Figure 2 shows the illustration of the proposed system. The system is composed of (1) digital phosphor oscilloscope (DPO); (2) arbitrary waveform generator (AWG); and (3) inductive couplers. AWG generates the reference signal and apply the signal to target cable though the inductive coupler. The inductive coupler uses the electromagnetic induction phenomenon to apply a signal to the cable without connection between the cable core and signal line. In this paper, we use three inductive couplers and coupler 1 is used to apply a signal to cable, couplers 2 and 3 acquire the signal. An incident signal propagates along the cable and was acquired in the oscilloscope through the couplers 2 and 3. For ease of understanding, the signals were numbered according to the order in which they were acquired. The transmitted signals acquired through the couplers 2 and 3, before the signal being reflected, were numbered 1 and 2 and located at
and
, respectively. The distance propagated based on the first acquired signal though coupler 2 becomes the position of the signal. The signals reflected from the cable termination were in turn acquired via couplers 3 and 2, which were located at
and
. In order to verify the variation of group velocity due to the dispersion, we converted the experimental system into an equivalent circuit model using the simulation tool. Through the tuning of loss factor in the simulation tool, we conducted the simulation to compare the signal passing through the lossy medium with lossless medium. The comparison results are shown in
Figure 3. The results consist of 3 types simulations: (a) lossless cable: 40 m; (b) lossy cable: 40 m; (c) lossy cable: 80 m. To verify the effect of loss factor, we compared the transmitted signal in lossy medium with that in lossless medium (simulations: (a) and (b)). Also, simulations were conducted with different cable lengths in order to analyze whether the group velocity of the signal varies due to the reflection (simulations: (b) and (c)). As shown in
Figure 3a, the incident signals of each simulation are identical, but the reflected signals of each did not match.
Figure 3b shows an enlarged view of the reflected signals. The highest peak in the time domain of the signal can be thought of as the highest energy point, and the group velocity of the signal can be determined by the time delay between these peak points, and this time delay is called the time of arrival of group velocity. Comparing the waveforms of (a) and (b), the time delay between the peak points of the reflected signal generated at
and
are
and
, respectively. If there was no change in group velocity depending on the travel distance, the time of arrival of group velocity of the reflected signal generated at
had to be
. More time delays mean slower group velocity. These results show that the group velocity is decreasing as the signal propagates. Since the incident signal is a positive chirp signal, the rear part of the signal in time domain contains a high frequency component. As shown in the reflected signals of red and black line of
Figure 3b, in the front part, the zero-crossing points of each signal are matched, but the zero-crossing points in the rear part do not coincide with each other. These results indicate that the higher frequency components of the signal are attenuated as the signal propagates in the lossy medium. Comparing the reflected signals generated at
, we verified that the reflection only affect the magnitude of signal, not group velocity.
Figure 4a shows the total acquired signal from the inductive couplers after signal restoration process [
9,
10]. As seen in the fourth signal in
Figure 4a, because the signal is difficult to distinguish from the noise, TFCC is used to roughly find the position of the transmitted signal. To evaluate the accuracy of proposed method, we solved the true value of group velocity using the time of arrival of group velocity of the signals in the time domain. However, the reflected signal at
is too small to find out due to the attenuation. Because of this, it is very difficult to obtain the group velocity through the time delay between the highest peak points of the signal in the time domain, and the group velocity in the farther than
can not be calculated. The TFCC graph is shown in the
Figure 4b. As the signal propagates, attenuation of the signal occurs, which slows the group velocity and increases the time delay.
Figure 4b depicts a TFCC graph based on the constant group velocity measured on 40 m. Therefore, the distance errors of 80 m and 120 m is getting larger. Based on unwrapping algorithm and Hilbert transform, the instantaneous phase of the transmitted signal can be derived and shown in
Figure 5a. The signal having the positive slope of the instantaneous phase was extracted, and the frequency band of the extracted signal was obtained by FFT algorithm. In
Figure 5a, the time duration of the signal is obtained by extracting the signal where the slope of the instantaneous phase is positive. The
Figure 5b shows the frequency band of second signal of total signal. The changed values, time duration and frequency region, were obtained and substituted into Equation (
7) to compensate the group velocity. As seen in
Table 1, the group velocity in TFCC method seems to be equal regardless of the propagation distance. On the contrary, in the proposed method, the shifted terms ,
, can be obtained from the center frequency,
, of the received signal from
Figure 1 and
Figure 5. The compensation time delay term is calculated according to Equation (
8). The travelling distance, D, is the known value of the cable length and is the integral of the velocity determined by the center frequency of the transmitted signal. Based on the derived compensation time delay,
, and the distance value,
D, the average velocity during propagation of the transmitted signal can be obtained and are shown in
Table 1. The measurement values in
Table 1 were calculated by the time of arrival of group velocity of signal in
Figure 5a. As the distance between each signal is set as
, the group velocity, measurement value, can be obtained by dividing the distance by the measured group delay. The accuracy values in
Table 1 were calculated by dividing the group velocity of proposed method by measurement value. As seen in
Table 1, when the signal propagates to a short distance, the existing method is highly accurate, but the existing method does not reflect the change in group velocity when the signal propagates over a long distance. On the other hand, the proposed method compensates the group velocity change due to the dispersion, so that the accuracy of group velocity is good regardless of the distance.