# A Matheuristic for Joint Optimal Power and Scheduling Assignment in DVB-T2 Networks +
## 2 DVB-T2 Network Design
To model the decision of setting the power emission of the transmitters and establishing the assignment of users to transmitters in the context of DVB-T2 networks, we define two families of decision variables:  
* a continuous decision variable \(p_{sd}\in[0,P_{\max}]\) that models the power emission of each transmitter \(s\in S\) in each direction \(d\in D\);
* a binary decision variable \(x_{ts}\in\{0,1\}\) modeling the assignment of a TP to a station and such that: \[x_{bs}=\left\{ \begin{array}{ll}1&\text{if $s\in S$ is the serving station of TP $t\in T$}\\ 0&\text{otherwise}\end{array} \right.\] for every TP \(t\in T\) and every \(s\in S\). Concerning the evaluation of service quality, the first consideration to be made is that every TP \(t\in T\) is able to pick up signals from all stations \(s\in S\) and the power \(P_{t}(s)\) that \(t\) obtains from \(s\) is equal to the power emission of \(s\) multiplied by a coefficient \(a_{ts}\in[0,1]\) (i.e., \(P_{t}(s)=a_{ts}\cdot p_{s\,\delta(s,t)}\)--note that here, we consider the specific direction \(\delta(s,t)\) on which \(s\) emits to \(t\)). The factor \(a_{ts}\) is commonly called _fading coefficient_ and summarizes the power reduction experienced by a signal that propagates from \(s\) to \(t\)[51]. We have written that a TP picks up signals from all stations. However, in the case of DVB-T networks, only a subset of signals is useful for guaranteeing the service, while all other signals are interfering and thus deteriorate the quality of service. Specifically, in DVB-T, a TP has to decide when to start a _time window_ for picking up signals: all signals received within the time window can be summed up and increase the quality of service obtained, whereas all signals received outside the window are interfering and decrease service quality. We refer the reader to [27; 52] for a detailed discussion of how the time window mechanism works. We note that the start of the time window associated with a TP can be positioned in a very high number of points along the time axis, depending on the time sensitivity of the DVB system. However, it is common to let the window start when the signal of a station is received [27; 32]. In the case of our optimization model, we follow this practice and thus the number of possible starts of the time window of a TP \(t\in T\) equals \(|S|\), i.e., the number of stations. We call _server_ or _serving station_ of \(t\), the station \(s\) whose signal reception is used as starting time of the time window of \(t\). For a given TP \(t\in T\) and serving station \(s\in S\) of \(t\), we denote by \(U(s,t)\subseteq S\) the subset of stations that emit useful signals for \(t\) and by \(I(s,t)\subseteq S\) the subset of interfering stations. These two subsets form a partition of the station set \(S\), i.e., \(S=U(s,t)\cup I(s,t)\) and \(U(s,t)\cap I(s,t)=\emptyset\). Once the subsets of useful and interfering signals are established, a TP \(t\) is considered served by station \(s\) if its Signal-to-Interference Ratio (SIR) is above a threshold \(\phi>0\). The SIR is the ratio of the sum of useful powers to the sum of interfering powers [51; 52]:  
\[SIR_{\text{IS}}(p)=\frac{\sum_{\sigma\in U(s,t)}a_{t\sigma}\cdot p_{\sigma\, \delta(\sigma,t)}}{N+\sum_{\sigma\in I(s,t)}a_{t\sigma}\cdot p_{\sigma\, \delta(\sigma,t)}}\ \geq\ \phi. \tag{1}\]In the previous inequality, \(N>0\) is the system noise. Every TP \(t\) that is covered with service generates a revenue \(r_{t}>0\) (typically, the higher the number of users located in \(t\) the higher \(r_{t}\)).