## 4 Results and Discussions
### Ejector Nozzle  
The first dataset analyzed involves microphone recordings near a micro turbojet in two configurations: baseline engine and engine with an ejector. Data were collected at 115,486 rpm (baseline) and 115,179 rpm (ejector) at a sampling frequency of \(f_{s}=48,000\) samples/s. Both signals, shown in Figure 6, represent 300 s of recording (the engine is started, reaches the maximum regime and stops). Figure 6: Pressure signal: (**a**) baseline—original signal, (**b**) baseline—trimmed signal, (**c**) ejector—original signal and (**d**) ejector—trimmed signal. To calculate acoustic metrics, the most stationary portions of these non-stationary signals were extracted during maximum regime of the engine: [130; 131; 132; 133; 134; 135; 136; 137; 138; 139; 140; 141; 142; 143; 144; 145; 146; 147; 148; 149; 150] seconds for the baseline and [100; 101; 102; 103; 104; 105] seconds for the ejector. The segments' durations differ, but each was selected to satisfy stationarity and ergodicity tests. Once verified, the extracted data were used for further processing, treating the signals as wide-sense ergodic stationary processes. The procedure to check the stationarity and ergodicity of a general sampled random signal, \(\left(p(t_{i})\right)_{i=1,N}\) on a given period of time, \(T_{0}=\frac{N}{f_{s}}\) is as follows:  
* Compute the mean, the variance, the standard deviation and the autocorrelation for the entire signal of length \(N\);
* Divide the signal into \(M\) overlapping segments (n.b. the so-called shorter realizations) with 50% overlapping. Each segment should be of a reasonable length to capture meaningful data behavior. One chooses the segment length in the range of \(N_{s}=\left[10^{4},10^{5}\right]\) samples, with step \(10^{4}\);
* Examine if the random signal is stationary if the conditions presented in the theory section are satisfied. The expected value, the variance and the autocorrelation functions of the random signal in a discrete formulation are considered uniformly distributed. Numerically, it means that across all \(M\) segments of size \(N_{s}\), at the same time that \(t_{k},\ k=1,N\), one calculates the mean and the variance with the same probability (n.b. independent of time): \[ \begin{split}\mu(t_{k})&=E[p(t_{k})]=\frac{1}{M} \sum\limits_{n=1}^{M}P_{n}(t_{k}),\\ \sigma^{2}(t_{k})&=E\Big{[}(p(t_{k})-\mu(t_{k}))^{2} \Big{]}=\frac{1}{M}\sum\limits_{n=1}^{M}[P_{n}(t_{k})-\mu(t_{k})]^{2}\end{split} \] (3) where \(p_{n}(t)\) represent the signal's value at the \(n\)-th realization of the random process \(p(t)\). In the acoustic field, the signal represents the fluctuation in the pressure with regard to standard atmospheric pressure. Thus, one expects that the constants for the expectation and variance of the random process to be zero or very close to zero. As it is impossible to get the value zero in a real case, one should verify if the mean and the standard deviation are below a certain tolerance \(\varepsilon\) for all the random variables at each time: \[t_{k}|\mu(t_{k})|\leq\varepsilon,\ \sigma(t_{k})\leq\varepsilon\ \ k=1,N\] (4)  
The examination continues with the discrete autocorrelation function for different time lags (\(N_{lag}\leq N_{s}\)) and for each segment (realization) \(n\) of size \(N_{s}\) given by  
\[R_{n}(\tau_{k+1})=\frac{1}{N_{s}-k}\sum\limits_{j=1}^{N_{s}-k}P_{n}\big{(}t_{j }\big{)}P_{n}\big{(}t_{j}+\tau_{k+1}\big{)},k=0,N_{lag} \tag{5}\]  
where the \(\tau_{k}=k/f_{s}\) is the time lag. The mean of the autocorrelation function, at each delay \(\tau_{k}\), across all \(M\) realizations, is given by  
\[R(\tau_{k+1})=\frac{1}{M}\sum\limits_{n=1}^{M}R_{n}(\tau_{k+1}),k=0,N_{lag}-1 \tag{6}\]  
and the condition to be a stationary process is expressed as follows:  
\[|R(\tau_{k+1})|=ct(k),k=0,N_{lag}-1 \tag{7}\]This condition should be understood in the following way: the mean of the autocorrelation function of a signal is constant for the same delayed version of itself. However, the constant could depend on different lags.