# Cross-Layer Optimization for Heterogeneous MU-MIMO/OFDMA Networks
## 4 Optimization Framework
### Constraints on MU-MIMO and OFDMA Operation
#### 4.2.1 Number of Spatial Streams  
Let us first define \(s_{1 tf}\) as the number of spatial streams allocated to flow \(f\) on link \(l\) at time slot \(t\). We can easily see that this is derived using \(z_{1 tfs}\) as follows: \(s_{1 tf}=\sum_{s}s\cdot z_{1 tfs}\). In the case of MU-MIMO, it is clear that the total number of spatial streams on a link cannot exceed the maximum number of spatial streams available on that link, while for OFDMA case, \(s_{1 tf}\) is simply bounded at one. We can then combine these constraints into one, as follows:  
\[m_{1 tf}+o_{1 tf}\leq s_{1 tf}\leq S^{\text{max}}_{l}\cdot m_{1 tf}+o_{1 tf}, \forall l,t,f \tag{28}\]  
where \(S^{\text{max}}_{l}\) is the maximum number of spatial streams for link \(l\), and determined as \(\min(S_{i},S_{j})\), where \(i\) and \(j\) are the source and destination for link \(l\), respectively.  
We now discuss the constraints on \(s^{x}_{itk}\) and \(s^{y}_{itk}\). First, in the case of OFDMA, since the number of spatial streams is limited to one, the existing variables \(o^{x}_{itk}\) and \(o^{y}_{itk}\) can be used as they are to denote this. We will therefore regard the variables \(s^{x}_{itk}\) and \(s^{y}_{itk}\) here as the number of spatial streams when MU-MIMO is used. Then, from the constraint on the maximum number of spatial streams, we have:  
\[\sum_{l\in L^{\text{int}}_{i},f}z_{1 tfk}\leq s^{x}_{itk}\leq o^{x}_{itk}+S_{i} \cdot m^{x}_{itk}, \forall i,t,k \tag{29}\] \[\sum_{l\in L^{\text{int}}_{i},f}z_{1 tfk}\leq s^{y}_{itk}\leq o^{y}_{itk}+S_{i} \cdot m^{y}_{itk}, \forall i,t,k. \tag{30}\]  
To model the requirement that all spatial streams of the same MU-MIMO transmission share the same bandwidth, we use the idea that the number of spatial streams allocated to each subchannel should be the same: that is, \(s^{x}_{itk}=\sum_{l\in L^{\text{int}}_{i},f}s_{1 tf}\) and \(s^{y}_{itk}=\sum_{l\in L^{\text{in}}_{i},f}s_{1 tf}\). This gives the following constraints:  
\[-U(1-m^{x}_{itk})\leq s^{x}_{itk}-\sum_{l\in L^{\text{int}}_{i},f}s_{1 tf} \leq U(1-m^{x}_{itk}), \forall i,t,k \tag{31}\] \[-U(1-m^{y}_{itk})\leq s^{y}_{itk}-\sum_{l\in L^{\text{in}}_{i},f}s_{1 tf} \leq U(1-m^{y}_{itk}), \forall i,t,k \tag{32}\]  
where \(U\) is a large constant that ensures the constraints are valid. Please note that these constraints are relaxed in the case of joint MU-MIMO and OFDMA operation, which will be discussed later.  
We also need to consider the effect on interfering streams. In the case of OFDMA, interfering streams can only be handled by allocating different channels, while in the case of MU-MIMO, even interfering streams coming through the same channel can be handled, if there exist remaining DoFs:\[\sum_{j\in N_{i}^{\text{int}}}s_{jtk}^{\text{x}}\leq S_{i}\cdot m_{ itk}^{\text{y}}+U(1-m_{itk}^{\text{y}}), \forall i,t,k \tag{33}\] \[\sum_{j\in N_{i}^{\text{int}}}o_{jtk}^{\text{x}}\leq o_{itk}^{\text{ y}}+U(1-o_{itk}^{\text{y}}), \forall i,t,k. \tag{34}\]  
It is worth noting that \(U\) in the above constraints plays the same role as \(U\) in constraints (31) and (32), but their values do not have to be the same. Considering the other parameters given, such as \(N\) and \(L\), it is possible to determine the \(U\) values for each constraint, which in turn helps to reduce the search space. Since it is straightforward to find appropriate \(U\) values for each constraint, this part will be omitted for simplicity in this paper.