## 2 Materials and Methods  
### Multipath System-Level Models  
A general system-level schematic for a RF channel emulator can be synthesized as in Figure 1.  
The system consists of a transmitter under test, a receiver and a channel emulation block having a number of key tasks to perform. The RF analog signal undergoes a down-conversion with the extraction of the in phase and quadrature components. The subsequent band-limited analog signal is sampled and processed in a specialized Digital Signal Processing unit (DSP) in order to introduce all the impairments typically present in a real propagation channel. These phenomena include the time-dependent attenuation, additive gaussian noise, jammers, Doppler spread and many others. The modified signal undergoes a digital-to-analog conversion and an up-conversion to be injected into the receiver. It has been investigated here with particular emphasis on the presence of time-variant delays due to a multipath propagation. Bearing in mind this aspect, with reference to Figure 1, the generic RF signal coming from the transceiver under test can be written as [2]  
\[s(t)=x(t)Cos[\omega_{o}t+\varphi(t)]\text{ with }x(t)\geq 0 \tag{1}\]  
in which appear the envelope module \(x(t)\), the instantaneous phase \(\varphi(t)\) of the modulating signal and the carrier angular frequency \(\omega_{o}\). The transit of the signal in Equation (1) through a delay line of \(\tau\) seconds generates a new signal \(y(t)\) of the type:  
\[y(t)=x(t-\tau)Cos[\omega_{o}(t-\tau)+\varphi(t-\tau)] \tag{2}\]  
The complex envelope of the signal in Equation (2), with respect to the angular frequency \(\omega_{o}\) is equal to:  
\[i(t)=x(t-\tau)e^{i\omega_{o}\tau+i\varphi(t-\tau)} \tag{3}\]  
Equation (3) allows the processing of the signal in Equation (1) using directly its equivalent low frequency representation. When multipath is present, it can be correctly represented at complex envelope level as in Figure 2, where the complex envelope of the signal affected by multipath propagation is seen as the sum of the contributions coming from \(n\) discrete independent paths, each affected by a time-dependent delay and a time-dependent attenuation. The direct path is characterized by an absolute delay starting from the initial observation time. The remaining paths are affected not only by the main delay, but also by differential delays (small) with respect to the latter:  
\[ \begin{array}{l}\tau_{d}(t)=\frac{R_{d}(t)}{c_{o}}\\ \Delta\tau_{1}(t)=\frac{\Delta R_{\tau_{1}}(t)}{c_{o}}\\...\\ \Delta\tau_{m}(t)=\frac{\Delta R_{m}(t)}{c_{o}}\end{array}  \tag{4}\]  
Figure 1: System-level schematic of a generic RF channel emulator. with  
1. \(R_{d}(t)\) equal to the distance (instantaneous) travelled by the direct path between transmitter and receiver;
2. \(c_{o}\)\(\sim\)\(3~{}\times~{}10^{8}\) m/s equal to the speed of light;
3. \(\Delta R_{k}(t)\) equal to the difference between the \(k\)-th path and the direct path. By correctly emulating the behavior of the attenuations and the delays over time, the diagram in Figure 2 is able to correctly reproduce both fading and Doppler shift over the entire signal band, assuming the existence of a _discrete_ number of independent paths [3]. In the perspective of a discrete-time realization, the scheme in Figure 3 considers the amplitudes and delays of each path as constants for sufficiently representative time intervals [4]. If we renounce to the complete emulation of the time-dependant delay of the direct path, losing therefore the absolute Doppler information, we arrive at the scheme in Figure 4.  
Figure 3: Representation of the multipath signal in the complex envelope domain with delays and time-discrete amplitudes. Figure 2: Representation of the multipath signal in the complex envelope domain. In this last scheme, the Doppler shift is the one related to the \(f_{o}\) frequency and depends directly on the law of variation of the direct path, with the addition of the spreading due to the variations of the delays of the reflected paths.