## 2 Materials and Methods
### Required Sampling Step Size for the Emulation of the Time Expansion  
The emulation of the Doppler effect according to the principle of time expansion, performed digitally, implies a continuous discrete rotation of the signal phases in the different paths (see Figure 3) according to the time evolution of the respective delays. This entails an in-depth examination of what has already been discussed. To try to quantify the phenomenon, take into consideration the operation of quantization of the delay \(\tau\) using a quantum equal to \(q\) seconds:  
\[\tau_{q}=q\cdot Round\left(\frac{\tau}{q}\right) \tag{5}\]  
The previous operation performs a uniform quantization of the original signal. For simplicity, we can imagine the following scenario: a fixed source transmits a pure tone at frequency \(f_{o}\), a receiver moving at constant speed \(V_{o}\) with a null initial distance, receives this signal with a variable delay \(\tau(t)\):  
\[\tau\left(t\right)=\frac{V_{o}\cdot t}{c_{o}} \tag{6}\]  
The quantization of the delay generates a discrete signal so that the error  
Figure 5: Normalized amplitude transfer function for a two-ray model and different differential delays. Figure 6: Bandwidth in MHz at minima for a two-ray model. \[\varepsilon=\frac{\left(\tau-\tau_{q}\right)}{q} \tag{7}\]  
i.e., the difference between the original and the quantized one, normalized to \(q\), is limited to \(\pm\frac{1}{2}\). The error pattern is a sawtooth signal with a period \(T=\frac{q\cdot\varepsilon_{o}}{V_{o}}\) (see Figure 7). The complex envelope of the received analog signal, with respect to the angular frequency \(\omega_{o}\), is:  
\[x(t)=\exp\left(-i\omega_{o}\tau\right)=\exp\left(-i\omega_{o}\frac{V_{o}\cdot t }{c_{o}}\right) \tag{8}\]  
Considering therefore the quantized delay, equal to \(\tau_{q}=\tau-\varepsilon\cdot q\), it is possible to calculate the complex envelope of the actual received signal obtaining:  
\[x_{q}(t)=\exp\left(-i\omega_{o}\tau_{q}\right)=\exp\left[-j\omega_{o}\left( \tau-\varepsilon\cdot q\right)\right]=\exp\left[-j\omega_{o}\tau+j\omega_{o} \varepsilon\cdot q\right] \tag{9}\]  
The error signal is periodic, therefore it is possible to write it as a Fourier series expansion:  
\[ \begin{array}{l}\varepsilon\left(t\right)=\sum_{q=1}^{\infty}A_{n}Sin\left( \frac{2\pi\cdot n\cdot t}{T}\right)\\ A_{n}=-\left(\frac{1}{\pi\cdot n}\right)\left(-1\right)^{n}\end{array}  \tag{10}\]  
Figure 8 shows Equation (10) limiting the series to the 100 th order. The complex envelope of the received signal thus results:  
\[x_{q}(t)=\exp\left(-i\omega_{o}\tau_{q}\right)=\exp\left[-j\omega_{o}\left( \tau-\varepsilon\cdot q\right)\right]=\exp\left[-j\omega_{o}\tau+j\omega_{o}q \cdot\sum_{n=1}^{\infty}A_{n}Sin\left(\frac{2\pi\cdot n\cdot t}{T}\right)\right] \tag{11}\]  
From Equation (11) it is evident how the quantization of the delay involves the generation of a non-linear phase modulation of the received signal. The associated real signal results:  
\[x_{ric}(t)=\mathcal{C}os\left[\omega_{o}\left(1-\frac{V_{o}}{c_{o}}\right) \cdot t+\omega_{o}q\cdot\sum_{n=1}^{\infty}A_{n}Sin\left(\frac{2\pi\cdot n \cdot t}{T}\right)\right] \tag{12}\]  
Figure 7: Normalized quantization error. The spurious phase modulation introduced may be neglected if \(\omega_{o}q<<1\). The time-dependent delay time, provided as input to the channel emulator, is still sampled at frequency \(f_{s}\) even if no longer quantized. The signal received is therefore also sampled at this frequency. For a given platform speed, it is theoretically possible to eliminate unwanted components by imposing \(q=\frac{V_{a}}{c_{o}}\frac{1}{f_{s}}\), resulting in a sampling of the sinusoidal component at integer multiples of the period.