# Simulation Framework for Detection and Localization in Integrated Sensing and Communication Systems
## 3 Main Elements of Abstraction: Channel and SNR Generation  
The channel generation leverages the Stochastic Cluttered Environment (SCE) model from [7], where scatterers are distributed randomly throughout the scenario and multiple signal hops are considered when generating the desired channels. Once the scatterers are located, the model facilitates the computation of channel matrices between various points within the system. For example, it models a Base Station (BS) functioning simultaneously as a transmitter and receiver, facilitating monostatic sensing applications. Additionally, it allows for the previous BS to serve as the transmitter and a distinct point to act as the receiver, thereby obtaining the communication channel between the two points. In this paper, we consider a monostatic Orthogonal Frequency Division Multiplexing (OFDM) ISAC system, which is composed of a BS, a target, and a set of scatterers generated as in [7]. Figure 2 shows a representation of this system for the two possible channels: (a) the sensing channel and (b) the communication channel. The Multiple Path Components (MPCs) of both channels are represented as arrows. Round-trip red arrows represent target-involved MPCs; black arrows depict other MPCs. The black round-trip and one-way arrows in Figure 2b enclose the inherent MPCs for the communication-only channel. We assume that the BS uses \(L\) transmitted OFDM symbols over \(K\) sub-carriers to sense the same scenario as the one represented in Figure 2a. Let \(M\) be the number of antennas at the BS. The received sensing signal at the BS after sensing the target and scatterers, for the \(l\)-th OFDM symbol and the \(k\)-th subcarrier, is  
\[\mathrm{y}_{l}^{\mathrm{sen}}(k)=\mathrm{H}_{\mathrm{sen}}(k)\mathrm{s}_{l}(k )+\mathrm{z}_{l}(k), \tag{1}\]  
where \(\mathrm{H}_{\mathrm{sen}}(k)\in\mathbb{C}^{M\times M}\) is the sensing channel matrix, \(\mathrm{s}_{l}(k)\in\mathbb{C}^{M}\) is the transmitted symbol vector, and \(\mathrm{z}_{l}(k)\in\mathbb{C}^{M}\) is the noise vector. Figure 2: System model for (**a**) the monostatic sensing channel from the BS, and (**b**) the BS–target communication channel if the target acts as a receiver. Arrows represent direct links. The red arrows in the sensing channel represent the links involving the target. To abstract the performance of the sensing applications, we need to use some measure of the channel quality that serves as a reliable indicator of this performance. In conventional communication systems, such measures are typically the SNR of the received signal. As depicted in Figure 2a, where the received signal of interest in (1) results from the sensing system, it is affected by all MPCs, even those not involved with the target (black arrows). Therefore, considering an SNR of the received sensing signal requires that the signal term account for the effect of all those MPCs. The primary challenge with this type of SNR is that certain MPCs devoid of interactions with the target may substantially contribute to the total signal power. For instance, scatterers near the BS can reflect a significant portion of the transmitted power. Consequently, the power of the sensing signal in (1) is not a good indicator of the performance of sensing applications. Alternative measures of channel quality can be explored to overcome this problem. For example, by assuming that the target behaves like a conventional User Equipment (UE) in a communication system, it becomes feasible to focus on the communication channel between the BS and UE (target). In this scenario, the system model depicting this channel is illustrated in Figure 2b. In the context of the communication channel, all MPCs terminate at the target, thereby ensuring interaction with the target for all components.