*1.2. Contributions*

Our study presented four new main contributions on the subject of Massive MIMO pilot sequence allocation problem:


Furthermore, we present the differences between our work and those presented in the previous section:

• Two different mathematical models of the same practical problem were addressed;


## *1.3. Text Organization*

This paper is organized as follows. In Section 2, the system model is described with the mathematical problem. In Section 3, the solution with optimization is applied with the evolutionary algorithms. In Section 4, the best parameters are presented and the results of simulations are shown. Finally, in Section 5, we offer the conclusion and final considerations.

## **2. System Model**

We consider a Massive MIMO system with *L* > 1 cells using the same spectrum band and *K*- mobile terminals connected with its single base station which has *M* >> *K*- antennas. To acquire channel state information (CSI) at each coherence interval, each user in the system must send a pilot sequence through the uplink channel. The pilot sequence uplink signal received by the base station - is a *M* × *Tp* matrix where *Tp* is the pilot sequence size, and it may be described as:

> Desired signal from cell users 9:

$$\begin{array}{rcl} \mathbf{R}\_{\ell}^{\mathrm{ul}} &=& \overbrace{\sum\_{k=1}^{K\_{\ell}} \sqrt{p\_{k,\ell}}}^{K\_{\ell}} \overbrace{\mathbf{s}\_{k,\ell,\ell}}^{H} \mathbf{s}\_{k,\ell}^{H} + \\ &+ \underbrace{\sum\_{j=1}^{L} \sum\_{k'=1}^{K\_{j}} \sqrt{p\_{k',j}} \mathbf{g}\_{k',j,\ell} \mathbf{s}\_{k',j}^{H}}\_{\text{valid with}} + \eta\_{\ell} \end{array} \tag{1}$$

Pilot Contamination / Interference

where *k*, -, *k* and *j* are the user, cell of interest, interfering users and adjacent cell indexers, respectively. Moreover, **R***u*- ∈ C*M*×*Tp* , *pk*,- is the transmission power of each user, **<sup>s</sup>***k*,- ∈ C*Tp*×<sup>1</sup> is the pilot sequence, whereas (·)*<sup>H</sup>* is the Hermitian operator and is equivalent to the transposed complex conjugate, *η*- ∈ C*M*×*Tp* is the noise matrix whose elements are complex Gaussian random variables with zero mean and variance equal to *N*0 *B* where *N*0 is the noise power spectral density (The noise PSD (*N*0) is equal to the Boltzmann Constant times the temperature, i.e., it is equivalent to approximately 4.11 × 10−<sup>21</sup> watts per hertz at 25 degrees Celsius) and *B* the system bandwidth. Finally, **g***k*,*j*,- ∈ C*M*×<sup>1</sup> is the channel gain between the *k*-th user from cell *j* and the base station of cell -, which represents the large-scale fading (*β<sup>k</sup>*,*j*,-) and the small-scale fading (**h***k*,*j*,-), i.e.,:

$$\mathbf{g}\_{k,\mathbf{j},\ell} = \sqrt{\beta\_{k,\mathbf{j},\ell}} \,\mathbf{h}\_{k,\mathbf{j},\ell} \tag{2}$$

where **<sup>h</sup>***k*,*j*,- ∈ C*M*×<sup>1</sup> are independent and identically distributed complex Gaussian random variables with zero mean and unit variance. Meanwhile, *β<sup>k</sup>*,*j*,- is the path loss and shadowing effects. We assume a simplified path loss model, i.e.,:

$$\beta\_{k,\vec{j},\ell} = \left(\frac{\lambda}{4\pi d\_0}\right)^2 \left(\frac{d\_0}{d\_{k,\vec{j},\ell}}\right)^\gamma \mathcal{S}\_{k,\vec{j},\ell} \tag{3}$$

where *λ* is the wavelength, *d*0 is the reference distance (Typically, *d*0 ∈ [1, 10] meters for indoor environments and *d*0 ∈ (10, 100) meters for outdoors), *dk*,*j*,- is the distance (in meters) from the *k*-th user of cell *j* to the base station in cell -. Finally, *γ* ∈ [2, 8] is the path loss exponent which is directly related to the scenario where the wireless communications take place, and <sup>S</sup>*k*,*j*,- is the shadowing log-normal distributed random variable with zero

mean and variance 10 *σ*2*s* 10 where *σ*2*s* ∈ [4, 13] for outdoor channels.

We suppose the use of an orthogonal variable spreading factor (OVSF) code to generate the pilot sequences which are designated to each user through the pilot sequence allocation hipermatrix **Φ** ∈ {0, 1}*<sup>K</sup>*-×*Tp*×*<sup>L</sup>* whose elements are defined as:

$$
\phi\_{k,\eta,\ell} = \left\{ \begin{array}{ll}
0 & \text{pilot sequence is not allocated} \\
1 & \text{pilot sequence is allocated}
\end{array} \right\} \tag{4}
$$

Moreover, according to [5], the uplink signal to interference plus noise ratio (SINR) of user *k* from cell - observed by its base station when the number of antennas grows indefinitely is:

$$\delta\_{k,\ell}^{\mathbb{B}} = \frac{\beta\_{k,\ell,\ell}}{N\_0 \, B \, \big|\, \sum\_{\substack{j \neq \ell \\ j=1}}^{K\_j} \sum\_{k'=1}^{K\_j} \Phi\_{k,\ell} \Phi\_{k',j}^T \beta\_{k',j,\ell}}^{T} \tag{5}$$

where *φ<sup>k</sup>*,- is a row from the hipermatrix **Φ**, i.e., it is a {0, 1}<sup>1</sup>×*Tp* array that designates which pilot sequence is allocated to user *k* of cell -. Whenever *φ<sup>k</sup>*,*<sup>φ</sup>Tk*,*<sup>j</sup>* = 1, users *k* and *k* from cell - and *j*, respectively, use the same pilot sequence and, therefore, interfere in each others signal.
