3.2. Effect of External Noise on Time Delay Estimation
In practical engineering, partial discharge signals usually occur in complex electromagnetic environments, resulting in a low SNR of the collected partial discharge signals, which affects the accuracy of time delay estimation. Simulation of Gaussian white noise and narrow bandwidth interference signals to analyze the effect of noise interference on the accuracy of time delay estimation of the PHAT–SCOT–GCC algorithm [
17].
The broad spectrum and random generation of Gaussian white noise are utilized to simulate the bottom noise generated by the oscilloscope and the background noise in the environment [
18]. The signal
x1(
t) adds Gaussian white noise with an SNR of −5 dB, as shown in
Figure 3.
According to Equation (9), each parameter in Equation (9) is the same as
x1(
t), and comparing
x1(
t) delay
τ is 200 points to obtain
x2(
t), and the absolute value of the delay of the two signals is 2 μs. In the following section, time delay estimation of simulated partial discharge signals will be performed using the generalized cross-correlation algorithm with an improved weighted function proposed. The results of time delay estimation using three algorithms (PHAT weighted function, SCOT weighted function, and improved generalized cross-correlation algorithm with joint weighted function) for the SNR of −5 dB are shown in
Figure 4:
Figure 4a–c presents the time delay estimation results obtained using PHAT–GCC, SCOT–GCC, and PHAT–SCOT–GCC. Comparing the simulation results of the three algorithms, it can be seen that the PHAT–GCC and SCOT–GCC algorithms have poor immunity to interference noise. The principal peak value of the correlation function is almost submerged in the noise interference, which produces several false peaks, and the error of their delay results is more significant than 0.5 μs. The PHAT–SCOT–GCC algorithm peak sharpening is obviously without the interference of false peaks, with a delay value of −2.01 us and an error of 0.01 us, which indicates that the delay accuracy is still higher than the PHAT–GCC and SCOT–GCC algorithms. Therefore, the PHAT–SCOT–GCC algorithm improves the delay estimation accuracy at low SNR and is more resistant to noise interference.
For far-field measurement of weak signals in complex electromagnetic environments, the SNR is set to decrease linearly from 0 dB to −10 dB in 5 dB steps, and 1000 Monte Carlo experiments are carried out to experimentally estimate the three kinds of weighted functions for each fixed SNR. Meanwhile, the distribution of delay results for the three weighting functions with an SNR of −10 dB is shown in
Figure 5.
Upon examining the distribution of delay results in
Figure 5a–c, the distribution of delay results of PHAT–GCC and SCOT–GCC is more irregular, mainly concentrated around −2 μs and 0 μs, with occurrence counts of −2 μs being 34 and 28 times, respectively. The delay results for the PHAT–SCOT–GCC algorithm mainly center around −2 μs, with 146 occurrences at this value. Therefore, the PHAT–SCOT–GCC algorithm demonstrates more accuracy than the other two algorithms. We consider that the results of each delay estimation will be contingent due to the introduction of random noise. Therefore, based on the characteristics of the distribution of delay results for each algorithm, the absolute value (1 μs) of the median value of −2 μs and 0 μs in the centralized distribution area is chosen as the basis for determining whether the results are correct. We set an interval from −3 μs to −1 μs around the theoretical value of −2 μs, and results outside the interval are considered estimation failures. As a result, an analysis of correctness and root mean square error (RMSE) is necessary. The correctness of each algorithm at different SNRs is shown in
Table 2.
To further investigate the stability of the PHAT–SCOT–GCC algorithm under Gaussian white noise conditions with an SNR of −10 dB to 0 dB, the SNR is set to decrease linearly from 0 dB to −10 dB in 1 dB steps, and 1000 experiments are carried out to experimentally validate three kinds of weighted functions at each fixed SNR. The RMSE of their time delay estimation is shown in
Figure 6.
As shown in
Table 2 and
Figure 6, when the SNR is large (SNR = 0 dB), the PHAT–GCC and SCOT–GCC time delay estimation algorithms are more than 80% correct, and the RMSE is below 0.05 μs. However, the performance of the algorithms of the PHAT–GCC and SCOT–GCC approaches decreases dramatically as the SNR decreases, and their correctness reduces by 20 percentage points at an SNR of −5 dB. While the SNR is further reduced, the performance deterioration is even more severe. The stability of the PHAT–SCOT–GCC algorithm proposed is significantly improved at low SNR. Typically, the correct rate of this algorithm reaches 76% (SNR = −10 dB), which is much higher than that of the PHAT–GCC and SCOT–GCC algorithms. Additionally, the RMSE is much lower than that of the PHAT–GCC and SCOT–GCC algorithms at low SNR. The results reveal that the proposed method effectively improves the stability of the algorithm and the accuracy of time delay estimation.
To simulate the impact of broadcasting, communication, and other periodic narrowband interference on the performance of the PHAT–SCOT–GCC algorithm proposed, we conducted simulations. We take the superposition of three sinusoids of different frequencies to simulate the periodic narrowband interference, represented as follows:
where
Ai is the signal amplitude;
i = 1, 2. F
i represents the oscillation frequency,
i = 1, 2, 3. The values of the parameters in Equation (10) are shown in
Table 3.
As shown in
Figure 2, a mixed interference signal of Gaussian white noise and periodic narrowband noise is superimposed on the original partial discharge signal.
where
x1(
t) is the simulated first partial discharge signal shown in
Figure 2,
n(
t) is the mixed superimposed interference signal, and
X(
t) is the measured signal affected by the simulation.
The SNR of the simulated analog actual signal can be expressed as:
With an SNR = −5 dB, the first simulation mimics the measured partial discharge signal, as shown in
Figure 7.
As mentioned above, the PHAT–SCOT–GCC will be utilized in the following section for time delay estimation of simulated partial discharge signals. The results of time delay estimation using three algorithms (PHAT weighted function, SCOT weighted function, and improved generalized cross-correlation algorithm with joint weighted function) for an SNR of −5 dB are shown in
Figure 8:
As shown in
Figure 8a–c, the PHAT–GCC and SCOT–GCC algorithms cannot accurately calculate the actual delay value when the mixed superposition noise SNR = −5 dB, and the error of their delay results in more than 2 μs. The delay error of the PHAT–SCOT–GCC algorithm is 0.01 μs, which can accurately calculate the delay results. The PHAT–GCC and SCOT–GCC methods increase the false peaks of the cross-correlation function in low SNR, seriously affecting the performance of the algorithm. However, the PHAT–SCOT–GCC, with its mountains of the cross-correlation function, is free from the influence of false peaks, which can be observed intuitively due to the time delay. Therefore, the PHAT–SCOT–GCC algorithm further verifies the accuracy of the time delay estimation results in complex noise environments and is more resistant to noise interference than the PHAT–GCC and SCOT–GCC algorithms.
Again, this is consistent with the conclusions of the above analysis of
Figure 5a–c. The SNR is set to decrease linearly from 0 dB to −10 dB in 5 dB steps, and 1000 experiments are conducted at each fixed SNR for time delay estimation of the three weighted functions. The correct rate (within 1 μs error) of time delay estimation is shown in
Table 4.
To further understand the performance of the PHAT–SCOT–GCC algorithm under mixed noise conditions with an SNR of −10 to 0 dB, the SNR is set to decrease linearly from 0 dB to −10 dB in the 1 dB step. The experimental validation of the three weighted functions is carried out 1000 times for each fixed SNR. The RMSE of time delay estimation is shown in
Figure 9.
From
Table 4 and
Figure 9, it can be seen that the algorithms of the PHAT–GCC and SCOT–GCC perform excellently when the SNR is large and the correct rate is above 90%. As the SNR decreases, the performance of all three algorithms is affected. The performance of the PHAT–GCC and SCOT–GCC algorithms reduces dramatically, with the correctness rate below 20% while the SNR = −10 dB. However, the PHAT–SCOT–GCC algorithm has significantly improved its stability at low SNR; the correct rate of this algorithm reaches 66% when SNR = −10 dB, much higher than the PHAT–GCC and SCOT–GCC. Meanwhile, the RMSE of the PHAT–SCOT–GCC algorithm is lower than the PHAT–GCC and SCOT–GCC algorithms under the SNR from −10 dB to 0 dB.
In summary, under different types of noise interference, the PHAT–SCOT–GCC algorithm has superior anti-interference performance compared with the PHAT–GCC and SCOT–GCC algorithms. The reduction in SNR, the time delay estimation accuracy, and the stability of the PHAT–SCOT–GCC are significantly improved compared with the traditional algorithm.