrm{target}\end{array})\end{array}\\ H_1:s(n)=x(n)+w(n)\begin{array}{cc},& (\begin{array}{cccc}\mathrm{the}& \mathrm{target}& \mathrm{is}& \mathrm{present}\end{array})\end{array}\end{array}$$

(19)

where $x(n)$ and $w(n)$ denote the target HRRP and clutter at the nth range cell, respectively.

Moreover, ${\lambda }_{EI}$ in Equation (18) obeys ${\chi }^2$ distribution with the degree of freedom $2N$, and its PDF is:

$$f_{{\lambda }_{EI}}(x)=\exp (-\frac{x}{2}){\displaystyle \sum _{i=0}^{+\infty }\exp }(-\frac{{\displaystyle \sum _{n=1}^N{\left|s(n)\right|}^2}}{2{\sigma }^2}){\displaystyle \sum _{i=0}^{+\infty }\frac{1}{i!}}{\left(\frac{{\displaystyle \sum _{n=1}^N{\left|s(n)\right|}^2}}{2{\sigma }^2}\right)}^i(\frac{x^i}{2^{i+1}i!})\begin{array}{cc},& x\ge 0\end{array}$$

(20)

Thus, the detection probability $P_d$ can be expressed as:

$$P_d={\displaystyle \underset{T}{\overset{+\infty }{\int }}{f}_{{\lambda }_{EI}}}(x)dx=\exp \left(-\frac{{\displaystyle \sum _{n=1}^N{\left|s(n)\right|}^2}}{2{\sigma }^2}\right)\exp (-\frac{T}{2}){\displaystyle \sum _{i=0}^{+\infty }\frac{1}{i!}}{\left(\frac{{\displaystyle \sum _{n=1}^N{\left|s(n)\right|}^2}}{2{\sigma }^2}\right)}^i{\displaystyle \sum _{k=0}^i\frac{1}{k!}}{(\frac{T}{2})}^k$$

(21)

In Equation (21), it can be easily found that the detection performance of the traditional EI detector is relevant to the detection window length (the range cell number $N$ in the detection window).

Generally, target HRRPs convey structure signatures, such as target sizes and their scattering center distributions. Due to the target motion, HRRP distributions vary from pulse to pulse; thus, the range cell number $N$ in the detection window would be changed in practice. However, the range cell number $N$ in the detection window of the traditional EI method is preset and fixed and is not adaptive during all the detection processes. As a result, the detection performance is degraded. Moreover, for the moving RSTs, due to the range walking during a long integration period, the detection window is generally longer than the supporting area of target returns. In this case, the traditional EI method would accumulate the excess clutter energy, resulting in integration loss and a decrease in SCR. To handle these problems, an IEI method is proposed that is optimized by the test statistics, adaptive threshold, and detection window.

3.2.2. IEI Method

3.

Optimized Test Statistic

The test statistic of the traditional EI method in Equation (18) can be optimized as follows:

$${\lambda }_{}^{’}=\frac{{\displaystyle \sum _{i=1}^N{\left|s(n)\right|}^2}}{N{\sigma }^2}$$

(22)

From Equation (18), we can see that the detection statistic ${\lambda }_{EI}$ of the traditional EI is seriously sensitive to the detection window length $N$. Moreover, since the stability of test statistics is beneficial for detection performance improvement, from Equation (22), it can be seen that the probability of wild values in the detection statistics will be reduced to a certain extent compared with that of the detection statistics of the traditional EI in Equation (18).

4.

Adaptive Threshold

In order to ensure detection performance, it is necessary to accumulate the echo energy of the target scatterings and reduce the accumulation of clutter as much as possible. After determining the detection window where the target is located, the average power $P$ of the clutter can be estimated with the echoes in the adjacent sliding window (assuming that there is no target in the adjacent sliding window). On the one hand, for HRRPs in the detection window, the range cells with power greater than the average power $P$ are considered the cells occupied by target scatterings and are involved in energy accumulation. In this way, the energy of small target scatterings may be accumulated to some extent. On the other hand, the range cells with a power smaller than the average power $P$ are regarded as the cells occupied by clutter and are not involved in energy accumulation, which could decrease the excess clutter energy accumulation and be beneficial to SCR improvement.

5.

Adaptive Detection Window

In practical radar applications, due to the target’s attitude relative to the radar, the distribution characteristics of the target scatterings in HRRPs may vary from pulse to pulse. Therefore, the traditional EI detector based on the principle that the threshold is independent of the detection window length is unreasonable; thus, the adaptive detection window is necessary.

Briefly, the adaptive detection window can be achieved by matching the detection threshold to the detection window length $N$. The adaptive detection window is composed of three main procedures, as follows:

• Step 1: For a certain type of target, set the sliding window length area and sliding step value on the assumption that the target size range is known, which does not require obtaining the precise target size.
• Step 2: Calculate the test statistics in all sliding windows, make the sliding window corresponding to the maximum test statistic the detection window, perform the adaptive threshold processing, and then save the maximum test statistic as the threshold $T$ and the corresponding detection window length $N$.
• Step 3: Make a decision. Under the hypothesis $H_1$ (with different SCR cases), if the test statistic ${\lambda }_{tar}>T$(the detection window length corresponding to the test statistic ${\lambda }_{tar}$ should be close to or equal to that of the threshold $T$), then the target is absent. It is noted that the detection statistic ${\lambda }_{tar}$ and the threshold $T$ are obtained in the same way, the only difference is that the echoes of the threshold $T$ do not contain the target signal but are pure clutter.

3.3. Detailed Procedure of the Proposed Method

The detection procedure of the proposed method for moving RSTs is presented in this section. First, the range migration during the CI interval is corrected via MKT, and SCR in HRRPs can be effectively improved. Then the IEI method is optimized by the test statistics, the adaptive detection window, and the threshold to achieve the moving RST detection by Monte Carlo tests, which decreases the sensitivity of the traditional EI detector to the detection window length and reduces the accumulation of clutter energy. In short, the proposed method combining MKT and IEI in this paper would provide a promising scheme for improving the moving RST detection performance in a clutter environment. The flowchart of the proposed method is given in Figure 2.

4. Experiments and Discussions

In this section, the simulation and the real sea clutter data collected by the research institute are used to verify the performance of the proposed method combining MKT and IEI. Firstly, the simulated clutter, the real pure sea clutter, and the simulated signals of moving RSTs are generated. Secondly, the validity of MKT is verified in detail, and its performance is compared with ACCF and traditional KT. Finally, the detection performance of the proposed method is compared with that of the EI detector combining the traditional RCM correction methods, such as ACCF and KT, with different Doppler ambiguity factors.

4.1. Experiment Analysis Based on Simulated Sea Clutter

In the simulation experiments, assuming that the LFM signal is employed as the transmitted signal in the radar system, the radar parameters are given as follows: Carrier frequency $f_c=34$ GHz, radar bandwidth $B=300$ MHz, sample frequency $f_s=2$ GHz, pulse repetition frequency $f_p=500$ Hz, pulse number during the CI period $M’=64$, the radial velocity of the target $v=30$ m/s, thus the real Doppler ambiguity factor of the target $F=2T_{p}v{f}_c/c=14$ ($T_p$ is the pulse repetition interval, $T_p=1/f_p=2$ ms), the number $L$ of range cells occupied by the target are 3. The sea clutter obeys the Weibull distribution, and the clutter shape parameter $\mu$ and scale parameter $\beta$ are 1.5 and 1, respectively. For the HPO algorithm, the maximum iteration number $MaxIt$ is 10, the search agent number $N_1$ is 30, the lower boundary $lb$ is 1, the upper boundary $\mu b$ is 100, and the dimension $d$ is 1. For conventional KT using entropy with iterative search, the search range of Doppler ambiguity factors is set $[1,100]$, the interval is 1. Moreover, SCR is defined as follows [32]:

$$\mathrm{SCR}=10\log 10\left(\frac{{\displaystyle \sum _{n=1}^L{\left|s(n)\right|}^2}}{L{\sigma }^2}\right)$$

(23)

Moreover, in the practical detection problem, the distribution of target scatterings is unknown: First, the target type is unknown; second, due to the target’s orientation and structure relative to the radar, the distribution of its scatterings in each range cell varies greatly from pulse to pulse, even for the same target. Thus, the detection window length should not be fixed. To be closer to the practical application, in the simulation, the amplitude of each target scattering is varied from pulse to pulse, and as a consequence, the length of the detection window also gets changed.

Assuming that the single-sample SCR after matched filtering is 4.77 dB. The relationships between waveform entropy and Doppler ambiguity factor F are shown in Figure 3. It is observed that the Doppler ambiguity factor F estimated by MKT is 14 (the real Doppler ambiguity factor of the moving target is 14), which verifies the effectiveness of MKT.

Then, we compare MKT with conventional KT based on entropy minimization, where the number of Doppler ambiguity factors is obtained by iterative search, and the computational costs of both methods are shown in

Table 1. Comparison results of time consumption.

SCR/dB	KT Based on Entropy Minimization	MKT
4.77	403.4 s	221.6 s
5.19	392.6 s	224.9 s
5.74	373.7 s	223.7 s

. The results show that the Doppler ambiguity factor estimated by MKT is the same as that of conventional KT based on entropy minimization with iterative search. More importantly, the average computational costs of MKT are significantly lower than those of conventional KT based on entropy minimization with iterative search, i.e., 224.9 s and 392.6 s, respectively (SCR = 5.19 dB), which indicates that the MKT proposed in this paper can greatly eliminate the RCM efficiency.

Figure 4a shows the target signal energy distribution after pulse compression. It is shown that the location of the moving RST energy migrates away from its initial position by 51 range cells, which indicates that RCM occurs during a long-time integration period, thus the energy of the moving RST cannot be well focused, and SCR in HRRPs after CI processing is low (SCR = 4.51 dB), as shown in Figure 4b.

To overcome the RCM effects for moving RST detection, KT, ACCF, and MKT are performed, and the integration results are shown in Figure 5, Figure 6 and Figure 7, respectively. More specifically, compared with the RCM results in Figure 4a, the results in Figure 5a indicate that RCM can be corrected by KT (F = 14). As a consequence, the target signal energy after CI processing can be effectively focused, and the SCR (7.02 dB) in HRRPs for the target is improved compared with the integration results in Figure 4b. In addition, compared with the integration results in Figure 4, the RCM of moving targets can be well eliminated by the ACCF method, and the target signal energy after CI processing can be effectively focused, thus improving the SCR (6.61 dB) in HRRPs, as shown in Figure 6. After the RCM correction by the MKT method, it is obviously shown that the range migration of moving targets can be effectively eliminated and that the SCR (7.02 dB) in HRRPs after CI processing can be improved, which is higher than that of ACCF, as shown in Figure 7.

Moreover, to illustrate the performance of MKT in detail, the curves of the input SCR versus the output SCR after RCM correction are given in Figure 8. It can be noticed that KT (F = 14) has the best performance, followed by MKT; MKT has almost identical coherent integration ability with KT (F = 14) when $\mathrm{SCR}\ge 4.1$ dB. However, the KT needs prior knowledge of Doppler ambiguity factor F to obtain a high detection performance.

Finally, the detection probabilities $P_d$ of the proposed method combining MKT and IEI are analyzed by Monte Carlo tests. Here, the motion parameters of RST and clutter data are retained. The RCM correction is achieved by three methods, i.e., KT, ACCF, and MKT, respectively. Under the condition of the false alarm rate $P_{fa}={10}^{-3}$, 1000 Monte Carlo simulations are carried out on each SCR, and the detection performance curves (SCR versus $P_d$) of different methods are given in Figure 9. We can observe that the detection performance of the EI detector combining KT (F = 14, the real Doppler ambiguity factor is 14) is the best. However, in practice, prior knowledge of the Doppler ambiguity factor of moving RSTs is unknown. Moreover, the RCM correction results of KT are highly influenced by the Doppler ambiguity factor, thus the robustness of the EI detector combining KT is limited; the EI detector combining ACCF demands a high SCR input; by comparing the EI detector combining MKT and the proposed method (IEI detector combining MKT), it can be seen that the IEI detector is superior to the EI detector. Hence, the proposed method in this paper outperforms its counterparts.

4.2. Experiment Analysis Based on Real Measured Clutter

The real measured clutter data collected by Ku-band radar is adopted to verify the proposed method, where carrier frequency $f_c=16$ GHz, bandwidth $B=64$ MHz, sample frequency $f_s=100$ MHz, pulse repetition frequency $f_p=40.3$ Hz, and pulse number during the CI period $M’=64$. The radial velocity $v$ of simulated moving RSTs is 16 m/s, thus the real Doppler ambiguity factor of the target is $F=2T_{p}v{f}_c/c=42$, and the number $L$ of range cells occupied by the target is 4.

The relationships between the waveform entropy and Doppler ambiguity factor F are shown in Figure 10. It can be found that the Doppler ambiguity factor estimated by MKT is equal to the real Doppler ambiguity factor of a target.

As shown in Figure 11a, we notice that the RCM occurs during the long-time integration period, which results in the defocusing effects of the target. As shown in Figure 11b, the SCR in HRRPs after CI processing is low (as shown in Figure 11c), which may degrade the detection performance of moving RSTs. To eliminate RCM, the KT, ACCF, and MKT algorithms are respectively performed, as given in Figure 12, Figure 13, Figure 14 and Figure 15. It can be seen that: (1) The RCM of moving RSTs can be effectively eliminated by ACCF and MKT when prior knowledge on the Doppler ambiguity factor is difficult to obtain, as shown in Figure 14a and Figure 15a. As a consequence, the energy of all the target scatterers has been coherently integrated into a strong peak point, as shown in Figure 14b and Figure 15b; (2) KT is seriously affected by the Doppler ambiguity factor and has poor robustness, as shown in Figure 12 and Figure 13; (3) MKT has the better SCR enhance ability compared with KT (F = 10), KT (F = 35), and ACCF, and then SCR in target HRRPs after CI processing, wherein RCM correction methods are respectively performed by the KT (F = 10), KT (F = 35), ACCF, and MKT are 8.79 dB, 10.63 dB, 11.18 dB, and 11.39 dB, respectively, as shown in Figure 12c, Figure 13c, Figure 14c and Figure 15c.

Moreover, to illustrate the performance of those methods (MKT, KT, and ACCF), the detailed SCR enhancement results after CI processing are given in Figure 16. It can be observed that the performance of MKT and KT (F = 42) is similar (SCR $\ge$ 3.5 dB). However, KT needs prior knowledge of the Doppler ambiguity factor, otherwise, the detection performance is degraded drastically.

The detection performance curves (SCR versus $P_d$) of the five methods are given, as shown in Figure 17. It can be observed that the EI detector combining KT (F = 42) outperforms the other methods, followed by the proposed method. However, in practical application, the prior information on Doppler ambiguity factor F is unknown. In addition, the proposed method demands a lower SCR input than the EI detector combining ACCF. For instance, the detection performance of the proposed method is improved by about 2.5 dB compared with the EI detector combining ACCF. The reasons may be as follows: On the one hand, the IEI detector can reduce the accumulation of clutter energy in target HRRPs after CI processing, which demands a lower SCR input compared with the traditional EI. Therefore, the proposed method (IEI detector combining MKT) is superior to the EI detector combining MKT, i.e., when SCR = 4 dB, the $P_d$ for the proposed method is higher than that of the EI detector combining MKT; On the other hand, MKT can achieve better coherent integration ability due to its ability to deal with the RCM during the CI period, as shown in Figure 16. As a consequence, the target signal energy can be well integrated, which is beneficial to SCR improvement. For instance, compared with the EI detector combining KT (F = 10) and the EI detector combining the ACCF, the EI detector combining MKT can detect moving RSTs better. Therefore, the detection results in Figure 17 demonstrate that the proposed method outperforms the EI detector combining KT and the EI detector combining ACCF under low SCR backgrounds.

5. Conclusions

In this paper, a novel detection method combining MKT and IEI for moving RSTs in a sea-clutter environment is proposed. The simulations and experiments using real sea clutter data verify the following advantages:

(1) The MKT algorithm can overcome the disadvantages of the traditional KT algorithm, which requires a priori knowledge of the Doppler ambiguity factor, which is more practicable;

(2) The MKT algorithm can eliminate the RCM more efficiently than the conventional KT algorithm based on entropy minimization and effectively retain the target signal energy, which is helpful to improve the detection performance of moving RSTs in low SCR conditions;

(3) The IEI detector can decrease the sensitivity of the detection performance of the traditional EI detector to the detection window length, reduce the clutter energy integration within the detection window, and achieve better detection performance.

In the future, we will further verify the detection performance of the proposed method for moving RSTs with complex motion characteristics in different clutter environments.