# Application of Improved Phase Lock Method in Weak Signal Extraction

Authors: Haiyang Lou; Rujiang Hao; Jianchao Zhang; Qiang Xue
Journal: Mathematical Problems in Engineering
 (2022)

---

## 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.


---

## Body

## 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 variablex is p(x), and its first characteristic function φx is as follows:(1)φx=Eejωx=∫−∞∞pxejωxdx,where E… is the expectation.The Taylor expansion of formula (1) is as follows:(2)φx=∫−∞∞∑k=0∞xkjωkk!px=∑k=0∞Exkjωkk!.The second characteristic function is obtained from the first characteristic function:(3)ϕx=lnφω=lnEexpjωx.From formulas (2) and (3), the Taylor expansion of formula (3) is as follows:(4)ϕx=∑k=0∞ckjωkk!.The first-, second-, third-, and fourth-order cumulants can be obtained by formula (4):(5)c1x1=Ex1,c2x1,x2=Ex1x2−Ex1Ex2,c3x1,x2,x3=Ex1x2,x3−Ex1Ex2,x3−Ex2Ex1,x3−Ex3Ex1,x2+2Ex1Ex2Ex3,c4x1,x2,x3,x4=Ex1x2x3x4−Ex1x2Ex2x3−Ex1x2Ex3x4−Ex1x3Ex2x4−Ex1Ex2x3x4−Ex2Ex1x3x4−Ex3Ex1x2x4−Ex4Ex1x2x3+2Ex1x2Ex3Ex4+2Ex1x3Ex2Ex4+2Ex1x4Ex2Ex4+2Ex2x3Ex1Ex4+2Ex2x4Ex1Ex3+2Ex3x4Ex1Ex2+6Ex1Ex2Ex3Ex4.where c1, c2, c3, and c4 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:(6)c2τ=Ext⋅xt+τ,c3τ1,τ2=Ext⋅xt+τ1⋅xt+τ2,c4τ1,τ2,τ3=Ext⋅xt+τ1⋅xt+τ2−c2τ1c2τ2−τ3−c2τ2c2τ3−τ1−c2τ3c2τ1−τ2,where τ1,τ2…τn 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:(7)c4k,l,m=1N∑n=0N−1xn⋅xn+k∗⋅xn+l∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+k∗∑n=0N−1xn+l∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+l∗∑n=0N−1xn+k∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+m∗∑n=0N−1xn+k∗⋅xn+l∗,where k,l,m 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 Figure1.Figure 1
Fourth-order cumulant analysis of noise and mixed signal.It can be seen from Figure1 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.Figure 2
Fourth-order cumulant analysis of noise and mixed signal.In Figure2, 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 Figure2 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.


## 2.1. Higher-Order Cumulants

It is assumed that the probability density function of a continuous scalar random variablex is p(x), and its first characteristic function φx is as follows:(1)φx=Eejωx=∫−∞∞pxejωxdx,where E… is the expectation.The Taylor expansion of formula (1) is as follows:(2)φx=∫−∞∞∑k=0∞xkjωkk!px=∑k=0∞Exkjωkk!.The second characteristic function is obtained from the first characteristic function:(3)ϕx=lnφω=lnEexpjωx.From formulas (2) and (3), the Taylor expansion of formula (3) is as follows:(4)ϕx=∑k=0∞ckjωkk!.The first-, second-, third-, and fourth-order cumulants can be obtained by formula (4):(5)c1x1=Ex1,c2x1,x2=Ex1x2−Ex1Ex2,c3x1,x2,x3=Ex1x2,x3−Ex1Ex2,x3−Ex2Ex1,x3−Ex3Ex1,x2+2Ex1Ex2Ex3,c4x1,x2,x3,x4=Ex1x2x3x4−Ex1x2Ex2x3−Ex1x2Ex3x4−Ex1x3Ex2x4−Ex1Ex2x3x4−Ex2Ex1x3x4−Ex3Ex1x2x4−Ex4Ex1x2x3+2Ex1x2Ex3Ex4+2Ex1x3Ex2Ex4+2Ex1x4Ex2Ex4+2Ex2x3Ex1Ex4+2Ex2x4Ex1Ex3+2Ex3x4Ex1Ex2+6Ex1Ex2Ex3Ex4.where c1, c2, c3, and c4 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:(6)c2τ=Ext⋅xt+τ,c3τ1,τ2=Ext⋅xt+τ1⋅xt+τ2,c4τ1,τ2,τ3=Ext⋅xt+τ1⋅xt+τ2−c2τ1c2τ2−τ3−c2τ2c2τ3−τ1−c2τ3c2τ1−τ2,where τ1,τ2…τn 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:(7)c4k,l,m=1N∑n=0N−1xn⋅xn+k∗⋅xn+l∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+k∗∑n=0N−1xn+l∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+l∗∑n=0N−1xn+k∗⋅xn+m∗−1N2∑n=0N−1xn⋅xn+m∗∑n=0N−1xn+k∗⋅xn+l∗,where k,l,m 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 Figure1.Figure 1
Fourth-order cumulant analysis of noise and mixed signal.It can be seen from Figure1 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.Figure 2
Fourth-order cumulant analysis of noise and mixed signal.In Figure2, 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 Figure2 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 assf, and the loading signal as is set as st:(8)sf=A1∗sinn1∗π∗f∗t+φ1,(9)st=A2∗sinn2∗π∗f∗t+φ2,When the loading signal is loaded to the characteristic signal, then(10)st∗sf=A1∗sinn1∗π∗f∗t+φ1∗A2∗sinn2∗π∗f∗t+φ2=12A1A2cosn1−n2∗π∗f∗t+φ1−φ2+cosn1+n2∗π∗f∗t+φ1+φ2.The spectrum diagrams ofsf and st∗sf are shown in Figures 3 and 4.Figure 3
The spectrogram ofsf.Figure 4
The spectrogram of load signal (st∗sf).It can be seen from the carrier method that after a certain frequency (n2∗f) signal is loaded on the target signal, if there is a periodic signal (frequency n1∗f) in the target signal, the frequencies of n1−n2∗f and n1+n2∗f 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 st is shown in the equation, then(11)st=A2+AT0/2t−T2≤t≤0,A2−AT0/2t0≤t≤T2..According to the Fourier decomposition formula,(12)ft=a0+∑n=1∞ancosnω1t+bnsinnω1t.Triangular waves are decomposed by Fourier transform.Sincest is an even function, bn=0, a0 and an are just calculated:(13)a0=1T∫−T0/2T0/2xtdt=2T∗AT04−AT04=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/22∗A2−2AT0tcosnω0tdt=−2An2π2cosnπ−1=2An2π2sin2nπ2=2An2π2n=1,3,5,…,0n=2,4,6….Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(14)Sω=2Aπ2cosω0t+132cos3ω0t+152cos5ω0t+….(2) Square Wave Signal. It is assumed that the square wave signal expression is as follows:(15)xt=A−T2≤t≤0,−A0≤t≤T2.According to equation (12), the square wave is decomposed by Fourier transform:(16)a0=1T∫−T0/2T0/2xtdt=1T∫−T0/20Adt−1T∫0T0/2Adt=A2−A2=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/2−Acosnω0tdt+2T∫−T0/20Acosnω0tdt=0,bn=2T∫−T0/2T0/2xtsinnω0tdt=2T∫0T0/2−Asinnω0tdt+2T∫−T0/20Asinnω0tdt=2AT∫−T0/20sinnω0tdt−∫0T0/2sinnω0tdt=2Anπ2−2cosnπ=0n=2,4,6,…,4Anπn=1,3,5,….Therefore, the expression after Fourier decomposition of the square wave is as follows:(17)xt=4Aπsinω0t+13sin3ω0t+15sin5ω0t+….(3) Periodic Shock Signal. It is assumed that the periodic shock signal expression is as follows:(18)δt=∑n=−∞n=∞δt−nT,n=0,±1,±2,….According to equation (12), periodic shock is decomposed by Fourier transform:(19)Xω=1T∫−τ/2τ/2δte−jnωtdt=1T.


#### 3.1.3. Aperiodic Signal

It is assumed that the rectangular pulse signal expression is as follows:(20)xt=At≤τ2,0t≥τ2.According to equation (12), rectangular pulse signal is decomposed by Fourier transform:(21)Xω=∫−τ/2τ/2Ae−jωtdt=2Aωsinωτ2.From the analysis of Section3.1, 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 signalst to periodically change, that is, the φ2 of equation (9) continuously changes within −nπ,nπ, if the φ2 continuously changes in −nπ,nπ in a fixed step h, as can be seen from equation (10), the φ1−φ2, and φ1+φ2 will change in the same step h in −nπ,nπ. That is, the phase change step of the carrier signal is always h.


#### 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.Figure 5
Time domain plot of the noisy signal.The Fourier transform result of the noisy signal is shown in Figure6.Figure 6
Fourier transform result of the noisy signal.It can be seen from Figure6 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 n1∗f in the noisy signal, after it is multiplied by a certain frequency n2∗f signal, the phases at n1−n2∗f and n1+n2∗f and the phase of the multiplied signal can be known from equation (10). Let the phase of the signal with frequency n2∗f be φ2, and the phase of the signal with frequency n1∗f be φ1, then from equation (10) we can know n1−n2∗f and n1+n2∗f phases are φ1−φ2 and φ1+φ2, 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 n1−n2∗f and n1−n2∗f frequency change with the phase change of frequency n2∗f, when the phase of the signal with frequency n2∗f changes from -2π to 2π. The step size is 0.1π, the phase change of the signal with frequencies n1−n2∗f and n1+n2∗f is shown in Figure 7.Figure 7
Then1−n2∗f and n1+n2∗f phases change at frequency.
(a)(b)Figure7(a) shows the phase change at the frequency of n1−n2∗f and n1+n2∗f as the phase of the loaded signal changes. Figure 7(b) shows load signal phase changes and the magnitude of the phase at the n1−n2∗f and n1+n2∗f.It can be seen from Figure7 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 n1−n2∗f and n1+n2∗f 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 atn1−n2∗f frequency is s1, the signal at n1+n2∗f frequency is s2, and the phases of s1 and s2 signals are as follows:(22)s1:φ1−φ2∈−2π,2π=φ1+φ2′∈2π,−2π,(23)s2:φ1+φ2∈−2π,2π.It can be seen from equations (22) and (23) that the signs of the increments of the changes in the signals s1 and s2 are opposite, so the signs of the phase amplitudes of s1 and s2 are basically opposite in the same period.If it is not thes1 and s2 signals, the signal is selected at any frequency (not the s1 and s2 frequencies) and the phase change of the s2 signal is compared, as shown in Figure 8.Figure 8
Phase change of the signal at any frequency and the phase of s2.
(a)(b)It can be seen from Figure8 that the phase change of the signal at any frequency (not s1 and s2) and the phase of s2 do not have similar regularity, and the phase change step at frequencies other than s1 and s2 does not change around 0.1π. There is no regularity.Any two frequency signals (nots1 and s2 frequencies) are selected to observe the changes in their phases, as shown in Figure 9.Figure 9
The phase change of nots1 and s2 frequencies.
(a)(b)It can be seen from Figure9 that the phase changes of any two frequency signals (not s1 and s2 frequencies) have no regularity, and since the phases of the not s1 and s2 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 s1 and s2 frequencies when the carrier is used, the frequency of the weak signal under the strong noise background can be detected.


## 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 assf, and the loading signal as is set as st:(8)sf=A1∗sinn1∗π∗f∗t+φ1,(9)st=A2∗sinn2∗π∗f∗t+φ2,When the loading signal is loaded to the characteristic signal, then(10)st∗sf=A1∗sinn1∗π∗f∗t+φ1∗A2∗sinn2∗π∗f∗t+φ2=12A1A2cosn1−n2∗π∗f∗t+φ1−φ2+cosn1+n2∗π∗f∗t+φ1+φ2.The spectrum diagrams ofsf and st∗sf are shown in Figures 3 and 4.Figure 3
The spectrogram ofsf.Figure 4
The spectrogram of load signal (st∗sf).It can be seen from the carrier method that after a certain frequency (n2∗f) signal is loaded on the target signal, if there is a periodic signal (frequency n1∗f) in the target signal, the frequencies of n1−n2∗f and n1+n2∗f 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 st is shown in the equation, then(11)st=A2+AT0/2t−T2≤t≤0,A2−AT0/2t0≤t≤T2..According to the Fourier decomposition formula,(12)ft=a0+∑n=1∞ancosnω1t+bnsinnω1t.Triangular waves are decomposed by Fourier transform.Sincest is an even function, bn=0, a0 and an are just calculated:(13)a0=1T∫−T0/2T0/2xtdt=2T∗AT04−AT04=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/22∗A2−2AT0tcosnω0tdt=−2An2π2cosnπ−1=2An2π2sin2nπ2=2An2π2n=1,3,5,…,0n=2,4,6….Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(14)Sω=2Aπ2cosω0t+132cos3ω0t+152cos5ω0t+….(2) Square Wave Signal. It is assumed that the square wave signal expression is as follows:(15)xt=A−T2≤t≤0,−A0≤t≤T2.According to equation (12), the square wave is decomposed by Fourier transform:(16)a0=1T∫−T0/2T0/2xtdt=1T∫−T0/20Adt−1T∫0T0/2Adt=A2−A2=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/2−Acosnω0tdt+2T∫−T0/20Acosnω0tdt=0,bn=2T∫−T0/2T0/2xtsinnω0tdt=2T∫0T0/2−Asinnω0tdt+2T∫−T0/20Asinnω0tdt=2AT∫−T0/20sinnω0tdt−∫0T0/2sinnω0tdt=2Anπ2−2cosnπ=0n=2,4,6,…,4Anπn=1,3,5,….Therefore, the expression after Fourier decomposition of the square wave is as follows:(17)xt=4Aπsinω0t+13sin3ω0t+15sin5ω0t+….(3) Periodic Shock Signal. It is assumed that the periodic shock signal expression is as follows:(18)δt=∑n=−∞n=∞δt−nT,n=0,±1,±2,….According to equation (12), periodic shock is decomposed by Fourier transform:(19)Xω=1T∫−τ/2τ/2δte−jnωtdt=1T.


### 3.1.3. Aperiodic Signal

It is assumed that the rectangular pulse signal expression is as follows:(20)xt=At≤τ2,0t≥τ2.According to equation (12), rectangular pulse signal is decomposed by Fourier transform:(21)Xω=∫−τ/2τ/2Ae−jωtdt=2Aωsinωτ2.From the analysis of Section3.1, 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.


## 3.1.1. Sinusoidal Signal

The characteristic signal as is set assf, and the loading signal as is set as st:(8)sf=A1∗sinn1∗π∗f∗t+φ1,(9)st=A2∗sinn2∗π∗f∗t+φ2,When the loading signal is loaded to the characteristic signal, then(10)st∗sf=A1∗sinn1∗π∗f∗t+φ1∗A2∗sinn2∗π∗f∗t+φ2=12A1A2cosn1−n2∗π∗f∗t+φ1−φ2+cosn1+n2∗π∗f∗t+φ1+φ2.The spectrum diagrams ofsf and st∗sf are shown in Figures 3 and 4.Figure 3
The spectrogram ofsf.Figure 4
The spectrogram of load signal (st∗sf).It can be seen from the carrier method that after a certain frequency (n2∗f) signal is loaded on the target signal, if there is a periodic signal (frequency n1∗f) in the target signal, the frequencies of n1−n2∗f and n1+n2∗f 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 st is shown in the equation, then(11)st=A2+AT0/2t−T2≤t≤0,A2−AT0/2t0≤t≤T2..According to the Fourier decomposition formula,(12)ft=a0+∑n=1∞ancosnω1t+bnsinnω1t.Triangular waves are decomposed by Fourier transform.Sincest is an even function, bn=0, a0 and an are just calculated:(13)a0=1T∫−T0/2T0/2xtdt=2T∗AT04−AT04=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/22∗A2−2AT0tcosnω0tdt=−2An2π2cosnπ−1=2An2π2sin2nπ2=2An2π2n=1,3,5,…,0n=2,4,6….Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(14)Sω=2Aπ2cosω0t+132cos3ω0t+152cos5ω0t+….(2) Square Wave Signal. It is assumed that the square wave signal expression is as follows:(15)xt=A−T2≤t≤0,−A0≤t≤T2.According to equation (12), the square wave is decomposed by Fourier transform:(16)a0=1T∫−T0/2T0/2xtdt=1T∫−T0/20Adt−1T∫0T0/2Adt=A2−A2=0,an=2T∫−T0/2T0/2xtcosnω0tdt=2T∫0T0/2−Acosnω0tdt+2T∫−T0/20Acosnω0tdt=0,bn=2T∫−T0/2T0/2xtsinnω0tdt=2T∫0T0/2−Asinnω0tdt+2T∫−T0/20Asinnω0tdt=2AT∫−T0/20sinnω0tdt−∫0T0/2sinnω0tdt=2Anπ2−2cosnπ=0n=2,4,6,…,4Anπn=1,3,5,….Therefore, the expression after Fourier decomposition of the square wave is as follows:(17)xt=4Aπsinω0t+13sin3ω0t+15sin5ω0t+….(3) Periodic Shock Signal. It is assumed that the periodic shock signal expression is as follows:(18)δt=∑n=−∞n=∞δt−nT,n=0,±1,±2,….According to equation (12), periodic shock is decomposed by Fourier transform:(19)Xω=1T∫−τ/2τ/2δte−jnωtdt=1T.


## 3.1.3. Aperiodic Signal

It is assumed that the rectangular pulse signal expression is as follows:(20)xt=At≤τ2,0t≥τ2.According to equation (12), rectangular pulse signal is decomposed by Fourier transform:(21)Xω=∫−τ/2τ/2Ae−jωtdt=2Aωsinωτ2.From the analysis of Section3.1, 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 signalst to periodically change, that is, the φ2 of equation (9) continuously changes within −nπ,nπ, if the φ2 continuously changes in −nπ,nπ in a fixed step h, as can be seen from equation (10), the φ1−φ2, and φ1+φ2 will change in the same step h in −nπ,nπ. That is, the phase change step of the carrier signal is always h.


### 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.Figure 5
Time domain plot of the noisy signal.The Fourier transform result of the noisy signal is shown in Figure6.Figure 6
Fourier transform result of the noisy signal.It can be seen from Figure6 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 n1∗f in the noisy signal, after it is multiplied by a certain frequency n2∗f signal, the phases at n1−n2∗f and n1+n2∗f and the phase of the multiplied signal can be known from equation (10). Let the phase of the signal with frequency n2∗f be φ2, and the phase of the signal with frequency n1∗f be φ1, then from equation (10) we can know n1−n2∗f and n1+n2∗f phases are φ1−φ2 and φ1+φ2, 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 n1−n2∗f and n1−n2∗f frequency change with the phase change of frequency n2∗f, when the phase of the signal with frequency n2∗f changes from -2π to 2π. The step size is 0.1π, the phase change of the signal with frequencies n1−n2∗f and n1+n2∗f is shown in Figure 7.Figure 7
Then1−n2∗f and n1+n2∗f phases change at frequency.
(a)(b)Figure7(a) shows the phase change at the frequency of n1−n2∗f and n1+n2∗f as the phase of the loaded signal changes. Figure 7(b) shows load signal phase changes and the magnitude of the phase at the n1−n2∗f and n1+n2∗f.It can be seen from Figure7 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 n1−n2∗f and n1+n2∗f 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 atn1−n2∗f frequency is s1, the signal at n1+n2∗f frequency is s2, and the phases of s1 and s2 signals are as follows:(22)s1:φ1−φ2∈−2π,2π=φ1+φ2′∈2π,−2π,(23)s2:φ1+φ2∈−2π,2π.It can be seen from equations (22) and (23) that the signs of the increments of the changes in the signals s1 and s2 are opposite, so the signs of the phase amplitudes of s1 and s2 are basically opposite in the same period.If it is not thes1 and s2 signals, the signal is selected at any frequency (not the s1 and s2 frequencies) and the phase change of the s2 signal is compared, as shown in Figure 8.Figure 8
Phase change of the signal at any frequency and the phase of s2.
(a)(b)It can be seen from Figure8 that the phase change of the signal at any frequency (not s1 and s2) and the phase of s2 do not have similar regularity, and the phase change step at frequencies other than s1 and s2 does not change around 0.1π. There is no regularity.Any two frequency signals (nots1 and s2 frequencies) are selected to observe the changes in their phases, as shown in Figure 9.Figure 9
The phase change of nots1 and s2 frequencies.
(a)(b)It can be seen from Figure9 that the phase changes of any two frequency signals (not s1 and s2 frequencies) have no regularity, and since the phases of the not s1 and s2 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 s1 and s2 frequencies when the carrier is used, the frequency of the weak signal under the strong noise background can be detected.


## 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 signalst to periodically change, that is, the φ2 of equation (9) continuously changes within −nπ,nπ, if the φ2 continuously changes in −nπ,nπ in a fixed step h, as can be seen from equation (10), the φ1−φ2, and φ1+φ2 will change in the same step h in −nπ,nπ. That is, the phase change step of the carrier signal is always h.


## 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.Figure 5
Time domain plot of the noisy signal.The Fourier transform result of the noisy signal is shown in Figure6.Figure 6
Fourier transform result of the noisy signal.It can be seen from Figure6 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 n1∗f in the noisy signal, after it is multiplied by a certain frequency n2∗f signal, the phases at n1−n2∗f and n1+n2∗f and the phase of the multiplied signal can be known from equation (10). Let the phase of the signal with frequency n2∗f be φ2, and the phase of the signal with frequency n1∗f be φ1, then from equation (10) we can know n1−n2∗f and n1+n2∗f phases are φ1−φ2 and φ1+φ2, 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 n1−n2∗f and n1−n2∗f frequency change with the phase change of frequency n2∗f, when the phase of the signal with frequency n2∗f changes from -2π to 2π. The step size is 0.1π, the phase change of the signal with frequencies n1−n2∗f and n1+n2∗f is shown in Figure 7.Figure 7
Then1−n2∗f and n1+n2∗f phases change at frequency.
(a)(b)Figure7(a) shows the phase change at the frequency of n1−n2∗f and n1+n2∗f as the phase of the loaded signal changes. Figure 7(b) shows load signal phase changes and the magnitude of the phase at the n1−n2∗f and n1+n2∗f.It can be seen from Figure7 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 n1−n2∗f and n1+n2∗f 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 atn1−n2∗f frequency is s1, the signal at n1+n2∗f frequency is s2, and the phases of s1 and s2 signals are as follows:(22)s1:φ1−φ2∈−2π,2π=φ1+φ2′∈2π,−2π,(23)s2:φ1+φ2∈−2π,2π.It can be seen from equations (22) and (23) that the signs of the increments of the changes in the signals s1 and s2 are opposite, so the signs of the phase amplitudes of s1 and s2 are basically opposite in the same period.If it is not thes1 and s2 signals, the signal is selected at any frequency (not the s1 and s2 frequencies) and the phase change of the s2 signal is compared, as shown in Figure 8.Figure 8
Phase change of the signal at any frequency and the phase of s2.
(a)(b)It can be seen from Figure8 that the phase change of the signal at any frequency (not s1 and s2) and the phase of s2 do not have similar regularity, and the phase change step at frequencies other than s1 and s2 does not change around 0.1π. There is no regularity.Any two frequency signals (nots1 and s2 frequencies) are selected to observe the changes in their phases, as shown in Figure 9.Figure 9
The phase change of nots1 and s2 frequencies.
(a)(b)It can be seen from Figure9 that the phase changes of any two frequency signals (not s1 and s2 frequencies) have no regularity, and since the phases of the not s1 and s2 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 s1 and s2 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 signalsXt1 和 Yt2 is(24)Rxyt1,t2=EXt1Yt2=∫∫−∞∞xyfXYx,t1,y,t2dxdy.where Rxy is the correlation coefficient, E… is the expectation, and fXY 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 Bsinωt+φ2 and the noise signal nt is 0, that is,(25)Rnyt=limT⟶∞1T∫0TntBsinωt+φ2dt≈0⊕.The correlation coefficient of two periodic signals with different frequencies is(26)Rxyt=limT⟶∞1T∫0TAsinω1t+φ1∗Bsinω2t+φ2dt=limT⟶∞1T∗AB2∫0Tcosω1+ω2t+φ1+φ2+cosω1−ω2t+φ1−φ2dt≈0.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 Xt1 is only related to the frequency of the input periodic signal Yt2 and is not related to the frequency of the random noise nt, and the characteristic signal Xt1 and the random noise nt are independent of each other, so the theoretical value of their correlation coefficient is zero, that is, Rnyt=0.It is specifically explained through the following analysis.The expressions for the measured signaldst and the test signal stt are shown in equations (27) and (28):(27)dst=DC+∑i=0NAisinωit+φi+noiset,(28)stt=B∗sinωst+φs.From equations (24)–(28), the correlation coefficient expression of the signal under test is as follows:(29)Rds,stt=limT⟶∞12T∫−TTdst∗sttdt=limT⟶∞12T∫−TTB∗sinωst+φs∗DC+∑i=0NAisinωit+φi+noisetdt=limT⟶∞12T∫−TTTB∗sinωst+φs∗DC+limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidt+limT⟶∞12T∫−TTB∗sinωst+φs∗noisetdt=Rds,DCt+Rds,st+Rds,noiset.The following is calculated asRds,DCt, Rds,st, and Rds,noiset, respectively:(30)Rds,DCt=limT⟶∞12T∫−TTB∗sinωst+φs∗DCdt=limT⟶∞B∗DC2T∫−TTsinωst+φsdt=0,Rds,st=limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidt=limT⟶∞12T∫−TTB∗sinωst+φs∗Assinωst+φsdt+limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0,i≠sNAisinωit+φidt=limT⟶∞BAs4T∫−TTcosφs−φi−cos2ωst+φs+φidt+limT⟶∞∑i=0,i≠sNBAs4T∫−TTcosωs−ωit+φs−φi−cosωs+ωit+φs+φidt=limT⟶∞BAs4T∫−TTcosφs−φi−cos2ωst+φs+φidt=limT⟶∞BAs4T∫−TTcosφs−φidt=BAscosφs−φi2,Rds,noiset=limT⟶∞12T∫−TTB∗sinωs∗t+φs∗noisetdt=0.So the result ofRds,stt is as follows:(31)Rds,stt=limT⟶∞BAs4T∫−TTcosφs−φidt=BAscosφs−φi2.Combining the cyclic phase method with the associated coefficient method,(32)co=∫−nπnπ∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidtdφ==∫−nπnπBAscosφs−φi2dφ=BAs2φs=φi,BAscosφs−φi2φs≠φi.Hence, the−1<cosφs−φi<1, co gets the maximum value when φs=φi.It can be seen from equation (32) that the value of Rds,stt depends on the amplitude and phase of the characteristic signal and the input periodic signal. From equation (32), we can see that when ω1=ω2=ω, φ1=φ2=φ, equation (15) becomes the following:(33)co=BAs2.It can be seen from equation (33) that when the input periodic signal amplitude B=1, co=As/2, so As=2∗co, then the magnitude of the corresponding As/2 amplitude phase is a weak signal phase.


### 4.2. Correlation Coefficient Method Experiment

In this part, the signal shown in Figure5 is the detected signal sc, the characteristic signal amplitude in sc 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 φϵ−π,π, 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.Figure 10
Weak signal amplitude and phase detection method results.Figure 11
Enlarged view of the highest and lowest points.In Figures10 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 −0.3π and 0.3π. 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.


## 4.1. Cross-Related Knowledge

The correlation function of the two signalsXt1 和 Yt2 is(24)Rxyt1,t2=EXt1Yt2=∫∫−∞∞xyfXYx,t1,y,t2dxdy.where Rxy is the correlation coefficient, E… is the expectation, and fXY 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 Bsinωt+φ2 and the noise signal nt is 0, that is,(25)Rnyt=limT⟶∞1T∫0TntBsinωt+φ2dt≈0⊕.The correlation coefficient of two periodic signals with different frequencies is(26)Rxyt=limT⟶∞1T∫0TAsinω1t+φ1∗Bsinω2t+φ2dt=limT⟶∞1T∗AB2∫0Tcosω1+ω2t+φ1+φ2+cosω1−ω2t+φ1−φ2dt≈0.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 Xt1 is only related to the frequency of the input periodic signal Yt2 and is not related to the frequency of the random noise nt, and the characteristic signal Xt1 and the random noise nt are independent of each other, so the theoretical value of their correlation coefficient is zero, that is, Rnyt=0.It is specifically explained through the following analysis.The expressions for the measured signaldst and the test signal stt are shown in equations (27) and (28):(27)dst=DC+∑i=0NAisinωit+φi+noiset,(28)stt=B∗sinωst+φs.From equations (24)–(28), the correlation coefficient expression of the signal under test is as follows:(29)Rds,stt=limT⟶∞12T∫−TTdst∗sttdt=limT⟶∞12T∫−TTB∗sinωst+φs∗DC+∑i=0NAisinωit+φi+noisetdt=limT⟶∞12T∫−TTTB∗sinωst+φs∗DC+limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidt+limT⟶∞12T∫−TTB∗sinωst+φs∗noisetdt=Rds,DCt+Rds,st+Rds,noiset.The following is calculated asRds,DCt, Rds,st, and Rds,noiset, respectively:(30)Rds,DCt=limT⟶∞12T∫−TTB∗sinωst+φs∗DCdt=limT⟶∞B∗DC2T∫−TTsinωst+φsdt=0,Rds,st=limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidt=limT⟶∞12T∫−TTB∗sinωst+φs∗Assinωst+φsdt+limT⟶∞12T∫−TTB∗sinωst+φs∗∑i=0,i≠sNAisinωit+φidt=limT⟶∞BAs4T∫−TTcosφs−φi−cos2ωst+φs+φidt+limT⟶∞∑i=0,i≠sNBAs4T∫−TTcosωs−ωit+φs−φi−cosωs+ωit+φs+φidt=limT⟶∞BAs4T∫−TTcosφs−φi−cos2ωst+φs+φidt=limT⟶∞BAs4T∫−TTcosφs−φidt=BAscosφs−φi2,Rds,noiset=limT⟶∞12T∫−TTB∗sinωs∗t+φs∗noisetdt=0.So the result ofRds,stt is as follows:(31)Rds,stt=limT⟶∞BAs4T∫−TTcosφs−φidt=BAscosφs−φi2.Combining the cyclic phase method with the associated coefficient method,(32)co=∫−nπnπ∫−TTB∗sinωst+φs∗∑i=0NAisinωit+φidtdφ==∫−nπnπBAscosφs−φi2dφ=BAs2φs=φi,BAscosφs−φi2φs≠φi.Hence, the−1<cosφs−φi<1, co gets the maximum value when φs=φi.It can be seen from equation (32) that the value of Rds,stt depends on the amplitude and phase of the characteristic signal and the input periodic signal. From equation (32), we can see that when ω1=ω2=ω, φ1=φ2=φ, equation (15) becomes the following:(33)co=BAs2.It can be seen from equation (33) that when the input periodic signal amplitude B=1, co=As/2, so As=2∗co, then the magnitude of the corresponding As/2 amplitude phase is a weak signal phase.


## 4.2. Correlation Coefficient Method Experiment

In this part, the signal shown in Figure5 is the detected signal sc, the characteristic signal amplitude in sc 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 φϵ−π,π, 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.Figure 10
Weak signal amplitude and phase detection method results.Figure 11
Enlarged view of the highest and lowest points.In Figures10 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 −0.3π and 0.3π. 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.Figure 12
Program flow chart.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 Figures7–9, 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:(34)y=0,x>τ,x,−τ<x<τ,0,x<−τ,where y is the data output after the threshold structure, x is the input data, and τ is the size of the threshold.The detailed flow chart of the weak signal frequency detection method is shown in Figure13.Figure 13
Weak signal frequency detection method.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 signalsc is as follows:(35)sc=A1∗sinω1∗t+φ1+A2∗sinω2∗t+φ2+A3∗sinω3∗t+φ3+noise.The periodic function of the input is as follows:(36)s=B∗sinω∗t+φ,where A1=0.01, A2=0.02, A3=0.03, B=1, f1=500Hz, f2=1000Hz, f3=1500Hz, φ1=0.6π, φ2=0.8π, and φ3=0.6π.The calculation formula of the signal-to-noise ratio is as follows:(37)SNR=10logPScPn.where PSc is the power of the mixed signal, and Pn 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 Tables1 and 2.Table 1
Tested data.
DataAmplitudeFrequency (Hz)PhaseData volumeWhite Gaussian noise400000Sine function 10.01200.6π400000Sine function 20.02400.8π400000Sine function 30.03600.6π400000Table 2
Test batch and SNR.
TestSNRTest 1−35Test 2−45Test 3−55


#### 5.1.2. Triangular Wave Test signal (Nonsinusoidal Signal)

In this section, the expression of the triangular wave signalst is given by equation (11), The triangular amplitude value is 0.025, ω0=40π. The signal-to-noise ratio is −35. The time domain diagram is shown in Figure 14.Figure 14
The time domain diagram of the triangular wave.Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(38)xt=0.025∗cos40πt+132cos120πt+152cos200πt+….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 signalssc and st are tested by the method proposed in this study, and the test results are shown in Figures 15–18.Figure 15
Signalsc analysis results with signal-to-noise ratio of −35.Figure 16
Signalsc analysis results with signal-to-noise ratio of −45.Figure 17
Signalsc analysis results with signal-to-noise ratio of −55.Figure 18
Signalsc analysis results with signal-to-noise ratio of −35.
(a)(b)(c)Figures15–17 are the analysis diagrams of signal sc when the signal-to-noise ratio is −35, −45, and −55, and Figure 18 is the analysis diagram of signal st 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.Table 3
Analysis results of signalssc and st.
Test batchSNRSine functionTheoretical valueTest valueAmplitudeFrequencyPhaseAmplitudePhaseTest 1−35Sine function 10.0120 Hz0.6π0.0080.65 piSine function 20.0240 Hz0.8π0.020.8 piSine function 30.0360 Hz0.6π0.030.6 piTest 1−45–55Sine function 10.0120 Hz0.6π0.0180.56 piSine function 20.0240 Hz0.8π0.0240.76 piSine function 30.0360 Hz0.6π0.0320.57 piTest 1Sine function 10.0120 Hz0.6π0.030.4 piSine function 20.0240 Hz0.8π0.0520.5 piSine function 30.0360 Hz0.6π0.020.4 piTest 4−35Triangular wave0.02220 Hz0.0110.5 pi0.00260 Hz0.000090.5 piIt can be seen from Table3 that for signal sc, when SNR≥−50, 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 st 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 2ω0t, 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:(39)x′=x−x3+Acosω0t+φ+Γt.The stochastic resonance model is normalized, and the transformed model is as follows:(40)z′=z−z3+ba3Acosω0aτ+φ+2Dba2wτ.where z=xb/aτ=atThe input signal is as follows:(41)sz=A1∗sinω1∗t+φ1+nt,where A1=0.01f1=500Hz, φ1=0.6π, nt is the noise signal, and the signal-to-noise ratio of sz is −35.The signalsz is input into the improved phase lock method and stochastic resonance model, respectively, and its analysis diagram is shown in Figure 19.Figure 19
The results of the comparative analysis.
(a)(b)As can be seen from Figure19, 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.


## 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 signalsc is as follows:(35)sc=A1∗sinω1∗t+φ1+A2∗sinω2∗t+φ2+A3∗sinω3∗t+φ3+noise.The periodic function of the input is as follows:(36)s=B∗sinω∗t+φ,where A1=0.01, A2=0.02, A3=0.03, B=1, f1=500Hz, f2=1000Hz, f3=1500Hz, φ1=0.6π, φ2=0.8π, and φ3=0.6π.The calculation formula of the signal-to-noise ratio is as follows:(37)SNR=10logPScPn.where PSc is the power of the mixed signal, and Pn 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 Tables1 and 2.Table 1
Tested data.
DataAmplitudeFrequency (Hz)PhaseData volumeWhite Gaussian noise400000Sine function 10.01200.6π400000Sine function 20.02400.8π400000Sine function 30.03600.6π400000Table 2
Test batch and SNR.
TestSNRTest 1−35Test 2−45Test 3−55


### 5.1.2. Triangular Wave Test signal (Nonsinusoidal Signal)

In this section, the expression of the triangular wave signalst is given by equation (11), The triangular amplitude value is 0.025, ω0=40π. The signal-to-noise ratio is −35. The time domain diagram is shown in Figure 14.Figure 14
The time domain diagram of the triangular wave.Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(38)xt=0.025∗cos40πt+132cos120πt+152cos200πt+….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 signalssc and st are tested by the method proposed in this study, and the test results are shown in Figures 15–18.Figure 15
Signalsc analysis results with signal-to-noise ratio of −35.Figure 16
Signalsc analysis results with signal-to-noise ratio of −45.Figure 17
Signalsc analysis results with signal-to-noise ratio of −55.Figure 18
Signalsc analysis results with signal-to-noise ratio of −35.
(a)(b)(c)Figures15–17 are the analysis diagrams of signal sc when the signal-to-noise ratio is −35, −45, and −55, and Figure 18 is the analysis diagram of signal st 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.Table 3
Analysis results of signalssc and st.
Test batchSNRSine functionTheoretical valueTest valueAmplitudeFrequencyPhaseAmplitudePhaseTest 1−35Sine function 10.0120 Hz0.6π0.0080.65 piSine function 20.0240 Hz0.8π0.020.8 piSine function 30.0360 Hz0.6π0.030.6 piTest 1−45–55Sine function 10.0120 Hz0.6π0.0180.56 piSine function 20.0240 Hz0.8π0.0240.76 piSine function 30.0360 Hz0.6π0.0320.57 piTest 1Sine function 10.0120 Hz0.6π0.030.4 piSine function 20.0240 Hz0.8π0.0520.5 piSine function 30.0360 Hz0.6π0.020.4 piTest 4−35Triangular wave0.02220 Hz0.0110.5 pi0.00260 Hz0.000090.5 piIt can be seen from Table3 that for signal sc, when SNR≥−50, 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 st 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 2ω0t, 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.


## 5.1.1. Sinusoidal Signal

In this section, sinusoidal signals are characteristic signals with three different frequencies added to Gaussian white noise, that is, the mixed signalsc is as follows:(35)sc=A1∗sinω1∗t+φ1+A2∗sinω2∗t+φ2+A3∗sinω3∗t+φ3+noise.The periodic function of the input is as follows:(36)s=B∗sinω∗t+φ,where A1=0.01, A2=0.02, A3=0.03, B=1, f1=500Hz, f2=1000Hz, f3=1500Hz, φ1=0.6π, φ2=0.8π, and φ3=0.6π.The calculation formula of the signal-to-noise ratio is as follows:(37)SNR=10logPScPn.where PSc is the power of the mixed signal, and Pn 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 Tables1 and 2.Table 1
Tested data.
DataAmplitudeFrequency (Hz)PhaseData volumeWhite Gaussian noise400000Sine function 10.01200.6π400000Sine function 20.02400.8π400000Sine function 30.03600.6π400000Table 2
Test batch and SNR.
TestSNRTest 1−35Test 2−45Test 3−55


## 5.1.2. Triangular Wave Test signal (Nonsinusoidal Signal)

In this section, the expression of the triangular wave signalst is given by equation (11), The triangular amplitude value is 0.025, ω0=40π. The signal-to-noise ratio is −35. The time domain diagram is shown in Figure 14.Figure 14
The time domain diagram of the triangular wave.Therefore, the expression after Fourier decomposition of the triangular wave is as follows:(38)xt=0.025∗cos40πt+132cos120πt+152cos200πt+….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 signalssc and st are tested by the method proposed in this study, and the test results are shown in Figures 15–18.Figure 15
Signalsc analysis results with signal-to-noise ratio of −35.Figure 16
Signalsc analysis results with signal-to-noise ratio of −45.Figure 17
Signalsc analysis results with signal-to-noise ratio of −55.Figure 18
Signalsc analysis results with signal-to-noise ratio of −35.
(a)(b)(c)Figures15–17 are the analysis diagrams of signal sc when the signal-to-noise ratio is −35, −45, and −55, and Figure 18 is the analysis diagram of signal st 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.Table 3
Analysis results of signalssc and st.
Test batchSNRSine functionTheoretical valueTest valueAmplitudeFrequencyPhaseAmplitudePhaseTest 1−35Sine function 10.0120 Hz0.6π0.0080.65 piSine function 20.0240 Hz0.8π0.020.8 piSine function 30.0360 Hz0.6π0.030.6 piTest 1−45–55Sine function 10.0120 Hz0.6π0.0180.56 piSine function 20.0240 Hz0.8π0.0240.76 piSine function 30.0360 Hz0.6π0.0320.57 piTest 1Sine function 10.0120 Hz0.6π0.030.4 piSine function 20.0240 Hz0.8π0.0520.5 piSine function 30.0360 Hz0.6π0.020.4 piTest 4−35Triangular wave0.02220 Hz0.0110.5 pi0.00260 Hz0.000090.5 piIt can be seen from Table3 that for signal sc, when SNR≥−50, 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 st 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 2ω0t, 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:(39)x′=x−x3+Acosω0t+φ+Γt.The stochastic resonance model is normalized, and the transformed model is as follows:(40)z′=z−z3+ba3Acosω0aτ+φ+2Dba2wτ.where z=xb/aτ=atThe input signal is as follows:(41)sz=A1∗sinω1∗t+φ1+nt,where A1=0.01f1=500Hz, φ1=0.6π, nt is the noise signal, and the signal-to-noise ratio of sz is −35.The signalsz is input into the improved phase lock method and stochastic resonance model, respectively, and its analysis diagram is shown in Figure 19.Figure 19
The results of the comparative analysis.
(a)(b)As can be seen from Figure19, 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 ofn1−n2∗f 和 n1+n2∗f 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%



---

*Source: 9137927-2022-02-25.xml*