## 3 Model Parameter Estimation
### Propulsion Noise
#### 3.2.1 Resampling  
The emission synthesizer uses detailed information about the engine orders. These parameters are obtained by a narrowband analysis, which is described in the following section. Although during the measurements, the engine speed was kept fairly constant, the instantaneous order frequencies slightly fluctuate as exemplarily shown in Figure 11. This figure shows the spectrogram of a recording made at the rear of a car with an inline, four cylinder engine idling at 1100 rpm. To be able to separate engine orders and broadband noise by the narrowband analysis, a preceding resampling of the slightly non-stationary signals is performed. In order to actuate the resampling process, the instantaneous ignition frequency, \(F_{\text{ign}}(t)\), of the engine is required. This data are extracted from the audio recordings. In the first step, the average ignition frequency is estimated. From the signal taken closest to the exhaust, the power spectral density (PSD) with a frequency resolution <1 Hz is calculated. Based on the rough indication of the engine speed taken from the car's tachometer, a first estimate of the ignition frequency, \(F_{\text{ign}}\), is obtained using Equation (17). The location of the maximum value of the PSD within a search range around this frequency yields a better, second estimate. Particularly, for low engine speeds, at which the ignition frequency can be as low as 20 Hz, this estimate is still not precise enough due to the low relative resolution at low frequencies. Thus, this estimate is further enhanced  
Figure 10: Signal analysis flowchart to obtain the synthesizer parameters of propulsion noise as described in Section 2.2.3 from audio recordings. by considering the double ignition frequency, \(2F_{\rm ign}\), (_i.e._, engine order \(\
u=4\) in Equation (16) for a four-cylinder engine) within a smaller range of the PSD. In the second step, this information is used to track the course of the ignition frequency, \(F_{\rm ign}(t)\). This task is generally known as pitch detection [61; 62; 63]. A wide variety of algorithms exist that work in the time or frequency domain or a combination of them. In our application, a spectral method was established, in which the course of one discrete frequency component (_i.e._, an engine order) is tracked in a spectrogram. The spectrogram \(S(t,f)\) expressed in decibels is computed by the short-time Fourier transform (STFT):  
\[S(t,f)=10\cdot\log\left(\left|STFT(t,f)\right|^{2}\right) \tag{32}\]  
STFT is calculated using the FFT. Windows of 200 ms with a 50% overlap, _i.e._, a temporal resolution of \(\Delta t=100\) ms, are multiplied by a Hann window function. To obtain a high frequency resolution of \(\Delta f<0.5\) Hz, the signals are zero padded. A section of such a spectrogram is depicted in Figure 11. Within the spectrogram, the \"highest cost\" path between time \(t=0\) and \(t=T\), within a certain frequency range around a reference frequency, \(F_{r}\), is sought. \(F_{r}\) is chosen to be the first multiple of the mean ignition frequency above 55 Hz. This is a compromise between signal power and frequency localization: typically, the power decreases for increasing even orders (see Figure 11b), but higher orders exhibit larger absolute frequency variations (see Figure 11a). In the example of Figure 11, \(F_{r}\) is 71 Hz (corresponding to the fourth engine order), as the mean ignition frequency lies at 35 Hz. The optimization task is solved by dynamic programming, which breaks the complex problem down into many simple subproblems. This method prevents taking possible wrong local decisions and guarantees that the best solution is found. A well-known algorithm that uses dynamic programming is dynamic time warping (DTW), which is often applied in, e.g., automatic speech recognition (ASR). Additionally, we make use of the _a priori_ knowledge that the engine speed does not change rapidly over time. This is introduced as a requirement on the slope of the optimal path, \(F_{\rm opt}\).