## 4 Results and Discussions
### Ejector Nozzle
#### 4.1.2 Time and Frequency Domain Analysis  
The comparison between (B) and (E) cases in Table 4 shows that the (E) reduces OXSPL and L\({}_{\text{Xeq}}\) for all weightings X = A, C, Z. Thus, the SPL(Z) (zero-weighted for all frequencies) shows a reduction of 0.51%, while in the human frequency sensitive hearing range the OASPL decreases by 0.74%. Similarly, the SPL(C) shows a smaller reduction of 0.41%, reflecting a decrease in low-frequency range noise. However, the peak SPL values (L\({}_{\text{Zmax}}\) and L\({}_{\text{Cmax}}\)) show slight increases of 0.27% and 0.39%, respectively, in the (E) case. This suggests that while the average noise levels decrease, occasional high-intensity noise spikes may still occur. The L\({}_{\text{Amax}}\) exhibits a minor reduction of 0.41% in peak noise perceived by humans. Overall, the ejector configuration effectively reduces general noise levels but may not completely eliminate peak sound events. Figures 8a,b and 9a,b show the time-variant sound pressure level \(L_{p}(t)\) and the three time-average sound pressure level, \(L_{\text{Aeq,T}}\)\(L_{\text{Ceq,T}}\) and \(L_{\text{Zeq,T}}\) considering a period of time \(T=0.125s\) for capturing short-term variations in sound levels. As a quantitative observation, the time evolution of \(L_{p}(t)\) provides no meaningful information, as it is nearly identical for both configurations. However, when looking at the time-equivalent X-weighted sound pressure levels, one can observe a significant improvement, as the values for the ejector are consistently just under 118 dB. Figures 8c and 9c depict the power spectrum using the periodogram to identify what frequencies remain along the test period. The PSD provides valuable information about the distribution of signal power across different frequencies, especially for understanding the frequency content of a signal. In Figures 8d,e and 9d,e, by means of the PSD, the spectrum SPL is computed in four different ways. For numerical implementation, the data are divided into \(N_{seg}\) overlapped segments (n.b. 50% overlapping between the segments), and the power spectral density is calculated in four different ways: (a) using the PSD provided by the spectrogram routine with a Hamming window of \(N_{f}\) number of samples, (b) using the short-time Fourier transform from the spectrogram routine with a Hamming window of the same sample size, \(N_{f}\), (c) using the Welch periodogram and (d) using an in-house routine. The discrete expression of PSD is given by  
\[PSD(f_{k})=\frac{1}{T}\Big{|}\Delta Tp_{T}(n)e^{-2\pi f_{k}n\Delta T}\Big{|}^ {2}=\frac{1}{N\Delta T}|\Delta TP_{T}(f)|^{2} \tag{13}\]  
where the time step is \(\Delta T=1/f_{s}\), and \(f_{s}\) is the frequency sampling. Equation (15) is adapted for each segment of length \(T_{w}\) having \(N_{w}\) samples. Thus, the total power spectral density averaged over \(N_{seg}\) segments is given by  
\[PSD(f)=\frac{1}{N_{seg}}\sum_{f=1}^{N_{seg}}\frac{\left|\Delta TX_{r}(f) \right|^{2}}{T_{w}}=\frac{1}{N_{seg}}\sum_{f=1}^{N_{seg}}\frac{\left|X_{r}(f) \right|^{2}}{f_{s}N_{w}U_{w}} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **SPL(Z)** & **OASPL** & **SPL(C)** & \multirow{2}{*}{**L\({}_{\text{Aeq}}\) [dB(A)]**} & \multirow{2}{*}{**L\({}_{\text{Ceq}}\) [dB(A)]**} & **L\({}_{\text{Zmax}}\)** & **L\({}_{\text{Amax}}\)** & **L\({}_{\text{Cmax}}\)** \\  & **[dB]** & **[dB(A)]** & **[dB(C)]** & & & **[dB(A)]** & **[dB(C)]** & **[dB(A)]** & **[dB(C)]** \\ \hline B & 118.49 & 117.33 & 118.08 & 117.34 & 118.08 & 119.59 & 118.15 & 119.27 \\ E & 117.90 & 116.47 & 117.60 & 116.47 & 117.60 & 119.91 & 117.66 & 119.73 \\ \% & 0.51\% & 0.74\% & 0.41\% & 0.74\% & 0.42\% & 0.27\% & 0.41\% & 0.39\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Acoustic time metrics for baseline and ejector engine. where \(U_{w}\) is the power of the window function needed for truncation. Note that for a rectangular window \(U_{w}=1\).