## 3 Proposed MDM-MIMO-IIR Photonic Radar Modeling
Figure 2: Radial intensity distribution profiles: (**a**) LG 20 and (**b**) LG 30. The parameters involved in the equation include the operating range denoted by \(R\), the beam divergence represented by \(\theta\), and the transmitter and receiver aperture diameters denoted by \(d_{T}\) and \(d_{R}\), respectively. The impact of atmospheric conditions is also taken into account, denoted by \(\alpha\). In the case of autonomous vehicles (AVs), fog and rain are important factors that can limit the detection and ranging capabilities. The attenuation values used in this study adhere to the International Visibility codes [24]. For instance, clear weather is assigned 0.14 dB/km, while light rain is assigned 2.5 dB/km, with 12.5 dB/km for strong rain, 25 dB/km for low fog, 50 dB/km for medium fog, and 75 dB/km for dense fog. The implementation of Multiple-Input Multiple-Output (MIMO) technology in the receiver section of photonic radar systems can enhance their overall performance. A 2 \(\times\) 2 MIMO scheme can be employed, which utilizes two transmitter antennas and two receiver antennas. This configuration offers improved resolution and accuracy, enabling the radar system to measure the location and velocity of targets with greater precision. By simultaneously transmitting two signals using two transmitter antennas, the system can obtain more detailed information about the target. In addition, the use of two receiver antennas allows the system to receive signals from various directions, minimizing the effects of interference and improving the sensitivity of the radar system. Furthermore, the incorporation of 2 \(\times\) 2 MIMO can enhance the radar system's ability to detect and track multiple targets simultaneously, even when they are in close proximity or moving in different directions. The system can differentiate signals from different targets and track them individually by leveraging multiple antennas to process the signals. Within the transmitter, the modulated output signal is split into two segments. The first segment is transmitted through the channel for detection, while the second segment is utilized as the local oscillator signal in the receiver section. The power of the received echo signal, denoted as \(P_{r}\), can be calculated using Equation (3) [22; 23]:  
\[P_{r}=\left\{ \begin{array}{ll}P_{t}\frac{\rho_{t}D^{2}\kappa_{opt}t_{4m}^{2} }{4R^{2}}&\text{for extended target}\\ P_{t}\frac{\rho_{t}A_{t}D^{2}\kappa_{opt}t_{4m}^{2}}{4R^{2}A_{4 ill}}&\text{ for any target}\end{array} \right. \tag{3}\]  
where the surface reflectivity of the target is \(\rho_{t}\), the size of the irradiated zone at the target is \(A_{ill}\), the range of the target from the FMCW photonic radar is \(R\), the diameter of the receiver aperture is \(D\), the area of the target is \(A_{t}\), the atmospheric loss factor is \(\tau_{atm}\), and the optical domain loss in transmission is \(\tau_{opt}\). When a signal is reflected off a moving target at a range distance \(R\) and velocity \(v\), it experiences a Doppler shift given by \(f_{d}\) = 2\(v/\lambda\), where \(\lambda\) is the wavelength of the signal. The reflected signal undergoes a delay, denoted by \(\tau\), which is equal to twice the distance \(R\) to the target divided by the speed of light \(c\). This delay is represented by the equation \(\tau\) = 2\(R\)/\(c\). This echoed signal, with both Doppler shift and delay, is collected by the telescope. The strength of the reflected signal \(E_{ref}(t)\) can be expressed using Equation (4) [22]:  
\[E_{ref}(t)=\sqrt{P_{r}}\cos(2\pi f_{start}(t-\tau))+\frac{\pi B}{T_{m}}(t- \tau)^{2}.e^{(j(w_{o}-w_{d})t+\theta_{o(t)})} \tag{4}\]  
To mix the local oscillator signal \(E_{lo}(t)\) with the echo signal power \(E_{ref}(t)\), a balanced photodetector is employed, along with an ideal 3 dB optical coupler. The optical coupler functions by multiplying the input electrical signals, allowing for the frequency conversion and coherent detection of the received signals.