Generalized Dechirp-Keystone Transform for Radar High-Speed Maneuvering Target Detection and Localization

: The multivariate range function of the high-speed maneuvering target induces modulations on both the envelop and phase, i.e., the range cell migration (RCM) and Doppler frequency migration (DFM) which degrade the long-time coherent integration used for detection and localization. To solve this problem, many long-time coherent integration methods have been proposed. Based on mechanisms of typical methods, this paper names two signal processing modes, i.e., processing uniﬁcation (PU) mode and processing separation (PS) mode, and presents their general forms. Thereafter, based on the principle of the PS mode, a novel long-time coherent integration method, known as the generalized dechirp-keystone transform (GDKT), is proposed for radar high-speed maneuvering target detection and localization. The computational cost, energy integration, peak-to-sidelobe level (PSL), resolution, and anti-noise performance of the GDKT are analyzed and compared with those of the maximum likelihood estimation (MLE) method and keystone transform-dechirp (KTD) method. With mathematical analyses and numerical simulations, we validate two main superiorities of the GDKT, including (1) the statistically optimal anti-noise performance, and (2) the low computational cost. The real radar data is also used to validate the GDKT. It is worthwhile noting that, based on closed analytical formulae of the MLE method, KTD method, and GDKT, several doubts in radar high-speed maneuvering target detection and localization are mathematically interpreted, such as the blind speed sidelobe (BSSL) and the relationship between the PU and PS modes.


Introduction
With the development of science technology, especially the stealth technique, hypersonic aircraft, high-precision tracking, and high-resolution imaging quality [1][2][3], radar high-speed maneuvering target detection and localization has been receiving a growing attention and significant research efforts in the airborne, spaceborne, ground-based and shore-based radar [4][5][6][7]. The high-speed maneuvering target usually shows characteristics of far-range, low-observability, and strong noise, that is, the signal-to-noise ratio (SNR) of its echo is low. Without changing system parameters, the long-time integration-based detection and localization is a good alternative to improve the SNR and has been widely used [5,6]. However, the range cell migration (RCM) and Doppler frequency migration (DFM) easily happen, and the slant range of the high-speed maneuvering target should be modeled as a multivariate function in terms of the illumination time and motion parameters [5,6,8,9]. When the RCM and DFM happen, the traditional moving target detection and localization method cannot work anymore [5,6].

Prior Work
In past decades, many excellent methods have been proposed for the long-time integration. Based on whether the phase information of echoes is used or not, the longtime integration methods can be categorized into incoherent and coherent methods. The incoherent methods do not require the strict coherence of the radar system and are easy to realize, such as the Hough transform [10][11][12]. Unfortunately, the incoherent methods have a poor SNR improvement and there is also an "SNR threshold" [10][11][12][13][14][15].
The coherent methods use the phase information to guarantee a higher SNR improvement without the "SNR threshold". The maximum likelihood estimation (MLE) method is a typical coherent method [16]. Afterwards, proposed generalized Radon-Fourier transform [17], Radon-fractional Fourier transform [18], and polynomial Radon-polynomial Fourier transform [19] can be seen as its successful extensions. This kind of processing method is statistically optimal, while it is normally computationally expensive [17][18][19]. The keystone transform (KT) can speed up the MLE method and its extensions [20][21][22][23][24][25]. Unfortunately, under many conditions, the KT may encounter the half-blind-velocity effect and Doppler spectrum coupling [26,27] which may induce the energy spread. So far, to the best of the author's knowledge, there is no effective method to solve these two problems without any approximation or prior knowledge [26,27]. In order to blindly compensate the nonlinear RCM, high-order KTs are also developed [25,28,29]. Given the problems of the aforementioned coherent methods, researchers have carried out many other kinds of work. Representations include the hybrid integration [30], product scaled periodic Lv's distribution [31], adjacent cross-correlation function-based method [32,33], and scaled inverse Fourier transform-based method [34,35]. These methods are computationally efficient and do not have problems with the KT-based methods, while they have the integration gain loss.

Contribution
In this paper, we deduce closed analytical formulae of two typical detection and localization methods, i.e., the MLE method and KT-dechirp (KTD) method, for a better understanding of the PU mode and PS mode. With the closed analytical formulae, we interpret several doubts in radar high-speed maneuvering target detection and localization, i.e., the BSSL and the relationship between the PU and PS modes. Thereafter, based on the principle of the PS mode, a novel coherent method, known as the generalized dechirp-KT (GDKT), is proposed for radar high-speed maneuvering target detection and localization. Mathematical analyses and numerical simulations demonstrate that the GDKT has a statistically optimal anti-noise performance and is computationally efficient. The real radar data is also used to validate the practicability of the GDKT.

Organization
The remainder of this paper is organized as follows. Section 2 gives the signal model and problem formulation. General forms of the PU and PS modes are presented in Section 3, where closed analytical formulae of the MLE method and KTD method are also deduced. A novel coherent method, known as the GDKT, is proposed in Section 4, where its implementation method is also presented. Section 5 gives the mathematical analyses and comparisons. The real radar data is used to validate the proposed GDKT in Section 6. Section 7 includes the conclusion and future work.

Signal Model and Problem Formulation
Suppose radar transmits the linear frequency modulated (LFM) signal waveform [36] is a rectangle window.t is the fast time, i.e., the intra-pulse sampling time. T p , f c and γ denote the pulse duration, carrier frequency and frequency modulation rate, respectively. The echo of K high-speed maneuvering targets after the down-conversion can be represented as where t m ∈ [−T m /2, T m /2] is the slow time, i.e., the inter-pulse sampling time. T m denotes the integration time. A k is the received signal voltage. τ del,k = 2R k (t m )/c, c is the speed of light, and R k (t m ) is the slant range between the kth target and radar at the instantaneous slow time t m . n t , t m denotes the additive stationary zero-mean complex white Gaussian noise of the power σ 2 n . According to the radar equation [17], we have the SNR for each target in Equation (2) as where P t , G and σ are the radar transmission power, antenna gain and backscattering radar cross section, respectively. λ = c/ f c denotes the wavelength. B c , T e , F, L s and R k0 denote the Boltzmann constant, system Kelvin temperature, receiver noise figure, system loss and initial slant range, respectively. Performing the range compression to Equation (2), we obtain the compressed signal in the fast time-slow time (t -t m ) domain as where sinc(x) = sin(πx)/(πx). n com t , t m denotes the noise. After the range compression, the SNR for each target in Equation (4) is During the illumination time, the time-varying range R k (t m ) could be modeled as a multivariate function with N + 2 unknown parameters (R k0 , v k0 , a k1 , · · · , a kN ) [17]. Since the illumination time is usually short, a second-order model is enough for realistic ap-plications. However, we cannot deny that the higher the order is, the more accurate the time-varying range is.
where v k0 denotes the radial velocity. a k1 , · · · and a kN denote radial high-order motions. Substituting Equation (6) into Equation (4) yields The radar high-speed maneuvering target detection and localization model presented in Equation (7) is also widely used in SAR and ISAR imaging [38,39]. If the long-time coherent integration is perfectly carried out for s com (t, t m ), the SNR for each target will be where PRF denotes the pulse repetition frequency. Unfortunately, target motions induce the RCM and DFM, and the signal energy cannot be coherently integrated via performing the Fourier transform on s com (t, t m ) along t m . For the long-time coherent integration, many successful coherent methods have been proposed [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Referring to the published literature and conducting analyses of Equation (7), we summarize interference factors to realize the long-time integration and list them below for further research.

RCM and DFM
The target motions (v k0 , a k1 , · · · , a kN ) influence the long-time coherent integration from two aspects, e.g., the envelop and Doppler spectrum. In Equation (7), the envelop offset is v k0 t m + a k1 t m 2 /2 + · · · + 2a kN (N+1)! t m N+1 . When the envelop offset exceeds the range resolution c/2B, the RCM happens [see Figure 1a]. The Doppler spread is 2a k1 t m + · · · + 2a kN N! t m N /λ in Equation (7). When the Doppler spread exceeds the Doppler resolution 1/T m , the DFM happens (see Figure 1b). The RCM and DFM degrade the longtime coherent integration. In Figure 1a,b, R max is the maximum value of R k (t m ), and N ki and Ψ correspond to the searching number and maximum searching range of the ambiguity integer, respectively.

Doppler Ambiguity Integer
The exponential term in Equation (7) can be rewritten as where f k0 = 2v k0 /λ, f k1 = a k1 /λ, If the velocity is ambiguous (this easily happens under the high speed, large carrier frequency or low PRF) [37], the instantaneous Doppler of the kth target should be represented as where f k0d and N k0d denote the ambiguous Doppler and ambiguity integer, respectively. Equation (10) indicates that, if the velocity is ambiguous, there will be an ambiguity integer. According to [36,37], the Doppler ambiguity integer makes the long-time coherent integration complicated, such as the BSSL (see Figure 1c) and the following two problems. It is worthwhile noting the BSSL appears mainly because the RCM compensation and the coherent integration along the slow time correspond to two different velocity resolutions. This will be theoretically interpreted in Section 3.
The equation f k0 = f k0d + N k0d PRF is based on the assumption that the bandwidth has no influence on the wavelength. However, a high bandwidth may influence the wavelength and f k0 may span several PRF bands when f k0d is close to PRF/2. We call this effect as the half-blind-velocity effect [26].
Under the Doppler spectrum coupling and the half-blind-velocity effect, the ambiguity integer is not constant anymore. However, KT-based methods are based on the assumption that the ambiguity integer is constant. Therefore, under this condition, the ergodic searching of the ambiguity integer cannot compensate the linear RCM clearly after the KT-based methods.

PU Mode and PS Mode
In this section, we deduce closed analytical formulae of two typical detection and localization methods, i.e., the MLE method and KTD method, for a better understanding of the PU mode and PS mode. With closed analytical formulae, several doubts in radar high-speed maneuvering target detection and localization will be interpreted.

PU Mode
The signal expression of s com (t, t m ) in the range frequency-slow time domain can be written as where f denotes the range frequency. n c ( f , t m ) denotes the noise in the range frequencyslow time domain. It is easily seen from Equation (11) that the linear RCM compensation and inter-pulse energy integration can be done via the ergodic searching of the unknown f k0 . This is the basic principle of the PU mode. Referring to the literature [5,6] and [16][17][18][19], we present the general form of the PU mode as wheref 0 denotes the searching variable of f k0 . · · · denotes other searching variables.
denote two operations, such as the dechirp, integration, and so on [5,6]. For a better understanding of the PU mode and its general form, we consider a typical method of the PU mode, i.e., MLE method. It can be presented as whereR 0 ,â 1 , · · · , andâ N denote searching variables of R k0 , a k1 , · · · , and a kN , respectively. The target is non-cooperative and we do not have its motion parameters. Therefore, in realistic applications, we set the searching range and interval forR 0 ,â 1 , · · · , andâ N to match R k0 , a k1 , · · · , and a kN . For the detailed implementation, we can refer to Section 4.2.
In the MLE method, the dechirp operation marked with the single underline corresponds to OP 1 [•] and the integration marked with double underline corresponds to OP 2 [•].
Substituting S c ( f , t m ) into Equation (13), we obtain the closed analytical formula of the MLE method as where mod f 0 − f k0 /PRF returns the remainder after division of f 0 − f k0 by PRF.n M f 0 ,R 0 ,â 1 , · · · ,â N denotes the noise. Note that the second, third, and fifth sinc functions are only approximate expressions.
The signal energy is coherently integrated at ( f k0 , R k0 , a k1 , · · · , a kN ) after the MLE method. Note thatf 0 appears in two sinc functions marked in Equation (14). The resolution off 0 is 3 f c /(BT m ) in the first sinc function, while it is 1/T m in the second sinc function. Since f c is larger than B, 3 f c /(BT m ) is larger than 1/T m . Therefore, alongf 0 , the first sinc function determines the outside contour and the second sinc function samples this outside contour with the interval PRF, i.e., 1/T m . Here, we use a numerical example to illustrate thef 0 dimension of Equation (14).  Figure 2, the top left figure illustrates the first sinc function marked in Equation (14), the top right figure illustrates the second sinc function marked in Equation (14), and the bottom figure illustrates the energy distribution alongf 0 . In the bottom figure of Figure 2, besides the peak atf 0 = 1040Hz, many high sidelobes appear and are right located around ambiguity integer times. They are known as BSSLs which are samples of the first sinc function via the second sinc function marked in Equation (14). Obviously, BSSLs may influence the target detection and localization when we use the constant false alarm rate (CFAR) technique. As we know, the peak-to-first sidelobe ratio along the range dimension is about 13 dB and the influence of the sidelobe can be ignored. Therefore, we refer to this phenomenon and set radar parameters to let the ratio between the peak and BSSLs higher than 13 dB. The width of a sinc function at −13 dB is about twice the resolution. In addition, the interval between two adjacent samples of the first sinc function is PRF. Therefore, if the following Equation (15) holds, the influence of BSSLs may be ignored via a CFAR method.
where T m = M/PRF and M denotes the number of the integrated pulses. Equation (15) indicates that we may eliminate the influence of BSSLs via adjusting M, f c and B.

Remark 1:
The reference [37] also analyze BSSLs and two approaches, i.e., system parameter design and weighting function, are proposed for the BSSLs suppression. Our work can be seen as an extension of [37]. Compared to its work, our work is characterized by (1) the clearer interpretation of the BSSLs, i.e., the searching variablef 0 has two different resolutions; and (2) the explicit system parameter design, i.e., the Equation (15).
Above is the analysis of the PU mode via the MLE method. The problem of BSSLs is clearly interpreted and an explicit system parameter design is presented also. In the following subsection, we will analyze the PS mode via the KTD method.

PS Mode
The first exponential term of Equation (11) can be divided into two exponential terms, one corresponds to the linear RCM and the other corresponds to the inter-pulse energy integration. Basing on this characteristic, we rewrite S c ( f , t m ) as where exp(j2πN k0d PRFt m ) = 1 is ignored. In Equation (16), the first exponential term corresponds to the linear RCM and the second exponential term corresponds to the inter-pulse energy integration. Therefore, we can use two different physical variables to separately perform the linear RCM compensation and inter-pulse energy integration. This is the basic principle of the PS mode. Referring to the literature [24][25][26] and [29][30][31][32][33][34][35], we present the general form of the PS mode as  [34,35],ˆ = 2v 0 and the actual searching variable isv 0 (i.e., the searching variable of v k0 ).
For a better understanding of the PS mode and its general form, we consider a typical method of the PU mode, i.e., the KTD method. It can be presented as where τ m denotes a scaled variable corresponding to t m . In the KTD method, the KT operation marked with the single underline corresponds to OP 3 [•], and the dechirp operation and integration marked with double underline correspond to (18) and obtain the closed analytical formula of the KTD method as where n K f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N denotes the noise. Note that the second, third, and fourth sinc functions are only approximate expressions. The signal energy after the KTD method is coherently integrated at (N k0d , f k0d , R k0 , a k1 , · · · , a kN ). Two parameter dimensions, i.e.,N 0d andf 0d , appear in the KTD method instead of one parameter dimension, i.e.,f 0 , in the MLE method. Comparing Equation (19) with Equation (14), we find that the signal energy distribution alongN 0d can be seen as extractions of thef 0 dimension at ambiguity integers and the signal energy distribution alongf 0d can be seen as extractions of thef 0 dimension in [−PRF/2, PRF/2). Here, we use a numerical example to illustrateN 0d andf 0d dimensions of Equation (19).

Example 2:
The radar and motion parameters are set as those of Example 1. The number of integrated pulses is also 200. In Figure 3, the top left figure shows the signal energy distribution alongN 0d , the top right figure shows the signal energy distribution alongf 0d , and the bottom figure illustrates the signal energy distribution in theN 0d −f 0d domain.
Extractions of thef 0 dimension at ambiguity integers are also plotted with the red line in the top left figure of Figure 3. As analyzed above, the signal energy distribution alongN 0d can be seen as extractions of thef 0 dimension at ambiguity integers. Based on Equations (14) and (19), and related analyses and results shown in Figures 2 and 3, it is not difficult to find that the KTD method and the MLE method have the same energy integration, while the searching range off 0d is fixed into [−PRF/2, PRF/2) and the searching procedure for f k0d can be speed up via the fast Fourier transform (FFT) [24][25][26][29][30][31][32][33][34][35]. That is, these two methods have the same energy integration, while the KTD method is more computationally efficient than the MLE method.

Remark 2:
Several published studies claim that their KT-based methods can eliminate BSSLs. Based on analyses above, we know that, in KT-based methods, the BSSLs disappear along the ambiguous Dopplerf 0d dimension, while they still exist along the ambiguity integerN 0d dimension.
Remark 3: Note that we use " ≈ " instead of " = " in Equation (19). This is because Equation (19) is conditional, i.e., the Doppler spectrum coupling and half-blind-velocity effect do not happen. If these two problems happen, the signal energy after the KTD method may spread. In Section 5, we will use numerical simulations to demonstrate drawbacks of the KTD method. Above is the analysis of the PU mode and PS mode with two typical methods, i.e., the MLE method and the KTD method. Compared to the PU mode, the PS mode is more computationally efficient. However, its representative method, i.e., the KTD method, may encounter anti-noise performance loss when the Doppler spectrum coupling and halfblind-velocity effect happen. In the following, we continue the research of the PS mode and propose a novel long-time coherent integration method, i.e., GDKT, which is statistically optimal and computationally efficient.

GDKT and Implementation
In this section, the GDKT belonging to the PS mode is proposed for radar high-speed maneuvering target detection and localization. A fast implementation method is also presented for the GDKT.

GDKT
The KT-based methods are based on the assumption that the ambiguity integer is constant. However, due to the Doppler spectrum coupling and half-blind-velocity effect, the ambiguity integer may not be constant and the ergodic searching of the unknown ambiguity integer cannot compensate for the linear RCM clearly [24][25][26][27]. Without doubt, if the influence of the Doppler spectrum coupling and half-blind-velocity effect can be eliminated before the KT, the proposed KT-based method will be statistically optimal and computationally efficient. Under such a guideline, we propose a novel coherent method, known as the GDKT.
In the GDKT, the dechirp operation marked with single underline corresponds to OP 3 [•] and the integration marked with double underline corresponds to OP 4 [•]. Note that the first exponential term used in the dechirp operation can eliminate the influence of the Doppler spectrum coupling and the second exponential term used in the dechirp operation can eliminate the influence of half-blind-velocity effect.
We substitute S c ( f , t m ) into Equation (20) and obtain the closed analytical formula of the GDKT as GDKT f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N = where n G f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N denotes the noise. Note that the second, third, and fourth sinc functions are only approximate expressions. The closed analytical formula of the GDKT method is the same as that of the KTD method, except " = ". The signal energy after the GDKT is coherently integrated at (N k0d , f k0d , R k0 , a k1 , · · · , a kN ). Compared to the KTD method, the GDKT is statistically optimal. Since example 2 does not take the influence of the Doppler spectrum coupling and half-blind-velocity effect into consideration, results shown in Figure 3 can also illustratev 0 andN 0d dimensions of Equation (21). Now, the CFAR technique can be used to do the target detection and localization.
Compare GDKT f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N with the threshold value η (can be easily calculated with the false alarm rate and noise power of reference cells). For more details of the CFAR, the readers can refer to [2][3][4].
GDKT f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N If GDKT f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N is larger than η, there will be a target. Meanwhile, all the motion and localization parameters can be directly estimated. On the contrary, if GDKT f 0d ,N 0d ,R 0 ,â 1 , · · · ,â N is smaller than η, there will be no target. Note that the BSSLs do exist in the GDKT and may influence the CFAR. In this paper, we base on the deduced Equation (15) and propose to eliminate the influence of the BSSLs by extending the integration time T m .
Compared to the MLE method and the KTD method, the GDKT has two main superiorities, including

•
The GDKT usesv 0 andf 0d to separately process the linear RCM and inter-pulse energy integration. Similar to the KTD method, the GDKT is more computationally efficient than the MLE method; • In the GDKT, the KT is employed after eliminating the influence of the Doppler spectrum coupling and half-blind-velocity effect. Therefore, same as the MLE method, the GDKT is statistically optimal.
Above is the proposed GDKT for radar high-speed maneuvering target detection and localization. In the following subsection, we will present an implementation method for the GDKT.

Implementation of the GDKT
The key steps of the GDKT are the dechirp and KT operations. In this subsection, we present an implementation method for the GDKT. Based on the sampling scheme of Claasen and Mecklenbrauker [38], the discrete form of can be expressed as where B k = A k exp(−j4πR k0 f c /c).∆ f and ∆ t m denote sampling intervals of f and t m , Step 1: Construct the N + 3-dimension matched filtering function.
Step 4: Perform the inverse FFT (IFFT) with respect to n f .
Step 5: Compare GDKT nˆt, n f tm ,N 0d , nâ 1 , · · · , nâ N with the threshold value η determined by the CFAR to complete the target detection and localization.
GDKT nˆt, n f tm ,N 0d , nâ 1 , · · · , nâ N H 1 Above is the proposed fast implementation method of the GDKT. Figure 4 gives its low chart. In the following, we use an example to illustrate how the GDKT works.

Example 3:
The GDKT is a linear method. Here, for the convenience of the image presentation and without loss of generality, we consider a uniformly accelerated target, i.e., Target1. The radar and target parameters are given in Table 1. Figure 5 shows simulation results. In this example, M is set to 1000 which satisfies the deduced Equation (15).    Figure 5a, i.e., the MTD method, we obtain the integration result in Figure 5b. From these two figures, we could see that the RCM and DFM happen, and the echo energy cannot be integrated via the MTD method. Figure 5c shows the trajectory in the range-slow time domain after the dechirp operation of Step 2. Although the RCM and DFM induced by the ambiguity integer and acceleration are eliminated, the RCM induced by the ambiguous Doppler still exists. Therefore, we perform the KT operation of Step 3, and the trajectory in the range-slow time domain is shown in Figure 5d. Performing the IFFT with respect to the slow time dimension of Figure 5d, we obtain the GDKT. Figure 5e shows the acceleration-ambiguity integer slice of the GDKT and Figure 5f shows the range-ambiguous Doppler slice of the GDKT. The GDKT eliminates the RCM and DFM and completes the echo energy integration perfectly. Now, we can use the CFAR to complete the target detection and localization.

Theoretical Comparisons and Numerical Illustrations
In this section, we compare the GDKT with the MLE method and the KTD method from five aspects: the computational cost, energy integration, resolution, PSL, anti-noise performance, and practicability under multi-component high-speed maneuvering targets [5,6,[32][33][34][35]. Theoretical derivations and numerical illustrations are used to demonstrate superiorities of the GDKT.

Computational Cost
In this subsection, we compare the computational cost of the GDKT to those of the MLE method and the KTD method. With the implementation method presented in SubSection 4.2, we obtain that the computational cost of the GDKT is in the order of Based on the implementation methods presented in [16] and [25], we obtain that computational costs of the KTD method and MLE method are in the order of O NN 0d Nâ 1 · · · Nâ N N f M log 2 M + MN f log 2 N f and O Nf 0 Nâ 1 · · · Nâ N N f M , respectively (where Nf 0 denotes searching times for the unknown parameter f k0 ). For easy comparison and without loss of generality, we assume N f = M [24][25][26][27] and obtain the ratio for computational costs of these three methods as Equation (30) indicates that computational costs of the GDKT and the KTD method are similar and much lower than that of the MLE method. In order to give a more intuitive comparison, Figure 6 shows the computational complexity ratio versus different numbers of integration pulses.

Energy Integration
In this subsection, we compare the energy integration of the MLE method, KTD method, and GDKT. The analyses above have indicated that due to the Doppler spectrum coupling and half-blind-velocity effect, the KTD method may encounter the defocusing of the energy integration. By contrast, the MLE method and GDKT do not encounter these two problems and could obtain the perfect energy integration. Here, we use an example to illustrate the energy integration of these three methods.
Example 4: These three methods are linear. For the convenience of the image presentation and without loss of generality, we consider a uniformly accelerated target, i.e., Target2. The radar and target parameters are given in Table 2. In this example, M is set to 200 to satisfy the deduced Equation (15). Figure 7 shows the simulation results of these three methods.  Figure 7a,b show the acceleration-ambiguity integer slice and range-ambiguous Doppler slice of the KTD method, respectively. Based on the radar and target parameters listed in Table 2, we separately analyze the Doppler spectrum coupling and half-blindvelocity effect in Equations (31) and (32) as Equation (31) indicates that the Doppler spectrum coupling makes the Doppler spectrum span eight PRF bands. Equation (32) indicates that the half-blind-velocity effect makes the Doppler spectrum span three PRF bands. Therefore, the KTD method encounters the defocusing of the energy integration. By contrast, in the GDKT, the KT is employed after eliminating the influence of the Doppler spectrum coupling and half-blind-velocity effect. Therefore, same as the MLE method in Figure 7e,f, the GDKT in Figure 7c,d do not encounter the defocusing of the energy integration. Note that, although in this example, M is set to 200 to satisfy the deduced Equation (15), the BSSLs are still obvious in Figure 7e and may influence the weak targets' detection and localization. Therefore, in the future, the BSSLs' elimination method is still necessary.

Resolution and PSL
The resolution and PSL play important roles in the adjacent targets' detection and localization. The KTD method encounters the defocusing of the energy integration. Therefore, we only compare the GDKT with the MLE method. Based on the closed analytical formulae of the MLE method and GDKT in Equations (14) and (21), we present their theoretical resolutions and PSLs as PSL MLE f 0_outline = PSL GDKT N 0d = −13.3dB (34) For the MLE method, we usef 0_mainlobe andf 0_outline to separately present the main lobe part and outline along the Doppler domain. Equations (33) and (34) indicate that theoretical resolutions and PSLs of the MLE method and GDKT are the same. Here, we use an example to illustrate the resolutions and PSLs of these two methods.

Example 5:
The radar and target parameters are the same as those listed in Table 2. Figure 8a shows simulation results along the range dimension. Figure 8b shows simulation results along the acceleration dimension. Figure 8c shows simulation results along the ambiguity integer dimension. Figure 8d shows the simulation results along the Doppler dimension. Figure 8e shows the simulation results along the ambiguous Doppler dimension. Figure 8f shows the zoom part of the first BSSL marked in Figure 8d.
With radar parameters and Equation (33), we could obtain that, for the MLE method and GDKT, theoretical resolutions along the range, ambiguous Doppler (orf 0_mainlobe ) and acceleration should be 3 m, 1 Hz, and 0.3 m/s 2 , respectively. Simulated resolutions in Figure 8 conform to theoretical resolutions. Simulated PSLs along the acceleration dimension is lower than the theoretical PSL given in Equation (34). This is because corresponding integration variables are different from those in the integrant and the sidelobe is elevated under the finite discrete signal [38,39]. The signal energy distribution alongN 0d can be seen as extractions of thef 0 dimension at ambiguity integers. When the extraction is just not the peak of the first BSSL, the influence of the BSSLs can be reduced. Figure 8f shows zoom part of the first BSSL marked in Figure 8d. Comparing Figure 8f to Figure 8c, we find that the extractions in the GDKT is not the peak of the first BSSL and the influence of the BSSLs is obviously reduced, which indicates that the GDKT benefits the PSL.

Anti-Noise Performance
This section focuses on the theoretical anti-noise performance analysis of the GDKT. The GDKT is linear. Without loss of generality, we consider the discrete noisy echo s e (nˆt, n t m ) of the kth high-speed maneuvering target. s e (nˆt, n t m ) = s s,e (nˆt, n t m ) + n e (nˆt, n t m ) (35) where s s,e (nˆt, n t m ) denotes the discrete echo of the kth target. n e (nˆt, n t m ) denotes the discrete noise. Via the Fourier transform along the fast time dimension of s e (nˆt, n t m ), we obtain the discrete noisy echo in the slow time-range frequency domain.
S e n f , n t m = S s,e n f , n t m + n S,e n f , n t m (36) Now, we use S trans n f (the transmitting waveform in the range frequency domain) to perform the range compression, and then perform the GDKT.
In order to analyze the theoretical detection and localization performance, we use the software "Wolfram Mathematica" and properties of the moment of zero mean complex Gaussian random variables to calculate the expect value at the peak point.
The output SNR after the GDKT is Equations (8) and (39) indicate that, same as the MLE method [16,17], the anti-noise performance of GDKT is statically optimal. In this subsection, we use a numerical example to validate analyses of the theoretical anti-noise performance.

Example 6:
The radar and target parameters are the same as those in Table 2. The simulated radar data is contaminated by the zero-mean white Gaussian noise and the input SNRs are set as [−40:1: 10] dB. 1000 trials are done for each input SNR value. Figure 9a-c shows the detection probabilities under the false alarm rates 10 −5 , 10 −6 , and 10 −7 , respectively. The MLE method, KTD method, and MTD method are used as comparisons.
Due to the RCM and DFM, the MTD has a bad anti-noise performance in Figure 9. As analyzed above, the Doppler spectrum coupling and half-blind-velocity effect influence the anti-noise performance of the KTD method seriously. Figure 9 shows that the anti-noise performance of the KTD is about 13 dB worse than those of the GDKT and the MLE method. On the contrary, the GDKT can compensate the RCM and DFM without the anti-noise performance loss. Therefore, under false alarm rates 10 −5 , 10 −6 , and 10 −7 , its detection probabilities are almost the same as those of the MLE method, which conforms to the theoretical anti-noise performance analyses.

Practicability under Multicomponent High-Speed Maneuvering Targets
The analyses above indicate that considering the computational cost, energy integration, resolution, PSL, and anti-noise performance, the GDKT is more suitable for the high-speed maneuvering target detection and localization than other methods. In this subsection, we use a numerical example to demonstrate the practicability of the GDKT under multi-component high-speed maneuvering targets. The MTD method, MLE method, and KTD method are used for comparisons.

Example 7:
These four methods are all linear. Therefore, for the convenience of the image presentation and the clear comparison, we consider three uniformly accelerated targets, i.e., Target3, Target4, and Target5, without noise. The target parameters are listed in Table 3 and radar parameters are the same as those in Table 2. Figure 10 shows the simulation results.    Figure 10a shows the integration result of the MTD method. The RCM and DFM occur and the echo energy cannot be integrated via the MTD method. Figure 10b,c show the range-ambiguous Doppler slice and acceleration-ambiguity integer slice of the GDKT, respectively. The GDKT eliminates the RCM and DFM completely and completes the echo energy integration. Figure 10d,e show the range-Doppler slice and acceleration dimension of the MLE method, respectively. As expected, the MLE method also eliminates the RCM and DFM completely and completes the echo energy integration. Figure 10f,g show the range-ambiguous Doppler slice and acceleration-ambiguity integer slice of the KTD method, respectively. Based on the radar parameters, motion parameters, and Equations (31) and (32), we could obtain that the Doppler spectrum coupling and halfblind-velocity effect make the Doppler spectrum of these three targets span several PRF bands. Therefore, the KTD method encounters the defocusing of the energy integration in Figure 10f,g. Combined with the computational cost analysis, we can say that the GDKT is more suitable for radar high-speed maneuvering target detection and localization than the MLE method and KTD method.
Here, based on mathematical analyses and numerical simulations, we provide a final table (see Table 4) comparing the performances of the MTD method, MLE method, KTD method, and GDKT. From this table, it is very easy to find that the GDKT is more suitable for radar high-speed maneuvering target detection and localization.

Real Radar Data Validation
In this section, we use the real radar data to validate the proposed detection and localization method. As we know, due to the complicated and unknown realistic environment, it is difficult to control the SNR and radial motion parameters. Therefore, in this paper, we only use the real radar data to validate the practicability of the GDKT instead of its superiorities. All processing of the real radar data is completed on a personal computer with an Intel (R) Xeon (R) CPU E5-1603 v4(3.70 GHz) processor and 64 GB memory.
The raw data were recorded by an X-band ground-based radar. This radar transmits the LFMCW waveform with the PRF of 50 Hz and a bandwidth of 500 MHz. In the following simulations, 216 effective pulses are extracted from the data set. Figure 11a shows the experiment scene and Figure 11b shows trajectories after the range compression. The GDKT takes the ambiguity integer searching into consideration. When the ambiguity integer is found correctly, the target energy can be integrated as much as possible in the range-ambiguous Doppler slice. Figure 11c shows the energy integration result when the ambiguity integer, k = 3, is correctly compensated in the range-ambiguous Doppler slice. To observe the effect of ambiguity number compensation on the energy integration, the energy focusing result in the ambiguous Doppler-ambiguity integer slice with the range gate number 455 is shown in Figure 11d. Obviously, if the ambiguity integer is not compensated correctly, the energy integration effect will become worse as the searching parameter is far away from the correct value. In order to show the superiority of the GDKT in the energy integration effect, the energy integration effect of the MTD algorithm is shown in Figure 11e. Because there is no correction of the RCM, the energy is scattered in different range units and the integration effect of MTD is very poor.

Conclusions and Future Work
For radar high-speed maneuvering target detection and localization, this paper names two signal processing modes, i.e., PU mode and PS mode, and then deduces closed analytical formulae of two typical methods, i.e., the MLE method and the KTD method, for a better understanding of these two modes. With the deduced closed analytical formulae, we interpret several doubts in radar high-speed maneuvering target detection and localization, such as the BSSL and the relationship between the PU and PS modes. Thereafter, based on the principle of the PS mode, the GDKT is proposed for radar high-speed maneuvering target detection and localization. Mathematical analyses and numerical simulations are used to demonstrate that the GDKT has a statistically optimal anti-noise performance and is computational efficiently.
We think that the largest significance of this paper is the interpretation of the PU and PS modes. Although the GDKT is statically optimal and computationally efficient, its computational cost is still a little high. Therefore, in the future, we will continue to study the PS mode and fast implementation of the GDKT. In addition, the resolution enhancement via the compressed sensing and the BSSL suppression method will also be studied [40,41]. Other signal waveforms, i.e., nonlinear LFM signal waveform, can also be used [42].

Conflicts of Interest:
The authors declare no conflict of interest.