A physical optics based method is presented for calculation of monostatic Radar Cross-Section (RCS) of a shell-shaped projectile. The projectile is modeled using differential geometry. The paper presents a detailed analysis procedure for RCS formulation using physical optics (PO) method. The shortcomings of the PO method in predicting accurate surface current density near the shadow boundaries are highlighted. A Fourier transform-based filtering method is proposed to remove the discontinuities in the approximated surface current density. The modified current density is used to formulate the scattered field and RCS. Numerical results are presented comparing the proposed method with conventional PO method. The results are also compared with published results of similar objects and found to be in good agreement.

Prediction and measurement of Radar Cross-Section (RCS) have been a significant area of research for scientists and engineers for many years. The widespread uses of radar technology since the Second World War have demanded accurate and efficient prediction of fields scattered by radar targets. The knowledge of echo characteristics of radar targets is of great importance for the design of high-performance radars, as well as low visibility stealth targets [

In radar technology, an antenna radiates electromagnetic (EM) energy. When an object is illuminated by the radar EM field, it reflects back some EM energy, which is received by an antenna. In monostatic radar system, the transmitting and reception of the EM energy are done by the same antenna or by multiple antennas located very close to each other [

Numerical simulation of RCS of an object requires calculation of the scattered field from the object for a given incident field. Several numerical methods exist for scattered EM field calculations. Pure numerical methods such as Method of Moments (MoM), Finite-Difference Time-Domain (FDTD) method, Fast Multipole Method (FMM), and Transmission-Line Matrix (TLM) have been successfully used in predicting RCS of radar targets [

Because of its relatively high accuracy, the PO method has been widely used for RCS formulation [

This paper presents a detailed description and procedure of RCS calculation method of an object using PO method and modified PO method. The procedure includes the geometrical modeling of the shell-shaped radar object, approximation of the induced surface current on the object, and formulation of the PO radiation integral in parametric space. Although the paper concentrates on RCS formulation of a specific shell-shaped radar target, the analysis procedure is general and can be applied to objects of any geometry. In spite of the presence of many research and review articles [

The paper is arranged as follows: in Section

The first step in simulating the RCS of an object is to accurately model the surface of the object. This paper concentrates on the RCS of a shell-shaped object. A shell-shaped object is selected as most projectiles represent this basic shape. A schematic diagram of the object and the three-dimensional coordinate system is shown in Figure

Three dimensional geometry of the shell-shaped projectile, and the coordinate system along with incident field vectors.

The origin of the coordinate system is taken at the center of the half sphere, and

Computer-generated three-dimensional model of the shell-shaped projectile.

Once the surfaces of the object are defined, it is necessary to define normal vectors on each point of the surface. These normal vectors are necessary for GO-, UTD-, or PO-based scattering formulations [

Normal vectors (blue lines) on the object surfaces.

To formulate the RCS of the object, the incident field must be defined first. As the radar and the target are usually very far away from each other, the incident field can be modeled as a plane wave, implying that the direction of the wave, the direction of the electric field, and the direction of the magnetic field are perpendicular to each other. With respect to a coordinate system defined at the source point of the wave, if the wave travels at

For monostatic RCS calculation, the reflected ray is

From

So, to increase the accuracy of the PO method, the discontinuity

To perform filtering operations to make

Now, the modified smoother current density,

It is noted that that as some components of the current densities are filtered out, the energy of the conventional PO current may not be equal to the energy of the modified PO current. This may not be consistent with conservation of energy. To rectify this problem, the filtered current densities must be multiplied by a constant scalar so that the resulting current densities have the same energy as the PO current. The value of the constant can easily be calculated by calculating the energy of the conventional PO current and the unscaled modified PO current. This scaling ensures that no changes have occurred in total energy of the surface currents.

Due to harmonic oscillating nature of Fourier transform,

For numerical analysis, the frequency of the incident field is taken to be 10 GHz. The radius of the sphere and cylinder,

Figures

Surface current density using PO method for an incident field

Surface current density using PO method for an incident field

Using the proposed modified PO method, the surface current densities are smoothed. Fourier transform is used with respect to both parameters. The resulting smoothed current densities are shown in Figures

Surface current density using modified PO method for an incident field

Surface current density using modified PO method for an incident field

To verify accuracy of the surface current densities approximated in the proposed filtering method, a simplified canonical problem of scattering from an infinite cylinder is considered. Considering an infinitely long cylinder with axis parallel to the

Surface current densities on an infinite cylinder surface at

The RCS of the shell-shaped object is formulated using both conventional PO and the modified PO method. The RCS at

Monostatic RCS of the shell-shaped projectile at 10 GHz as a function of

RCS is expressed in dB with respect to 1 m^{2} area. This unit is represented as dB m^{2} or dB sm [

The RCS of the projectile at 5.5 GHz as a function

Monostatic RCS of the shell-shaped projectile at 5.5 GHz as a function of

For the angles where RCS is high, the results of both methods are almost identical. The deviation arises for angles where RCS value is low. The higher degree of accuracy obtained from MoM comes at a cost of complex computation and extensive simulation time. The errors produced by the proposed method are relatively small and acceptable for many applications. In most cases the accuracy from PO method is sufficient, and the proposed method is expected to be more accurate than the PO method.

The RCS of the shell-shaped projectile for frequencies 3 GHz, 6 GHz, 10 GHz, and 14 GHz as a function

Monostatic RCS of the projectile at 3 GHz, 6 GHz, 10 GHz, and 14 GHz as a function of

The computer simulation was performed on an Intel Core i5-2430M 2.4 GHz CPU with 2.94 GB usable RAM. The simulation time of conventional PO and modified PO is compared in Table

Comparison of CPU simulation time.

Frequency (GHz) | Simulation time for PO method (sec) | Simulation time for modified PO method (sec) |
---|---|---|

3 | 16.83 | 26.09 |

6 | 32.09 | 50.79 |

10 | 53.19 | 87.01 |

14 | 74.15 | 144.23 |

From Table

RCS formulation procedure using PO method is described in detail in this paper. The paper covers geometrical modeling of shell-shaped projectile using differential geometry, formulation of surface current density, evaluation of the scattered field integral in parametric space, and monostatic RCS formulation. A modified PO method is presented which approximates surface current density more accurately. The additional computational steps require Fourier transform and inverse Fourier transform only. These can be easily incorporated in computer code using well-known Fast Fourier Transform (FFT) algorithm. Thus the accuracy is increased without significant increase in computational complexity. The obtained results using the modified PO method are consistent with similar results found in literature.