# Validation of an RF Image System for Real-Time Tracking Neurosurgical Tools
## 2 Materials and Methods
### System Hardware
#### 2.1.2 Antenna Time Domain Analysis  
The signals used in medical imaging systems are usually broadband, i.e., they involve remarkably short pulses in time covering a considerably broad frequency spectrum. Since the pulses are narrow, they are greatly affected by dispersion. As a result, the incoming pulse at the antenna will never be the same as the outgoing pulse. For this reason, a time domain analysis of the transmitted pulses was completed, in order to predict the distortion inherent to the system.  
Figure 3: Measured and simulated return losses for the proposed antenna.  
To do that, the transfer function for a system in which a pulse is transmitted and received by the same single antenna (like the system shown here) can be measured and characterized by an equivalent system in which there is a direct transmission between two identical antennas (i.e., the receiving antenna is exactly the same as the transmitting antenna). This assumption considerably simplifies the analysis and measurement of the transfer function of the original system, and it will be therefore taken in this section. Please note that this assumption is taken for characterization purposes only. This scheme uses two identical antennas vertically oriented in a face-to-face manner and separated by 40 cm, distance sufficient to ensure far field transmission. This way, the system's transfer function reduces to the \(S_{21}\) parameter. Therefore, the two antennas were located inside an anechoic chamber and the \(S_{21}\) parameter was measured using the VNA, with one antenna acting as a transmitter while the other as a receiver. The experiment was replicated in silico by means of free-space transmission simulations. The information obtained after this frequency-domain analysis was post-processed to obtain the time domain signals.  
From the \(S_{21}\), some metrics can be used to analyze the performance of the antenna, such as the group delay (i.e., the phase derivative for the \(S_{21}\) response). The experimentally measured and simulated group delays for the above-mentioned antenna arrangement are plotted in Figure 5. It can be seen that it is fairly flat within the antenna bandwidth (shadowed area), with values between 0.20 and 0.25 ns. These results suggest that the system will show a low distortion for signals whose bandwidth falls within the antenna bandwidth. The best way to quantify this effect is to use the so-called System Fidelity Factor (SFF), which is a measurement of the correlation between the transmitted and received pulses. This factor calculates the ratio between the energy of the convolution between the transmitted and received pulses and the energy of each pulse separately [26]. The SFF is defined as:  
\[\text{SFF}=max_{n}\left|\frac{\int_{-\infty}^{+\infty}T_{S}(t)R_{S}(t+\tau)d \tau}{\sqrt{\int_{-\infty}^{+\infty}\left|T_{S}(t)\right|^{2}}dt\cdot\sqrt{ \int_{-\infty}^{+\infty}\left|R_{S}(t)\right|^{2}}dt}\right| \tag{1}\]  
where \(t\) is the time, \(T_{S}\) is the transmitted pulse, and \(R_{S}\) is the received pulse, which is computed from the standard \(S_{21}\) parameter (which means that SFF takes into consideration the distortion induced by both antennas). Thus, for the proposed antennas the SFF value obtained is 95.29% if the theoretical transfer function is used, and 96.97% if the measured \(S_{21}\) is used, which gives an idea of the high signal integrity achieved for the transmitted signals.  
Figure 4: Measured antenna radiation pattern at specific frequency points: (**a**) E-plane; (**b**) H-plane.