# Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches
## 2 System Model
\right\} \tag{4}\]  
Moreover, according to [5], the uplink signal to interference plus noise ratio (SINR) of user \(k\) from cell \(\ell\) observed by its base station when the number of antennas grows indefinitely is:  
\[\delta^{\text{B}}_{k,\ell}=\frac{\beta_{k,\ell,\ell}}{N_{0}\,B+\sum\limits_{j =1}^{K_{j}}\sum\limits_{k^{\prime}=1}^{K_{j}}\Phi_{k,\ell}\boldsymbol{\phi}_ {k^{\prime},j}^{T}\,\beta_{k^{\prime},j,\ell}} \tag{5}\]  
where \(\boldsymbol{\phi}_{k,\ell}\) is a row from the hipermatrix \(\boldsymbol{\Phi}\), i.e., it is a \(\{0,1\}^{1\times T_{p}}\) array that designates which pilot sequence is allocated to user \(k\) of cell \(\ell\). Whenever \(\boldsymbol{\phi}_{k,\ell}\boldsymbol{\phi}_{k^{\prime},j}^{T}=1\), users \(k\) and \(k^{\prime}\) from cell \(\ell\) and \(j\), respectively,  
the binary allocation matrix representation found in Equation (5) and is mathematically defined as:  
\[\underset{\boldsymbol{\Phi}}{\text{maximize}} \mathcal{J}_{1}(\boldsymbol{\Phi})\;=\;\sum_{\ell=1}^{L}\sum_{k=1}^ {K_{\ell}}\log_{2}\!\left(1+\delta_{k,\ell}^{\text{B}}\right)\] (9) subject to \[\sum_{q=1}^{T_{p}}\phi_{k,q,\ell}\leq 1,\;\forall\;k\;\text{and}\;\;\ell \tag{10}\] \[\sum_{k=1}^{K_{\ell}}\phi_{k,q,\ell}\!=1,\;\forall\;q\;\text{and} \;\;\ell\] (11) \[\phi_{k,q,\ell}\in\{0,1\},\;\forall k,\;q,\;\ell \tag{12}\]  
where Equation (10) assures that no more than one pilot sequence is assigned to each user, while Equation (10) guarantees that each pilot sequence is used by only one user in each cell, and Equation (12) imposes that the decision variable is binary. Meanwhile, optimization problem 2 (OP2), as in Equation (13), uses the integer index representation introduced in (8) and may be described as:  
\[\underset{\boldsymbol{\Theta}}{\text{maximize}} \mathcal{J}_{2}(\boldsymbol{\Theta})\;=\;\sum_{\ell=1}^{L}\sum_{k= 1}^{K_{\ell}}\log_{2}\!\left(\delta_{k,\ell}^{\text{Z}}\right)\] (13) subject to \[\theta_{\ell,k}\
eq\theta_{\ell,k},\;\forall\;\ell,\;k\;\text{and }k^{\prime} \tag{14}\] \[\theta_{\ell,k}\in\{1,\ldots,T_{p}\},\;\forall\;\ell\;\text{and}\;k \tag{15}\]  
where Equation (10) assures that the same pilot sequence is used more than one time within a cell, avoiding intra-cell interference, while Equation (11) assures that only the available pilot sequences are allocated to each user in the cell. Note that the constraints in OP2 are an integer adaptation of the same constraints in OP1. To solve the optimization problem in (9), we use two different metaheuristic approaches. First, the binary version of the Particle Swarm Optimization (PSO) algorithm is considered. Later, a binary version of the Genetic Algorithm (GA) is also used to achieve a centralized pilot sequence allocation. Furthermore, we use the Smallest Position Value (SPV) technique along the PSO algorithm and Variable Neighbourhood Search (VNS) to solve the second optimization problem in (13). It is worth noting that although (9) and (13) have different domains, they represent the same physical problem and the conversion from one domain to another follows the rule presented by Equation (6).