An Adaptive SVD-Based Approach to Clutter Suppression for Slow-Moving Targets

Abstract

The presence of strong clutter remains a critical challenge for radar system target detection. Traditional clutter suppression techniques such as Doppler-based filters often fail to extract low-velocity targets from clutter. To address this limitation, this paper proposes an adaptive singular value decomposition (A-SVD) method utilizing support vector machines (SVM). The proposed approach leverages the augmented implicitly restarted Lanczos bidiagonalization (AIRLB) algorithm to decompose echo matrices into different subspaces, which are then characterized in relation to Doppler frequency, energy, and correlation. These features are employed to classify the clutter subspaces using an SVM classifier, which solves the problem of selecting the SVD threshold. The clutter subspaces are suppressed by zeroing out corresponding singular values, and the matrix is then recomposed by the rest of the subspaces to recover the echo. Experiments on simulated and real datasets show that the proposed method achieves an average improvement factor (IF) above 40 dB and reduces runtime by over 85% in most scenarios.

1. Introduction

Low, slow, and small (LSS) targets such as drones, small aircraft, and low-flying objects pose a growing challenge for modern radar systems due to their weak echo signals and low visibility in cluttered environments [1,2,3]. Although a variety of advanced detection techniques have been explored across different radar configurations [4,5,6,7], clutter suppression remains a prerequisite for enhancing signal conditions and enabling reliable detection, especially under complex and dynamic scenarios.

Radar systems are frequently deployed in environments with substantial clutter, including urban areas, dense forests, and open seas. These complex environments introduce strong background reflections that can easily overwhelm weak echoes from objects of interest. Such clutter not only masks meaningful signals but also increases the difficulty of information extraction. Despite widespread radar use in fields such as air traffic management, meteorology, and intelligent transportation, effective clutter suppression remains a persistent challenge for robust signal analysis over extended ranges.

Conventional clutter suppression techniques such as moving target indicator (MTI) and moving target detection (MTD) [8] rely on Doppler frequency differences between targets and clutter, assuming that clutter has a near-zero compensated Doppler frequency. These methods are effective for fast-moving objects, but often fail when the target velocity is low because the Doppler shift overlaps with clutter, making the systems unreliable.

To address these limitations, narrowband filtering approaches have been explored. The Kalmus filter [8,9], for instance, is designed with the No. 0 and No. N-1 components of an equivalent Fourier filter bank to isolate slow-moving targets. While effective under specific conditions, it suffers from performance degradation near 0.5 pulse repetition frequency (PRF), showing instability in certain Doppler ranges and leaving residual clutter that may cause false detections. Additionally, its static filter configuration reduces adaptability to dynamic clutter environments, which limits its robustness.

Subsequently, subspace-based methods [10,11,12,13] have been introduced for radar clutter suppression. Eigenvalue decomposition (EVD) [14] and singular value decomposition (SVD) [11] are most commonly employed for this. SVD was first introduced in radar signal processing by Martin W.Y. Poon [11] by composing the Hankel matrix from the echo of one pulse. This approach showed excellent target-keeping performance in all velocity ranges. Subsequently, SVD-based methods have gained popularity in such various radar applications as ground-penetrating radar (GPR) [15], laser radar [16], and through-the-wall radar (TWR) [17,18] thanks to their high effectiveness. For multiple-pulse systems, the echo matrix can be directly employed for decomposition [19]. Nevertheless, accurately partitioning the subspaces using SVD remains a major challenge. The boundary between clutter and non-clutter subspaces is often unclear, which can lead to either the loss of target echoes or the retention of clutter. Previous research has typically relied on singular values to differentiate between subspaces, using the numerical characteristics of singular values to identify clutter, signal, and noise. A common approach is to employ either the amplitude of the singular values [20] or their difference spectrum [21,22]. This involves calculating the difference between consecutive singular values to identify the point with the largest difference, which is assumed to separate clutter from signals and noise. However, the question of optimally estimating the boundary between the different subspaces remains unanswered. Additionally, clutter characteristics can vary significantly depending environmental features such as urban, maritime, or forested areas, further complicating the identification process. This requires a more robust and adaptive method for clutter suppression that can effectively differentiate clutter from signals under various operating conditions. In [23], the authors proposed a spatiotemporal singular value decomposition method for clutter rejection of ultrasonic data acquired at ultrafast frame rates. In [24], the matrix features of the SVD algorithm were analyzed using the degree of resemblance of spatial singular vectors to discriminate between static tissue and moving blood flow in a medical imaging context. For radar signal processing, selection of the singular values threshold (SVT) has not been discussed in detail in the literature. Meanwhile, the execution efficiency of SVD is an another consideration. Traditional QR-based SVD algorithms [25] compute the full set of singular values and vectors; however, in applications where only the leading components are required, such methods are inefficient. In contrast, the AIRLB algorithm is based on iteration within a Krylov subspace framework, making it more suitable for partial SVD in large-scale radar echo processing.

To address the above-mentioned limitations, we adopt the augmented implicitly restarted Lanczos bidiagonalization (AIRLB) algorithm, which efficiently computes only the leading singular values and vectors without requiring full decomposition. This approach extends complexity reduction of SVD-based clutter suppression to large echo matrices. To further improve the separation of clutter and target subspaces, we introduce a support vector machine (SVM) classifier which learns a robust decision boundary based on multidimensional subspace features. Unlike fixed-threshold methods, SVM provides adaptive classification that accounts for variations in Doppler, energy, and correlation characteristics. Based on these two modules, an adaptive framework named adaptive SVD (A-SVD) is developed, integrating partial SVD via AIRLB and subspace classification via SVM. The effectiveness of the proposed A-SVD method is validated on both simulated and real-world radar clutter datasets. Experimental evaluations indicate that the proposed method consistently enhances detection performance for slow-moving targets, with the average improvement factor (IF) exceeding 40 dB in typical scenarios. In terms of efficiency, A-SVD achieves over 85% runtime reduction in most test cases. These results support the method’s suitability for real-time applications and highlight its advantages over traditional clutter suppression techniques.

The main contributions of this paper are summarized as follows:

• The AIRLB algorithm is employed to accelerate the computation of the SVD algorithm. By focusing on the leading singular values and vectors, this method significantly reduces the computational complexity, making it suitable for large-scale radar datasets.
• This research dissects the subspace features related to the energy distribution, Doppler frequency, and temporal-spatial correlation in order to enhance subspace classification accuracy. These features collectively form a multidimensional space, which improves the classifier’s robustness in complex scenarios.
• An SVM classifier is introduced to divide clutter and non-clutter subspaces. By leveraging an SVM to discriminate between classes, the proposed method achieves precise subspace partitioning, thereby mitigating the dependency on unclear energy-related thresholds in conventional SVD-based methods.
• Experimental results verify the total performance of the proposed method on both simulated and real-world radar datasets. The method in this paper demonstrates significant improvements in clutter suppression and slow-moving target detection under challenging conditions.

The rest of this paper is organized as follows: Section 2 discusses the materials and methods used in the paper, including the received signal model and the rationale behind the proposed method; Section 3 describes the experimental results, including the simulated and real clutter datasets used for evaluation, then compares the performance of A-SVD with existing clutter suppression algorithms; finally, Section 4 concludes the paper and outlines future research directions.

2. Materials and Methods

2.1. Clutter Suppression with SVD

The transmitted signal is modeled as a linear frequency modulated (LFM) signal, and its complex form is given as

$$s\left(t\right)=\mathrm{rect}({\displaystyle \frac{t}{T_{\mathrm{e}}}})exp(\mathrm{j}2\pi f_{0}t+\mathrm{j}\pi \mu t^2),0\le t\le T,$$

(1)

where $\mathrm{rect}(·)$ is the rectangular function, $T_{\mathrm{e}}$ is the pulse width, $f_0$ is the carrier frequency, $\mathrm{j}$ is the imaginary unit, and $\mu$ is the frequency modulation rate. The echo of one target is given as

$$s_{\mathrm{target}}\left(t\right)=s(t-\mathrm{\Delta }t)exp\left[\mathrm{j}2\pi f_{\mathrm{d}}(t-\mathrm{\Delta }t)\right],$$

(2)

where $\mathrm{\Delta }t$ is the delay time of the target echo. Then, the complete received signal for multiple targets is expressed as

$$s_{\mathrm{r}}\left(t\right)=\sum _{m=1}^M{A}_{m}s(t-\mathrm{\Delta }{t}_m)exp\left[\mathrm{j}2\pi f_{\mathrm{d},m}(t-\mathrm{\Delta }{t}_m)\right]+c\left(t\right)+n\left(t\right),$$

(3)

where $\mathrm{\Delta }{t}_m$ is the delay time of the mth target, M is the number of targets, $c\left(t\right)$ denotes clutter, and $n\left(t\right)$ denotes Gaussian noise. The clutter echo is the main interference of the target echo, which is much stronger than the target echo; thus, suppression of the clutter is a significant problem to be solved, especially for detection of weak and slow targets.

The received signal is sampled and the echo matrix $\mathit{A}\in {\mathbb{C}}^{N\times L}$ is obtained. Every row of the matrix is composed of echo signals of the pulses (fast-time), while every column denotes the time-varying signal of every range cell (slow-time). $\mathit{A}$ is given as follows:

$$\mathit{A}=\left[\begin{array}{cccc}{s}_{\mathrm{r}1}\left(0\right)& s_{\mathrm{r}1}\left(1\right)& \dots & s_{\mathrm{r}1}\left(L\right)\\ s_{\mathrm{r}2}\left(0\right)& s_{\mathrm{r}2}\left(1\right)& \dots & s_{\mathrm{r}2}\left(L\right)\\ ⋮& ⋮& \ddots & ⋮\\ s_{\mathrm{rN}}\left(0\right)& s_{\mathrm{rN}}\left(1\right)& \dots & s_{\mathrm{rN}}\left(L\right)\end{array}\right]$$

(4)

where $s_{\mathrm{r},\mathit{i}}\left(j\right)$ is the received echo of the jth range cell in the ith pulse, N is the number of pulses transmitted in a coherent processing interval (CPI), and L is the number of range cells. In typical systems, the number of pulses is significantly smaller than the number of range cells, i.e., $N\ll L$.

In the fast-time dimension, pulse compression is commonly used to improve the signal-to-noise ratio (SNR) of target echoes, typically achieved through a matched filter. This filter, being the conjugate of the transmitted signal, optimizes the SNR for a single pulse; however, clutter, caused by strong reflections from static objects, clouds, rain, or ocean sprays, is also matched with the transmitted signal, and is consequently amplified by the same filter. In the slow-time dimension, the echoes from each range cell typically have the same amplitude but different phases. Fourier transformation is often used to accumulate the target energy from pulses within a CPI, creating a peak in both the range and Doppler domains.

Unlike EVD, SVD exists for every matrix, which is considered to possess the ability to extract the information from all dimensions of the matrix [23]. Here, the echo matrix $\mathit{A}$ of a CPI is chosen to be processed, and its SVD is provided as follows:

$$\mathit{A}=\mathit{U}\mathbf{\Sigma }{\mathit{V}}^{\mathrm{H}}$$

(5)

where $\mathit{U}$ denotes the left singular matrix and $\mathit{V}$ denotes the right singular matrix, both of which are unitary matrices, while $\mathbf{\Sigma }$ is the matrix of singular values, which is diagonal; meanwhile, $\mathit{V}$ and $\mathit{V}$ are the eigenvector matrix of ${\mathit{AA}}^{\mathrm{H}}$ and ${\mathit{A}}^{\mathrm{H}}\mathit{A}$, and the form of $\mathbf{\Sigma }$ is as follows:

$$\mathbf{\Sigma }=\left[\begin{array}{cc}\mathbf{S}& \mathbf{0}\end{array}\right]$$

(6)

where $\mathbf{S}=\mathrm{diag}({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r)$ are the singular values, which are ranked by their size, while r demotes the number of singular values. The main diagonal elements are non-zero, which means that the left part of $\mathbf{\Sigma }$ is non-zero and the rest is zero as $N<L$. In this way, the dimensions are divided; they are considered to contain the Doppler frequency, energy, and correlation information of the echo [23]. The columns of $\mathit{U}$ and $\mathit{V}$ are called temporal (left) and spatial (right) singular vectors of $\mathit{A}$, respectively, both of which are orthogonal matrices. In this way, for two singular matrices we have

$$\left\{\begin{array}{c}{\mathit{u}}_i^T{\mathit{u}}_j=0\hfill \\ {\mathit{v}}_p^T{\mathit{v}}_q=0,\hfill \end{array}\right.$$

(7)

where

$$1\le i,j,p,q\le r$$

(8)

and $p\ne q$, $u\ne v$. Naturally, every singular value and its corresponding left and right singular vectors form a subspace of the echo, which is represented by

$${\mathit{A}}_i={\sigma }_i{\mathit{u}}_i{\mathit{v}}_i^{\mathrm{H}},$$

(9)

where ${\mathit{A}}_i$ is the ith subspace. From a physical perspective, the orthogonal subspace decomposition property of SVD enables structural separation of different signal components within radar echoes. In high-frequency radar returns, sea or ground clutter generally exhibits strong energy and high average correlation. Target echoes have weaker energy and lower average temporal correlation. Noise is a low-power random process that is uniformly distributed throughout the matrix. These differences allow SVD to project each component into distinct subspaces. Clutter dominates the leading high-energy subspaces associated with the largest singular values, while target signals are typically distributed across the subsequent few subspaces and noise is spread throughout all subspaces with approximately uniform energy. This separation mechanism provides a solid theoretical foundation for clutter suppression and target enhancement under complex conditions.

To mitigate strong clutter, the singular values of clutter subspaces are set to zeros [11], and the echo matrix ${\mathit{A}}^{\prime }$ is recomposed as follows:

$${\mathit{A}}^{\prime }=\mathit{U}{\mathbf{\Sigma }}^{\prime }{\mathit{V}}^{\mathrm{H}}=\sum _{i=s}^r{\sigma }_i{\mathit{u}}_i{\mathit{v}}_i^{\mathrm{H}}$$

(10)

where ${\mathbf{\Sigma }}^{\prime }=\left[\begin{array}{cc}{\mathbf{S}}^{\prime }& \mathbf{0}\end{array}\right]$, ${\mathbf{S}}^{\prime }=\mathrm{diag}(0,\dots ,0,{\sigma }_{s+1},\dots ,{\sigma }_r)$ and s is the SVT order of the clutter and signal subspaces. In this way, the first s subspaces are for clutter and the $(s+1)$th to rth subspaces are for non-clutter. After the clutter subspaces are removed, the clutter is suppressed and the target echoes remain.

2.2. Rationale of A-SVD

This section presents the basic processes of A-SVD, which consists of efficient SVD, feature extraction, and classification. Specifically, the AIRLB algorithm is employed to efficiently compute the first k subspaces of SVD, generating singular triplets: the left singular matrix $\mathit{U}$, the singular values matrix $\mathbf{\Sigma }$, and the right singular matrix $\mathit{V}$. These components are then utilized as the basis for feature extraction, focusing on the three key aspects of energy, Doppler frequency, and correlation. To effectively distinguish between clutter and non-clutter subspaces, an SVM classifier is employed to categorize the extracted features into distinct groups. The following sections provide a detailed analysis of how these components are integrated to enhance clutter suppression and improve detection accuracy.

2.2.1. Subspace Extraction via AIRLB

The Lanczos-based iterative algorithm [26] was first proposed to quickly compute k outlier eigenvalues of a matrix, and was then developed into the AIRLB to compute the largest or the smallest k singular values of a matrix [27]. This approach does not take other singular values into account, and is rather appropriate for SVD clutter suppression. The AIRLB method is an iterative approach for computing the largest or the smallest k singular values and singular vectors of large-scale matrices. For a given input $N\times L$ matrix $\mathit{A}$, the m steps of partial Lanczos bidiagonalization are provided as follows:

$$\mathit{A}{\mathit{P}}_k={\mathit{Q}}_k{\mathit{B}}_k,{\mathit{A}}^T{\mathit{Q}}_k={\mathit{P}}_k{\mathit{B}}_k^T+r_k{\mathit{e}}_k^T$$

(11)

where ${\mathit{P}}_k$ and ${\mathit{Q}}_k$ are orthonormal bases, ${\mathit{B}}_k$ is bidiagonal, and $r_k$ is the residual. Then, augmented implicit restarting is implemented by adding a small number of Ritz or harmonic Ritz vectors to improve the accuracy of the approximation and reduce computational complexity.

Compared to traditional direct SVD, which has a computational complexity of $\mathcal{O}(N{L}^2+N^3)$ for an $N\times L$ matrix, AIRLB focuses only on the leading k singular values, reducing the complexity to $\mathcal{O}\left(kNL\right)$, where $k\ll min(N,L)$. This method reduces the high computational complexity of traditional SVD to the level of matrix-vector multiplications. This significant reduction in computational cost makes AIRLB particularly effective for applications of SVD-based methods, where isolating the principal singular subspaces is crucial for distinguishing between signal and clutter components.

2.2.2. Energy

As is common with most SVD-based methods, singular values are employed to identify subspaces based on their respective energy levels. As demonstrated in Equation (9), the left and right singular vectors are unitary and the singular values assign varying weights to each subspace, thereby reflecting their relative energy magnitudes. The energy of the subspaces can be calculated as the sum of the squares of the singular values, which is denoted as feature $f_1$:

$$f_{1,i}={\sigma }_i,1\le i\le r$$

(12)

where i is the subspace number and r is the total number of subspaces. Typically, the initial subspaces tend to capture more energy, particularly in scenarios where clutter reflections dominate due to static or large-area reflectors; however, this is not always the case, as multiple closely spaced clutter returns or overlapping Doppler frequencies may result in singular values that do not strictly correspond to clutter strength. Hence, energy-based separation alone may lead to misclassification. The differences between the singular values are calculated as follows:

$$f_{2,i}={\sigma }_{i+1}-{\sigma }_i,1\le i\le r-1.$$

(13)

Given that clutter subspaces contain the bulk of the energy and that their values should be much higher than other parts of the echoes, they can be identified by analyzing the differences between singular values. Figure 1 shows the singular values and the difference spectrum of the echo. The first three singular values have more energy than the rest, and the difference between the third and forth singular values is the largest. The largest difference between singular values [28] is typically used as the threshold to separate clutter and non-clutter subspaces; however, this division is not always well-defined. In certain instances, the position of the largest difference may not coincide with the actual boundary, and targets may still persist in some adjacent subspaces. Because the largest differences are not always sufficient to accurately classify the subspaces, the use of additional features is required in order to enhance the classification process. The energy ratio of the ith and first subspace is then introduced, which is given as

$$f_{3,i}={\displaystyle \frac{{\sigma }_i^2}{{\sigma }_1^2}},1\le i\le r.$$

(14)

This feature is used to quantify the relative energy distribution across subspaces, providing a normalized measure of subspace. This is significant because when the energy of a subspace is close to the first subspace, it is likely to be considered to be clutter. Additionally, the singular value spectrum slope is another feature that reflects the decay rate of singular values across subspaces, provided as follows:

$$f_{4,i}={\displaystyle \frac{{\sigma }_1-{\sigma }_i}{i-1}},2\le i\le r.$$

(15)

The slope characterizes the degree of the singular value decay, with a steeper slope indicating a more concentrated energy distribution in the primary subspaces.

2.2.3. Doppler Frequency

The Doppler frequency is the most typical feature used by clutter suppression methods. The Doppler frequency of clutter is generally low or close to zero, whereas the signal and noise subspaces exhibit a range of unexpected Doppler frequencies due to target movement and the randomness of noise. The Doppler frequency of the subspaces can be determined using the left singular matrix $\mathit{U}$, which is considered to represent the Doppler frequency characteristics of the subspaces [23]. Specifically, feature $f_5$ is the Doppler frequency, which is calculated between each pair of consecutive left-singular vectors ${\mathit{u}}_i$ by the following expression:

$$f_{5,i}=f_{\mathrm{d}}\left(i\right)={\displaystyle \frac{1}{2\pi T_{\mathrm{r}}}}\arctan {\displaystyle \frac{\mathrm{Im}\left[{\mathit{R}}_{\mathit{i}}\left({\mathit{T}}_{\mathrm{r}}\right)\right]}{\mathrm{Re}\left[{\mathit{R}}_{\mathit{i}}\left({\mathit{T}}_{\mathrm{r}}\right)\right]}},1\le \mathrm{i}\le \mathrm{N}$$

(16)

where $f_{\mathrm{d}}\left(i\right)$ is the main Doppler frequency of the ith subspace, $T_{\mathrm{r}}$ is the pulse repetition time (PRT), and $R_i$ is the correlation function of pulses. Then, $R_i$ is calculated by

$$R_i\left(T_{\mathrm{r}}\right)={\displaystyle \frac{1}{N-1}}\sum _{n=1}^N{\mathit{A}}_{i,n}{\mathit{A}}_{i,n+1}^{*},1\le n\le N-1,$$

(17)

where ${\mathit{A}}_{i,n}$ is the nth row of the ith subspace. However, some energy of targets may also leak into the first few subspaces, which causes severe deviation to zero in a random subspace. In this case, the Doppler frequency cannot be the only feature used to identify the subspaces. Therefore, the standard deviations of the sequence of Doppler frequencies are adopted to illustrate the variation of the Doppler frequency as $f_6$, which is calculated as follows:

$$f_{6,i}={\sigma }_{\mathrm{d}}\left(i\right)=\sqrt{{\displaystyle \frac{1}{i}}\sum _{q=1}^i{[f_{\mathrm{d}}\left(q\right)-\overline{f_{\mathrm{d}}}\left(q\right)]}^2},1\le i\le N$$

(18)

where $\overline{f_{\mathrm{d}}}\left(i\right)$ is the mean value of the sequence of Doppler frequencies for the first i subspaces. Figure 2 shows an example of computed Doppler frequencies of the subspaces and their standard deviations. Feature $f_6$ sees the highest variation between the third and fourth subspaces, which indicates the presence of clutter subspaces.

2.2.4. Correlation

Correlation is usually considered in terms of two aspects, namely, range correlation and time correlation. The mean autocorrelation coefficient [29] is defined as

$$\rho \left(k\right)={\displaystyle \frac{M}{M-k}}\times {\displaystyle \frac{{\sum }_{m=0}^{M-k-1}x(m+k)x^{*}\left(m\right)}{{\sum }_{m=0}^{M-1}{\left|x\left(m\right)\right|}^2}},k\ge 0,$$

(19)

where k is the lagged number, $x\left(m\right)$ is the sequence to be processed, and M is the length of $x\left(m\right)$.

The range correlation is determined by analyzing the received signals from consecutive range cells within a single pulse. However, for clutter characteristic analysis, the range correlation is generally low; in most cases, the range correlation coefficients decline rapidly, often falling below $1/e$ (approximately equals to $0.37$) within the range resolution [29], which is considered to be the boundary value between correlation and non-correlation. This rapid decay indicates that clutter is largely uncorrelated across range cells at this scale. Similarly low levels of range correlation can be exhibited by clutter subspaces as well as the signal and noise subspaces, suggesting that they can be considered uncorrelated in the time domain. Figure 3 illustrates the computed range correlation between range cells in a real-world clutter dataset, demonstrating the characteristic rapid drop in correlation across the range resolution.

In contrast, clutter typically exhibits high time correlation, while the signal and noise subspaces tend to have much lower correlation. This is because clutter is often composed of stationary or slowly moving objects, and as such tends to persist across multiple pulses, leading to stronger correlation over time; in contrast, the signal and noise subspaces are influenced by target motion and the randomness of noise, resulting in weaker time correlation.

The time correlation of the subspaces can be calculated using the received signal from each pulse, as follows:

$$\begin{array}{c}\hfill {\rho }_{\left(v\right)}^{\left(u\right)}\left(k\right)={\displaystyle \frac{N}{N-k}}\times {\displaystyle \frac{{\sum }_{n=0}^{N-k-1}{s}_{n+k}^u\left(v\right){s_n^u}^{*}\left(v\right)}{{\sum }_{n=0}^{N-1}{\left|s_n^u\left(v\right)\right|}^2}}\end{array}$$

(20)

where $s_n^u\left(v\right)$ represents the received signal for the nth pulse of the vth subspace and k is the time lag or delay between pulses. The time correlation coefficient ${\rho }_{\left(v\right)}^{\left(u\right)}\left(k\right)$ measures the degree of correlation between pulses of the vth range cell separated by a lag of k in the uth subspace, where $k\ge 0$, $1\le u\le N$, $1\le v\le L$. Then, the average time correlation coefficient of different range cells is collected by

$${\overline{\rho }}_u\left(k\right)={\displaystyle \frac{1}{L}}\sum _{v=1}^L{\rho }_{\left(v\right)}^{\left(u\right)}\left(k\right),1\le u\le N.$$

(21)

Figure 4a shows the time correlation of the first six subspaces; only the coefficients before the first local minimum are plotted, as the time coefficients would show fluctuations after they first decrease to a local minimum. The coefficients before the first local minimum help to illustrate the variation of the time correlation among the subspaces. As the order of the subspaces grows, the correlated pulse number obviously decreases. For clutter subspaces, the time correlation is typically high, indicating strong persistence of clutter over multiple pulses; in contrast, the signal and noise subspaces usually exhibit lower time correlations than clutter. Understanding this distinction is crucial for time-domain signal processing and clutter suppression, as it helps in distinguishing clutter from target signals. Then, the number of correlated pulses in every subspace is collected to form another feature, $f_7$, for further characterization. Figure 4b shows the average and the standard deviation of the number of correlated pulses, for which data were collected from 100 CPIs with different SCRs. The first subspace shows a dominant level of eleven pulses, which is usually considered as the clutter subspace. The next three subspaces show a lower level, after which the rest of the subspaces show a very low level, with only 1–3 pulses being correlated.

All of the features mentioned above are then combined to tackle the problem of subspace identification. In this way, the features naturally form several different dimensions. Subsequently, a classifier can be employed to label the subspaces into groups.

2.2.5. Classifier

SVM is a practical supervised learning model used for binary classification and regression tasks [30,31]. The primary goal of this algorithm is to find the optimal hyperplane that best separates the classes by maximizing the distance between the hyperplane and the data points of different classes.

Each of the features is normalized to the range [0,1] in preparation for SVM training, then arranged into a feature matrix $[{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,\dots ,{\mathit{x}}_m]$, denoted as $\mathit{X}$, where ${{\mathit{x}}_i=[f_{1,i},f_{2,i},\dots ,f_{7,i}]}^T$, $1\le i\le m$. Each column of $\mathit{X}$ represents a sample corresponding to a particular subspace, while the columns represent the extracted features. This matrix is used to train the SVM model for subspace classification. After SVD and feature computation, the resulting feature matrix $\mathit{X}$ serves as the input to the classifier, where the number of rows corresponds to the number of features and the number of columns represents the total number of samples.

The separating hyperplane can be formulated as

$${\mathit{w}}^T\mathit{x}+b=0,$$

(22)

where $\mathit{w}$ is the normal vector of the hyperplane and b is the bias term relative to the origin of the coordinates.

The SVM algorithm seeks to maximize the margin between the data points of different classes, which is defined as the distance between the closest data points of each class and the hyperplane. The problem can be formulated as the following optimization question:

$$\begin{array}{cc}\hfill & \underset{\mathit{w},b}{min}{\displaystyle \frac{1}{2}}{\mathit{w}}^T\mathit{w}+C\sum _{i=1}^m{\xi }_i\hfill \\ \hfill s.t.y_i({\mathit{w}}^T{\mathit{x}}_i+& b)\ge 1-{\xi }_i,{\xi }_i\ge 0,i=1,\dots ,m\hfill \end{array}$$

(23)

where C is the regularization parameter, ${\xi }_i$ are the slack variables, and $y_i\in \{-1,+1\}$ are the class labels. The slack variables ${\xi }_i$ allow some data points to violate the margin constraints, providing a way to handle noisy data and outliers.

The solution to this optimization problem provides the optimal hyperplane characterized by $\mathit{w}$ and b. The decision function for a new sample $\mathit{w}$ is provided by

$$f\left(\mathit{x}\right)={\mathit{w}}^T\mathit{x}+b.$$

(24)

If $f\left(\mathit{x}\right)\ge 0$, then the sample is classified as class $+1$; otherwise, it is classified as class $-1$.

The features extracted via SVD are used to train the classifier model, which is designed to improve its ability to classify data into distinct groups. The model is trained to classify the primary features extracted from the decomposed matrices, thereby identifying which subspaces correspond to clutter and non-clutter signals.

Overall, the model can utilize the most conspicuous features to enhance its classification capability. By focusing on the most relevant features, SVM improves its accuracy and robustness, making it a powerful tool for weak target detection. This integrated approach leverages the strengths of both techniques, resulting in a model that can effectively handle complex datasets and deliver superior performance.

3. Results

3.1. Experiments on Simulated Radar Echoes

To evaluate the effectiveness of the proposed method, we simulated echoes of clutter, targets, and noise. The system parameters were set as shown in

Table 1. Simulation parameters.

Parameters	Value
Carrier Frequency	3 $\mathrm{G}$$\mathrm{Hz}$
Bandwidth	30 $\mathrm{M}$$\mathrm{Hz}$
Range Resolution	5 $\mathrm{m}$
Pulse Repetition Interval	625 $\mathrm{\mu }$$\mathrm{s}$
Pulse Duration	40 $\mathrm{\mu }$$\mathrm{s}$
Velocity Detection Range	[−40, 40] $\mathrm{m}/\mathrm{s}$
Num of Range Cells	2000
Number of Pulses in a CPI	64

. The simulated radar echo matrices had a size of $64\times 2000$, corresponding to 64 pulses and 2000 range cells, which was consistent across the clutter, signal, and noise components. The following subsections provide a detailed description of the simulated dataset, highlighting the specific conditions and parameters used to generate realistic scenarios. Following this, the results obtained using the proposed method are presented, showcasing its ability to separate clutter, signals, and noise in the subspace domain. These results are then compared with those produced by the conventional SVD to assess the improvements in performance. Finally, an analysis of the outcomes is provided, focusing on key performance metrics such as classification accuracy, comparison of related methods, improvement of SCR, and complexity under different input scales.

3.2. Clutter and Target Generation

The clutter echoes were generated using the zero-memory nonlinearity (ZMNL) algorithm, followed by the K-distribution [32,33] in amplitude and the Gaussian distribution in power spectrum. The expressions are

$$\begin{array}{cc}\hfill f\left(x\right)& ={\displaystyle \frac{4c}{\mathrm{\Gamma }\left(v\right)}}{\left(cx\right)}^v{K}_{v-1}\left(2cx\right),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill S\left(f\right)& =S_{0}exp[{\displaystyle \frac{{(f-f_{\mathrm{d}})}^2}{2{\sigma }_f^2}}],\hfill \end{array}$$

(26)

where v is the shape parameter, c is the scale parameter, $K_{v-1}(·)$ is the modified Bessel function of the second kind, $\mathrm{\Gamma }(·)$ is the gamma function, $S_0$ is the average power of the clutter, $f_d$ is the center frequency of the clutter, and ${\sigma }_f$ is the spectral width of the clutter. The generated clutter maintains the same dimension as the defined parameters ($64\times 2000$). Figure 5 denotes the generated clutter data. Figure 5a shows the amplitude probability density function (PDF) of the simulated clutter, while Figure 5b shows its power spectrum density (PSD). For the K-distribution, the shape parameter v and scale parameter c are respectively set to 1 and 0.5. The spectral width of clutter is set to 60 $\mathrm{Hz}$.

3.3. Training and Evaluation

By decomposing the simulated signal matrices, samples of the subspaces are obtained from the echoes of every CPI, forming a dataset with binary classification (clutter subspace and non-clutter subspace). The singular value matrix after decomposition has a rank of 64, corresponding to 64 orthogonal subspaces, each associated with one singular value and its left and right singular vectors. Slow targets with different SCR values are added to the echoes to ensure the dataset covers a wide range of scenarios. Only the first 20 subspaces of generated CPIs are considered among the 64 subspaces in total. The dataset includes 1000 subspace samples from 50 independently generated CPIs with SCR values ranging from $-30dB$ to $0dB$.

The linear kernel function is used in the SVM classifier. The penalty parameter C is set to 1, which was tested and found to have a satisfactory performance. Then, K-fold cross-validation is implemented to evaluate the general performance of the classifier, where K is set to 5. The dataset is randomly divided into five independent groups. Each group is used as the testing set in turn, and the rest of the samples in the other four groups are used as the training set. After five rounds of training and testing, the classification precision of each group is calculated and the average precision is obtained. The results are shown in

Table 2. K-fold validation performance.

Order of Training	Precision
1st Training	98.44%
2nd Training	98.96%
3rd Training	100.00%
4th Training	95.83%
5th Training	97.66%
Average Precision	98.18%

.

3.4. Performance of A-SVD

Given the promising performance of the classifier, A-SVD is further compared with other widely adopted clutter suppression algorithms to comprehensively evaluate its overall effectiveness. Targets 1 and 2 are added to the range window in different range cells, which have respective radical velocities of 12 $\mathrm{m}$/$\mathrm{s}$ and $1.5$ $\mathrm{m}$/$\mathrm{s}$.

Figure 6a–d illustrate the results of the MTI, Kalmus filter, traditional SVD, and A-SVD methods, respectively. It can be observed that the MTI filter effectively suppresses clutter and retains target 1; however, target 2 is significantly impaired. The Kalmus filter also preserves target 1, yet the amplitude of target 2 is greatly reduced and noticeable clutter residues remain. The conventional SVD algorithm, which relies on the difference spectrum of singular values, preserves both target peaks but leaves residual clutter. In contrast, the proposed A-SVD method effectively suppresses clutter while maintaining the integrity of both targets. Figure 7a–d presents the coherent accumulation results of the four methods. Compared to the original outputs in Figure 6, these plots show how each method affects the two targets. The MTD (MTI after FFT) method suppresses clutter near zero, but also leads to the disappearance of target 2, as its radial velocity falls within the suppression range of the filters. The Kalmus filter shows limited improvement after accumulation, and the amplitude of target 2 is greatly reduced. Its narrow-notch design often leaves residues, leading to unexpected false alarms. The traditional SVD method improves the visibility of the two targets; however, residual clutter is still present. In contrast, the proposed A-SVD method retains clear signatures for both targets while effectively reducing clutter residues, demonstrating its superior performance in clutter suppression.

IF is used to evaluate the performance of the clutter suppression methods by measuring the improvement of the SCR after the clutter suppression. The IF is defined as

$$\mathrm{IF}={\left({\mathrm{SCR}}_{\mathrm{o}}\right)}_{\mathrm{dB}}-{\left({\mathrm{SCR}}_{\mathrm{i}}\right)}_{\mathrm{dB}}$$

(27)

where ${\mathrm{SCR}}_{\mathrm{i}}$ and ${\mathrm{SCR}}_{\mathrm{o}}$ are the SCR input and the output of the clutter suppression, respectively.

The IF results of the different methods (Kalmus filter, SVD, and A-SVD) for slow-moving targets with various velocities are shown in

Table 3. Average improvement factor (IF) for targets with different velocities.

Methods	1 m/s	3 m/s	5 m/s	7 m/s	13 m/s	19 m/s	25 m/s
Kalmus Filter	$26.74dB$	$36.01dB$	$40.96dB$	$44.31dB$	/	/	/
SVD	$19.37dB$	$25.10dB$	$28.59dB$	$31.00dB$	$32.89dB$	$34.44dB$	$35.70dB$
A-SVD	$33.86dB$	$39.09dB$	$42.15dB$	$44.32dB$	$45.46dB$	$46.60dB$	$47.67dB$

. The Kalmus filter performs well when the target velocity is below 7 $\mathrm{m}$/$\mathrm{s}$, with IF values ranging from approximately 26 dB to 44 dB; however, its performance becomes unstable beyond this range. The SVD clutter suppression algorithm demonstrates relatively consistent performance across different velocities, maintaining an IF around 30 dB. In contrast, the IFs of the proposed A-SVD algorithm are above 40 dB for most velocities, showing superior overall performance compared to both the Kalmus filter and traditional SVD methods. Notably, A-SVD also outperforms the Kalmus filter for low-velocity targets, demonstrating its enhanced clutter suppression capability under these conditions.

However, when the target’s radial velocity is as low as 1 $\mathrm{m}$/$\mathrm{s}$ (approaching zero), the limited number of pulses restricts the SVD method’s ability to completely separate the target from clutter during subspace decomposition. In this scenario, most of the clutter energy resides in the first subspace, and both the proposed method and SVD only suppress this first subspace. Consequently, the IF values for both methods are relatively lower compared to higher velocity targets. Increasing the number of pulses could improve this situation. The Kalmus filter shares this limitation, but its performance cannot be directly enhanced due to the difficulty of modifying its amplitude–frequency response. Nevertheless, the A-SVD algorithm still performs better than the Kalmus filter in this case.

3.5. Computational Complexity

The complexities of the two SVD approaches, as previously presented in Section 2.2.1, are $\mathcal{O}(N{L}^2+N^3)$ and $\mathcal{O}\left(kNL\right)$, respectively. For conventional SVD, after decomposition, calculating the difference and recovering the the remaining subspaces requires $\mathcal{O}\left(N\right)$ and $\mathcal{O}\left(NL\right)$ operations. Therefore, the overall complexity of the SVD approach is $\mathcal{O}(N{L}^2+N^3+N+NL)$, which is approximately $\mathcal{O}\left(N{L}^2\right)$, because usually $L\gg N\gg k$. For A-SVD, the complexity of the subspace decomposition AIRLB is $\mathcal{O}\left(kNL\right)$, where k is the number of required subspaces. Features $f_1$, $f_2$, $f_3$, $f_4$, and $f_6$ share a complexity of $\mathcal{O}\left(k\right)$, while features $f_5$ and $f_7$ have complexities of $\mathcal{O}\left(kL\right)$ and $\mathcal{O}\left(kN\right)$, respectively. Additionally, the complexity of the classifier is given as $\mathcal{O}\left(m\right)$, where m is the dimension number of the features. Therefore, the overall complexity of A-SVD is $\mathcal{O}(kNL+kN+kL+m)$, which is approximately $\mathcal{O}\left(kNL\right)$, considering that the required number of subspaces k is much smaller than N and L.

The methods were then implemented on an Intel Core i5-12400F processor using the MATLAB 2022b platform, with different input sizes used to analyze their execution times. Based on the simulated classification results, k can be set as $\lceil 0.1\times N\rceil$, where $\lceil ·\rceil$ is the ceiling function. The running time results are presented in

Table 4. Execution time results of different methods.

Input Scale	SVD	A-SVD	Time Reduction
$64\times 2000$	218 $\mathrm{m}$$\mathrm{s}$	20 $\mathrm{m}$$\mathrm{s}$	$90.9$%
$64\times 5000$	1456 $\mathrm{m}$$\mathrm{s}$	38 $\mathrm{m}$$\mathrm{s}$	$97.4$%
$64\times 10,000$	5411 $\mathrm{m}$$\mathrm{s}$	68 $\mathrm{m}$$\mathrm{s}$	$98.7$%
$128\times 2000$	246 $\mathrm{m}$$\mathrm{s}$	48 $\mathrm{m}$$\mathrm{s}$	$80.5$%
$128\times 5000$	1632 $\mathrm{m}$$\mathrm{s}$	108 $\mathrm{m}$$\mathrm{s}$	$93.3$%
$128\times 10,000$	6067 $\mathrm{m}$$\mathrm{s}$	218 $\mathrm{m}$$\mathrm{s}$	$96.4$%
$256\times 2000$	301 $\mathrm{m}$$\mathrm{s}$	182 $\mathrm{m}$$\mathrm{s}$	$39.5$%
$256\times 5000$	1964 $\mathrm{m}$$\mathrm{s}$	475 $\mathrm{m}$$\mathrm{s}$	$75.8$%
$256\times 10,000$	7350 $\mathrm{m}$$\mathrm{s}$	998 $\mathrm{m}$$\mathrm{s}$	$86.4$%

. As the number of pulses increases, the efficiency improvement of the algorithm runtime diminishes. Conversely, as the number of distance units increases, the efficiency improvement of the runtime enhances.

3.6. Realistic Radar Dataset

The proposed method was tested on a real radar dataset featuring strong sea clutter. The carrier frequency was in the S band, with a bandwidth of 40 $\mathrm{M}$$\mathrm{Hz}$ and a range resolution of $3.75$ $\mathrm{m}$. The pulse repetition interval was 690 $\mathrm{\mu }$$\mathrm{s}$ and the pulse duration was 40 $\mathrm{\mu }$$\mathrm{s}$. The velocity detection range was [−40,40] $\mathrm{m}/\mathrm{s}$, and each CPI consisted of 80 pulses. The scenario included several targets with varying velocities, providing a challenging environment for effective clutter suppression and target detection.

The echoes were extracted and rearranged into a matrix of dimensions $80\times 2200$, corresponding to the number of pulses and range cells. The signals underwent initial preprocessing steps to improve the visibility of differences between of the methods, including pulse compression and coherent integration.

The results obtained using the realistic dataset are illustrated in Figure 8. Figure 8a presents the original accumulated echoes, while Figure 8b,c shows the processed results of the Kalmus filter and the SVD method, respectively. These methods are unable to maintain the four original targets due to their respective limitations. In contrast, Figure 8d demonstrates the effectiveness of the proposed clutter suppression method, which not only suppresses the clutter but also successfully detects all four targets. Notably, the proposed method preserves slow-moving targets with different velocities. These findings underscore our method’s ability to achieve substantial clutter reduction while retaining the integrity of target information, showcasing its robustness in challenging environments.

The successful suppression of clutter and retention of slow-moving target echoes validate the effectiveness of our method under realistic conditions. These results indicate that the proposed approach not only provides an enhanced signal-to-clutter ratio but also improves the detection performance of targets with low RCS and slow velocities. This demonstrates its potential for practical applications in scenarios where traditional clutter suppression techniques might struggle to differentiate between targets and clutter, especially in environments with complex and strong clutter sources.

4. Conclusions

This study provides a detailed analysis of the subspace characteristics in SVD when applied to radar echoes, revealing structural differences between clutter and target components. Based on this analysis, we propose a clutter suppression method combining partial SVD with SVM-based subspace classification, which we name A-SVD. Extensive experiments on simulated and real-world datasets confirm the effectiveness of the proposed method. A-SVD achieves an improvement factor exceeding 40 dB in typical low-velocity target scenarios and reduces runtime by more than 85% in most cases. These results demonstrate its potential for real-time radar applications along with its robustness under various clutter conditions.

5. Discussion

Future work will focus on refining the subspace separation strategy for better clutter–target discrimination, especially under nonstationary clutter conditions such as dynamic urban or maritime environments. In addition, we plan to investigate the construction of dedicated detectors based on the current subspace framework and to explore hybrid approaches that integrate classical signal processing with lightweight learning-based models to further enhance the adaptability and intelligence of A-SVD.