Received 19 January 2024, accepted 3 February 2024, date of publication 7 February 2024, date of current version 22 February 2024. Digital Object Identifier 10.1109/ACCESS.2024.3363242 Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity ANATOLII A. KONONOV , (Member, IEEE), AND MIN-HO KA , (Member, IEEE) School of Integrated Technology, Yonsei University, Seoul 21983, Republic of Korea Corresponding author: Min-Ho Ka (kaminho@yonsei.ac.kr) This work was supported by the National Research Foundation of Korea (NRF) through Ministry of Science and ICT (MSIT) under Grant 2021R1A2C2006025. ABSTRACT This paper introduces a Doppler domain localized (DDL) implementation of the adaptive matched filter (AMF) for radar target detection in severely heterogeneous clutter environments with limited training data. The proposed detector uses the concept of a region of possible target detection (RPTD), a small set of Doppler bins that captures most of the target signal power. This RPTD-based DDL-AMF detector outperforms an earlier suggested DDL implementation of the generalized likelihood ratio (GLR) test, which employs the region of detection improvement (RODI) concept. Unlike the RODI-based DDLGLR detector, the proposed DDL-AMF detector requires no information on clutter spectrum parameters and no measurements to determine the number and locations of RODIs. Moreover, the RODI-based DDLGLR detector’s performance falls far below the optimum when the target Doppler frequency is unknown. In contrast, the RPTD-based DDL-AMF detector ensures rapid adaptive detection with near-optimum performance under unknown target Doppler frequency and multimodal clutter spectra. INDEX TERMS Adaptive detection, adaptive matched filter (AMF), constant false alarm rate (CFAR), Doppler domain localized (DDL) adaptivity, generalized likelihood ratio (GLR), heterogeneous clutter. I. INTRODUCTION This paper addresses the challenge of achieving near-optimum detection performance for radar targets with unknown velocity in nonstationary and nonhomogeneous clutter environments with complex spectra. This is a fundamental problem for airborne and coastal surveillance radars, which must detect targets of interest amidst multiple strong interferences that vary in time and space. Reliable target detection in such scenarios is difficult due to the limited availability of training data for estimating unknown clutter statistics. For simplicity, we focus on adaptive detectors for a monostatic pulse Doppler radar having a single antenna with no adaptive pattern control. We assume a coherent processing interval (CPI) with N pulses, where N must be sufficiently large even for the optimum detector (Neyman-Pearson’s likelihood ratio test) to deliver a reliable detection of weak targets embedded in the strong clutter with complex spectra. In Gaussian interference, the classical adaptive detection algorithms, The associate editor coordinating the review of this manuscript and approving it for publication was Chengpeng Hao VOLUME 12, 2024 . e.g., GLR test [1] and AMF [2], possess the desired CFAR property and may deliver near-optimum detection performance if they use at least as large N as the optimum detector. Besides, they require a training data set containing at least 4N independent and identically distributed vectors [3]. However, even for moderate N , such training data are often unavailable in nonstationary and heterogeneous clutter environments. Thus, directly implementing classical adaptive CFAR detectors does not solve the problem under consideration. To address this challenge, [4] proposed a Doppler domain localized generalized likelihood ratio (DDL-GLR) detector that combines the localized adaptivity principle in the Doppler domain with the GLR test. Implementing the DDL adaptive processing requires the Fourier transform (weighted or unweighted) must be first applied to the original time-domain data sampled at the output of the receiver I/Q channel to confine the signal and interference power to small regions in the Doppler domain. According to [4], the key idea for implementing data-efficient adaptive detectors is to apply the adaptivity only to those areas of the Doppler domain where some 2024 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 25411 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity detection improvement is necessary and possible. It proposed dividing the Doppler domain into two types of regions to identify such areas. The first type refers to the flat or nearly flat spectrum regions, and the second one refers to the sharply changing spectrum regions. Suppose the target Doppler frequency is within the first type of spectrum regions and far away from the clutter spectrum peaks. Then, the conventional processor employing a windowed DFT bank of filters followed by the traditional CA CFAR detectors is expected to provide near-optimum detection performance. In contrast, when the target Doppler falls on sharply changing spectrum regions or regions near the clutter spectrum peaks, the conventional detector performance may fall far below the optimum. Therefore, [4] suggested improving detection performance for such regions using adaptive processing. These regions are called the regions of detection improvement (RODI). Each RODI covers a small set of n adjacent Doppler bins, with n ≪ N . As shown in [4], a RODI-based DDL-GLR detector may achieve near-optimum detection for N ≫ 1 even when the RODI order is 4 − 6. This small order results in an essentially smaller training data size than that for the time-domain GLR detector of order N . The DDL-GLR detector also has a lower computational load than its time-domain counterpart and allows parallel/distributed implementation. However, this detector has several flaws preventing its practical applications. First, as pointed out in [4], a challenging problem is determining the number and locations of the RODIs to achieve near-optimum detection performance. Auxiliary clutter measurements are required to obtain the RODI parameters from the received data, making implementing the RODIbased DDL-GLR detector complicated and computationally intensive. Additionally, this detector has the following disadvantages: i) Determining the number and locations of the RODIs is overly complex when multiple clutter types (e.g., ground, sea, and weather) appear in the cell under test. ii) Significant departures from optimum detection performance may occur even for known target Doppler frequencies, regardless of whether they take on or off DFT grid values. iii) These departures prevent the implementation of the RODI-based DDL-GLR detector under unknown Doppler frequencies by using test statistics associated with the regular DFT grid, as is done by a traditional CFAR algorithm when detecting targets embedded in thermal noise. The present paper aims to develop a new adaptive DDL detector that is free of the disadvantages of the RODI-based DDL-GLR detector. The new adaptive DDL detector must ensure rapid and near-optimum detection of targets in training data-deficient scenarios without using RODIs, without information on clutter spectrum parameters, and without measurements for determining the number and locations of RODIs. It must also ensure reliable target detection in clutter environments with multimodal power spectral density (PSD), such as when multiple clutter types (ground, sea, 25412 rain, etc.) are simultaneously present in the cell under test. Additionally, it must provide near-optimum detection under unknown Doppler frequencies of targets of interest for any target frequency within the interval of unambiguous Doppler measurements. This paper proposes a new DDL implementation of the classical time-domain AMF (TD-AMF) detector. The fundamental idea for this implementation is to employ the concept of a region of possible target detection (RPTD), a small set of Doppler bins that captures most of the target signal power. The proposed RPTD-based DDL-AMF detector is free of the disadvantages of an earlier suggested RODI-based DDL-GLR detector. The former ensures rapid adaptive detection with near-optimum performance under unknown target Doppler frequency and multimodal clutter spectra. It should be noted that since the publication of [4], we have found only one work [5] on a similar topic in open literature. In addition, [6] extended the principle of adaptive DDL processing to the adaptive spatial-temporal (angle-Doppler) processing. One of the reasons for the lack of publications on adaptive DDL detection is that from the early 90s, much attention has been paid to the problem of adaptive target detection in sea clutter governed by compound-Gaussian (CG) distributions [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and references therein confirm the existence of plenty of publications on this topic. The present paper considers the DDL-based adaptive radar detection under Gaussian interference for simplicity of discussion. Extending the theory of classical adaptive detection to the Doppler domain is straightforward since Gaussian distributions are closed under affine transformations. Building the theory of adaptive target detection in the Doppler domain under interference governed by a complex elliptically symmetric distribution (CES) is also simple: CES distribution is closed under affine transformations [21]. It should also be noted that the class of CES distributions includes CG distributions as a particular case. The remaining part of this paper is organized as follows. Section II describes the data representation for adaptive DDL detectors. Section III analyzes the disadvantages of the RODI-based DDL-GLR detector. Section IV introduces the concept of the region of possible target detection (RPTD). The new RPTD-based DDL-GLR and DDL-AMF detectors are then described in Section V. These are the detectors under investigation in the present paper. Section VI analyzes the detection performances of the detectors under investigation in several clutter scenarios under known and unknown target Doppler and discusses their advantages and disadvantages. Section VII evaluates the computational load of the RPTDbased DDL-AMF detector relative to its classical counterpart. The conclusion is given in Section VIII. II. DATA REPRESENTATION FOR DDL DETECTORS This section describes the data representation for adaptive DDL processing. To clarify implementing adaptive VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 1. Data arrangement for adaptive detectors: (a) time domain and (b) Doppler domain. DDL detectors, it considers a data arrangement for two-dimensional pulse Doppler radar as a simple example. Fig. 1 (a) illustrates a typical data arrangement for adaptive target detection in the time domain. The input data are organized into a complex-valued range-pulse (fast/slow time)   data matrix X = xij of size M × N ; M is the number of range cells, and N is the number of pulses in a current CPI. The entries in this matrix represent baseband demodulated, sampled, digitized, and pulse-compressed radar returns received in the CPI. In Fig. 1 (a), white circles mark the cell under test (CUT) samples that represent a CUT vector x = [xi1 xi2 . . . xiN ]T corresponding to the i-th range cell to be tested for the presence of targets. The shaded circles represent the samples in the training vectors xq = [xiq 1 xiq 2 . . . xiq N ]T , where iq is the range cell index the q-th training vector is associated with, q = 1, 2, . . . , KT , and KT is the number of vectors. It is assumed that xq are target-free, mutually independent, and share the same covariance matrix with the CUT vector x under hypothesis H0 (target absent). Fig. 1 (b) shows a similar data arrangement in the Doppler domain for adaptive target detection based on the DDL adaptivity principle (DDL principle   for brevity). The range-Doppler data matrix Ỹ = ỹij , M ×N results from the row-wise DFT applied to the data matrix X in T Fig. 1 (a). In particular, the vector ỹ = ỹi1 ỹi2 . . . ỹiN of size N ×1 is a Doppler domain image of the corresponding T time domain CUT vector x =  T [xi1 xi2 . . . xiN ] . The n×1 vector ỹm = ỹid1 ỹid 2 . . . ỹid n , n ≪ N , is a DDL CUT vector to be tested by a DDL detector for the presence of an m-th presumable target with unknown Doppler frequency that may present in the i-th range cell. The vector ỹm comprises the n entries of the vector ỹ corresponding to a set of Doppler bins d1 , d2 , . . . , dn , where the essential portion of the m-th VOLUME 12, 2024 presumable target’s power is concentrated. The procedure for determining such a set of Doppler bins will be introduced further. The shaded T samples in the training  circles mark the vectors ỹmk = ỹik d1 ỹik d 2 . . . ỹik d n , where ik is the range cell index the k-th training vector is associated with, k = 1, 2, . . . , K , and K (K ≪ KT ) is the number of vectors to be used for estimating an unknown DDL covariance matrix 8̃m of size n × n. The entries in the training vectors are associated with the Doppler bins d1 , d2 , . . . , dn . The vectors ỹmk are target-free, mutually independent, and share the same DDL covariance matrix 8̃m with the DDL CUT vector ỹm under hypothesis H0 . It is straightforward to extend the theory of classical adaptive detection to the Doppler domain since Gaussian distributions are closed under affine transformations. This property means that for any complex-valued Gaussian vector x, x ∼ CN (µ, 6), a new vector ỹ = Bx + b also
follows a Gaussian distribution, ỹ ∼ CN Bµ + b, B6BH , for all nonsingular B ∈ CN ×N and b ∈ CN . The Discrete Fourier Transform (DFT) is a linear transformation given by ỹ = Fx, with F being the DFT matrix. Hence, for a time domain vector c∼CN (0,6), its Doppler domain image is a vector c̃∼CN 0,6̃ , with 6̃ = F6FH . Thus, to build the theory of adaptive signal detection in the Doppler domain, one should follow the corresponding time domain theory and apply proper symbol substitutions for parameters, vectors, and matrices by their corresponding Doppler domain equivalents. This theory operates on DFT images of size N ×1 for vectors and N × N for matrices. The theory of adaptive signal detection based on the DDL principle operates on vectors and matrices that represent Doppler domain data associated with a small set of Doppler bins d1 , d2 , . . . , dn , n ≪ N . Hence, the DDL-based theory 25413 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity of adaptive detection is a particular case of the adaptive detection theory in the Doppler domain. We first define the Doppler domain representations (DFT images) corresponding to the time domain data. These are the N ×1 received data vector x = [x1 x2 . . .xN ]T , the N ×1 target steering vector s = [1 ej2πF . . . ej2π(N −1)F ]T , F is the known normalized Doppler frequency, and the known disturbance covariance matrix 6 of size N × N (disturbance stands for noise plus clutter). This matrix is given by 6 = Pc C0 +Pn I (1) where Pc and  P  n represent the clutter and thermal noise power, C0 = cij is the N × N normalized clutter covariance matrix (cii = 1, i = 1, 2, . . . , N ), and I is the N × N identity matrix. In the Doppler domain, we use a zero-Doppler-centered data format. Therefore, the DFT images ỹ and s̃ of size N ×1 associated with the vector x and the vector s are given by ỹ = [ỹ1 ỹ2 . . .ỹN ]T =fftshift(fft(x)), (2) T (3) s̃ = [s̃1 s̃2 . . .s̃N ] =fftshift(fft(s)), where fft and fftshift stand for MATLAB functions, respectively implementing the FFT and the data shift needed to put the zero-frequency component in the middle of the DFT spectrum. Accordingly, the DFT image 6̃, N × N , corresponding to the disturbance covariance matrix 6 is given by 6̃=fftshift (fft (6)) (4) Having defined the DFT images of size N ×1 (vectors) and of size N × N (matrices), it is straightforward to determine the DDL representations for vectors and matrices associated with a given set of Doppler bins d1 , d2 , . . . , dn . The n× 1 DDL steering vector t̃ = [t̃1 t̃2 . . .t̃n ]T associated with this set is generated by extracting from the N ×1 vector s̃ the entries located at the dk -th positions, i.e., t̃k = s̃dk , k = 1, 2, . . . , n. Similarly, the n×1 DDL vector of received data τ̃ = [τ̃1 τ̃2 . . .τn ]T is generated by extracting from the N ×1 vector ỹ the entries located at the dk -th positions, i.e., τ̃k = ỹdk , k = 1, 2, . . . , n. The n × n disturbance covariance matrix 8̃ associated with the set of Doppler bins d1 , d2 , . . . , dn is represented by the entries located at the intersection of the dk -th rows and dl -th  columns, k, l = 1, 2, . . . , n in the N × N matrix 6̃= σ̃mn , m, n = 1, 2, . . . , N . Thus, the DDL covariance matrix 8̃ is generated by extracting the corresponding entries in the matrix 6̃ as given below   8̃ = ϕ̃kl , ϕ̃kl = σ̃dk dl , k, l = 1, 2, . . . , n (5) To estimate the unknown covariance matrix 6 classical adaptive detectors employ a sample covariance matrix 6̂ that is calculated using the set of training vectors xq , q = 1, 2, . . . , KT as XKT xq xH (6) 6̂ = (1/KT ) q q=1 25414 By analogy, an adaptive DDL detector of order n employs ˆ of size the corresponding sample DDL covariance matrix 8̃ n × n as an estimate of the exact DDL covariance matrix 8̃ associated with the set of Doppler bins d1 , d2 , . . . , dn . ˆ n × n, is calculated The sample DDL covariance matrix 8̃, using the collection T of the DDL training vectors ỹmk =  ỹik d1 ỹik d 2 . . . ỹik d n , k = 1, 2, . . . , K as ˆ = (1/K ) XK 8̃ k=1 ỹmk ỹH mk (7) III. RODI-BASED DDL-GLR DETECTOR As mentioned in Introduction, the RODI-based DDL-GLR detector [4] assumes that the entire Doppler domain may be divided into two types of regions. The first type refers to the flat or nearly flat spectrum regions, and the second one refers to the sharply changing spectrum regions. According to [4], if the target Doppler frequency is within the first type of spectrum regions and far away from the clutter spectrum peaks, the conventional processor employing a windowed FFT bank of filters followed by the traditional CA CFAR detectors is expected to provide nearly optimum detection performance (at least under Gaussian clutter). In contrast, when the target Doppler falls on sharply changing spectrum areas or regions near the clutter spectrum peaks, the traditional detector performance may fall far below the optimum. Therefore, [4] recommends using the adaptive DDL processing for improving detection performance only for such spectrum regions called the regions of detection improvement (RODI). The samples extracted from the Doppler bins in the l-th RODI jl1 jl2 . . . jlnl are used to form the corresponding nl × 1 CUT and training vectors to feed the DDL-GLR detector of the order nl associated with the l-th RODI. The order of the l-th RODI nl is the number of Doppler bins covered by this RODI, nl ≪ N . Let ỹl = [ỹijl1 ỹijl2 . . .ỹijln ]T be the CUT vector, nl × 1, l associated with the l-th RODI at the i-th range cell in the range-Doppler matrix Ỹ (detection matrix). Similarly, let ỹlk = [ỹik jl1 ỹik jl2 . . .ỹik jln ]T be the k-th vector, with ik being l the k-th range cell index the k-th training vector is associated with, k = 1, 2, . . . , K in a set of training data associated with the ỹl . Let s̃l = [s̃jl1 s̃jl2 . . .s̃jlnl ]T be the nl × 1 DDL steering vector at some normalized Doppler frequency F ′ from the interval corresponding to the l-th RODI. The entries in s̃l are extracted from the positions jlm , m = 1, 2, . . . , nl in the N × 1 target steering vector in the Doppler domain s̃=[s̃1 s̃2 . . .s̃jl1 s̃jl2 . . .s̃jlnl . . .s̃N −1 s̃N ]T (8) The vector s̃ in (8) is the result of the FFT transform (unweighted or weighted) applied to the N × 1 time domain ′ ′ ′ steering vector s = [1ej2πF ej2π2F . . . ej2π(N −1)F ]T . For the l-th RODI jl1 jl2 . . . jlnl , the DDL-GLR detector of the order nl computes nl test statistics ηlm covering the VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity Doppler bins jlm , m = 1, 2, . . . , nl , as given by −1 ηlm 2 s̃H lm 6̂ l ỹl   = , −1 H 6̂ −1 ỹ /K s̃H 6̂ s 1+ ỹ lm l l lm l l 1 (9) P H where 6̂ l = (1/K ) K k=1 ỹlk ỹlk , nl × nl , is the interference covariance matrix estimate associated with the l-th RODI. In (9) above, the m-th vector s̃lm = [s̃1m s̃2m . . .s̃nl m ]T , m = 1, 2, . . . , nl , is the DDL steering vector associated with the m-th Doppler bin in the l-th RODI. The entries in s̃lm are extracted from the vector s̃ as specified by (8). For each m, the vector s̃ in (8) is the result of the FFT applied to the time domain vector scomputed for the on-grid Doppler frequency F = F lm = jlm N associated with the m-th Doppler bin of the l-th RODI. Hence, no calculations are needed to determine DDL steering vectors associated with Doppler bins: all entries in the m-th vector s̃lm , m = 1, 2, . . . , nl are zeros except the m-th one, which is unity. Each test statistic ηlm , m = 1, 2, . . . , nl in (9) is then compared with the threshold ηl to complete the hypothesis test for the l-th RODI. The tests for Doppler bins not in any RODI can be performed by some conventional CFAR method. The false alarm probability of the DDL-GLR at any bin of the l-th RODI relates to the detection threshold ηl as given by   ηl K −nl +1 PFAl = 1 − (10) nl It should be noted that the GLR statistic ηlm in (9) assumes that the target signal does not contain unknown parameters. In [1], Kelly pointed out that if the signal steering vector includes one or more unknown parameters (such as target Doppler), the test statistic has to be maximized over these parameters. The maximization over target Doppler generally cannot be carried out explicitly. A remedy to this problem is a standard technique [1]: approximating the test statistic by computing it on a discrete set of Doppler frequencies. This technique is equivalent to forming a filter bank and declaring target presence if any filter output exceeds the threshold. Reference [4] analyzes the detection performance of the RODI-based DDL-GLR detector only for the non-fluctuating target model (the target signal amplitude is a constant) with a known target Doppler. This detector may provide reliable target detection under these assumptions even when the RODI order nl is essentially smaller than N . Presented numerical results demonstrate that even for nl = 4 − 6, it may achieve near-optimum detection performance delivered by the optimum detector of the order N = 64. An example discussed in [4, p. 535, Fig. 2] shows the RODI-based DDL-GLR detector’s performance on the DFT grid for the known target Doppler. The strong clutter is assumed to have a Gaussian-shaped PSD. The detector employs two adjacent RODIs of the 4-th order placed next to the PSD peak; these RODIs cover the Doppler bins VOLUME 12, 2024 from 11 to 18 (non-zero-Doppler-centered axis). The total number of pulses in the CPI is N = 64, with K = 24 available training samples. The clutter PSD is centered at the normalized Doppler frequency Fcp = 0.15 (peak location) with the spread parameter σc = 0.0025. The clutter-tonoise ratio (CNR) is set to 60 dB. The signal-to-noise ratio (SNR) is 3 dB, which results in the signal-to-disturbance ratio SDR = –57 dB at the detector’s input. This example shows near-optimum detection for the DDL-GLR detector at each RODI-associated Doppler bin, except for a slight departure from the optimum at bin 15. An essential performance departure from the optimum at both on-grid and off-grid Doppler frequencies is demonstrated by the frequency response (PD versus Doppler frequency) for the 4th order RODI-based DDL-GLR detector [see 4, p. 536, Fig. 3]. Analyzing the frequency response shows that there is a need to improve the detection performance around some off-grid frequency point (‘‘bad’’ frequency point). To remedy this problem, [4] suggests using an additional target steering vector corresponding to the middle of the nearest on-grid points. However, the radar processor needs accurate information on these ‘‘bad’’ frequency points to apply this remedy. In real radar systems, such information is unavailable and can only be obtained by measurements from the received data. These measurements complicate the RODI-based DDL-GLR implementation, especially in multiple clutter situations. In turn, measurement errors will inevitably result in performance degradation. This section complements the results in [4] with the detection performance analysis for the Swerling I target model, assuming a known target Doppler. We will show that the RODI-based DDL-GLR detector may suffer essential detection performance departures from the optimum. These departures may occur even for the known target Doppler frequency, no matter whether it takes on or off DFT grid values. The said detection performance degradations prevent the implementations of the RODI-based DDL-GLR detector under unknown Doppler frequency by using DDL-GLR test statistics associated with the DFT grid. For the known target Doppler frequency, the time domain GLR and AMF (TD-GLR and TD-AMF) detectors’ equations describing the detection performance for the Swerling I target model are given in [2]. Following [2], Appendices B and C derive similar DDL-GLR and DDL-AMF detectors’ equations. The corresponding equations for optimum detectors are given in Appendix A. Fig. 2 shows the on-grid detection performance of the RODI-based DDL-GLR detector as PD -vs-F plots (probability of detection PD versus normalized Doppler frequency F). In this figure, the number of pulses and training samples equals those in [4]: N = 64 and K = 24, respectively. Other settings are the Gaussian-shaped clutter PSD centered at Fcp = 0.15 or Fcp = 0.156 with σc = 0.0025, CNR = 60 dB, and SNR = 10 dB, which gives the input SDR = -50 dB. The two adjacent RODIs of the 4-th order next to the clutter PSD peak are RODI1 = [43, 44, 45, 46] and 25415 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 2. On-grid detection performance of RODI-based DDL-GLR detector for Swerling I target. FIGURE 4. Target signal power distribution across Doppler bins in RODIs. FIGURE 3. On- and off-grid detection performance of RODI-based DDL-GLR detector for Swerling I target. RODI2 = [47, 48, 49, 50] (we use a zero-Doppler-centered axis). As seen in Fig. 2 (a), the PD value (PD = 0.7536) at the Doppler bin 47 is noticeably below the optimum, while it keeps near the optimum for the neighboring Doppler bins. Fig. 2 (b) illustrates the performance degradation for Fcp = 0.156; this Fcp value is just a 4% shift relative to Fcp = 0.15 in Fig. 2 (a). Comparing Fig. 2 (a) and Fig. 2 (b) shows that the performance of the RODI-based DDL-GLR detector is quite sensitive to the location of clutter PSD peak. This sensitivity is especially pronounced in Doppler regions with a sharp change in the clutter spectrum. Moreover, it will also result in inevitable performance degradation due to errors in measuring clutter spectrum parameters. Fig. 3 shows the probability of detection PD at the three off-grid Doppler frequencies for the DDL-GLR detector associated with RODI1 and RODI2 previously used in Fig. 2. These off-grid frequencies are 25416   F1 = 13 N + 0.25 N = 0.2070,   F2 = 13 N + 0.5056 N = 0.2110,   F3 = 13 N + 0.75 N = 0.2148. As can be seen, there are essential departures from the optimum at the off-grid frequencies F2 and F3 . To explain this effect, we consider the signal power distribution across the Doppler bins in the corresponding RODIs for each of these specific Doppler frequencies. Figs. 4 (a) through (d) respectively plot the normalized (unit norm) DFT power spectra for the target signals whose Doppler frequencies are F46 , F1 , F2 , and F3 . Analyzing these spectra yields that the PD value approaches closer to the optimum when the portion of the total signal power in the corresponding RODI becomes more significant. In Fig. 4 (a),  for the signal at frequency F46 = 13 N , the total signal power is in the RODI1 = [43, 44, 45, 46] and, correspondingly, the associated PD = 0.8796 is close to the optimum. In Fig. 4 (b), for the signal at frequency F1 = 0.2070, the portion of its total power in the RODI1 = [43, 44, 45, 46] exceeds 0.8, and the corresponding PD = 0.7916 is not too far from the optimum. In contrast, in Fig. 4 (c), for the VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 5. SPC and CPC regions and associated with them clusters in Doppler profiles. signal at F2 = 0.2110, the portion of its total power in the RODI2 = [47, 48, 49, 50] is about 0.5, and the corresponding PD = 0.1268 is far from the optimum. For the signal at F3 = 0.2110, the portion of its total power in the RODI2 is equal to that for the signal at F1 in the RODI1 . However, the PD value at F3 is less than that at F1 . This is because the clutter samples in RODI2 are less correlated than in RODI1 ; therefore, the clutter associated with the RODI2 is less suppressed than that in RODI1 . Inspecting Fig. 4 (c) and (d) leads to the conclusion that choosing new RODI2 = [45, 46, 47, 48] would bring the PD values at frequencies F2 and F3 near the optimum. However, this choice is based on the knowledge of clutter PSD parameters. Unfortunately, they are unknown in the actual radar operational environment. As pointed out in [4], determining the number and locations of RODIs required to achieve near-optimum detection performance at arbitrary target Doppler frequency is a challenging problem. However, [4] suggests no practical solutions to this problem; moreover, we have not found answers in the open literature. Analyzing the effect of the signal power distribution across RODI on DDL-GLR performance yields that using the RODI concept in DDL processing does not maximize the signal power portion within RODI as required to achieve near-optimum detection at arbitrary target Doppler frequency. Thus, the shortcomings of the RODI-based DDL-GLR detector [4] make its practical use questionable. IV. REGION OF POSSIBLE TARGET DETECTION The analysis of the signal power distribution within a limited Doppler region on the DDL-GLR detection performance reveals that the DDL detector has to capture as much signal power as possible to have its detection performance closer to the optimum. This observation also suggests that VOLUME 12, 2024 determining the limited Doppler region for a DDL detector must be based on the target signal spectrum rather than the clutter spectrum to capture the maximum possible portion of the target signal power. This idea leads to the Region of Possible Target Detection (RPTD) concept. As shown below, the RPTD-based DDL detectors can provide near-optimum detection performance at arbitrary target Doppler, no matter whether it is known or not. For a DDL detector of order n, the region of possible target detection (RPTD) is such a set of n Doppler bins D = {d1 , d2 , . . . , dn } that with high probability contains the maximal portion of the power of a target signal that may presumably be present in the cell under test associated with a given range cell in the range-Doppler data matrix (or the detection matrix) Ỹ. As is well known, the DFT concentrates an essential portion of the signal power within some narrow Doppler region around the peak value in the signal power distribution. for Fig. 5 (b) illustrates the power distribution    clutter and target signals in the matrix Z̃ = z̃ij = |ỹij |2 , M × N , calculated from the corresponding entries in the complex-valued   detection matrixỸ = ỹij , M × N . The detection matrix Ỹ is the result of the row-wise FFT transform applied to the original CPI data matrix shown in Fig. 5 (a). For simplicity, Fig. 5 (a) shows only one target-associated sample (maximum magnitude) on the range axis. In Fig. 5 (b), for each range cell that contains target signal data, we call the region of signal power concentration (SPC) a set of Doppler bins SPC = {d1 , d2 , . . . , dn } , n ≪ N , where some essential portion of the signal power is confined. Similarly, we denote CPC = {d1 , d2 , . . . , dm } , m ≪ N , the region of clutter power concentration (CPC). Fig. 5(b) shows two isolated regions SPCi1 = {d7 d8 d9 } and CPCi1 = {d3 d4 d5 }, at the i1 -th range bin, and two overlapping regions SPCi2 = {d2 d3 d4 } and CPCi2 = {d3 d4 d5 }, at the i2 -th range bin. Since SPC and CPC 25417 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity regions may overlap, the data associated with CPC regions may also contain the target signal power. Fig. 5 (c) illustrates the Doppler profiles associated with the range cells that contain SPC and CPC regions depicted in Fig. 5 (b). These Doppler profiles represent magnitude-squared data samples from the corresponding rows in the matrix Z̃. The data samples in each Doppler profile can be grouped in clusters. We define a cluster as a group of samples consisting of the maximum sample (local peak) and its neighboring samples up to the nearest minimum sample on the right and left of the local peak. Among all possible clusters in a Doppler profile, clusters associated with SPC and CPC regions are power-dominant since the signal-to-noise ratio (SNR) and clutter-to-noise ratio (CNR) increase after pulse compression and coherent integration (FFT). Samples in such SPC/CPC-associated clusters are shadowed in Fig. 5 (c). However, which clusters are due to targets is unknown a priori. Therefore, the data in the detection matrix Ỹ associated with each cluster must be tested for the target presence. The local peak value in a target-associated cluster with a high probability is the closest to the maximum value of the target signal power spectrum. Hence, to determine the RPTD of order n, we need to locate the peak value in a cluster and select a group of n − 1 bins closest to the peak-associated Doppler bin to maximize the portion of the signal power captured by this group. index dm associated with the maximum value in p̃ dm = arg max (p̃i , i = 1, 2, . . . , N ) i 
 p̃dm = max p̃1 p̃2 . . . p̃N (12) In the second step, the procedure may be essentially simplified. One can find dm using the simple equation do = round (FN ) + N2 + 1, where the function round (x) returns the nearest integer to x. The value of do needs correction considering the DFT periodicity − if do < 1, then dm = do + N ; if do > N , then dm = do − N ; otherwise, dm = do . Finally, the third step computes the RPTD associated set D = {d1 d2 . . . dn } using an algorithm shown in Fig. 6. This algorithm uses two closest neighboring samples p̃d − and p̃d + computed considering the DFT spectrum periodicity from the following equations ( a, if a > 0 − where a = dm − 1 d = N , otherwise (13) ( b, if b ≤ N where b = dm + 1 d+ = 1, otherwise The RPTD identification procedure described above can also be used when the target Doppler frequency is unknown. In this case, one should substitute an unknown value of the target Doppler F with its estimate F̂. This estimate can be obtained using fine Doppler estimators [22]. V. RPTD-BASED DDL DETECTORS FIGURE 6. RPTD computing algorithm. To introduce the RPTD identification procedure, we assume that the target Doppler F is known (|F| < 0.5). This procedure comprises the following steps. First step computes the target steering vector s = [1 ej2πF . . . ej2π(N −1)F ]T , its DFT image s̃= [s̃1 s̃2 . . .s̃N ]T =fftshift(fft(s)), and the corresponding Doppler profile p̃   p̃ = p̃1 p̃2 . . . p̃N , p̃i = |s̃i |2 , i = 1, 2, . . . , N (11) The vector p̃ represents the steering vector’s power distribution in the Doppler axis. The second step finds the Doppler 25418 In the case of an unknown target Doppler frequency, the standard technique is to test the outputs of a filter bank associated with a fixed DFT grid. For the adaptive DDL processing, the present paper proposes a non-uniform grid whose elements are Doppler frequency estimates associated with local peaks (maxima) in a given Doppler profile. We refer to the Doppler grid defined in such a way as the maximum likelihood Doppler (MLD) grid. Before calculating the MLD grid, we generate a set of representative range cells. To this end, we use the data in Doppler T lines which are the columns   [z̃1j2z̃2j . . . z̃Mj ] , j = 1, 2, . . . , N , of the matrix Z̃ = z̃ij = |ỹij | , M × N . First, we find the positions of all local range peaks (maxima) in each Doppler line of the matrix Z̃. For example, this location procedure can be done using the Matlab function ‘‘findpeaks.’’ Then we compose a comprehensive set of representative range cell indices as the set R = {ik , k = 1, 2, . . . , MR }, where the unique MR indices ik ∈ {1, 2, . . . , M }, MR < M , are all those range cell indices for which there exists a local range peak at least in one out of the N Doppler lines. Having obtained the set R, we compute power Doppler profiles associated with the representative range cells. This is done by applying an Nfft -point FFT to each vector x = [xik 1 xik 2 . . . xik N ]T with the subsequent elementwise magnitude-squaring. The vectors x are extracted from the ik -th row of the CPI data matrix X, where ik ∈ R. In computing these profiles, we use the Nfft -point FFT with VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity Nfft = qN , q ≥ 2 because it provides favorable initial conditions for accurate Doppler measurements with a simple non-iterative fine Doppler estimator operating on magnitude data [22]. This computationally efficient estimator uses a peak and two neighboring samples for each local maximum in a given Doppler profile. In addition, such qN -point FFT avoids gross measurement errors at Doppler frequencies near ±0.5. Let [ṽik 1 ṽik 2 . . . ṽik Nfft ] be a power Doppler profile associated with the ik -th representative range cell, k = 1, 2, . . . , MR . To obtain the MLD grid, we first find the positions of all local peaks in the vector [ṽ1 ṽ2 . . . ṽNfft ] representing this Doppler profile (for simplicity, the range cell index ik for the samples ṽik j will be omitted). Since we have to consider the periodicity of the Doppler profile, a local peak identification procedure uses a 1 × (Nfft + 2) vector ṽNfft ṽ1 ṽ2 . . . ṽNfft ṽ1 that is a given Doppler profile vector appended by the element ṽNfft on the left and the element ṽ1 on the right. Fig. 7 explains the simple procedure for identifying local peaks and shows its output for the profile of length 16 (q = 2, N = 8). Having obtained the positions jkm , m = 1, 2, . . . , Pk of Pk local peaks in a power Doppler profile associated with the ik -th range cell, ik ∈ R, we calculate the corresponding MLD grid by applying a fine Doppler estimator to each data set associated with the m-th local peak, m = 1, 2, . . . , Pk . A simple fine Doppler estimator [22] uses a closed-form equation based on quadratic interpolation. This estimator measures the Doppler frequency estimate fˆm as the peak location of the interpolating parabola fitted through the m-th local peak z̃jm and two neighboring samples z̃j− and z̃j+ . The normalized Doppler frequency estimate is calculated as F̂m = fˆm /Nfft . It should be noted that some of the local peaks in a Doppler profile corresponding to the ik -th representative range cell, ik ∈ R represent range-Doppler peaks Pm (ik , jkm ) associated with two-dimensional (range-Doppler) point target responses. These responses are due to radar returns from individual specular points (scattering centers) of various objects within a radar scene represented in the range-Doppler data matrix Ỹ. Each point target response consists of a group of correlated samples. Some of these groups are due to targets of interest that may present among the objects in the radar scene. Therefore, the data in such groups are of primary interest for detection hypothesis testing. We use the normalized Doppler frequency estimates F̂m , m = 1, 2, . . . , Pk , from the MLD grid associated with the ik -th range cell, ik ∈ R, with Pk being the number of local peaks, in determining the m-th RPTD Dm = {d1 , d2 , . . . , dn } associated with the m-th local Doppler peak. The elements in the set Dm are the indices of columns in the detection matrix Ỹ, from which the Doppler data are extracted for testing the target presence in the m-th RPTD. Fig. 8 shows the location of the DDL data (CUT and training vectors) in the range-Doppler detection matrix Ỹ. VOLUME 12, 2024 FIGURE 7. Local peaks identification procedure. FIGURE 8. CUT and training data for RPTD-based DDL detector. The n × 1 vector ỹm = [ỹik d 1 ỹik d 2 . . . ỹik d n ]T is the DDL CUT vector formed by the entries at the columns associated with the m-th RPTD Dm = {d1 , d2 , . . . , dn }, (n ≪ N ) in the ik -th row of the range-Doppler data matrix Ỹ. Similarly, the n × 1 vector ỹmq = [ỹiq d 1 ỹiq d 2 . . . ỹiq dn ]T is the q-th DDL training vector (q = 1, 2, . . . , K ) associated with the same m-th RPTD Dm . The data for the q-th training vector are extracted from the corresponding iq -th row of Ỹ. ˆ be the estimate of the exact DDL covariance Let 8̃ m matrix 8̃m associated with the m-th RPTD Dm . We compute this estimate as the sample DDL covariance matrix ˆ = 1 XK ỹ ỹH (14) 8̃ mq mq m q=1 K ˆ is not singular, It is assumed that K > n; hence, the matrix 8̃ m ˆ −1 exists. so its inverse 8̃ m Let ˆt̃m = [t̃ˆ1 t̃ˆ2 . . .t̃ˆn ]T be the DDL steering vector estimate formed respectively by the entries s̃ˆd1 , s̃ˆd2 , . . . , s̃ˆdn that are extracted from the estimate of the Doppler domain steering vector s̃ˆm = [s̃ˆ1 s̃ˆ2 . . .s̃ˆN ]T , where s̃ˆm = fftshift(fft(ŝm )) with ŝm being the time domain steering vector estimate: ŝm = [1 ej2π F̂m ej2π2F̂m . . . ej2π(N −1)F̂m ]T is entirely determined by 25419 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity the Doppler frequency estimate F̂m associated with the local maximum of the m-th RPTD. For the m-th RPTD Dm , the DDL-GLR detector of an order n is given by 3̃GLRm = 1 ˆ −1 ˆt̃H 8̃ m m ỹm ˆ˜ −1 ˆH ˆ −1 ˆ 1+(ỹH m 8m ỹm )/K t̃m 8̃m t̃m 2 H1 ≶ λ̃GLR (15) H0 where λ̃GLR is the detection threshold computed for the given order n, the training sample size K , and the specified probability of false alarm. A simple alternative to the GLR detector is the AMF detector. These detectors possess the CFAR property and have comparable detection performance for target signals aligned with the assumed steering vector [2]. However, as shown in [23], the GLR detector provides essential rejection for signals misaligned with the assumed steering vector, whereas the AMF detector does not [2]. Because of measurement errors, the DDL steering vector estimate ˆt̃m is not aligned with the actual DDL steering vector t̃m corresponding to the target signal. Therefore, for the detector (15), one can expect essential performance degradation. The present paper also considers the DDL implementation of the AMF detector. Similarly, for the m-th RPTD Dm , the DDL-AMF detector of an order n is given by 3̃AMFm = ˆ −1 ˆt̃H 8̃ m m ỹm ˆ −1 ˆ ˆt̃H 8̃ m m t̃m 2 H1 ≶ λ̃AMF (16) H0 where λ̃AMF is the corresponding detection threshold computed for the given order n, the training sample size K , and the specified probability of false alarm. Under hypothesis H1 , the DDL CUT vector ỹm associated with the m-th RPTD Dm should contain an essential portion of the total signal power carried by the corresponding time domain CUT vector x of size N × 1 even when n ≪ N . Hence, one can expect a minor performance degradation in the DDL-AMF detector relative to its time domain counterpart TD-AMF detector. For the DDL-AMF detector, calculating the detection threshold λ̃AMF in (16) from (B9) given in Appendix B entails time-consuming numerical iterations. This approach is not acceptable for a real-time implementation of the DDL-AMF detector. Appendix D provides a computationally simple and accurate approximating formula for computing λ̃AMF . VI. PERFORMANCE OF RPTD-BASED DDL DETECTORS In this section, we analyze the performance of the detectors under investigation: the RPTD-based DDL-GLR detector (15) and the RPTD-based DDL-AMF detector (16). We compare their performance to the optimum detector of order N and the optimum DDL detector of order n. This analysis includes known and unknown target Doppler frequency scenarios for the Swerling I target model. Appendices A, B, and C provide equations for calculating 25420 FIGURE 9. On-grid PD -vs-F plots comparison of RPTD- and RODI-based DDL detectors. detection performance under known target Doppler. We use Monte-Carlo simulations to evaluate detection performance for unknown target Doppler frequency. The FFT of size Nfft = 4N is used to compute Doppler profiles required to identify local range-Doppler peaks and perform target Doppler measurements. A Matlabembedded function, ‘‘findpeaks,’’ is exploited for identifying local peaks. The fine Doppler estimator from [22] is used for Doppler measurements. In determining the RPTD, we employ the RPTD identification procedure described in Section IV. We first compare the detection performance of the DDL-GLR detector that employs the RODI concept proposed in [4] with that of the detectors under investigation for the scenario with known target Doppler. Fig. 9 compares the ongrid PD -vs-F plots (F is the normalized Doppler frequency) of the RPTD-based DDL-GLR and the RODI-based DDLGLR detectors in the presence of strong clutter under the settings identical to that in Fig. 3. In Fig. 9, it can be seen that both detectors exhibit similar performance at frequencies on the DFT grid corresponding to bins in the RODI regions. However, the RPTD-based DDL-GLR detector outperforms the RODI-based DDL-GLR one at the off-grid points. The advantage of the former against the latter is evident in Fig. 10, which compares their PD -vs-F plots for continuous target Doppler frequency (on- and off-grid performance comparison). Fig. 11 shows the PD -vs-n plots (n is the order of a DDL detector) for the probability of false alarm Pfa = 10−9 . The plots are calculated for the known Doppler frequency scenario (F = 0.25) at the signal-to-noise ratio SNR = 15dB (input SDR = −45 dB). The strong clutter (CNR = 60 dB) is assumed to have a zero-centered Gaussianshaped PSD, which spectral width σc = 0.0025. As can be seen, both of the detectors under investigation exhibit comparable and near-optimum detection performance when the target Doppler frequency is known. Fig. 12 plots the loss in the detection probability for the detectors under investigation relative to the optimum detector VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 10. On- and off-grid PD -vs-F plots comparison of RPTD- and RODI-based DDL detectors. FIGURE 11. PD -vs-n plots for RPTD-based DDL-GLR and DDL-AMF detectors (known Doppler frequency). FIGURE 12. Loss in PD versus n for RPTD-based DDL-GLR and DDL-AMF detectors (known Doppler frequency). of order N with different choices of N . As seen, for n ≥ 3, the loss in the detection probability does not exceed 5% for N = 64 and 128. However, for n ≥ 4, this loss is below 5% for N = 32 and does not exceed 2.5% for N = 64 and 128. VOLUME 12, 2024 FIGURE 13. PD -vs-n plots for RPTD-based DDL-GLR and DDL-AMF detectors. FIGURE 14. Loss in PD versus n for RPTD-based DDL-GLR and DDL-AMF detectors. The probability of detection for the optimum detector of order N is PD = 0.9897. Fig. 13 compares the PD -vs-n plots estimated using 10,000 Monte-Carlos for the unknown target Doppler frequency (marked UnKnown) with those for the known target Doppler (marked Known) scenario. As can be seen, the RPTD-based DDL-AMF detector is robust to Doppler measurement errors, while the RPTD-based DDL-GLR one severely degrades. Indeed, as seen in Fig. 14, when the target Doppler frequency is unknown, the PD loss for the DDL-AMF detector does not exceed 5% for n ≥ 4, while for the DDL-GLR one, it is about 48% for n = 4 and 25% even for n = 8. The PD losses are calculated relative to PD = 0.9897 for the optimum detector of order N . For the detectors under investigation of order n = 4, Fig. 15 shows the PD -vs-F plots in the case of the known target Doppler embedded in the strong clutter with a zero-centered Gaussian PSD and spectral width σc = 0.0025. For the Swerling I target, the input signal-to-noise ratio is supposed to be SNR = 15 dB (input SDR = −45 dB). The training sample size is KT = 5N = 320 for the optimum detector of order N 25421 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 15. Comparison of PD -vs-F plots for RPTD-based DDL-GLR and DDL-AMF detectors, optimum detectors, and Taylor windowed FFT followed by CA CFAR processor (known target Doppler). FIGURE 17. PD -vs-F plots comparison at different Nfft for fine Doppler measurements. FIGURE 18. PD -vs-F plots for detectors under investigation in multiple-clutter scenario. FIGURE 16. PD -vs-F plots comparison for known and unknown target Doppler at Nfft = 4N. and K = 5n = 20 for the DDL detectors. As can be seen, both the detectors under investigation have minor ripples in their frequency responses (PD -vs-F plots). However, they exhibit near-optimum detection performances. Fig. 15 includes the PD -vs-F plot for a detector employing Taylor windowed N -point FFT followed by a CA CFAR that uses Nref = K = 20 reference samples. As one can see, this traditional detector does not provide near-optimum performance over the flat spectrum regions associated with the Doppler intervals −0.5 < F < −0.06 and 0.06 < F < 0.5. Moreover, the frequency response of this detector exhibits noticeable ripples due to the inevitable straddling loss. Thus, the critical assumption in [4] that adaptive DDL detection should be applied only to sharply changing spectrum regions is generally incorrect. Fig. 15 shows that the probability of detection for conventional detectors may fall far below the optimum even when the target Doppler is far away from the clutter spectrum peaks. Thus, adaptive DDL 25422 detection should generally be applied over the entire Doppler domain. Fig. 16 compares the PD -vs-F plots (estimated using 2,000 Monte-Carlos) for the unknown target Doppler frequency against those for the known target Doppler shown in Fig. 15. As one can see, the DDL-AMF detector’s performance for the unknown target Doppler frequency is close to that for the case of known Doppler. The minor ripples in the detection curve do not deprive its near-optimum behavior. However, the DDLGLR detector’s performance is severely impaired. Thus, the former significantly outperforms the latter in detection capability in scenarios with unknown target Doppler. Fig. 17 compares the PD -vs-F plots of the RPTD-based DDL-AMF detector of order n = 4 evaluated for the unknown target Doppler frequency (using 2,000 Monte Carlos) at two different lengths Nfft = 4N and 2N of the FFT for fine Doppler measurements. As expected, better detection performance is achieved for the former. Fig. 18 shows that the RPTD-based DDL-AMF detector of order n = 4 can maintain reliable detection performance in VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity FIGURE 19. PD -vs-SDR plots for detectors under investigation at different target Doppler. three clutters even for the unknown target Doppler frequency. In this multiple-clutter scenario, the RPTD-based DDL-GLR detector also degrades when the target Doppler frequency is unknown, as in situations with a single clutter. For scenarios with the known and unknown target Doppler frequency F, Fig. 19 (a) and (b) show the PD -vs-SDR plots (SDR stands for the signal-to-disturbance ratio) evaluated for F = 0.15 and F = 0.17, respectively. These plots confirm the significant advantage of the DDL-AMF detector against the DDL-GLR one under the unknown target Doppler frequency. VII. COMPUTATIONAL LOAD This Section evaluates and compares the computational load of the conventional TD-AMF detector and the RPTD-based DDL-AMF detector proposed in this paper. This evaluation considers the total amount of dominant calculations per CPI. For the TD-AMF detector, the CPI data are represented by the fast/slow time (range-pulse) matrix X of size M × N . For the DDL-AMF detector, the CPI data are represented by the M × N Range-Doppler matrix Ỹ, which results from the row-wise FFT applied to the matrix X. We assume that the number of representative range cells MR and the average number of local Doppler peaks per each representative range cell ND are known. Hence, the number of local range-Doppler peaks per CPI is Np = ND M R . For the TD-AMF detector implementation, the N × N sample covariance matrix (SCM) must be calculated, which requires about KT N 2 complex-valued floating-point operations (flops); hence, KT = 4N incurs 4N 3 flops. Computing the inverse of the SCM requires about 2N 3 flops. Thus, calculating the SCM and the inverse matrix involve about 6N 3 flops per cell under test (CUT). Since the TD-AMF operates on data in the matrix X, the CUT vectors for this detector are those rows in X whose indices are in the set of representative range cell indices R. Hence, these operations involve CL1TD = 6N 3 MR flops per CPI. VOLUME 12, 2024 Because the target Doppler frequency is unknown, the TD-AMF must calculate its test statistic ND times (on average) per CUT or exactly Np times per CPI. These calculations require CL2TD = 2N 2 Np flops per CPI. Calculating each test statistic requires the N ×1 steering vector estimate associated with the corresponding local range-Doppler peak. In turn, this estimate uses the corresponding Doppler frequency estimate. The fine Doppler estimator needs data only from one Nfft -point FFT to obtain these Doppler frequency estimates (ND times on average) for each representative range cell. One such FFT incurs (3/2) Nfft log2 Nfft flops. Hence, estimating the steering vectors incurs CL3TD = MR × (3/2) Nfft log2 Nfft flops per CPI. Thus, the total dominant computation load per CPI for the TD-AMF detector is CLTD = CL1TD + CL2TD + CL3TD (17) For the RPTD-based DDL-AMF detector, calculating the corresponding SCM of size n × n requires about Kn2 flops; hence, K = 4n incurs 4n3 flops. Computing the inverse matrix involves about 2n3 flops. Calculating DDL test statistics requires 2n2 flops per CUT. Therefore, calculating the SCM, the inverse matrix and test statistics involves about 6n3 + 2n2 flops per CUT or CL1DDL = Np × (6n3 + 2n2 ) flops per CPI because each DDL CUT is associated with a local range-Doppler peak. To evaluate the dominant computational load for the DDL-AMF detector, we must also consider the FFT operations involved in its implementation. First, these are about CL2DDL = M ×(3/2)N log2 N flops per CPI for the N -point FFT that calculates the M × N detection matrix Ỹ. Second, the fine Doppler estimator employs the Nfft -point FFT that incurs additional (3/2)Nfft log2 Nfft flops per representative range cell or CL3DDL = MR ×(3/2)Nfft log2 Nfft flops per CPI. Third, calculating the DFT images for the estimated target steering vectors incurs CL4DDL = Np ×(3/2)N log2 N flops per CPI. Thus, the total dominant computation load per 25423 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity TABLE 1. DDL gain in computational load. detection in compound-Gaussian clutter and extended to spatial-temporal adaptive processing for airborne surveillance radar systems. APPENDIX A OPTIMUM DETECTORS A1. TIME-DOMAIN OPTIMUM DETECTOR OF ORDER N CPI for the DDL-GLR detector is CLDDL = CL1DDL + CL2DDL + CL3DDL + CL4DDL (18) Using (17) and (18), we calculate the DDL gain GDDL in the computational load as GDDL = CLTD /CLDDL (19) Before evaluating the order of magnitude of achievable DDL gain, we note that the upper bound for the number of representative range cells per CPI is sup MR = M . Therefore, assuming that an actual MR value makes up a definite γ percent of its maximum is reasonable. Thus, MR = round (0.01γ M ). We evaluate the DDL gain in computational load using (19) for n = 4, 5 and 6, at N ∈ {64, 128, 256}, Nfft = 4N , M = 8,000, γ = 90%, and ND = N . This setting for ND is reasonable for Nfft ≥ 2N . Table 1 summarizes the GDDL values rounded to the nearest smallest integer. As can be seen, the RPTD-based DDL-AMF detector can provide faster adaptive detection than its time domain counterpart due to a much lower computational load. This is another advantage that makes the former the preferable detector for actual implementation in radar systems. VIII. CONCLUSION This paper has proposed a new Doppler domain localized (DDL) adaptive implementation of the classical AMF detector. The proposed DDL-AMF detector employs the concept of a region of possible target detection (RPTD), a small set of Doppler bins that captures most of the target signal power. This concept is a central contribution to the proposed DDL-AMF detector and distinguishes it from earlier RODI-based DDL implementations. This new approach to adaptive detection ensures near-optimum detection of targets in various scenarios, including training data-deficient scenarios, clutter environments with multimodal power spectral density, and unknown Doppler frequencies of targets. The computational load required to implement the proposed DDL-AMF detector is essentially lower than that of the classical AMF detector. The proposed rapidly adaptive DDL-AMF detector offers a promising solution for detecting radar targets embedded in strong nonhomogeneous clutter. Future work may extend the RPTD-based adaptive DDL processing principle to target detection under interferences governed by complex elliptically symmetric distribution. This approach can be developed for adaptive target 25424 From the Neyman-Pearson lemma, the optimum strategy for detecting a coherent signal embedded in Gaussian disturbance is based on the whitening-matched filter, also known as the filter maximizing the output signal-to-disturbance ratio (SDR). For a given disturbance covariance matrix 6, N × N , with disturbance being Gaussian clutter plus thermal noise, the N ×1 weight vector of the optimum filter is calculated as [24] w = κ6 −1 s (A1) where κ is a scalar that does not affect the output SDR, s = [1 ej2πF ej2π2F . . .ej2π(N −1)F ]T is the N × 1 desired (or steering) signal vector for a known Doppler frequency F (the superscript T stands for the matrix transposition). We specify the disturbance covariance matrix 6 as 6 = Pc C0 +Pn I (A2) where Pc and Pn represent the clutter and thermal noise power, respectively, I is an N × N identity matrix, and C0 is an N × N normalized clutter covariance matrix defined by i h 2 (A3) C0 = [cmn ] = e−2(π σc (m−n)) +i(m−n)2πFcp In (A3), the normalized Doppler frequency Fcp , Fcp < 0.5, defines the center of the clutter spectrum, and σc is the parameter controlling the clutter spectrum bandwidth. The output SDR of the optimum filter is given by [24] γo = Ps · sH 6 −1 s (A4) where Ps is the average signal power. Using the normalized disturbance covariance matrix 6 0 = 6/(Pc + Pn ), for which [6 0 ]kk = 1, k = 1, 2, . . . , N , yields   γo = γ in sH 6 −1 s (A5) 0  where γin = Ps (Pc + Pn ) is the input SDR. Using the matrix 6 0 allows the following presentation for (A1) w = κ6 −1 0 s (A6) To test the received data vector x for the target presence, the optimum detector first calculates the optimum filter out−1 put Y = wH x =κsH 6 0 x and the module-squared value 2 −1 2 |Y |2 = κ sH 6 0 x . Then, it compares |Y |2 against the detection threshold η H1 |Y |2 ≶ η (A7) H0 VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity and declares that the hypothesis H1 (target present) is true if |Y |2 ≥ η, otherwise it declares that the hypothesis H0 (no target) is true. For the optimum detector of order N given by (A7), the probability of detection PD in the case of the Swerling I target is calculated as [2] 1 1+γo PD = PFA (A8) h i where the matrix 6̃ 0 = σ̃ij0 , i, j = 1, 2, . . . , N is the DFT image corresponding to the time-domain normalized disturbance covariance matrix 6 0 through the equation 6̃ 0 =6̃/(Pc + Pn ). Following the theory of the time-domain optimal detector yields that for the m-th RPTD, the n×1 vector of optimum weights (optimum DDL filter) maximizing the RPTDassociated SDR is given by −1 where PFA is the required false alarm probability. A2. OPTIMUM DDL DETECTOR Gaussian distributions are closed under the DFT. Hence, the optimum DDL detector theory can be built by analogy with the time-domain optimum detector. To design an optimum DDL detector, we first define the DFT images for the received time domain data vector x = [x1 x2 . . .xN ]T , the steering vector s = [1 ej2πF . . . ej2π(N −1)F ]T , and the disturbance covariance matrix 6 of size N ×N , assuming that the target Doppler is known. Because the present paper uses zero-Doppler-centered DFT data, the DFT image ỹ associated with the data vector x is ỹ= [ỹ1 ỹ2 . . .ỹN ]T =fftshift(fft(x)). Similarly, the DFT image s̃ associated with the steering vector s is given by s̃= [s̃1 s̃2 . . .s̃N ]T =fftshift(fft(s)). Accordingly, for the matrix 6, we have 6̃=fftshift (fft (6)). Using formula 6 = (Pc + Pn ) 6 0 yields the DFT image associated with the normalized disturbance covariance matrix 60 as 6̃ 0 =fftshift (fft (6 0 )) = 6̃/(Pc + Pn ). Since the target Doppler frequency F is known, one can easily define the RPTD region corresponding to the m-th local range-Doppler peak using the identification procedure given in Section IV. This m-th RPTD region is represented by an associated set of Doppler bins D = {d1 d2 . . . dn }. Having defined the m-th RPTD set D = {d1 d2 . . . dn }, it is straightforward to obtain the n×1 DDL steering vector t̃m = [t̃m1 t̃m2 . . .t̃mn ]T associated with this RPTD by extracting from the N ×1 vector s̃= [s̃1 s̃2 . . .s̃N ]T the entries located at the dk -th positions, i.e., t̃mk = s̃dk , k = 1, 2, . . . , n. Similarly, we obtain the n×1 DDL vector of received data ỹm = [ỹm1 ỹm2 . . .ỹmn ]T by extracting from the N ×1 vector ỹ= [ỹ1 ỹ2 . . .ỹN ]T the entries located at the dk -th positions, i.e., ỹmk = ỹdk , k = 1, 2, . . . , n. The n × n disturbance covariance matrix 8̃m associated with this RPTD comprises the entries located at the intersection of the dk -throws and dl -th columns, k, l = 1, 2, . . . , n in the matrix 6̃= σ̃mn , m, n = 1, 2, . . . , N . Thus, the matrix 8̃m associated with the set D = {d1 d2 . . . dn } is generated by simply extracting the corresponding entries in the matrix 6̃ as given below   8̃m = ϕ̃kl , ϕ̃kl = σ̃dk dl , k, l = 1, 2, . . . , n (A9) Similarly, the n × n DDL matrix 8̃0m associated with the N × N matrix 6̃ 0 is generated as h i (A10) 8̃0m = ϕ̃kl0 , ϕ̃kl0 = σ̃d0k dl , k, l = 1, 2, . . . , n VOLUME 12, 2024 w̃m = κ 8̃m t̃m (A11) where 8̃m is the n × n disturbance covariance matrix and t̃m is the n×1 DDL steering vector that both are associated with m-th RPTD, and κ is an arbitrary scalar that does not affect the output SDR. The SDR at the output of the optimum DDL filter of order n is given by −1 γ̃o = Ps · t̃H m 8̃m t̃m (A12) where Ps is the average signal power (the superscript H stands for the Hermitian transposition). Using the matrix 8̃0m , yields   −1 (A13) γ̃o = γ in t̃H m 8̃0m t̃m  where γin = Ps (Pc + Pn ) is the input SDR. Using the matrix 8̃0m we get for the optimum DDL vector (A11) −1 w̃m = κ 8̃0m t̃m (A14) To test the DDL vector of the received data ỹm for the target presence, the optimum DDL detector first calculates H −1 the optimum DDL filter output Ỹm = w̃H m ỹm =κ t̃m 8̃0m ỹm and the module-squared value |Ỹm |2 . Comparing it against the detection threshold η̃ H1 |Ỹm |2 ≶ η̃ (A15) H0 results in declaring the hypothesis H1 is true if |Ỹm |2 ≥ η̃, otherwise the hypothesis H0 is claimed. For the optimum DDL detector (A15), the probability of detection in the case of the Swerling I target is 1 1+γ̃o PD = PFA (A16) where γ̃o is given by (A13). APPENDIX B EVALUATION OF THE PROBABILITY OF FALSE ALARM B1. GLR DETECTOR B1.1 CONVENTIONAL TIME DOMAIN (TD) GLR DETECTOR The TD-GLR Detector is given by [1], [2] sH 6̂ 1 H 6̂ 1+ x −1 KT x −1 sH 6̂ 2 x −1 H1 ≶ λGLR = KT ξT s (B1) H0 where s is the N × 1 known target steering vector (since the target Doppler frequency is known), and x is the N × 1 vector 25425 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity associated with the cell under test (CUT). In this equation, 6̂ is the estimate of the interference covariance matrix 6 (unknown). This estimate is the sample covariance matrix based on a set of the independent training vectors xk , k = 1, 2, . . . , KT , KT > N . The probability of false alarm for the TD-GLR detector in (B1) is given by [2] 1 PFA = (B2) (1 + αT )L where L = KT − N + 1 and αT = ξT /(1 − ξT ) with ξT being the factor for the threshold in (B1). B1.2 DOPPLER DOMAIN LOCALIZED (DDL) GLR DETECTOR The DDL-GLR detector of the order n associated with the m-th RPTD is given by (15) as ˆ −1 ˆt̃H 8̃ m m ỹm 1 1+ ˆ −1 ỹH m 8̃m ỹm K ˆ −1 ˆ ˆt̃H 8̃ m m t̃m 2 H1 ≶ λ̃GLR = K ξ̃ (B3) H0 ˆ is the sample DDL covariance matrix, see (14). where 8̃ m The DFT transform is a linear invertible transform that preserves Gaussianity with the only change of the received data vector, the signal steering vector and the disturbance covariance matrix. Therefore, the probability of false alarm equation for the DDL-GLR detector (B3) is identical to that for the TD-GLR detector. The only modification one must apply is the corresponding substitutions for the parameters. Thus, the probability of false alarm for the DDL-GRL detector is given by 1 (B4) PFA = (1 + α̃)Q where Q = K − n + 1, and α̃ = ξ̃ /(1 − ξ̃ ) with ξ̃ being the factor for the threshold in (B3). B2. AMF DETECTOR B2.1 CONVENTIONAL TIME DOMAIN AMF DETECTOR −1 sH 6̂ 2 x −1 H1 ≶ λAMF = KT ζT s H1 ˆ −1 ˆ ˆt̃H 8̃ m m t̃m ≶ λ̃AMF = K ζ̃ (B8) H0 Since the DFT preserves Gaussianity, the equation for the probability of false alarm for the DDL-AMF detector (B8) is identical to that for the detector (B5). Hence, the corresponding substitutions for the parameters yield Z 1 fβ (ρ; Q + 1, n − 1) dρ (B9) PFA = (1 + α̃)Q 0 where Q = K − n + 1, and α̃ = ζ̃ /(1 − ζ̃ ) with ζ̃ being the factor for the threshold in (B8). One should use numerical iterations to find the scalar α̃ from (B9) for a given PFA . APPENDIX C EVALUATION OF THE PROBABILITY OF DETECTION C1. GLR DETECTOR C1.1 CONVENTIONAL TIME DOMAIN (TD) GLR DETECTOR For the conventional TD-GLR Detector given by (B1), the probability of detection PD is provided by the following integral expression [2] PD = Z1  0 1 + γo ρ 1 + αT + γo ρ L fβ (ρ; L + 1, N − 1) dρ (C1) where γo = γin sH 6 −1 s is the optimum filter output SDR, with γin being the SDR at the input (see Subsection A1). Other parameters are: L = KT − N + 1, the scalar αT is determined from (B2) for the specified probability of false −1/L alarm PFA as αT = PFA − 1, and fβ (x; n, m) is the central Beta density function given by (B7). (B5) Since the DFT preserves Gaussianity, the probability of detection for the DDL-GLR detector given by (B3) follows from (C1) after proper symbol substitutions as H0 where the symbols s, x, and 6̂ are defined above in B1.1. The probability of false alarm for the TD-AMF detector in (B5) is given by [2] Z 1 fβ (ρ; L + 1, N − 1) dρ (B6) PFA = (1 + αT )L 0 where L = KT − N + 1, αT = ζT /(1 − ζT ) with ζT being the factor for the threshold in (B5), and the central Beta density function is (n + m − 1)! n−1 fβ (x; n, m) = x (1 − x)m−1 (B7) (n − 1)! (m − 1)! For a given PFA , one can find the scalar αT from (B6) using numerical iterations. 25426 2 ˆ −1 ˆt̃H 8̃ m m ỹm C1.2 DOPPLER DOMAIN LOCALIZED (DDL) GLR DETECTOR The TD-AMF Detector is given by [2] sH 6̂ B2.2 DOPPLER DOMAIN LOCALIZED (DDL) AMF DETECTOR The DDL-AMF detector of the order n associated with the m-th RPTD is given by (16) as PD = Z1  0 1 + γ̃o ρ 1 + α̃ + γ̃o ρ Q fβ (ρ; Q + 1, n − 1) dρ (C2) −1 where γ̃o = γ in (t̃H m 8̃0m t̃m ) is the optimum DDL filter output SDR, with γin being the SDR at the input (see Subsection A2). Other parameters are: Q = K − n + 1, the scalar α̃ is determined from (B4) for the specified probability −1/Q of false alarm PFA as α̃ = PFA − 1. C2. AMF DETECTOR C2.1 CONVENTIONAL TIME DOMAIN (TD) AMF DETECTOR For the conventional TD-AMF Detector given by (B5), the probability of detection is provided by the following integral VOLUME 12, 2024 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity expression [2] PD = Z1  0 1 + γo ρ 1 + (αT + γo ) ρ L fβ (ρ; L + 1, N − 1) dρ (C3) where γo = γin sH 6 −1 s is the optimum filter output SDR, with γin being the SDR at the input (see Subsection A1). Other parameters are: L = KT − N + 1 and the scalar αT is determined from (B6) for the specified probability of false alarm PFA using numerical iterations. C2.2 DOPPLER DOMAIN LOCALIZED (DDL) AMF DETECTOR Since the DFT transform preserves Gaussianity, the probability of detection for the DDL-AMF detector given by (B8) follows from (C3) after proper symbol substitutions as PD = Z1  0 1 + γ̃o ρ 1 + (α̃ + γ̃o ) ρ Q fβ (ρ; Q + 1, n − 1) dρ (C4) −1 where γ̃o = γ in (t̃H m 8̃0m t̃m ) is the optimum DDL filter output SDR, with γin being the SDR at the input (see Subsection A2). Other parameters: Q = K − n + 1, the scalar α̃ is determined from (B9) for the specified probability of false alarm PFA using numerical iterations. 3̃a (c) = [λ̃a1 (c) λ̃a2 (c) . . . λ̃aP (c)] represents the approximate DDL-AMF thresholds, where the q-th value λ̃aq (c), q = 1, 2, . . . , P is computed from (D1) for the corresponding PFAq values in P. Figure 20 compares the exact thresholds 3̃ and the
corresponding optimum approximate thresholds 3̃a copt computed as functions of the probability of false alarm for n = 4, K = 20 and the reference vector P of size 1 × 41 that is defined over the interval [10−16 , 10−6 ] as P = unique([logspace (-16, -15, 5), logspace(-15, -14, 5), logspace(-14, -13, 5), logspace(-13, -12, 5), logspace(-12, 11, 5), logspace(-11, -10, 5), logspace(-10, -9, 5), logspace(9, -8, 5), logspace(-8, -7, 5), logspace(-7, -6, 5)]) (D3) where ‘‘unique’’ and ‘‘logspace’’ are MATLAB functions. Under these settings, the optimum coefficients are found to be (in double precision format) c1opt = 1.137593213858974, c2opt = 15.875828450315428, c3opt = 1.168722472188503. Figure 20 shows that the exact and optimum DDL-AMF thresholds are in excellent agreement (their plots are APPENDIX D DDL-AMF DETECTOR: SIMPLE AND ACCURATE THRESHOLD CALCULATION For the DDL-AMF detector, calculating the detection threshold λ̃ from (B9) entails time-consuming numerical iterations. This approach is not acceptable for a real-time implementation of DDL-AMF detectors. This Appendix provides an efficient and accurate approximating formula for computing λ̃. The approximating equation for a given PFA is (−1/c2 ) λ̃a (c) = c1 PFA − c3 (D1) where λ̃a (c) is the approximate DDL-AMF detector’s threshold and c = [c1 , c2 , c3 ] is the vector of approximating coefficients to be sought. We define the optimum vector c =copt as the solution to the following minimax problem   copt = arg minmax 3̃ − 3̃a (c) ./3̃ FIGURE 20. Comparison of exact and approximate DDL-AMF thresholds. TABLE 2. Optimum approximating coefficients for computing DDL-AMF thresholds. c co = [1, K , 1] (D2) where the symbols ./ represent the elementwise division, and co is the initial approximation, with K being the number of training vectors. In (D2), the vector 3̃ = [λ̃1 λ̃2 . . . λ̃P ] is the vector of exact DDL-AMF thresholds precomputed using (B9) for the vector of the reference PFA values P = [PFA1 PFA2 . . . PFAP ] with P being the length of the vector P. The vector VOLUME 12, 2024 25427 A. A. Kononov, M.-H. Ka: Rapid Adaptive Matched Filter for Detecting Radar Targets With Unknown Velocity not visually distinguished) because the relative error 1λ̃ between  them does not exceed the minimax error 100 × max 3̃ − 3̃a (copt ) ./3̃ = 0.07513%. The relative error 1PFA between the exact PFA values and approximate PFA values computed from (D1) using approximate thresholds from the vector 3̃a (copt ) is also small – it does not exceed 1.07276%. For n = 4 and 5, Table 2 summarizes the optimum approximating coefficients for computing corresponding DDL-AMF thresholds. The coefficients are calculated for K = 3n, 4n, and 5n using the reference vector in (D3). [20] E. Conte, M. Longo, M. Lops, and S. L. Ullo, ‘‘Radar detection of signals with unknown parameters in K-distributed clutter,’’ IEE Proc., F, Radar Signal Process., vol. 138, no. 2, pp. 131–138, 1991. [21] E. Ollila, D. E. Tyler, V. Koivunen, and H. V. Poor, ‘‘Complex elliptically symmetric distributions: Survey, new results and applications,’’ IEEE Trans. Signal Process., vol. 60, no. 11, pp. 5597–5625, Nov. 2012. [22] M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern Radar: Basic Principles, vol. 1. Raleigh, NC, USA: SciTech Publishing, 2010, ch. 14, p. 519. [23] E. J. Kelly, ‘‘Performance of an adaptive detection algorithm; rejection of unwanted signals,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 25, no. 2, pp. 122–133, Mar. 1989. [24] P. Lombardo, M. Greco, F. Gini, A. Farina, and J. B. Billingsley, ‘‘Impact of clutter spectra on radar performance prediction,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 3, pp. 1022–1038, Jul. 2001. REFERENCES [1] E. J. Kelly, ‘‘An adaptive detection algorithm,’’ IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, no. 2, pp. 115–127, Mar. 1986. [2] F. C. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, ‘‘A CFAR adaptive matched filter detector,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, May 1992. [3] D. Boroson, ‘‘Sample size considerations for adaptive arrays,’’ IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, no. 4, pp. 446–451, Jul. 1980. [4] H. Wang and L. Cai, ‘‘A localized adaptive MTD processor,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 3, pp. 532–539, May 1991. [5] H. Wang, M. Wicks, and Y. Zhang, ‘‘A new Doppler processing technique for detection performance improvement in existing airborne radars,’’ in Proc. IEEE Radar Conf., Alexandria, VA, USA, 1995, pp. 72–76. [6] H. Wang and L. Cai, ‘‘On adaptive spatial-temporal processing for airborne surveillance radar systems,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 3, pp. 660–670, Jul. 1994. [7] Z. Wang, Z. He, Q. He, and J. Li, ‘‘Adaptive CFAR detectors for mismatched signal in compound Gaussian sea clutter with inverse Gaussian texture,’’ IEEE Geosci. Remote Sens. Lett., vol. 19, 2022, Art. no. 3502705. [8] J. Wang, Z. Wang, Z. He, and J. Li, ‘‘GLRT-based polarimetric detection in compound-Gaussian sea clutter with inverse-Gaussian texture,’’ IEEE Geosci. Remote Sens. Lett., vol. 19, 2022, Art. no. 4028005. [9] J. Xue, S. Xu, J. Liu, M. Pan, and J. Fang, ‘‘Bayesian detection for radar targets in compound-Gaussian sea clutter,’’ IEEE Geosci. Remote Sens. Lett., vol. 19, pp. 1–5, 2022. [10] J. Xue, M. Ma, J. Liu, M. Pan, S. Xu, and J. Fang, ‘‘Wald- and rao-based detection for maritime radar targets in sea clutter with lognormal texture,’’ IEEE Trans. Geosci. Remote Sens., vol. 60, 2022, Art. no. 5119709. [11] J. Xue, S. Xu, and J. Liu, ‘‘Persymmetric detection of radar targets in nonhomogeneous and non-Gaussian sea clutter,’’ IEEE Trans. Geosci. Remote Sens., vol. 60, 2022, Art. no. 5103709. [12] Z. Wang, Z. He, Q. He, B. Xiong, and Z. Cheng, ‘‘Persymmetric adaptive target detection with dual-polarization in compound Gaussian sea clutter with inverse gamma texture,’’ IEEE Trans. Geosci. Remote Sens., vol. 60, 2022, Art. no. 5118117. [13] X. Liang, P.-L. Shui, and H.-T. Su, ‘‘Bi-phase compound-Gaussian mixture model of sea clutter and scene-segmentation-based target detection,’’ IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 14, pp. 4661–4674, 2021. [14] K. J. Sangston, F. Gini, and M. S. Greco, ‘‘Adaptive detection of radar targets in compound-Gaussian clutter,’’ in Proc. IEEE Radar Conf. (RadarCon), Arlington, VA, USA, May 2015, pp. 0587–0592. [15] K. J. Sangston, F. Gini, and M. S. Greco, ‘‘Coherent radar target detection in heavy-tailed compound-Gaussian clutter,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 48, no. 1, pp. 64–77, Jan. 2012. [16] K. J. Sangston, F. Gini, and M. S. Greco, ‘‘New results on coherent radar target detection in heavy-tailed compound-Gaussian clutter,’’ in Proc. IEEE Radar Conf., Arlington, VA, USA, May 2010, pp. 779–784. [17] E. Conte and A. De Maio, ‘‘Mitigation techniques for non-Gaussian sea clutter,’’ IEEE J. Ocean. Eng., vol. 29, no. 2, pp. 284–302, Apr. 2004. [18] F. Gini and A. Farina, ‘‘Vector subspace detection in compound-Gaussian clutter. Part I: Survey and new results,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 4, pp. 1295–1311, Oct. 2002. [19] K. J. Sangston and K. R. Gerlach, ‘‘Coherent detection of radar targets in a non-Gaussian background,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 2, pp. 330–340, Apr. 1994. 25428 ANATOLII A. KONONOV (Member, IEEE) received the M.Sc. degree in electrical engineering from Odesa National Polytechnic University, Odesa, Ukraine, in 1975, and the Ph.D. degree in electrical engineering from Electrotechnical University ‘‘LETI,’’ Saint Petersburg, Russia, in 1983. From 1975 to 2002, he was with Odesa National Polytechnic University as a Research Engineer and an Associate Professor with the Department of Radio Engineering. From 2002 to 2005 and from 2010 to 2022, he was a Senior Researcher with the Research Center, STX Engine, Yongin, South Korea. From 2006 to 2008, he was with the Department of Electronic Engineering, Tech University of Korea (TU Korea), Siheung, South Korea, as an invited Professor. From 2008 to 2010, he was a Senior Researcher with the Department of Earth and Space Sciences, Chalmers University of Technology, Gothenburg, Sweden. Since 2023, he has been with the School of Integrated Technology, Yonsei University, Seoul, South Korea, as a Senior Researcher. His main research interests include radar signal processing and system analysis and design. MIN-HO KA (Member, IEEE) received the B.S. and M.S. degrees in electronics engineering from Yonsei University, Seoul, Republic of Korea, in 1989 and 1991, respectively, and the Ph.D. degree in radio engineering from the Moscow Power Engineering Institute, Moscow, Russia, in 1997. From 1997 to 2000, he was with the Agency for Defence Development (ADD), Ministry of Defence, South Korea, for the development of spaceborne and airborne synthetic aperture radars (SAR). From 2002 to 2010, he was a Full Professor with the Department of Electronic Engineering, Tech University of Korea (TU Korea), Siheung, South Korea, where he was the Dean of the Department and the Head of Planning Office. He is currently a Full Professor with the School of Integrated Technology, College of Computing, Yonsei University. His research interests include the system design and development of radar sensors, systems, and SARs. He is the Chairman of APSAR Korea and was the Chairman of the Radar Research Group, Korean Institute of Electromagnetic Engineering and Science (KIEES). VOLUME 12, 2024