## 2 Model Development
### Propagation Filtering
#### 2.3.1 Effects Due to Source Motion and Propagation Delay  
Due to the travel time of sound and the movement of the source, the source and the receiver have differing time axis. By neglecting wind and turbulence, the warped time axis at the receiver is given by:  
\[t^{\prime}=t+\Delta t(t)=t+\frac{r_{\text{dir}/\text{gr}}(t)}{c_{0}} \tag{21}\]  
where \(r_{\text{dir}/\text{gr}}\) denotes the sound propagation distance of the direct sound or the ground reflected sound, respectively, and \(c_{0}\) is sound speed in still air. A constant sound speed of \(c_{0}=340\) m/s is assumed. Since the receiver signal is supposed to have a constant sampling rate of \(f_{s}\), the corresponding times, \(t_{s}\), on the emission time axis have to be found. This is achieved by linear interpolation of Equation (21). The emission signals \(x\) for the direct and the ground-reflected path, respectively, with respect to the receiver time \(t^{\prime}\) are:  
\[x_{\text{dir}/\text{gr}}(t^{\prime})=s(t)\cdot\mathcal{D}^{2}(t) \tag{22}\]  
with the Doppler factor:  
\[\mathcal{D}\equiv\frac{f^{\prime}}{f}=\frac{dt}{dt^{\prime}} \tag{23}\]  
Equation (22) describes the kinematic and the aerodynamic effect of source motion. The former is known as the Doppler effect, _i.e._, the Doppler frequency shift and amplification. The latter is known as convective amplification. The exponent two of the Doppler factor indicates that a Lighthill [47] monopole [48] and/or dipole source [49] is assumed. The change of the time axis in Equation (22) realizes the propagation delay, as well as the Doppler frequency shift. For digital signals, this change corresponds to an asynchronous resampling process. It can by implemented using a variable delay-line with delay \(\Delta t\)[50]. If \(\Delta t\) is just rounded to the nearest sample, audible artifacts occur, so-called \"zipper noise\". Therefore, an interpolation strategy has to be used. As we are only interested in sequential access to the emission signal, a fractional delay filter can be used [51]. In [18] and [50], a linear interpolator is proposed. This however produces high frequency attenuation, as well as strong nonlinear distortions due to aliasing. Therefore, here, we introduce a band-limited interpolation or, respectively, a windowed sinc interpolation [51]:  
\[s(k_{s})=\sum_{k=[k_{s}]-b+1}^{[k_{s}]+b}s[k]K(|k_{s}-k|) \tag{24}\]  
with the floor function \(\lfloor.\rfloor\), the integer sample index \(k\), the non-integer sample index \(k_{s}=t_{s}f_{s}\) and the Hamming kernel:  
\[K(m)=\left\{ \begin{array}{ll}\left[0.54+0.46\cos(\pi m/b)\right]\;\mathrm{ sinc}(m)&\text{if }0\leq m<b\\ 0&\text{otherwise}\end{array} \right. \tag{25}\]  
with an integer \(b\) describing the filter length. To keep the computational effort low, in the implementation of Equation (24), values of the kernel \(K\) are stored in a look-up table. The Doppler factor \(\mathcal{D}\) in Equation (22) is implemented by approximating the derivative in Equation (23) by finite differences as:  
\[\mathcal{D}(k_{s,i})\cong k_{s,i+1}-k_{s,i} \tag{26}\]  
with index \(i\). In order to validate different implementations of Equation (22), numerical simulations were performed. Figure 5 compares the signal attenuation introduced by different interpolation schemes. The high frequency attenuation of the linear interpolation can be improved by a windowed sinc interpolation and controlled by parameter \(b\). Figure 6 shows spectrograms of receiver signals calculated by the same three interpolation schemes. As an extreme case, a virtual source emitting a 1 kHz pure tone travels at constant speed \(v=150\) km/h and passes a static receiver at a distance of \(D=7.5\) m. Figure 6 shows that by introducing a windowed sinc interpolation of sufficient filter length, artifacts due to aliasing can be significantly reduced compared to a linear interpolation (a).