## 8 Conclusions  
In this paper, we have proposed the regularization factor for recursive adaptive estimation. The regularization factor needs no prior knowledge of the true channel impulse response. We have also reformulated the recursive estimation algorithm as \(l_{1}\)-RTLS type. This formulation is robust to the uncertainty in the regularization factor without a priori knowledge of the true channel impulse response. Simulations show that the proposed regularization factor and \(l_{1}\)-RTLS algorithm provide good performance comparable to \(l_{1}\)- RLS with the knowledge of the true channel impulse response.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sparsity (S)** & **Algorithm** & **MSD** \\ \hline \multirow{4}{*}{4} & \(l_{1}\)-RLS with the true channel & \(-40.6\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-37.8\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) method & \(-38.5\) dB \\ \cline{2-3}  & Oracle-RLS & \(-50.4\) dB \\ \hline \multirow{4}{*}{8} & \(l_{1}\)-RLS with the true channel & \(-39.5\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-28.4\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) method & \(-38.5\) dB \\ \cline{2-3}  & Oracle-RLS & \(-46.9\) dB \\ \hline \multirow{4}{*}{16} & \(l_{1}\)-RLS with the true channel & \(-38.4\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-18.2\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) method & \(-37.6\) dB \\ \cline{2-3}  & Oracle-RLS & \(-43.6\) dB \\ \hline \multirow{4}{*}{32} & \(l_{1}\)-RLS with the true channel & \(-37.6\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-9.1\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) method & \(-37.3\) dB \\ \cline{2-3}  & Oracle-RLS & \(-40.6\) dB \\ \hline \hline \end{tabular}
\end{table}
Table 2: MSD (mean square deviation) comparison.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reverberation Time (T60)** & **Algorithm** & **MSD** \\ \hline \multirow{4}{*}{100 ms} & \(l_{1}\)-RLS with the true channel & \(-38.5\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-34.7\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) & \(-35.4\) dB \\ \cline{2-3}  & Oracle-RLS & \(-45.3\) dB \\ \hline \multirow{4}{*}{400 ms} & \(l_{1}\)-RLS with the true channel & \(-32.1\) dB \\ \cline{2-3}  & \(l_{1}\)-RLS with the new \(\rho\) method & \(-20.9\) dB \\ \cline{2-3}  & proposed \(l_{1}\)-RTLS with the new \(\rho\) method & \(-30.1\) dB \\ \cline{2-3}  & Oracle-RLS & \(-36.0\) dB \\ \hline \hline \end{tabular}
\end{table}
Table 3: MSD (mean square deviation) comparison in sparse RIR estimations.  
**Author Contributions:** Conceptualization, J.L.; Methodology, J.L.; Validation, J.L. and S.L.; Formal analysis, J.L.; Investigation, J.L. and S.L.; Writing--original draft preparation, J.L.; Writing--review and editing, S.L.; Visualization, J.L.; Project administration, J.L.; Funding acquisition, J.L. and S.L.  
**Funding:** This research received no external funding.  
**Acknowledgments:** This research was supported by Agency for Defense Development (ADD) in Korea (UD160015 DD).  
**Conflicts of Interest:** The authors declare no conflict of interest.