### I. Introduction

*P*

*and*

_{t}*P*

*denote the transmitted and received powers, respectively, and*

_{r}*G*

*and*

_{t}*G*

*refer to the gain of the transmitting and receiving antennas, respectively. Furthermore,*

_{r}*σ*is the radar cross section (RCS) of a target in m

^{2}and

*A*

*indicates the effective area of the receiving antenna, which usually becomes*

_{er}*G*

_{r}*λ*

^{2}/(4

*π*). In addition,

*λ*is the transmitted wavelength,

*R*is the distance between the radar and the target, and

*R*

*≒*

_{t}*R*

*≡*

_{r}*R*for the quasi-monostatic radar. The most significant feature of (1) is that the received power decreases as the fourth power of

*R*. However, this equation is valid only under conditions where the receiving antenna is situated far from the target and the target is also located at a substantial distance from the transmitting antenna.

### II. Effect of Large Target

*D*of an antenna is larger than its wavelength, the regions of the electromagnetic field around the antenna can be divided into the reactive near-field, radiative near-field (Fresnel), and far-field (Fraunhofer) regions. The most well-known boundaries between these three regions are 0.62(

*D*

^{3}/λ)

^{0.5}and 2

*D*

^{2}/λ [8]. Additionally, the

*D*

^{2}/4λ and 0.6

*D*

^{2}/λ criteria have also been frequently cited in estimating the extent of near-field and transition (Fresnel) regions [9].

*J*

*on the transmitting antenna results in forward propagation of the target located in the far-field region, it can be considered that a uniform plane wave is incident on the target. This incident wave induces current*

_{t}*J*

*on the target, after which the induced current generates a scattered field. These sequences can also be calculated using the surface equivalence principle and radiation integrals [8]. In this context, it should be noted that the target in this example acts as an antenna with dimension*

_{s}*D*

*due to the induced current*

_{s}*J*

*. Therefore, even when the target is located in the far-field region of the transmitting antenna, the receiving antenna may not be within the far field of the target due to the target’s dimensions.*

_{s}*W*based on a fixed simulation dimension, the electric field distribution was calculated with respect to the distance from the target. Fig. 3(b) and 3(c) show the total field (i.e., both the incident and the scattered fields) covered by PEC rectangular plates with widths of 100 mm and 10 mm, respectively. In this context, it should be noted that the total field patterns of the two cases are entirely different. As expected, the electric field generated by the larger scatterer exhibits fluctuating characteristics in the reactive near-field region, as well as a characteristic decrease after a slight increase in the radiative near-field region. This further indicates that the entire analysis region is in the near-field region. In contrast, the small scatterer exhibited a very short near-field region.

*W*= 50, 100, and 200 mm are in their near-field regions, their scattered fields fluctuate around 0 dB and remain fairly constant at various distances from the scatterer. Similar results have been reported in [12]. In contrast, the scattered field created by the small scatterer with

*W*= 10 mm decreases proportionally with the inverse-square of distance (1/

*R*

^{2}) from around five wavelengths after fluctuation, which is closer to 0.6

*D*

^{2}/λ than 2

*D*

^{2}/λ.

*G*

*(*

_{t}*θ*)=

*G*

*(*

_{r}*θ*), while a constant false alarm was used to detect the target against background noise and clutter. Table 1 presents the measured MDR results obtained for several radar module heights.

*R*′ were regarded as

*σ*and MDR

*R*, respectively. Meanwhile, the angle of direction to the target registered a maximum 4° difference depending on the radar height. Since automotive radars generally possess a narrow beam width at an elevation angle to achieve high antenna gain, the detectable range changes in relation to the radar module height can primarily be attributed to antenna gain in the direction of the target.

*E*is the far-field strength in V/m and

*η*represents the wave impedance. Furthermore, if the classic radar equation is effective in this situation, the received power scattered from the target located at the MDR can be formulated using (1) and (2), as follows:

*E*(

*R*

_{max},

*θ*)| must be identical at all MDRs, since

*P*

*and*

_{t}*σ*are also constant.

*E*(

*R*

_{max},

*θ*)| at the MDRs, only the target was removed from the setup presented in the actual experiment, after which the electric field from the radar antenna was simulated using the CST Microwave Studio. Fig. 6 illustrates the simulated electric field strength at the MDRs listed in Table 1. At the center frequency of 79 GHz, the electric field strength presents different values for the MDRs. This indicates that the classic radar equation is unsuitable for a large target. Therefore, a new radar equation is required for situations in which the receiving antenna is located in the near-field region of the target.

### III. Near-Field Radar Equation

### 1. Formulation

*T*

*and height*

_{w}*T*

*in an infinite ground plane and a plate of width*

_{h}*T*

*and height*

_{w}*T*

*in free space are complementary structures, as shown in Fig. 7(b). Therefore, the scattered field can be obtained from field*

_{h}*E*

_{0}on the aperture instead of the current density

*J*

*on the metal plate.*

_{s}*x*−

*x*′)

^{2}+ (

*z*−

*z*′)

^{2}<<

*y*

^{2}. Therefore, the distance from the origin of the aperture can be approximated as follows:

*E*

_{0}(

*x*′,

*z*′) in the aperture represents a part of the incident field from the transmitter. Therefore, from (2),

*E*

_{0}(

*x*′,

*z*′) can be expressed as:

*E*

_{0}(

*x*′,

*z*′) is uniformly distributed in the aperture, it becomes a constant value of

*E*

_{0}. In such a case, the scattered field can be formulated as follows:

*F*(

*z*) and parameter

*q*can be defined as:

*R*away from the aperture can be presented as follows:

*q*

*= (2/λ*

_{R}*R*)

^{0.5}. In this context, it should be noted that the derived near-field radar equation in (13) would be valid only when the distance between the observation point and the target is greater than the target dimension, i.e., in the Fresnel region, due to (4).

*R*from the plate at a height of 1.72 m, bearing a width of 0.55 m. As depicted in Fig. 8, the received power declines up to approximately 70 m, after which the square of distance begins to decrease stably until the fourth of the distance reaches about 400 m, which is closer to 0.6

*D*

^{2}/λ than 2

*D*

^{2}/λ. Notably, the results obtained for the near-field region contrast sharply with those of the classic radar equation in the far-field region. In addition, the near-field radar equation in (14) can be applied to both far-field and near-field regions. Notably, if the receiving antenna is located in the far-field region of the target, the equation in (14) will become the classic radar equation presented in (1). This transformation process is detailed in Appendix B.

### 2. Verification with Simulated and Experimental Data

*R*

_{max1}and

*R*

_{max2}can be expressed as follow:

##### (15)

*E*(

*R*

_{max},

*θ*)|

^{2}and |

*E*(

*R*

_{max1},

*θ*)|

^{2}·|

*F*(

*q*

_{R}_{max},

*T*

_{w}*/*2)|

^{2}the right side of the equation in (15), respectively.