*Alternative Representation*

An alternative to the allocation variable binary representation is to use an integer representation, i.e., instead of using a binary pilot sequence allocation hipermatrix for the set of cells, one may use a single integer pilot sequence allocation **Θ** ∈ 11, . . . , *Tp*2*<sup>L</sup>*×*<sup>K</sup>* such that:

$$
\theta\_{\ell,k} = \mathbf{x} \tag{6}
$$

if the pilot sequence number *x* ∈ {0, . . . , *Tp*} is allocated to user *k* of cell -.

In this representation, we introduce a new function which is responsible to verify if user *k* from cell *j* is using the same pilot sequence as user *k* from cell -. Therefore, consider *<sup>F</sup>*(*k*, -, *k*, *j*) : Z*K*×*L*×*K*×*<sup>L</sup>* → {0, 1} such that:

$$F(k, \ell, k', j) = \begin{cases} \ 0, & \text{if } \theta\_{\ell, k} \neq \theta\_{j, k'} \\ 1, & \text{otherwise} \end{cases} \tag{7}$$

Furthermore, we may rewrite Equation (5) using (7):

$$\delta\_{k,\ell}^{\mathbb{Z}} = \frac{\beta\_{k,\ell,\ell}}{N\_0 \, B \, \big|\, \begin{array}{c} \mathcal{S}\_{k,\ell,\ell} \\ \sum\_{j \neq \ell} \sum\_{k'=1}^{K\_j} F(k,\ell,k',j) \, \upbeta\_{k',j,\ell} \end{array}}{\text{(8)}} \tag{8}$$

It is clear from Equations (5) and (8) that we may derive two equivalent optimization problems which are described in the next section.

## **3. Optimization Problems**

The pilot sequence allocation optimization problem discussed here aims to maximize the total system spectral efficiency, simultaneously satisfying the constraints of pilot sequence orthogonality within the same cell as well as the fact that no users should be assigned different pilot sequences at the same time.

Moreover, it is important to differentiate the two mathematical models of allocation into two optimization problems: one optimization problem (OP1), as in Equation (9), uses the binary allocation matrix representation found in Equation (5) and is mathematically defined as:

$$\underset{\Phi}{\text{maximize}}\qquad\mathcal{J}\_1(\Phi) \;= \sum\_{\ell=1}^{L} \sum\_{k=1}^{K\_\ell} \log\_2 \left(1 + \delta\_{k,\ell}^{\mathbb{B}}\right) \tag{9}$$

$$\text{subject to } \sum\_{q=1}^{T\_p} \phi\_{k,q,\ell} \le 1, \forall k \text{ and } \ell \tag{10}$$

$$\sum\_{k=1}^{K\_{\ell}} \phi\_{k,q,\ell} = 1 \; \forall \; q \; \text{and} \; \ell \tag{11}$$

$$
\phi\_{k,q,\ell} \in \{0,1\}, \forall \, k, \, q, \, \ell \tag{12}
$$

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:

$$\underset{\mathbf{O}}{\text{maximize}} \qquad \mathcal{J}\_2(\mathbf{O}) \quad = \sum\_{\ell=1}^{L} \sum\_{k=1}^{K\_\ell} \log\_2 \left( \delta\_{k,\ell}^{\mathbb{Z}} \right) \tag{13}$$

$$\text{subject to} \qquad \theta\_{\ell,k} \neq \theta\_{\ell,k'} \; \forall \; \ell, k \; \text{and} \; k' \tag{14}$$

$$\theta\_{\ell,k} \in \{1, \ldots, T\_p\}, \; \forall \; \ell \text{ and } k \tag{15}$$

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).
