# 6G Goal-Oriented Communications: How to Coexist with Legacy Systems?
## 3 Coexistence of a Goal-Oriented and a Data-Oriented Communication System
### System Setup  
The system of interest is illustrated in Figure 1. A GO transmitter-receiver pair (which we, in short, term after as GO system), consisting of a single-antenna UE, denoted as \(UE_{g}\) and referred to as _GO user_, and an AP, denoted as \(AP_{g}\) and referred to as _GO AP_, equipped with \(M_{g}\) antennas, fully shares the available radio frequency spectrum of bandwidth \(W\) (Hz) with a DO transmitter-receiver pair (which we, in short, term after as DO system) of the same characteristics, i.e., a single-antenna UE, denoted as \(UE_{d}\) and referred to as _DO user_ and an AP, denoted as \(AP_{d}\) and referred to as _DO AP_, equipped with \(M_{d}\) antenna elements. It is assumed that \(AP_{g}\) and \(AP_{d}\) are interconnected via a backhaul link characterized by high capacity and low delay. In addition to radio infrastructure, it is assumed that an MEH is collocated with \(AP_{g}\), where the interconnection delay can be considered as negligible. Generalizing this would be straightforward and is left for future investigations.  
Uplink data communication takes place for both systems during operation time. For the GO system, uplink data communication is carried out to offload a batch of data patterns (e.g., images, either received or generated at \(U\!E_{g}\)) to the MEH to obtain classification information as output by means of a downlink transmission (whose delay is considered negligible in this work). In contrast to the GO system, the aim of DO system communication is for \(U\!E_{d}\) to provide an incoming data stream, e.g., a video feed, to \(AP_{d}\). Storage and distribution of this content are not investigated in this paper and are left for future investigations.  
At the physical layer, we consider an uplink Single-Input Multiple-Output (SIMO) system. We denote by \(\mathbf{h}_{ij,t}\in\mathbb{C}^{M\times 1}\) the uplink channel vector between \(U\!E_{j}\) (with \(j=g,d\)) and \(AP_{i}\) (with \(i=g,d\)) at time slot \(t\), which can be written as  
\[\mathbf{h}_{ij,t}=\frac{\sqrt{\beta_{ij,t}^{-1}}}{\sqrt{(K+1)}}\Big{(}\sqrt{K }\mathbf{h}_{ij,t,\text{LOS}}+\mathbf{h}_{ij,t,\text{NLOS}}\Big{)}, \tag{6}\]  
where the \(t\)-th component of the Line-Of-Sight (LOS) component vector reads as \(\mathbf{h}_{ij,\text{LOS}}(t)=e^{-\frac{j\pi d_{ij,t}}{\lambda}}\) with \(\lambda\) the wavelength, \(d_{mij,t}\) the distance from \(U\!E_{j}\) single antenna to the \(m\)-th antenna of \(AP_{i}\), while the Non-LOS (NLOS) component follows a circularly symmetric Gaussian distribution, i.e., \(\mathbf{h}_{ij,t,\text{NLOS}}\sim\mathcal{CN}(0,1)\). Path loss between \(U\!E_{j}\) and \(AP_{i}\) is represented by \(\beta_{ij,t}\), while \(K\) is the Rician factor, strongly dependent on the scenario (e.g., indoor, outdoor, etc.).  
In this reference setting, both system communication and computing resources are involved. From a computation point of view, we assume an MEH hosting a relevant ML model (e.g., a Deep Neural Network--DNN, that is assumed to be already trained), which dynamically assigns resources to the GO system, based on its current availability. Therefore, on a per-slot basis, the MEH communicates with \(AP_{g}\) to select an inferencing pattern offloading policy, based on current connect-compute conditions, and based on the goal cost and goal effectiveness definition.