Abstract
The existing method of the weak signal under the background of strong noise (signal amplitude relative to the noise amplitude is much smaller) analysis is not very ideal. This study proposes an improved phase lock method to analyze the weak periodic signal under Gaussian white noise. Experimental results show that this method can better analyze weak signal-related information than the stochastic resonance method in a certain range of signal-to-noise ratio, and the amount of calculation is small, and the correlation theory is simple and adapted to the rapid detection of the signal; in order to further reduce the amount of calculation, the introduction of a high-order accumulation of the signal presence detection, experiments verify that the high-order accumulation can detect whether there is a weak signal in the Gaussian noise. The combination of high-order cumulant and improved phase lock method has a good effect on the analysis of weak periodic signals under strong noise backgrounds.
1. Introduction
In actual engineering, the Fourier algorithm is often used for signal spectrum analysis, but the results of the Fourier algorithm analysis of signals collected in some environments are not ideal, such as vibration and sound signals, when shield machines are working, and locomotive running parts. Due to the large background noise energy of vibration and sound signals, etc., the valuable signal energy is relatively small. The analysis result of the signal is mainly dominated by noise. At this time, Fourier analysis is powerless for such signals, so many scholars are aiming at a lot of research that has been performed on this kind of signal, and the algorithms for weak signal extraction are mainly divided into two types of algorithms: linear detection method and nonlinear detection method.
The linear detection method mainly suppresses noise to realize the weak signal detection. The main methods are multivariate time series [1], K-means clustering [2], and variational mode decomposition [3]; time-series predictive analysis is to use the characteristics of the signal in the past period of time to predict the characteristics of the signal in the future period of time. Its data analysis methods are mainly traditional methods and machine learning methods. The K-means algorithm is a clustering method based on partitioning. In the dataset, K points are selected as the initial center of each cluster according to a certain strategy, and then, the remaining data are observed, and the data are divided into the clusters closest to the K points. After calculating the center point of each cluster for many times, it is divided until the result of each division remains unchanged. VMD is a completely nonrecursive variational mode decomposition method. It decomposes a signal into some discrete component signals, and most of each mode is closely around the center frequency and then uses the quadratic penalty function term and the Lagrange multiplier operator to get an unconstrained problem is solved, and finally the problem is solved to realize the signal analysis.
The nonlinear detection method mainly realizes the weak signal detection through the interaction between the established system and the weak signal [4, 5]. The main methods include stochastic resonance and chaotic oscillator method [6], the stochastic resonance method passes the signal through the interaction with the nonlinear bistable system potential function that produces a synergistic phenomenon, so that the entire system finally runs with a weak signal frequency, and then, the frequency detection is achieved through Fourier analysis, the stochastic resonance method is used in practical applications, and the a and b parameters of the nonlinear bistable system potential function will seriously affect the accuracy of detection and also require the intensity of background noise, which limits its application. The chaotic operator method mainly uses the numerical analysis method for weak signal detection, but it also has a large amount of calculation and is easily affected by noise changes.
In this study, we propose an improved phase lock method for weak signal analysis. The linear method mentioned above is mainly to classify the signal, and there are certain limitations in the specific analysis of the signal. The nonlinear method is theoretically complex and easily affected by the parameters of the differential equation in the model. The method proposed in this study is mainly to analyze the weak signal under the background of strong noise under certain restrictions, which provides a basis for subsequent data classification and fault diagnosis.
The organization of this study is as follows: the second part introduces the knowledge of high-order cumulants. The third part introduces the carrier method, and the carrier method and the cyclic phase method commonly used in the communication system are combined to judge the frequency of the characteristic signal. The fourth part introduces the content of using correlation coefficient and cyclic phase method to judge the amplitude and phase of the characteristic signal. The fifth part is the experimental part. The sixth part summarizes the experimental conclusions and the assumptions and limitations of using this method.
2. Higher-Order Cumulants
Theoretically, the higher-order cumulant [7–11] of the Gaussian random process is 0. The higher-order cumulant used in this study is described below.
2.1. Higher-Order Cumulants
It is assumed that the probability density function of a continuous scalar random variable is p(x), and its first characteristic function is as follows:where is the expectation.
The Taylor expansion of formula (1) is as follows:
The second characteristic function is obtained from the first characteristic function:
From formulas (2) and (3), the Taylor expansion of formula (3) is as follows:
The first-, second-, third-, and fourth-order cumulants can be obtained by formula (4):where , , , and are the 1st-, 2nd-, 3rd-, and 4th-order cumulative moment of the real-valued random variable x.
For a steady-state random process with zero mean, the second, third, and fourth-order cumulants are defined as follows:where are time shifts.
In practical applications, for a one-dimensional array with N sampling points, the formula for calculating the fourth-order cumulant is as follows:where are offsets.
2.2. The Use of Fourth-Order Cumulants
The theoretical value of the fourth-order cumulant of Gaussian white noise is zero. In order to verify the function of the fourth-order cumulant in this study, a noise signal and a sinusoidal signal are generated. The signal-to-noise ratio of the sinusoidal signal and noise is −35, which is verified by experiments. When the signal-to-noise ratio reaches above −30, the Fourier algorithm has no meaning for the analysis results of the characteristic signal. At the same time, the Fourier algorithm cannot detect whether there is a characteristic signal in the signal. Although the fourth-order cumulant in this study cannot be used for a specific analysis of characteristic signals, it can be used to detect the existence of characteristic signals according to the nature of the fourth-order cumulant. Now the fourth-order cumulant analysis is performed on the mixed signal of noise, sinusoidal signal, and noise (signal-to-noise ratio −35), and the analysis result is shown in Figure 1.
It can be seen from Figure 1 that the results of the fourth-order cumulant analysis of noise and mixed signals are not strictly zero. Further analysis shows that if the noise amplitude is large in practical applications, the value of the fourth-order cumulant will also become larger. At this time, if the analysis value of the fourth-order cumulant is expected to approach zero, the noise and characteristic weak signal energy value can be reduced in the same proportion. However, in this study, the value of the fourth-order cumulant that tends to zero is not strictly required. The application of the fourth-order cumulant is mainly used to detect the existence of characteristic signals. It can be seen from Figure 1 that the fourth-order cumulant values of noise and mixed signals are different. In order to further illustrate this difference, a fourth-order cumulant analysis is performed by combining the noise signal and the mixed signal. The result is shown in Figure 2.
In Figure 2, the red mark is the result of the fourth-order cumulant analysis of the mixed signal, and the nonred mark is the result of the fourth-order cumulant analysis of the noise. It can be seen from Figure 2 that the fourth-order statistics can be used to detect the collected signal whether there is a characteristic weak signal, thereby reducing the amount of data for subsequent analysis.
However, it can also be seen from the analysis of Figure 2 that high-order statistics cannot achieve accurate measurement of weak signal frequency, phase, and amplitude in weak signal analysis. In order to accurately measure the above three parameters, this study uses an improved phase lock method to measure the characteristics of the signal analyzed.
3. Feature Weak Signal Frequency Detection Method
3.1. Carrier Method
The carrier is to load the test signal on the characteristic signal of a certain frequency [12, 13]. When the test signal is not loaded, the amplitude of the characteristic signal is fixed. After loading, the amplitude, phase, and frequency of the characteristic signal change with the change in the loaded signal. For the test signal, the determination of its frequency should be based on the sampling theorem and the frequency estimation of the measured characteristic signal, according to the sampling theorem, if the frequency of your test signal is too large, or if the frequency of the characteristic signal after your operation cannot be reproduced due to the limitations of the sampling theorem, it will lead to inaccurate operations, so a high-frequency signal should not be tried to be chosen as a test signal. This section analyzes the frequency of carrier detection in different signal cases.
3.1.1. Sinusoidal Signal
The characteristic signal as is set as , and the loading signal as is set as :
When the loading signal is loaded to the characteristic signal, then
The spectrum diagrams of and are shown in Figures 3 and 4.
It can be seen from the carrier method that after a certain frequency () signal is loaded on the target signal, if there is a periodic signal (frequency ) in the target signal, the frequencies of and have related effects on the amplitude and phase. This study will use this feature to detect weak signal frequency under strong noise background.
3.1.2. Nonsine Periodic Signal
For nonsinusoidal periodic signals, this section analyzes two typical signals, namely, the triangular wave signals and square wave signals.
(1) Triangular Wave Signal. Suppose the triangular wave signal expression is shown in the equation, then
According to the Fourier decomposition formula,
Triangular waves are decomposed by Fourier transform.
Since is an even function, , and are just calculated:
Therefore, the expression after Fourier decomposition of the triangular wave is as follows:
(2) Square Wave Signal. It is assumed that the square wave signal expression is as follows:
According to equation (12), the square wave is decomposed by Fourier transform:
Therefore, the expression after Fourier decomposition of the square wave is as follows:
(3) Periodic Shock Signal. It is assumed that the periodic shock signal expression is as follows:
According to equation (12), periodic shock is decomposed by Fourier transform:
3.1.3. Aperiodic Signal
It is assumed that the rectangular pulse signal expression is as follows:
According to equation (12), rectangular pulse signal is decomposed by Fourier transform:
From the analysis of Section 3.1, it can be seen that for sine signals, the application of the carrier method is of no problem, for nonsinusoidal periodic signals, according to the Fourier decomposition, because the nonsine periodic signal can be decomposed into the superposition of multiple sinusoidal signals, the carrier method application and the sine signal are not different but for nonsinusoidal periodic pulse signals, because they are decomposed as a periodic constant, it is not suitable for the conditions of carrier method application, for nonperiodic signals, the spectrum after Fourier decomposition is continuous, and it is also suitable for the carrier method analysis under certain conditions.
3.2. Frequency Measurement by Carrier and Cyclic Phase Method
3.2.1. Cyclic Phase Method
This study uses a combination of carrier method and cyclic phase method to measure the frequency, the cyclic phase method is to make the phase of the detection signal to periodically change, that is, the of equation (9) continuously changes within , if the continuously changes in in a fixed step , as can be seen from equation (10), the , and will change in the same step in . That is, the phase change step of the carrier signal is always .
3.2.2. Frequency Measurement
Assuming that the characteristic signal has been given by equation (8), noise is added to the characteristic signal, and the signal-to-noise ratio is -35. After the characteristic signal is added to the noise, the time domain image is shown in Figure 5.
The Fourier transform result of the noisy signal is shown in Figure 6.
It can be seen from Figure 6 that the Fourier transform results of weak signals under strong noise backgrounds are not ideal. Any method that uses the difference in energy to extract weak signals is not ideal under strong noise backgrounds. Based on the above analysis, this study’s carrier method is used to detect the weak signal frequency. The specific method is as follows: this study multiplies the noisy signal by a sine signal of a certain frequency. According to the relevant knowledge of the carrier, if there is a characteristic weak signal with a frequency of in the noisy signal, after it is multiplied by a certain frequency signal, the phases at and and the phase of the multiplied signal can be known from equation (10). Let the phase of the signal with frequency be , and the phase of the signal with frequency be , then from equation (10) we can know and phases are and , but in practice due to the influence of noise, the phase value does not strictly follow the theoretical value above, but the phase values of and frequency change with the phase change of frequency , when the phase of the signal with frequency changes from -2π to 2π. The step size is 0.1π, the phase change of the signal with frequencies and is shown in Figure 7.
(a)
(b)
Figure 7(a) shows the phase change at the frequency of and as the phase of the loaded signal changes. Figure 7(b) shows load signal phase changes and the magnitude of the phase at the and .
It can be seen from Figure 7 that because the step size of the loading signal phase change is 0.1π, according to the relevant knowledge of the loading signal mentioned earlier in this study, the step length of the phase change at the and should be 0.1π, but due to the large noise intensity, the influence of the noise phase cannot be ignored, but even if there is the influence of the noise phase, the step length changes around 0.1π, and the two signs of the phase amplitudes at two frequencies are opposite in the same time period. The opposite of the phases at the two frequencies is analyzed below.
It is assumed that the signal at frequency is , the signal at frequency is , and the phases of and signals are as follows:
It can be seen from equations (22) and (23) that the signs of the increments of the changes in the signals and are opposite, so the signs of the phase amplitudes of and are basically opposite in the same period.
If it is not the and signals, the signal is selected at any frequency (not the and frequencies) and the phase change of the signal is compared, as shown in Figure 8.
(a)
(b)
It can be seen from Figure 8 that the phase change of the signal at any frequency (not and ) and the phase of do not have similar regularity, and the phase change step at frequencies other than and does not change around 0.1π. There is no regularity.
Any two frequency signals (not and frequencies) are selected to observe the changes in their phases, as shown in Figure 9.
(a)
(b)
It can be seen from Figure 9 that the phase changes of any two frequency signals (not and frequencies) have no regularity, and since the phases of the not and frequencies have not theoretically changed, the change step is 0. This can be seen from Figure 9(a), and the change step is basically located near the 0 point. According to the regular changes of the phases of the not and frequencies when the carrier is used, the frequency of the weak signal under the strong noise background can be detected.
4. Detection Method of Weak Signal Amplitude and Phase
4.1. Cross-Related Knowledge
The correlation function of the two signals 和 iswhere is the correlation coefficient, is the expectation, and is the joint probability density.
It can be seen from equation (24) that due to the relative independence between the periodic signal and the noise signal, the theoretical value of the correlation coefficient between the periodic signal and the noise signal is 0, that is,
The correlation coefficient of two periodic signals with different frequencies is
This study extracts the amplitude and phase of the weak signal according to equations (24)–(26). After the periodic signal and the periodic signal mixed with noise are multiplied and integrated, the signal is detected from the noise, and the purpose of reducing the influence of the noise through the cross-correlation operation is achieved. Because of the various signal components contained in the measured signal, the frequency of the characteristic signal is only related to the frequency of the input periodic signal and is not related to the frequency of the random noise , and the characteristic signal and the random noise are independent of each other, so the theoretical value of their correlation coefficient is zero, that is, .
It is specifically explained through the following analysis.
The expressions for the measured signal and the test signal are shown in equations (27) and (28):
From equations (24)–(28), the correlation coefficient expression of the signal under test is as follows:
The following is calculated as , , and , respectively:
So the result of is as follows:
Combining the cyclic phase method with the associated coefficient method,
Hence, the , gets the maximum value when .
It can be seen from equation (32) that the value of depends on the amplitude and phase of the characteristic signal and the input periodic signal. From equation (32), we can see that when , , equation (15) becomes the following:
It can be seen from equation (33) that when the input periodic signal amplitude , , so , then the magnitude of the corresponding amplitude phase is a weak signal phase.
4.2. Correlation Coefficient Method Experiment
In this part, the signal shown in Figure 5 is the detected signal , the characteristic signal amplitude in is 0.02, the initial phase angle is 0.3π, the description of the input periodic signal is equation (9), its amplitude is 1, the phase angle variation range is , the step size is 0.1π, and the weak signal amplitude and phase detection method are used. The detection results are shown in Figures 10 and 11.
In Figures 10 and 11, the abscissa is the number of steps, and the ordinate is the amplitude. It can be seen from Figure 11 that the minimum and maximum values appear at the number of steps 3 and 13, and the phases at the number of steps 3 and 13 are and . The amplitudes are −0.0098 and 0.0098.
According to the theory of weak signal amplitude and phase detection method, the characteristic signal amplitude is 0.0196 and the phase is 0.3π; the actual amplitude of the characteristic signal is 0.02 and the phase angle is 0.3π, so the detection structure has relatively high accuracy.
5. Experiments
This study combines the weak signal existence detection method, the weak signal frequency detection method, and the weak signal amplitude and phase detection method in the big data analysis program [14–16]. The program flow chart is shown in Figure 12.
A soft threshold is added to the weak signal frequency detection method in the program. Since there are abnormal points in the frequency detection process, as shown in Figures 7–9, the threshold structure is added, but the threshold size is not fixed. According to the detection abnormality, the size of the value varies, and the threshold value of different analysis data is different so that the frequency can be accurately detected. The threshold expression is as follows:where is the data output after the threshold structure, is the input data, and is the size of the threshold.
The detailed flow chart of the weak signal frequency detection method is shown in Figure 13.
Due to the difference in signal-to-noise ratio, sometimes the amplitude and phase detection will deviate from the true value. Therefore, in the actual application of this program, the amplitude and phase detection method can be reduced by multiple detections to reduce the error between the detected value and the true value.
This section is divided into two parts. The first part analyzes sinusoidal and nonsinusoidal signals with the improved phase lock method, and the second part compares the improved phase lock method with the stochastic resonance method.
5.1. Experimental Analysis of Sinusoidal and Nonsinusoidal Signals
5.1.1. Sinusoidal Signal
In this section, sinusoidal signals are characteristic signals with three different frequencies added to Gaussian white noise, that is, the mixed signal is as follows:
The periodic function of the input is as follows:where , , , , , , , , , and .
The calculation formula of the signal-to-noise ratio is as follows:where is the power of the mixed signal, and is noise power.
In this section, three groups of test signals test 1, test 2, and test 3 are selected, with signal-to-noise ratios of −35, −45, and −55, respectively. The specific analysis data are shown in Tables 1 and 2.
5.1.2. Triangular Wave Test signal (Nonsinusoidal Signal)
In this section, the expression of the triangular wave signal is given by equation (11), The triangular amplitude value is 0.025, . The signal-to-noise ratio is −35. The time domain diagram is shown in Figure 14.
Therefore, the expression after Fourier decomposition of the triangular wave is as follows:
It can be seen from equation (22) that when the sinusoidal function is input into the system, it is equivalent to a series of sinusoidal functions input into the system, and the amplitude, frequency, and phase of the signal can still be determined by the method proposed in this study.
5.1.3. Signal Test
The signals and are tested by the method proposed in this study, and the test results are shown in Figures 15–18.
(a)
(b)
(c)
Figures 15–17 are the analysis diagrams of signal when the signal-to-noise ratio is −35, −45, and −55, and Figure 18 is the analysis diagram of signal when the signal-to-noise ratio is −35. Figure 18(a) is an analysis image of 20 Hz, Figure 18(b) is an analysis image of 60 Hz, and Figure 18(c) is an analysis image of 100 Hz. However, because the energy of 100 Hz signal is too small, the detection of frequency and amplitude is failed. The specific analysis results are shown in Table 3.
It can be seen from Table 3 that for signal , when , the detection result accuracy is quite high, and the maximum error is about 20%, but the probability of maximum error is not very large; when the noise is added to the nonsinusoidal signal to make its signal-to-noise ratio reach −35, the input system has high accuracy for the sinusoidal component with amplitude of 0.022 and frequency of 20 Hz, and there is a large error for the detection results of other components. When the frequency is greater than , the amplitude becomes very small, and its signal-to-noise ratio far exceeds −50. However, it can be seen from Figure 18(b) that there is no problem in 60 Hz frequency detection.
5.2. Data Analysis and Comparison Experiments of the Two Models
In this section, the stochastic resonance model [17–22] and the improved phase lock method are used for comparative analysis. The stochastic resonance model [6, 23] used in this section is as follows:
The stochastic resonance model is normalized, and the transformed model is as follows:where
The input signal is as follows:where , , is the noise signal, and the signal-to-noise ratio of is −35.
The signal is input into the improved phase lock method and stochastic resonance model, respectively, and its analysis diagram is shown in Figure 19.
(a)
(b)
As can be seen from Figure 19, Figure 19(a) can accurately test the characteristic signal, as can be seen from Figure 19(b), but when the signal-to-noise ratio reaches −35, the analysis result of the stochastic resonance model is not satisfactory, which is because the stochastic resonance model has high requirements on noise and parameters, and it is not universal.
6. Conclusions
The improved phase lock method proposed in this study has obvious advantages in the analysis of noisy signals: the theory is clear, and the implementation is simple; first, the detection of the existence of weak signals is realized by the fourth-order cumulant's different calculation results of Gaussian white noise and periodic function; second, carrier theory analyzes the phase change and change step length of 和 frequency to determine the weak signal frequency; third, correlation analysis is used to determine weak signal amplitude and phase; combined with big data and neural network programs, it is found through experiments that it has better detection results. It is suitable for weak periodic signal extraction under strong noise background and has certain practical significance.(1)A method of analyzing weak signal with phase parameter is proposed, which overcomes the disadvantage that weak signal amplitude is not easy to be detected from noise amplitude(2)The carrier method is applied to the analysis of weak signals, which transforms the analysis of a weak signal of a certain frequency into a dual-frequency analysis, which is beneficial to reduce the error of the analysis of a weak signal of a certain frequency(3)The cyclic phase method is proposed to overcome the interference of the noise phase to the weak signal phase(4)The correlation coefficient method uses the principle that the correlation coefficient of the same frequency is not 0, and the correlation coefficient of different frequencies is 0 to analyze the weak signal amplitude, and proposed method combining the cyclic phase method and the correlation coefficient method solves the problem that the amplitude and phase of weak signals are not easy to determine(5)When the signal-to-noise ratio SNR ≥ −50, the accuracy of detection results can be controlled below 20%
Data Availability
The data of the manuscript is already included in this manuscript, or the data can be obtained through https://github.com/loubishi/The-repository-of-Jason.
Conflicts of Interest
The authors declare that they have no conflicts of interest.