## 2 Model Development
### Emission Module
#### 2.2.3 Sound of Propulsion  
The structure of the emission synthesizer for the sound of propulsion is depicted in Figure 4. The sound pressure emission signal of the sound of propulsion is assumed to consist of a deterministic signal representing the most important engine orders and a quasi-stochastic signal:  
\[s_{\text{prop}}(t)=s_{\text{prop,ord}}(t)+s_{\text{prop,noise}}(t) \tag{13}\]  
Enginee order \(\
u\) corresponds to an event taking place \(\
u\) times per engine revolution. The engine order signal is composed of the sum of the engine orders \(\
u\), which are generated using additive synthesis [29; 30]. The engine order signal is thus calculated by [6]:  
\[s_{\text{prop,ord}}(t)=\sum_{\
u}p_{0}10^{t}\
icefrac{{t}}{{\text{prop,ord, $\
u$}}}(t)/20\cdot\sqrt{2}\cos(\beta_{\
u}(t)) \tag{14}\]  
A proper selection of the essential orders \(\
u\) strongly depends on the specific vehicle type and its condition. In the context of sound design, it is known that at least orders up to \(\
u=18\) are relevant [23]. Further, the sound characteristics can be influenced by half-orders [23]. In this model, it was decided to synthesize orders \(\
u=1\) to 30 in half-order steps, resulting in a total of 59 orders. This somewhat arbitrary, but safe choice leaves room for optimization. In Equation (14), \(L^{\ddagger}_{\text{prop,ord,$\
u$}}\) denotes the order level and the instantaneous order phase:  
\[\beta_{\
u}(t)=\phi^{\ddagger}_{\
u}(t)+2\pi\int_{-\infty}^{t}F_{\
u}(\tau)d\tau \tag{15}\]  
with the order phase \(\phi^{\ddagger}_{\
u}\) and the order frequency:  
\[F_{\
u}(t)=\
u\cdot n(t)/60. \tag{16}\]  
Listening tests revealed that in this application, the order phase is a relevant synthesizer parameter. For a four-stroke engine with \(N_{\text{cyl}}\) cylinders, the engine order corresponding to the ignition, and mostly the predominant order, is \(\
u_{\text{ign}}=N_{\text{cyl}}/2\)[23]. Thus, the ignition frequency reads [5]:  
\[F_{\text{ign}}(t)=N_{\text{cyl}}/2\cdot n(t)/60 \tag{17}\]  
Figure 4: Signal flow chart of the synthesizer for the sound of propulsion. For time-discrete signals, Equations (14) and (15) can be interpreted as a modified numerically-controlled oscillator (NCO) [28], whereas Equation (15) corresponds to the phase accumulator (PA), and the phase-to-amplitude converter (PAC) is realized in Equation (14). This formulation concurrently implements a frequency modulation by \(F\) and a phase modulation by \(\phi\). The noise signal component of the sound of propulsion is synthesized similarly as the sound of tires (Equation (3)) by:  
\[{}_{\text{prop,noise}}(t)=\sum_{i=1}^{N_{b}}p_{0}10^{L_{\text{prop,noise},i}( t)/20}\cdot\xi_{i}^{\prime}(t) \tag{18}\]  
A total of \(N_{b}=29\) bands from 20 Hz to 12.5 kHz are used. The \(1/3\) octave band level function is formulated as:  
\[L_{\text{prop,noise},i}(t)=L_{\text{eq,prop,noise},i}^{\dagger}+\sigma_{i}^{ \dagger}\cdot R(t)-K(\sigma_{i}^{\dagger}) \tag{19}\]  
with the level \(L_{\text{eq,prop,noise},i}^{\dagger}\), a level standard deviation \(\sigma_{i}^{\dagger}\) and a level fluctuation function \(R(t)\) with zero mean and unit power. The constant \(K\) ensures that despite the level fluctuations, the equivalent continuous level (Leq) is not altered. This level modulation simulates the rattling sound component that elicits a roughness sensation, which is particularly characteristic for low engine speeds and diesel engines. Motivated by measurement data that showed the strongest level fluctuations at the ignition frequency, \(R\) is modeled by a quasi-periodic function with period \(1/F_{\text{ign}}(t)\). The first half-period of \(R\) is composed of a Hann window, whereas the second half-period is held constant. In summary, the presented synthesizer needs about 180 input parameters to generate a stationary signal for the sound of propulsion. However, during a pass-by, the sound of propulsion may considerably vary, and so do these parameters.