Result
Before using the filtered noise in our program, we tested their stationarity and correlation as described in the theory section. All of them have passed the ADF test. Their autocorrelations were also plotted and checked. One of the autocorreation plots was included in below.
Figure 3. Autocorrelation functions of two different orders of Chebyshev type I filtered noise.
1. Computer Simulation
In all computer simulations, we were able to obtain the onset time of the signal exactly. To see whether we were able to extract the correct shape of the signal or not, we calculated the mean squared error of the extracted signal from the test signal. In addition, we also plotted the extracted signal versus the actual signal and did a linear fit to obtain an R2 value. Typically these were rather high (around 0.9). In the following example taken from one of the simulations, the onset time was set to be at data point 4000. Two following plots of the signal and noise-corrupted signal show that the signal was well buried in noise. It would be hard to tell it apart from the noise in both time and frequency domain.
Figure 4. The plot on the left is the time series data of both noise corrupted signal and the test signal. The plot on the right is the power spectrum of both signal and noise from point 4000 to 4500
After running our AR model program with time lag 20 on the data, the error plot in below was obtained.
Figure 5. The plot of error with different magnifications obtained from subtracting the prediction of the noise off the noise corrupted signal. The rightmost is the most zoomed-in plot. The blue line is the error, orange line is the original signal.
As shown in the above figure, the presence of the error peak at 4020 gave us an estimation of the onset at point 4000 which is exactly the onset time of our test signal. Moreover, the extracted signal is very close to the test signal both in shape and in magnitude. From this simulation, we obtained an R2 of 0.94, and root mean squared error of 0.1523. These values gave us a quantitative justification that our extracted signal is indeed quite close to the test signal.
2. Audio Experiment
The same type of analysis as in the computer simulation was applied.
Figure 6. The time series plot of the noise-corrupted signal and the recorded signal (left); the power spectrum of the noise-corrupted signal and the recorded signal (right). The blue line represents the noise-corrupted signal, the orange line is the recorded signal. As we can see, the signal was well buried in noise in both time and frequency domain.
We applied an AR model with time lag 20, and obtained the following error plot.
Figure 7. The plot of error from audio experiments with different magnifications. The orange line is the error, the blue line is the recorded signal.
We obtained an onset time for this signal of 3.502s, which was within the standard deviation of 2ms of the actual onset time of 3.5s. The presence of two error peaks is a side effect of using sine wave instead of exponentially decaying sine. Notice that the noise alone, the signal (pure sine wave) alone, and the combination of both noise and signal are all stationary. However, the transition among the three states are not. It was the transition that was responsible for the error peaks in this experiment. Since the stationarity was violated twice: one by the introduction of the signal, the other by the removal of the signal, there should be two error peaks which corresponded to the start and end time of the signal. Nevertheless, the signal was still extracted with shape close to the recorded signal. When plotting the extracted signal verses the recorded signal and doing a linear fit, we obtained an R2 value of 0.82, which indicates that our extracted signal has a roughly similar shape to the recorded signal.