## 3 Model Parameter Estimation
### Propulsion Noise
#### 3.2.3 Noise Analysis  
The noise levels and their short-term level fluctuations are obtained by a series of filtering operations. Starting with the resampled signal, in a first attempt, the engine orders are suppressed using cascaded notch filters. These filters are designed analogously to the engine order filter bank from the previous section, except that instead of bandpass filters, band-stop filters are generated (see Figure 12). Figure 15 shows two power spectral densities, which illustrate the effect of the order suppression filter. After this operation, the signal is split into sub-bands for further analysis. The signal is therefore decomposed into 1/3 octave bands using a 1/3 octave band filter bank. Each of the \(N_{b}\) filters yields a signal \(q_{i}(t)\).  
Figure 14: Comparison of sound pressure signals of a recording (**a**) and the corresponding synthesis consisting of engine orders with estimated phases (**b**). For the purpose of illustration only, the four dominant engine orders (colored lines) are used. The recording was conducted at the rear of a Ford Focus 1.8i at 1000 rpm and full load.  
Figure 15: Power spectral densities illustrating the effect of the order suppression filter, which is applied to a recording of an inline, four-cylinder engine idling at 4000 rpm.  
From \(q_{i}(t)\), the noise levels, \(L_{\text{eq,prop,noise,}i}\) in Equation (19), are calculated as Leqs. Moreover, from \(q_{i}(t)\), using a moving average filter, smoothed level-time curves:  
\[L_{q,i}(t)=10\cdot\log\left(\frac{1}{K}\int_{0}^{K}\frac{q_{i}^{2}(t+\tau)}{p_{ 0}^{2}}d\tau\right) \tag{42}\]  
are calculated using a window length of \(K=4\) ms. Subsequently, from \(L_{q,i}\), the mean value is subtracted to obtain a DC-free level fluctuation signal:  
\[\Psi_{q,i}(t)=L_{q,i}(t)-\overline{L_{q,i}(t)} \tag{43}\]  
Figure 16a exemplifies such a fluctuation signal for the 2.5-kHz band recorded at the front of a diesel engine car. The periodic structure is clearly visible. Following [6], the autocorrelation function (ACF) is used to estimate the standard deviations \(\sigma_{i}\) (used in Equation (19)) of the level fluctuations with period \(1/F_{\text{ign}}\) by:  
\[\sigma_{i}^{2}=\text{ACF}_{\Psi_{q,i}}(1/F_{\text{ign}}) \tag{44}\]  
Figure 16b shows the square root of the ACF of the level fluctuation signal depicted in Figure 16a. Clear peaks can be observed at lag zero and multiples of the ignition period of 34 ms. The standard deviation \(\sigma\) amounts to about 5 dB. The fact that a higher peak appears at the double ignition period, at 68 ms, indicates that the signal contains an additional level modulation with a modulation frequency equal to the half ignition frequency. This can also be observed in Figure 16a in which every second peak is about 5 dB higher than the previous one.  
Figure 17 shows measured spectra of the standard deviations \(\sigma_{i}\). The measurements were performed in front of five cars idling at low engine speeds. It can be seen that diesel cars feature higher values compared to gasoline engine cars. This finding corresponds to the increased rattling sound noticed in the field.  
Figure 16: Level fluctuation signal (**a**) of the 2.5-kHz 1/3 octave band and its square root of the autocorrelation function (**b**) from a recording taken at the front of an idling four-cylinder diesel engine at 870 rpm, corresponding to an ignition period of 34 ms. The right plot indicates that for this band, the level standard deviation, \(\sigma\), amounts to about 5 dB.