# Compensation for Group Velocity of Polychromatic Wave Measurement in Dispersive Medium
## 2 Group Velocity Measurement Based on Electromagnetic Wave Theory  
The proposed group velocity measurement method consists of two main parts. The first part comprises the transmitted signal detection method using TFCC. After the first step, the time offset and shifted center frequency of the transmitted signal are used to compensate for the group velocity. In electromagnetic theory, the group velocity of an electromagnetic plane wave can be derived as follows [1]:  
\[v_{p}(f)=\frac{c}{\sqrt{(e(f)\cdot\mu(f))}}=\frac{c}{(e_{\infty}+\frac{e_{ \infty}-e_{\infty}}{1+\frac{f}{f_{r}}})^{1/2}\cdot(1+\frac{\mu_{s}-1}{(1+j\frac {f}{f_{m}})^{2}})^{1/2}}, \tag{1}\]  
where \(f_{r},f_{r},f_{m}\) are the operating frequency, relaxation frequency at which the imaginary part of the complex permittivity reaches a maximum, and resonant frequency, respectively. \(e_{\infty},e_{s}\) are the permittivity at infinity frequency and static permittivity, respectively. \(\mu_{s}\) is the static permeability. According to (1), the group velocity depends on the operating frequency. When the polychromatic wave comprising several frequencies propagates in the dispersive media, the wave is distorted and a group velocity error occurs because of the frequency dependency of group velocity. Especially when the signal is transmitted over a long distance, the more high frequency components are attenuated, and velocity error is induced. In other words, as the signal propagates, the group velocity changes continuously with the propagation distance. In this paper, the incident signal is the signal that we injected into the cable, the reflected signal is generated at the cable termination, and transmitted signals are signals that are acquired by the oscilloscope through the inductive couplers as the incident signal flows along the cable, and the signals acquired after being reflected at the cable termination are classified as reflected signals, and the reflected signal is included in the transmitted signal. We use the Gaussian linear chirp signal as the incident polychromatic wave, and the incident signal is represented as follows:  
\[s(t)=\left\{\begin{array}{ll}Ae^{-\alpha t^{2}/2+j(0.5\xi_{1}t^{2}+\omega t )},&\text{t= 0}\leq\text{t}\leq T_{1}\\ 0,&\text{otherwise}.\end{array}\right. \tag{2}\]  
where \(T_{1}\) is the duration of the incident signal, \(A\) is the amplitude, \(\xi_{1}\) is the normalized angular frequency sweep rate, and \(\omega\) is the normalized angular frequency. The transmitted signal is expressed as follows:  
\[r(t-d)=\left\{\begin{array}{ll}\eta\cdot Ae^{-\alpha t^{2}/2+j(0.5\xi_{2}(t-d )^{2}+\omega(t-d)+\phi)},&\text{d}\leq\text{t}\leq\text{d}+T_{2}\\ 0,&\text{otherwise}.\end{array}\right. \tag{3}\]where \(\eta\) is the magnitude of the attenuation coefficient at the travelling distance, and \(d\) is the time delay of the transmitted signal. Also, because the transmitted signal has high frequency attenuation, we define the changed parameters \(\xi_{2}\) and \(T_{2}\) as frequency sweep rate and time duration of transmitted signal. The normalized cross-correlation between the incident signal and the transmitted signal is used to detect the transmitted signal from the cable termination. This can be expressed as follows [3]:  
\[C_{sr}(t)=\sum_{k=0}^{N-1}\frac{s_{k}\bigotimes\limits_{r_{k-t}}r_{k-t}}{ \sqrt{E_{s}E_{r}}} \tag{4}\]  
where \(E_{s}\) is the energy of the incident signal in Wigner-Ville distribution, \(E_{r}\) is the energy of the transmitted signal, and \(\bigotimes\) is the correlation operator. Through the normalized cross-correlation process, the transmitted signal can be extracted from the total acquired signal.