## 2 Efficient Estimation of Residual OFDM BER for Small Cells  
### General Error Floor Due to Time Dispersion  
Let us first review the OFDM BER floor for propagation environments ranging from indoor to small-cell outdoor, where mostly strong signals and thus large SNRs can be expected in various 4G/5G scenarios for residual channel estimation. Consider an OFDM symbol \(\sum\limits_{m=1}^{M}s_{m}e^{j\varphi_{m}}\cdot e^{im\cdot\frac{2\pi}{Mt_{S}} \tau}\) at an arbitrary sampling instant \(\tau\), as comprising \(M\) original symbols \(s_{m}e^{j\varphi_{m}}\) of duration \(T_{\text{S}}\) each. On the other hand, the multipath channel is represented by the sum of complex delta functions, each with a Rayleigh-weighted amplitude \(A_{i}\), uniformly distributed phase, and certain delay \(\tau_{i}\), \(i\in[1,2,\ldots,N]\). Then the OFDM BER floor for the gray-mapped \(m\)-quadrature amplitude modulation (\(m\)-QAM) formats applied in 4G/5G is [12]:  
\[ \begin{array}{l}BER=\ \frac{k_{m}}{2\sqrt{n}}\cdot\left(\sqrt{W^{-}E\left[ \left(\frac{\Delta\tau^{-}_{i}}{\tau_{s}}\right)^{2}\cdot\Delta s^{2}_{n/n+1} \right]}+\sqrt{W^{+}E\left[\left(\frac{\Delta\tau^{+}_{i}}{\tau_{s}}\right)^{2} \cdot\Delta s^{2}_{n-1/n}\right]}-\frac{\sqrt{W^{-}E\left[\left(\frac{\Delta \tau^{-}_{i}}{\tau_{s}}\right)^{2}\cdot\Delta s^{2}_{n/n+1}\right]}\sqrt{W^{-}E \left[\left(\frac{\Delta\tau^{+}_{i}}{\tau_{s}}\right)^{2}\cdot\Delta s^{2}_{n/n +1}\right]}}{\sqrt{W^{-}E\left[\left(\frac{\Delta\tau^{-}_{i}}{\tau_{s}}\right)^ {2}\cdot\Delta s^{2}_{n/n+1}\right]+\sqrt{W^{+}E\left[\left(\frac{\Delta\tau^{+ }_{i}}{\tau_{s}}\right)^{2}\cdot\Delta s^{2}_{n-1/n}\right]}}}\right)\\ k_{m}=\left\{\begin{array}{lcl}1.000;&m=4&\text{for}&4-\text{QAM}\\ 0.750;&m=16&\text{for}&16-\text{QAM}\\ 0.583\ ;&m=64&\text{for}&64-\text{QAM}\end{array}\right.\end{array}  \tag{1}\]  
where \(\Delta\tau^{-}_{i}\) and \(\Delta\tau^{+}_{i}\):  
\[\Delta\tau^{-}_{i}=\tau-\tau^{-}_{i}\quad;\quad\Delta\tau^{+}_{i}=\tau-\tau^{+ }_{i} \tag{2}\]  
are relative delays of the channel response impulses, with respect to \(\tau\) that distinguishes the preceding (\"\(-\)\") from the delayed (\"\(+\)\") multipath echoes. Furthermore, the normalized \"\(-\)\" and \"\(+\)\" rms delay spreads in Equation (1) are defined as:  
\[\sqrt{E\left[\left(\frac{\Delta\tau^{-}}{T_{s}}\right)^{2}\right]}=\sqrt{ \frac{\sum\limits_{i=1}^{N}\left(A^{-}_{i}\right)^{2}\left(\frac{\Delta\tau^{ -}_{i}}{T_{s}}\right)^{2}}{\sum\limits_{i=1}^{N}\left(A^{-}_{i}\right)^{2}}}; \sqrt{E\left[\left(\frac{\Delta\tau^{+}}{T_{s}}\right)^{2}\right]}=\sqrt{ \frac{\sum\limits_{i=N^{-}+1}^{N}\left(A^{+}_{i}\right)^{2}\left(\frac{\Delta \tau^{+}_{i}}{T_{s}}\right)^{2}}{\sum\limits_{i=1}^{N^{-}}\left(A^{+}_{i} \right)^{2}}} \tag{3}\]  
respectively, whereas the mean aggregate \"\(-\)\" and \"\(+\)\" echoes' powers, relative to the total mean power, \(W=\sum\limits_{i=1}^{N}A^{2}_{i}=1\), are:  
\[W^{-}=\sum\limits_{i=1}^{N^{-}}\left(A^{-}_{i}\right)^{2},\qquad W^{+}=\sum \limits_{i=N^{-}+1}^{N}\left(A^{+}_{i}\right)^{2};\qquad W^{-}+W^{+}=1 \tag{4}\]  
respectively, where \(N^{-}\) denotes the count of preceding echoes (out of \(N\) overall) that, timewise, corresponds to the sampling instance. Moreover, the variances of differences between the \(n\)-th and the (\(n\)+1)-th OFDM symbol in the observed data sequence, and between the (\(n\)-1)-th and the \(n\)-th OFDM symbol \(E\left[\Delta s^{2}_{n/n+1}\right]\) and \(E\left[\Delta s^{2}_{n-1/n}\right]\), respectively [12]:  
\[E\left[\Delta s^{2}_{n/n+1}\right]=\sum\limits_{m=1}^{M}\frac{s^{2}_{m,n}+s^{2 }_{m,n+1}}{M^{2}}\quad,\quad\quad E\left[\Delta s^{2}_{n-1/n}\right]=\sum \limits_{m=1}^{M}\frac{s^{2}_{m,n-1}+s^{2}_{m,n}}{M^{2}} \tag{5}\]  
can reasonably be expected to be small, as the difference between neighboring OFDM symbols does not change fast; rather it can be regarded as almost constant for the observed (\(n\)-indexed) triplet of symbols.