## 2 Efficient Estimation of Residual OFDM BER for Small Cells
### Optimal Sample Delay for Least BER Floor  
The optimal sample time instant \(\tau_{\rm{opt}}\) which minimizes the BER floor, can be found by deriving Equation (8):  
\[\frac{d}{d\tau}\left(\sqrt{W^{-}}\cdot\sqrt{\left(\tau-\tau^{-}\right)^{2}}+ \sqrt{W^{+}}\cdot\sqrt{\left(\tau-\tau^{+}\right)^{2}}\right)=0 \tag{9}\]  
Tedious testing of the second derivation is not necessary to be presented here, as no maximal \(P(\mathrm{e})\) value exists, so the first derivation is sufficient with this regard. Furthermore, let us preselect the expected optimal sampling region: \(\tau_{i}(N^{-})<\tau<\tau_{i}(N^{-}+1)\) of the power delay profile where the \"borderline\" value of \(N^{-}\) between the \"\(-\)\" and \"\(+\)\" echoes mutually balances \(W^{-}\) and \(W^{+}\), and thus maximizes the numerator of the second term in Equations (1) and (6), and consequently minimizes BER. Moreover, if (without much loss of accuracy) we consider that both \(W^{-}\) and \(W^{+}\) are independent of \(\tau\), then after making derivation in Equation (9) we obtain:  
\[\sqrt{W^{-}}\cdot\frac{\overline{\tau-\tau^{-}}}{\sqrt{\left(\tau-\tau^{-} \right)^{2}}}+\sqrt{W^{+}}\cdot\frac{\overline{\tau-\tau^{+}}}{\sqrt{\left( \tau-\tau^{+}\right)^{2}}}=0 \tag{10}\]  
Solving the implicitly expressed Equation (10) provides the optimal sample delay \(\tau=\tau_{\rm{opt}}\). Moreover, considering balanced \"\(-\)\" and \"\(+\)\" rms delay spreads, both denominators in Equation (10) take mutually close values, so it reduces to:  
\[\sqrt{W^{-}}\cdot\left(\tau-\overline{\tau^{-}}\right)+\sqrt{W^{+}}\cdot\left( \tau-\overline{\tau^{+}}\right)\approx 0 \tag{11}\]  
enabling a simple approximate explicit expression for the optimal sample time:  
\[\tau_{\rm{opt}}\approx\frac{\sqrt{W^{-}}\cdot\overline{\tau^{-}}+\sqrt{W^{+} }\cdot\overline{\tau^{+}}}{\sqrt{W^{-}}+\sqrt{W^{+}}} \tag{12}\]As it is obvious from Equation (12), generally, the optimal sample delay is not at the mean delay:  
\[\overline{\tau}=\frac{W^{-}\cdot\overline{\tau^{-}}+W^{+}\cdot\overline{\tau^{+}} }{W^{-}+W^{+}}=W^{-}\cdot\overline{\tau^{-}}+W^{+}\cdot\overline{\tau^{+}}=\sum _{i=1}^{N}\left(A_{i}\right)^{2}\tau_{i} \tag{13}\]  
Specifically, for symmetrical delay profiles, such as, for example, the two-delay model [9], it is: \(W^{-}\approx W^{+}\approx 0.5\), and: \(\overline{\tau^{-}}=-\overline{\tau^{+}}\), implying that Equations (12) and (13) are identical in that the optimal sample delay is at the centerline of the power delay profile. Thereby, as the mean delay alike, \(\tau_{\text{opt}}\) is also explicit in Equation (12), and determined by interpretable, though somewhat modified, delay dispersion parameters (distinct for \"\(-\)\" and \"\(+\)\" echoes of the power delay profile). However, apart from the general BER floor expression in Equation (6) that is really rather complex, from the perspective of low-performance network end-points, even Equation (12) is still somewhat more demanding for implementation than Equation (13); specifically as sampling at the mean delay further simplifies Equation (8), so that the residual BER becomes proportional to delay variance:  
\[BER=\frac{k_{m}}{2\sqrt{\pi M}T_{\text{s}}}\cdot\sqrt{\left(\tau-\overline{ \tau}\right)^{2}}=\frac{k_{m}}{2\sqrt{\pi M}T_{\text{s}}}\cdot\sqrt{\text{var} (\tau)} \tag{14}\]  
This, as well the formal similarity between Equations (12) and (13), raise the question of near-optimality of the mean delay (as the simplest to \"tune-in\") (i.e., what will be the BER penalty for choosing the mean delay in Equation (13) instead of the optimal sample time in Equation (12). Looking analytically, we can comment that the weighting factors of \"\(-\)\" and \"\(+\)\" mean delays in Equation (12) are \"\(-\)\" and \"\(+\)\" normalized aggregate rms values of the profile impulses' amplitudes, whereas the powers themselves have the weighting role in Equation (13). This implies somewhat greater sensitivity of the mean delay on the imbalance between the \"\(-\)\" and \"\(+\)\" parameters, while in Equation (13) it is a bit smoother.