# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 4 Numerical Experiments and Performance Analysis
### Results and Discussion  
The simulation results show that coefficient clustering mechanism during impulse response reconstruction (in methods **R** and **G**) has a generally positive influence on the transmission quality. The reference classic LS estimate has the worst performance in terms of MSE and BER among tested methods, as might be expected, but this result was an entry point of those modifications. The DFT-based approach, by simply removing the noise part from the LS impulse response, clearly improves estimation and transmission quality. The gain is about 1.5-2.0 dB for BER = 0.01 in relation to LS method. However, this simplest solution (**C**) can be improved. Both authors' propositions (**R** and **G**) have the competitive quality measures in low and mid range of SNR (up to about 20 dB). Their performance results are graphically depicted in Figures 4-7. The performance of the proposed method **G** shows quality improvement comparable to the one existing between **C** and **N**. This is more apparent for the MSE measure. Looking at BER, the additional gain in regard to the **N** \begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel** & **Average Relative Paths Delays [ns]** & **Average Paths Gains [dB]** \\ \hline Indoor A [20] & [0, 50, 110, 170, 290, 310] & [0, \(-\)3, \(-\)10, \(-\)18, \(-\)26, \(-\)32] \\ Indoor B [20] & [0, 100, 200, 300, 500, 700] & [0, \(-\)3, \(-\)6, \(-\)7, \(-\)10.8, \(-\)18, \(-\)25.2] \\ Indoor B Shift & [50, 150, 250, 350, 450, 750] & [0, \(-\)3, \(-\)6, \(-\)7, \(-\)10.8, \(-\)18, \(-\)25.2] \\ Outdoor-To-Indoor A [20] & [0, 110, 190, 410] & [0, \(-\)9.7, \(-\)19.2, \(-\)22.8] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Chanel models. is about 0.2-0.3 dB, which translates into approximately 5-7 percent saving in transmission energy. More specific values for BER = 0.1 and 0.01 are listed in Table 4.  
Figure 4: MSE and BER measures for Indoor A channel model. Figure 5: MSE and BER measures for Indoor B channel model. Figure 6: MSE and BER measures for modified Indoor B channel model. The main part of the proposed method is a moving average filter with dynamic adjusted length \(U\). Table 5 depicts median values of \(U\) obtained during the simulations for the particular SNR and channel models. As transmission conditions improve, the filter length \(U\) increases. The maximum possible value is \(U_{max}=M\). In good transmission conditions (high SNR values), the estimation quality for \(\mathbf{G}\) and \(\mathbf{N}\) solutions are roughly similar. Because the \(\mathbf{N}\) method does not have to implement the additional averaging filter (detailed complexity issues are elaborated in next section), a noise-controlled automatic switching between these methods could be considered, but such a procedure is out of the scope of this study. Another interesting conclusion can be drawn comparing the simulation results between the proposed and \(\mathbf{R}\) methods. The estimation quality is almost the same in the entire tested SNR range. Moreover, the performance of \(\mathbf{R}\) can be slightly better for a specific channel, as presented in Table 6. Nevertheless, a decisive advantage of the proposed method is the lower computational complexity manifested in particular by the absence of multiplication operations. \begin{table}
\begin{tabular}{c c c c c} \hline
**BER** & **Indoor A** & **Indoor B** & **Indoor B Shift** & **Outdoor-to-Indoor A** \\ \hline
0.1 & 0.3 dB & 0.2 dB & 0.2 dB & 0.3 dB \\
0.01 & 0.1 dB & 0.2 dB & 0 dB & 0.2 dB \\ \hline \end{tabular}
\end{table} 
Table 4: SNR gap between \(\mathbf{N}\) and \(\mathbf{G}\) methods (\(SNR_{N}-SNR_{G}\)) for the particular BERs and channel models with a resolution 0.1 dB. \begin{table}
\begin{tabular}{c c c c c} \hline
**SNR** & **Indoor A** & **Indoor B** & **Indoor B Shift** & **Outdoor-to-Indoor A** \\ \hline
0 dB & 5 & 5 & 7 & 4 \\
5 dB & 6 & 5 & 9 & 4 \\
10 dB & 8 & 6 & 11 & 5 \\
15 dB & 11 & 7 & 14 & 6 \\
20 dB & 14 & 7 & 15 & 7 \\
25 dB & 15 & 8 & 17 & 8 \\ \hline \end{tabular}
\end{table}