## 3 Simulation Results and Discussion
### ANC Simulation Results and Discussion
In contrast, the noise residue of the proposed method (represented by the blue line) has small fluctuations along the amplitude of the reference noise at 1.6, 2.2, and 2.6 s. The LSTM neural networks predict the frequency-modulated signal better than the amplitude modulation. As shown in Figure 7b, the noise level increases overall in all the frequency bands after the engine starts. The FxLMS algorithm does not show good performance for impulsive sound control during engine starts. Approximately 1.5 s after the engine starts, the noise level decreases. However, when ANC is performed using the LSTMs, the noise in the high-frequency band increases slightly, according to the amplitude change of the noise. This increase does not affect performance. The original noise, FxLMS, and LSTM equivalent continuous noise levels for a total of 6 s from white noise to the engine start and steady state are 69.1, 48.2, and 39.8 dB, respectively. The LSTM and FxLMS algorithms show the highest performance compared to the previous simulation results for harmonic signals. Figure 7c shows the power spectral density of noise and ANC results for 3 to 6 s after the ANC algorithm converges. After starting the engine and entering the steady-state, the performance of the two algorithms was similar in the low frequency band as in the previous simulation results, but the performance of the LSTM was better in the high frequency band. The developed algorithm is robust to transient sounds, thus validating that the ANC performance for the impulsive noise from the LSTM controller is also better than that from the conventional FxLMS algorithm. As a result of the ANC simulations for the noise generated from the diesel engine, the controller of the proposed LSTM neural network shows a performance improvement in terms of convergence and accuracy relative to the conventional FxLMS approach. In practice, noise signals are rarely composed of single-frequency components. Therefore, noise control for impulsive and frequency-modulated noises is essential. The proposed ANC method using the LSTM controller is expected to provide rapid and precise ANC performance for the presented engine sound excitations. Before proceeding with ANC, an appropriate process for training the LSTM layers must be conducted. Recently, training neural networks has become easier and faster as hardware and software performance has increased. The elapsed time for calculating time series output data of length 600 through FxLMS and LSTM in the ANC process using a 3.2 GHz core computer with 16 GB RAM was 0.3486 s and 0.3857 s, respectively. The software used for the analysis was MATLAB R2021a. Through these results, it can be indirectly confirmed that the computational cost does not increase significantly compared to FxLMS, even when deep neural networks composed of LSTM is used. Although this study only considers diesel engine sounds, LSTM neural networks can contribute to noise reduction in other complex situations, including road and factory environments. Through the proposed algorithm, it was confirmed that the LSTM is capable of noise cancellation for diesel engine noise different from the training data (different fundamental frequency, superposed harmonic sound, engine start noise) in the linear system. Recently, there are various studies on advanced ANC algorithms using deep neural networks [14; 15; 26; 27]. Comparing the performance of advanced ANC algorithms in practical applications is a future task. Figure 6: ANC simulation result using FxLMS and LSTM for superposed harmonic sounds with main frequencies of \(33,35,50\), and \(60\) Hz measured from diesel engines. (**a**) Time data of diesel engines noise and ANC results; (**b**) short-time Fourier transform analysis of original noise and noise residue; (**c**) power spectral density of the diesel engine noise and ANC result in the period when the error signal is minimized.