## 7 Simulation Results  
This section confirms the performance of the proposed algorithm in sparse channel estimation. In the first experiment, the channel estimation performance is compared with other algorithms using randomly generated sparse channels. In this simulation, we follow the same scenario in the experiments as [17]. The true system vector \(\mathbf{w}_{true}\) is 64 dimensions. In order to generate the sparse channel, we set the number of the nonzero coefficients, S, in the 64 coefficients and randomly position the nonzero coefficients. The values of the coefficients are taken from an \(N(0,1/\text{S})\) distribution, where \(N(\ )\) is the normal distribution. In the simulation, we estimate the channel impulse response by the proposed algorithms that are \(l_{1}\)-RLS using the \(\rho\) in Table 1 and \(l_{1}\)-RTLS using the \(\rho\) in Table 1. For the comparison, we estimate the channel impulse response by \(l_{1}\)-RLS using the true channel response; in addition, we also execute the regular RLS algorithm in an oracle setting (oracle-RLS) where the positions of the true nonzero system parameters are assumed to be known. For the estimated channel results, we calculate the mean standard deviation (MSD), where \(\text{MSD}=E\Big{(}|\hat{\mathbf{w}}-\mathbf{w}_{true}|^{2}\Big{)}\), \(\hat{\mathbf{w}}\) is the estimated channel response and \(\mathbf{w}_{true}\) is the true channel response. For the performance evaluation, we simulate the algorithms in the sparse channels for S = 4, 8, 16, and 32. Figure 3 illustrates the MSD curves. For S = 4, Figure 3a shows that the estimation performance of \(l_{1}\)-RTLS using the regularization factor with the \(\rho\) in Table 1 is almost the same as the \(l_{1}\)-RLS using regularization with a true channel impulse response. However, the performance of \(l_{1}\)-RLS using the regularization factor with the \(\rho\) in Table 1 is gradually degraded and shows a kind of uncertainty accumulation effect. In the other cases of S, we can observe the same trend in the MSD curves. Therefore, we can confirm that the new regularization factor selection method and the new modeling for \(l_{1}\)-RLS can estimate the sparse channel as good as \(l_{1}\)-RLS using the regularization with the true channel impulse response. In all the simulation scenarios, oracle RLS algorithm produces the lowest MSD as expected. Table 2 summarizes the steady-state MSD values as varying S from 4 to 32. The results show that the proposed \(l_{1}\)-RTLS with the new \(\rho\) is comparable to \(l_{1}\)-RLS with the true channel. In the second experiment, we compare channel estimation performance using room impulse response. The size of the room is (7.49, 6.24, 3.88 m). The position of the sound source is (1.53, 0.96, 1.12 m) and the position of the receiver is (1.81, 5.17, 0.71 m), respectively. T60 is set to 100 ms and 400 ms. The impulse response of the room is generated using the program in [31]. We focus on the direct reflection part and the early reflection part in the RIR because the direct reflection and early reflection part of the RIR has a sparse property. This is the part that is estimated in the AEC applications [32]. This part is also related to localization and clarity in room acoustics [33; 34; 35]. Comparing the impulse response (IR) generated by setting T60 = 100 ms to the channel with 65 coefficients used in the first experiment, it is equivalent to S = 4 in the channel with 65 coefficients. In the same manner, the IR generated by setting T60 = 400 ms is equivalent to S = 10. Figure 3: Steady-state MSD for S = 4, 8, 16, and 32 when applying the new \(\rho\) method to the regularization factor (-\(\omega\): \(l_{1}\)-RLS with the true channel response, -\(\times\): \(l_{1}\)-RLS with the new \(\rho\) method, -*-: proposed \(l_{1}\)-RTLS with the new \(\rho\) method, -\(\infty\): oracle-RLS). Table 3 summarizes the steady-state MSD values.