# BER Aided Energy and Spectral Efficiency Estimation in a Heterogeneous Network
## 2 Analysis
### Spectral and Energy Efficiency Model  
The tiers are sorted in ascending order according to the access points' density: \(\lambda_{1}\leq\lambda_{2}\ldots\lambda_{k-1}\leq\lambda_{k}\). For a certain \(\lambda_{k}\), the count of tier \(k_{i}\) (\(i\) = 1,2,...,\(N_{T}\)) access points within the covered area \(\mathcal{A}\) [m\({}^{2}\)] is a Poisson random variable with mean value of \(\mathcal{A}\cdot\lambda_{k}\), being independent of other tiers. Furthermore, all \(k\)-tier access points transmit with power \(P_{k}\). Each downlink is modeled by Rayleigh fading channel, with the BS-transmitted power \(P_{i}^{tx}\) and the UE-received power \(P_{i}^{rx}\) at \(R_{i}\) distance from BS. In this model, we have chosen the path-loss exponent to be equal to 4 [2], while macro BSs do not transmit during the Almost Blank Subframes (ABS) [3]. For each tier, we consider the frequency reuse factor of unity, and the RF band of one channel skipped between the two same-standard tiers, which implies that for a particular UE being connected to tier \(k\), all of the interfering BSs are within that tier (\(k\)), with the exception of the serving one. In the considered scenario, each UE is granted access only to the specific BSs in tiers 1, 2,..., \(K_{open}\) from Open Access (OA) macro-/femtocells, while the Closed Subscriber Group  
Figure 5: Graphical illustration of AWGN to non-AWGN trade-off for 16 QAM modulation. (CSG) femtocells normally do not provide service to the considered users [8]. So, a certain HetNet is represented by the counts of tiers: \(N_{T}=3\) and OA tiers: \(N_{open}=2\), respectively, where tier 1, tier 2, and tier 3 represent the macro cells, the OA femtocells, and the CSG femtocells, respectively. Moreover, we presume the maximal allowed BS-transmitted power (for the actual tier). Now, let us analyze the above-explored relation of the network \(SE\) to the total power, so that the distribution of BSs within each tier follows the PPP model. In addition, we suppose that any particular BS \(b_{k}\) of a serving tier \(k_{i}\) transmits only the users' subset \(U_{b}\) served by \(b_{k}\in\Phi_{k}\). Now, consider the SINR \(\Gamma(u_{b})\) for the specific user \(u_{b}\in U_{b}\), expressed by \(BER\), according to Equation (5). Then, the spectral efficiency \(SE_{k}\) of the link from \(b_{k}\) to any target \(u_{b}\) is:  
\[SE_{k}=\mathrm{E}\{\log_{2}\cdot[1+\Gamma(u_{b})]\}\approx\mathrm{E}\Bigg{\{} \log_{2}\cdot\Bigg{[}1+\frac{M-1}{3}\cdot\Bigg{(}Q^{-1}\bigg{(}\frac{BER\cdot \log_{2}M}{4}\bigg{)}^{2}\Bigg{)}\Bigg{]}\Bigg{\}} \tag{11}\]  
\[b\in\Phi,\ P\{U_{b}=u_{b}\}=\frac{1}{|U_{b}|},u_{b}\in U_{b}\]  
The analytical model derived here presents the spectral efficiencies \(SE_{k}\) and \(SE_{\text{TOT}}\) for individual tiers (\(k=1\ldots N_{T}\)) and for the whole HetNet, respectively. Furthermore, the selection of serving or candidate-serving cells according to the LTE-A standard is mostly centered around the picocell BSs range extension that enables traffic load balancing, preventing inter-cell radio interference in those areas with evident or expected signal overlapping coverage [13]. The mean levels of the UE-received pilot, originating by the candidate-serving macro and pico BSs, were used for selecting the optimal small-cell tier, which is to serve a particular UE, following two schemes:  
Firstly, we consider the macro tier \(i\) to be the serving tier, and the pico tier \(j\) to be the candidate-serving tier, otherwise it is the pico tier \(j\) to serve the UE, whereas the macro tier \(i\) is the candidate-serving tier [8]. In the following, with \(R_{i}\) and \(R_{j}\), we denoted the distances of the UE to the candidate-serving (i.e., the nearest) macro BS and the femto BS, respectively. As we plan to simply model the HetNet \(SE\), we adopt that the power of the instantaneous transmitted signal of any macro BS is considered a random variable close to zero during the ABS state or to \(P_{1}^{\text{tx}}\) otherwise.