I. INTRODUCTION
Ballistic missiles, owing to their long range, high speed, and capacity to carry multiple warheads, including nuclear ones, represent a highly threatening weapon system [1]. As a result, numerous countries are striving to develop ballistic missile defense systems [1, 2]. The effective operation of such defense systems requires determining interception priorities, which in turn necessitates identifying the type of ballistic missile. Previous studies on ballistic missile identification have primarily proposed decision-making methods based on flight information, such as range, altitude, and velocity, along with launch site data [3]. Additionally, motion characteristics after warhead separation have been analyzed to distinguish between warheads and decoys [4]. However, since ballistic missiles can adjust their range and altitude based on the operator’s intent, certain missile types with similar flight profiles may be difficult to classify based solely on these parameters. Furthermore, motion-based classification after warhead separation is only applicable to medium- and long-range ballistic missiles, where separation actually occurs. Given the destructive power of intercontinental ballistic missiles (ICBMs), accurately distinguishing warheads from decoys and prioritizing warhead interceptions is of critical importance. Consequently, most ballistic missile identification research has focused on classifying warheads and decoys rather than differentiating missile types.
However, for countries in close proximity to potential adversaries, short-range ballistic missiles (SRBMs) may pose a more immediate threat. Thus, additional research must be conducted to identify specific SRBM types. Moreover, since SRBMs usually do not undergo warhead separation, a different approach is required for their identification. In this study, considering the characteristic rotational motion of ballistic missiles along a stable elliptical trajectory, an inverse synthetic aperture radar (ISAR) is employed for ballistic missile identification. ISAR is a radar technique that transmits electromagnetic waves from a stationary radar to a rotating object and then processes the reflected data to locate the object’s scattering centers [5, 6]. Constructing these scattering centers enables object identification. Notably, extensive research has been conducted to utilize this capability. However, ISAR research on airborne objects has primarily focused on the identification of drones and aircraft [7, 8]. These studies investigated formation flight scenarios and explored methods for effectively identifying multiple targets in close proximity. However, studies on the identification of ballistic missile types using ISAR alone are difficult to find. Other ISAR-related studies include noise reduction for image quality enhancement and the generation of 3D ISAR images [9].
This study focuses on SRBMs with similar flight patterns to investigate methods for identifying missile types using ISAR technology. However, the direct application of conventional ISAR methods for ballistic missile identification has several limitations. Missiles following a consistent trajectory provide only partial scattering data at specific observation angles. Furthermore, strong scattering from certain parts of the missile may overshadow smaller structural features, thereby reducing visibility [10]. To address these limitations, alternative methods that enable the effective identification of SRBM types are necessary.
The distinguishing features of missiles primarily involve the presence or absence of canards (head fins) and the shape of tail fins. While body shape may also be examined for identification, significant variations from the traditional missile body shape are uncommon. The challenge, in this context, lies in the fact that the shapes of canards and tail fins are not clearly depicted in traditional ISAR images. To demonstrate this, four missile models were created. The first model is an approximately 12-m-long missile without canards featuring a delta-shaped tail fin. The second features the same size and body shape as the first but includes canards and has square-shaped tail fins. The third model is a scaled-down version of the first but with a length of approximately 9 m and a body shape similar to the first two models. The fourth model is also approximately 9 m in length but has a slightly different body shape and warhead. Additionally, the size and position of the canards in the fourth model are distinct from those of the third model. Fig. 1 presents the four ballistic missile models created using the commercial software Feko. For convenience, these models are referred to as Types A, B, C, and D.
Fig. 2 shows the ISAR images generated using only the VV-polarized scattering data acquired from a limited observation angular range of 70°–100°, as would be obtained during flight. The scattering data spans a bandwidth of 1–5.5 GHz and was obtained using physical optics (PO). Notably, the frequency bandwidth was set to reflect operational radar frequencies, which were determined based on the frequency ranges specified by weapon system manufacturers for ballistic missile detection [11]. The detailed parameters used for ISAR image generation are presented in Table 1. In this scenario, the canards and tail fins were not clearly distinguishable, making it evident that differentiating between the four models was challenging.
To overcome these limitations, this paper proposes a method for directly estimating scattering centers using combinations of the six range profiles obtained for various observation angles as the ballistic missile moves along its trajectory. Furthermore, a technique for generating ISAR images by transforming these scattering center locations into the frequency domain is demonstrated. Merging the transformed frequency domain data with the original scattering data results in ISAR images in which the scattering centers appear more distinctly. This approach also facilitates the potential integration of data obtained from various radar sources into the obtained ISAR images, thereby significantly enhancing the effectiveness of ballistic missile identification.
II. METHOD FOR ENHANCING ISAR IMAGE VISIBILITY
1. Concepts of the ISAR and the High-Resolution Range Profile
ISAR is a technique that processes scattered waves obtained from the multiple observation angles of a target to project its scattering centers onto a two-dimensional image projection plane (IPP) consisting of the range and Doppler (or cross-range) dimensions. Range represents the distance along the radar’s line of sight (LOS), estimated based on the received signal over time, while cross-range refers to the distance perpendicular to the range direction, which is derived from Doppler frequency variations caused by the target’s rotation. The scattering center is closely related to the phenomenon by which electromagnetic waves striking an object are scattered in multiple directions [6]. Notably, this scattering does not occur uniformly across the entire object. Instead, certain regions exhibit stronger scattering due to interference effects. These specific points are referred to as scattering centers.
The total reflected wave from an object can be approximated as the sum of the reflected waves reemitted from a finite number of scattering centers. Fig. 3 illustrates the concept of scattering centers, depicting only three scattering centers for brevity. It should be noted that a greater number of scattering centers may exist in actual target trajectories. A commonly used method for this two-dimensional projection is the 2D inverse Fourier transform (IFT). An ISAR image can be obtained by applying 2D IFT to k-space data composed of transmission frequencies and the observation angles of the target. Eq. (1) formulates a complex scattered wave as a function of its frequency and angle. Eq. (2) provides the formulation for deriving ISAR results via 2D IFT, using data converted from the polar coordinates of Eq. (1) into the Cartesian co-ordinates (kx, ky).
(2)
In Eqs. (1) and (2), A denotes the amplitude of the scattered wave, k is defined as 2π divided by the wavelength λ, k⃗ is the wave number vector in the propagation direction, while x̂ and ŷ represent the unit vectors in the x and y directions, respectively. Furthermore, ⇉ refers to the vector from the origin to the scattering center, while f and θ denote the frequency and angle, respectively. In addition, i represents the i-th scattering center, while x and y denote the coordinates of the scattering center. kx and ky represent the spatial frequencies along the x and y directions, respectively. c is the speed of light, K is the number of scattering centers, and δ refers to the delta function, which has a nonzero value only at the location of the scattering center.
Meanwhile, the range profile presents the results obtained by transforming the frequency-domain scattered wave data of a single angle into the distance domain using 1D IFT. In other words, the range profile reveals the separation distance between the target and the radar. Therefore, it can be regarded as the time-domain response of the target to a radar pulse. To provide an in-depth understanding of this factor, Fig. 4 illustrates the frequency spectrum and the corresponding range profile obtained using IFT and FT. To express this mathematically, it is assumed that there are K-th scattering centers located at different positions. Eq. (3) represents the sum of scattered waves from the scattering centers at a given frequency, and Eq. (4) demonstrates the process of generating the range profile [6].
(4)
Here, Ai is the scattering amplitude of the i-th scattering center, and
k = 2 π f c a = 2 f c
(5)
Here,
a H = 2 f H c a L = 2 f L c f c = f L + f H 2 k c = 2 π f c c
(6)
Here, B denotes the bandwidth, and c is the speed of light. When the frequency bandwidth is sufficiently wide, resolution is enhanced, allowing for the identification of individual scattering centers and their distances from the radar. This process is known as high-resolution range profile (HRRP) [12]. However, a single HRRP only provides the separation distance along the radar’s line-of-sight, offering no information on the perpendicular distance. Therefore, to locate scattering centers directly, at least two HRRPs pertaining to different observation angles are necessary. In other words, a minimum of two observation angles is required.
2. Scattering Center Estimation and ISAR Image Generation through Frequency Domain Transformation
By acquiring data using a broadband frequency band for multiple observation angles of the target, it is possible to generate HRRP for each angle. Notably, extracting the peak points of two HRRPs allows for the direct estimation of scattering center positions using the following equation:
Here, x and y denote the two-dimensional coordinates of the m-th scattering center, hn,m represents the location of the m-th peak in the n-th HRRP, θ1 refers to the angle between the first incident wave and the origin (0,0), while θ2 represents the angle between a second incident wave and the origin (0,0). Eq. (7) implies that knowledge of the peak positions of two HRRPs from different observation angles allows for the determination of all possible coordinates at which a scattering center may exist. However, the exact position of the scattering center cannot be accurately determined based on only two HRRPs. Although all intersections between two HRRPs represent potential locations of scattering centers, these intersections do not guarantee actual scattering center positions. In this study, the actual scattering points among potential locations are termed true scattering centers, while points that do not physically exist are termed false scattering centers. Notably, false scattering centers can be eliminated using additional HRRPs. Fig. 5(b) illustrates the process of removing false scattering centers using three HRRPs. In Fig. 5(a), two HRRPs are used to determine potential locations. In contrast, Fig. 5(b) demonstrates that the inclusion of one additional HRRP enables the distinction between true and false scattering centers [13].
However, since scattering centers on ballistic missiles are often densely clustered in close proximity, a single additional HRRP alone may not be sufficient to completely eliminate false scattering centers. In such cases, additional HRRPs may be incorporated to reduce the number of false scattering centers. The first step of this process involves excluding the false scattering centers eliminated by the first additional HRRP. Next, an additional HRRP is employed to identify the remaining false scattering centers. If more HRRPs are available, the previously identified false scattering centers are iteratively excluded, and the process continues. By sequentially filtering out false scattering centers, only the true scattering centers ultimately remain. This iterative filtering procedure is illustrated in Fig. 6. Fig. 6(a) depicts a scenario in which the scattering centers are more numerous and arranged in a more complex configuration than in Fig. 5, due to which two HRRPs are used to determine all potential locations of the scattering centers. Fig. 6(b) depicts the identification of false scattering centers using three HRRPs. Fig. 6(c) illustrates the elimination of previously identified false scattering centers by using an additional HRRP to further refine the selection. Fig. 6(d) and 6(e) represent cases involving five and six HRRPs, respectively.
In this study, six HRRPs were used at a time to estimate scattering centers. This configuration was selected because it yielded the fewest false scattering centers in the experimental results. Furthermore, considering the rotational motion of a ballistic missile along its trajectory, it was determined that acquiring HRRPs from at least six different aspect angles is feasible.
Once the scattering center positions were estimated, they were transformed into the frequency domain for visualization as an ISAR image. This was achieved using the inverse matrix property of Eq. (7), as expressed below:
Eq. (8) indicates that the HRRP for a given angle can be obtained using the two observation angles and the scattering center coordinate matrix. Furthermore, even when only a limited number of HRRPs are available for specific angles, it is possible to create the effect of having HRRPs at 360 different angles by means of Eq. (8). These generated data are referred to as the generated HRRP in this study for convenience. Notably, generating HRRPs for exactly 360 angles is not mandatory. However, in this paper, data acquisition was assumed to be performed at 1° intervals, leading to the generation of 360 HRRPs. Moreover, this approach facilitated matrix representation when integrating the actual frequency-domain data with the generated frequency-domain data in subsequent processes.
As previously mentioned, HRRP should ideally be represented by a delta function. However, due to limitations imposed by the frequency bandwidth, it is practically expressed as a sinc function. In the case of this paper, HRRP was generated using the delta function to ensure clear visibility of the sparsely distributed scattering centers in the ISAR image. Fig. 7 illustrates the HRRPs generated using the delta and sinc functions for a specific observation angle. It is evident that while the HRRP generated using the delta function maintains a uniform amplitude, the HRRP generated based on the sinc function varies based on the density of the scattering centers. In this regard, it must be noted that since ISAR images are formed by overlapping the HRRPs of all angles, when varying amplitudes overlap, as is the case with the sinc function-based HRRP, smaller regions become less distinct compared to larger ones in the ISAR image. Therefore, since the primary objective of this study is not to perfectly reconstruct the HRRP but to ensure that all scattering centers are clearly visible in the ISAR image, the delta function was employed.
Generating HRRPs for 360 angles produced a matrix in the range–angle domain. By performing 1D FT on the data pertaining to each individual angle, this matrix was transformed into the frequency–angle domain. Effectively, a frequency–angle matrix suitable for creating ISAR images was generated. This process is illustrated in Fig. 8.
Furthermore, by implementing 2D IFT on the frequency–angle matrix generated in the previous procedure, an ISAR image displaying only the estimated scattering centers was obtained. Such an image can be used for identification purposes since it allows for the precise localization of scattering centers. However, to further enhance identification, a merging process with the conventional ISAR image was undertaken. By inserting the scattered wave data from the measured angles into the corresponding angles of the generated frequency–angle matrix and performing a 2D IFT, a merged ISAR image was produced. Notably, for this process, scaling of both the actual and generated data had to be performed beforehand. This process is demonstrated in Fig. 9. Fig. 10 depicts the two ISAR images— one displaying only the estimated scattering centers obtained from the 2D IFT of the generated matrix, and the other showing the merged ISAR image created after incorporating the actual scattered wave data.
Moreover, when the target is limited to missiles, their inherent structural symmetry can be utilized to improve the visibility of the final ISAR image. Missiles exhibit rotational symmetry along their flight axis, thereby allowing for the replication of scattered waves on the opposite side of the axis using only partial angle data. Fig. 11 illustrates the symmetrical characteristics of a missile, while Fig. 12 shows the ISAR image obtained after the application of symmetry enhancement.
The improvement in ISAR image visibility based on the number of HRRPs used, as demonstrated in Fig. 6, is evident in Fig. 13. Fig. 13(a), 13(b), 13(c), and 13(d) correspond to cases where three, four, five, and six HRRPs are employed, respectively, for each scattering center estimation attempt. When only three HRRPs are used, the scattering centers are inaccurately estimated. However, with six HRRPs, a significant reduction in false scattering centers is observed. Additionally, Fig. 14 shows the results obtained using the sinc function. Compared to Fig. 13(d), which was generated using the delta function, Fig. 14 shows that the weaker intensity regions disappear and the scattering centers are more spread out.
A notable advantage of the proposed method is the minimal increase in processing time. Each additional observation angle allows for the estimation of more scattering centers, with approximately 2 seconds required to estimate the scattering centers from each angle combination. Furthermore, since this process involves performing a 1D IFT on the frequency domain to convert it into the range domain and comparing the peaks of HRRPs, the overall time required is minimal. Additionally, generating the final matrix of estimated scattering centers in the frequency–angle domain takes around 10 seconds. This increase in time is negligible, even when accounting for the missile’s total flight time. If necessary, the total processing time can be further reduced by limiting the overall number of observation angles. Fig. 15 provides a representation of the overall timetable for the proposed method.
3. Performance Analysis
For simulation, the identification feasibility of the proposed method was tested based on a total of 24 missiles of Types A, B, C, and D in flight. For verification, principal component analysis (PCA)—a technique that reduces high-dimensional data into lower dimensions by projecting it onto a basis that best captures the variance in the data structure [14]—was conducted. PCA employs singular value decomposition (SVD) to compute eigenvectors and eigenvalues from the original data, arranging them in descending order to extract features that have the greatest impact on the original variance. For the conventional ISAR method, a test dataset was used, consisting of 24 images each generated from its own unique, randomly selected set of angles within the 70°–100° range. The PCA results for this dataset displayed significant overlap among missiles with similar characteristics, implying that establishing a clear differentiation is difficult. However, the PCA results for the ISAR images generated from the same data using the proposed method exhibited distinct classification of the four missile types. These results are presented in Fig. 16, where Fig. 16(a) presents the PCA results obtained using conventional ISAR images, and Fig. 16(b) shows the results for the images generated using the proposed method. These results demonstrate that the proposed method facilitates the easier identification of different types of missiles.
4. Measurement Results of a Scaled Missile Model
A scaled missile model was fabricated to obtain the measurement results of the proposed method. The missile model was designed to incorporate various missile characteristics within a single prototype, including canards, a body with a diameter that gradually increases up to the midpoint, and a delta-shaped tail fin. After 3D printing the model, aluminum coating was applied to provide it with properties similar to those of a perfect electric conductor (PEC). The model’s dimensions were set to a length of 0.84 m and a height of 0.25 m, considering the limitations of the indoor measurement space. Fig. 17 presents the missile model.
The indoor measurements were conducted in a building corridor. The measurement equipment was configured as shown in Figs. 18 and 19. Table 2 provides the detailed parameters considered for the measurement. As shown in Fig. 18, the setup comprises an antenna, a support structure for holding the target object, and a vector network analyzer (VNA). The support structure is capable of rotating 360° horizontally. Assuming the missile’s nose direction to be 0° and its tail direction to be 180°, the data were collected at 1° intervals between 70° and 100°, as shown in Fig. 20. Additionally, Fig. 21 includes photographs taken during the actual measurement, showing the target object at 0°, 90°, 180°, and 270°.
To minimize the negative impact of external noise, absorbers were placed around the measurement area, and a time-gating technique was applied after measurement to isolate the scattered waves from the model’s position [15]. The time-gating procedure is illustrated in Fig. 22.
The parameters used to generate the ISAR image using the measurement equipment are provided in Table 3. In the conventional ISAR image generated using scattered wave data from partial observation angles (70°–100°), the shapes of the canards and tail fin were not fully visible. In contrast, upon using the proposed method, an ISAR image clearly reflecting the shapes of the canards and tail fin was obtained, as illustrated in Fig. 23. Furthermore, to verify whether the delta-shaped tail fins had been accurately reconstructed in the obtained ISAR image, a new simulation model was designed. As depicted in Fig. 24, the new model shares the same fuselage shape as the original but lacks canards and has rectangular tail fins. The ISAR reconstruction results for the new model using the proposed method are presented in Fig. 25. For comparison, Fig. 26 illustrates the differences in the reconstructed tail fin shapes between the measurement model and the newly generated model when using the proposed method. In the case of the measurement model with delta-shaped tail fins, three lines appear when connecting the fin tips. In contrast, for the model with rectangular tail fins, only two lines are observed. This confirms that the tail fin shape can be easily inferred from the ISAR reconstruction results.
III. CONCLUSION
This study presents a method for identifying the external shapes of SRBMs using ISAR images. Conventional ISAR methods face constraints in capturing distinct structural features, such as those of ballistic missiles, especially when angle data acquisition is restricted. Additionally, previous studies on ballistic missile identification have primarily focused on distinguishing between warheads and decoys in medium- and long-range ballistic missiles, rather than analyzing the types of SRBMs. In contrast, this study aims to enhance the visibility of ISAR images and thereby facilitate the classification of SRBMs, where warhead separation does not occur. By directly estimating scattering centers using the HRRPs pertaining to each angle, transforming the scattering centers into the frequency domain, and merging them with conventional ISAR images, visibility is significantly improved. The proposed method also achieved improved identification results for the automated classification of missile types using the PCA method. A notable advantage of this approach is its ability to accurately identify features, such as the presence of canards and the shape of tail fins, while minimizing increases in processing time. In addition, the proposed method was applied to a scaled fabricated missile model to confirm that the shapes of the canards and tail fins can be visually identified in the resulting ISAR images.
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