Application of Optimized Adaptive Chirp Mode Decomposition Method in Chirp Signal

: The adaptive chirp mode decomposition method has a good effect on processing chirp signals. The parameter α controls the smoothness of the output signal. Too small an α will cause a smooth output signal. The parameter β controls the instantaneous frequency (IF). If too small a β value is used, the output IF will be very smooth. However, rapidly changing IFs require a relatively large β . However, the choice of , αβ is artificially set, and there are errors in practical applications. Therefore, it employs the state transition algorithm to adaptively optimize , αβ to improve the signal-to-noise ratio (SNR) and resolution of the signal. First, as the species number of the state transition algorithm method is set artificially and has a long running time, this paper proposes a Rastrigin optimization test equation to test the optimization time of different species and determine the number of optimal species; second, the state transition algorithm determined by the number of species is employed to adaptively find the , αβ in the adaptive chirp mode decomposition algorithm; finally, the optimized adaptive chirp mode decomposition method is applied to the simulation signal and chirp signal from marine animals to verify the proposed method.

EMD is sensitive to noise and lacks mathematical models, it has been widely used in various fields. On the basis of EMD, the methods of EMD and ensemble empirical mode decomposition (EEMD) have been proposed; however, these approaches only address part of the problem, and some challenging issues remain to be addressed. For instance, these improved methods based on EMD have a poor resolution for split-off mode.
To further understand the characteristics of NC signals, it is necessary to extract each nonlinear chirp model from the signal data. In the past few decades, many methods have been developed to decompose NCs. These methods implement signal decomposition in the time, frequency, and time-frequency (TF) domains. From the TF plane, we can observe time-varying features and perform signal decomposition. However, due to the limitation of the Heisenberg uncertainty principle, the TF representation generated by conventional methods is usually fuzzy, and it is impossible to provide accurate TF representation for time-varying signals. The disadvantages of traditional TF methods severely limit their application in actual signal processing. In the past decades, TF has gradually developed many advanced methods, such as the reassignment method (RM) [14,15], synchrosqueezing transform (SST) [16,17], demodulated SST (DSST) [18,19], and high-order SST [20,21]. RM technology stimulates post-processing of traditional TF methods and is an effective way to obtain clearer TF results. However, the RM framework is based on a spectrogram, which means that RM results lose the ability to reconstruct the signal. The SST method not only enhances the TF resolution but also allows the signal to be reconstructed. On the contrary, when processing signals that change over time (such as a linear frequency-modulated signal or a non-linear frequency-modulated (FM) signal), SST cannot generate a concentrated TF result.
The time-frequency transform maps the chirp signal to the time-frequency domain. It is easy to observe the rule of frequency changes with time, which is conducive to signal detection and parameter estimation. The time-frequency transform can be used to describe the energy density of the signal at different times and frequency by constructing the joint function of time and frequency. At present, time-frequency transforms commonly used are the short-time Fourier transform (STFT) [22][23][24][25], Wigner-Ville distribution (WVD) [26,27], fractional Fourier transform [28,29], cubic phase function [30], and Zak transform (ZT) [31,32]. In addition, as the chirp signal usually presents an impact function of frequency linearly varying with time in the time and frequency domain, the linear detection and extraction method in image processing can be combined with the commonly used time-frequency transformation to estimate the initial frequency and frequency modulation slope. Chen [33] applied the VMD method to nonlinear chirp signals and achieved good results. However, as it processes the signals based on VMD, it is still limited to the determination of the number of signal decomposition layers. Chen [34] uses the matching tracking algorithm of the greedy algorithm to estimate each signal adaptively on the basis of Reference [33], which overcomes the defect of Variational nonlinear chirp mode decomposition (VNCMD). In reference [34], two important parameters , α β control the components of the output signal and instantaneous frequency (IF), respectively. However, as the selection of α and β is artificially set, there are errors in practical applications.
Based on the above problems, this paper proposes to use the state transition algorithm (STA) [35] to adaptively determine α and β to improve the resolution of the output signal. As the number of species in the STA is an artificial error, this paper proposes the Rastrigin optimization test equation to optimize the number of different species, and detects the time of each optimization (the number of species is 1-100, in which the rotation operator, translation operator, expansion operator, and axesion operator are unchanged, and are set to =1, =1, =1, =1 α β γ δ , respectively). In order to accurately determine the number of species, this paper takes the average value of 10,000 times the operation time of the same species optimization and compares the average optimization time of each species to determine the number of species corresponding to the minimum optimization time. Finally, the adaptive chirp mode decomposition (ACMD) method optimized in this paper is applied to the chirp signal from marine animals, which verifies the feasibility of the proposed method.

ACMD
ACMD was proposed by Chen on the basis of VNCMD, and they used a matching pursuit algorithm in the greedy algorithm to adaptively estimate each signal. The i-th signal should be defined as follows: is the residual signal from which the current component signal is removed, and 0 α > is the weight of the coefficient. Like matching pursuit, ACMD greedily looks for signal components that extract the most energy from the input signal. Assuming that the signal is discrete in time series, the discrete expression of Equation (1) is as follows: and Ω denotes the second-order difference matrix; , , [ ] where ( ) ( )  (2) can be solved by an iterative algorithm, which can update the demodulated signal and frequency function in turn. For the j-th iteration, the update of the vector i u is as follows: The frequency function iterations. In addition, the parameter α controls the smoothness of the output signal. At this time, the method is a TF band-pass filter, and the bandwidth of the filter is determined by α . Too small an α will cause the filter to have a narrow bandwidth (i.e., a smooth output signal). The signal components can be estimated as follows: Employing the demodulated signal obtained in Equation (6), the frequency increase can be estimated as follows: IF is a smoothing function and the growth of frequency should be smooth enough. Therefore, in practice, in order to reduce the influence of noise, a low-pass filter is used to process the signal growth. The IF may eventually be updated as follows: where , I is the identity matrix, Ω is the second-order difference matrix, and ( ) denotes a low-pass filter. If too small a β is used, the output IF will be very smooth. However, rapidly changing IFs require a relatively large β . Vector and there are errors in practical applications. Therefore, this paper employs the state transition algorithm (STA) to adaptively optimize , α β to improve the signal-to-noise ratio (SNR) and resolution of the signal.

State Transition Algorithm
The STA [35] can optimize several parameters at the same time, and it is an efficient random optimization algorithm. This algorithm treats the result of the optimization problem as a state, and regards the process of searching in the search space as an evolutionary algorithm of the process of state transition. This algorithm uses four state operators of rotation, translation, expansion, and axesion to solve the continuous optimization problem. The algorithm has strong global search ability, fast convergence speed, and high optimization precision.
The STA can be written as: To solve the optimization problem of Equation (10), the four core operation operators are as follows: Rotation operator: x as the center and α as the radius, and the radius α satisfies Translation operator: β is a positive number and becomes a translation operator. t R R ∈ is a random number with a uniform distribution between [ ] 0,1 . The operator searches along the positive direction of the x gradient with a maximum search step of β .
Expansion operator: γ is a positive number and becomes an expansion operator.
n n e R R × ∈ is a diagonal matrix that follows a Gaussian distribution. The operator can optimize the whole search space. Axesion operator: δ is a positive number and becomes an axesion operator.
is a diagonal matrix that follows a Gaussian distribution, and the matrix has only one random position with elements other than zero. The basic flowchart of the STA is shown in Figure 1. In Figure 1, Best is the optimal individual in the species, SE is the number of individuals in the search species, and , , , α β γ δ are the rotation operator, translation operator, expansion operator, and axesion operator, respectively. In Figure 1, three steps of expansion, rotation, and axesion are performed, and the translation operation is embedded in these three steps.

Identify problem goals and variables
The initial population is formed and the STA algorithm parameters are set Calculate the fitness of each individual in the species Whether the termination condition is satisfied Specific steps are as follows: Step1: Determine the problem objectives and variables. The number of searches SE for initializing the state transition algorithm and the value of each operation factor , , , α β γ δ .
Step2: Calculate the fitness value of each individual in the species.
Step3: Best ← expansion ( ) , , , Step4: Copy oldBest into a species with SE individuals, and expand to obtain species state . Calculate the individual fitness value in state . At this time, the best individual in the group is newBest , and the fitness value is gBest .
Step5: If and execute Step6; otherwise, execute Step7. Step6: , each copied SE times, paired in pairs. Species state is obtained after performing the transfer operation according to Equation (12). Calculate the individual fitness value in state . At this time, the optimal individual of the group is newBest , and the fitness value is gBest .
Step7: If Step8: Output Best as an expansion operation.
Step9: Best ← rotation ( ) , , , Best SE β δ are similar to Step3-Step8. Step10: Whether the final result satisfies the termination condition, if it is satisfied, the optimal result is output; otherwise, Step 2 to Step 9 are continued.

Optimized ACMD Method
Like other algorithms, the STA is based on the iterative search of individuals in the neighborhood. The difference is that each individual in the search species uses the group optimal value, that is, the elite individual. The search process is only related to the group optimal value of the previous generation, and the search is carried out within the group optimal neighborhood. For example, when performing the expansion operation, the optimal individual of the group is first taken as an input, the optimal individual is then copied to the number of individuals SE of the search species, and the expansion operation is performed on these optimal individuals. The optimal individual in the group is calculated by performing the expansion operation, and the optimal individual is used as the input for the next operation.
The state transition algorithm has five important parameters, that is, the number of individuals in the search species SE , rotation operator α , translation operator β , expansion operator γ , and axesion operator δ . In the case where the number of individuals SE is constant, the larger the value of , , , α β γ δ , the larger the search range of the corresponding operator, and the larger the search range, which is beneficial to the global search. When its value is small, the search accuracy is high, which is advantageous for the local search. In References [36] and [37], the species SE is set to where i x is the i-th element of the n -dimensional Euclid space element.
This paper uses the improved STA method to optimize in the ACMD method. The optimization process is shown in Figure 2.

ACMD processing
Fault signal STA algorithm with the best species The best α,β Step1: Determine the species of the STA when the average optimization time is the shortest through multiple detections. Step2: Bring the number of optimal species groups into the STA algorithm.
Step3: Employ the optimized STA algorithm to find the optimal parameter. Step4: The optimal parameter is input into ACMD, and the optimized ACMD is used to process the chirp signal.

STA
This article applies the intelligent algorithms of Ant Lion Optimizer (ALO), Grey Wolf Optimizer, Lightning Search Algorithm (LSA), Particle Swarm Optimization (PSO), STA, and Whale Optimization Algorithm (WOA) to the optimization process of Rastrigin optimization test equations. The number of functions is set to 100, the number of iterations is set to 500, and the operation is repeated 30 times. The total running time is shown in Table 1. The fastest speed of the STA algorithm is 13.196 s. Because the number of species of STA affects the speed and accuracy of the algorithm, before the study in this paper, the species number was set artificially and there were human errors. This paper proposes the Rastrigin optimization test equation to optimize the number of different species, and detects the time of each optimization (the number of species is 1-100, in which the rotation operator, translation operator, expansion operator, and axesion operator are unchanged and are set to =1, =1, =1, =1 α β γ δ , respectively) and obtains the result shown in Figure 3. In Figure 3, the X-coordinate is the number of different species, and the Y-coordinate is the optimization time. After three runs, it is clearly observed in Figure 3 that the number of optimal species for the first run was 32, and the optimization time was 0.05793 s. The number of optimal species in the second run was 9, and the optimization time was 0.06298 s. The number of optimal species in the third run was 36, and the optimization time was 0.06833 s. It can be predicted that the optimal number of species is different in each run, so in order to determine the optimal number of species more accurately, this paper will optimize each species 10,000 times and record the time of each run. As depicted in Figure 3, the number of optimal species groups is uncertain for each run. In order to determine the species number accurately, this paper takes the average value of 10,000 times the running time of the same species optimization. Then, 1-100 species are taken and run 10,000 times to give Figure 4. The X-coordinate is the number of species and the Y-coordinate is the average running time of 10,000 times for the same species optimization. Figure 4 clearly shows that when the species is

Parameter Optimization
In order to accurately compare with ACMD, it is proved that the method proposed in this paper optimizes the , α β value in the ACMD. In this paper, the same simulation signal is constructed as ACMD (two signals with large IFs fluctuation):  The simulated signals were processed by ACMD to obtain the final estimated signal IFs. The SNRs of IFs1 and S1 (t), and IFs2 and S2 (t) are calculated, respectively, and The figure in this paper is the simulation display of Matlab2016a version, and the data in Figure 7a,b are the approximations obtained by rounding. This article exports 100 detailed data, as shown in Tables 2 and 3, where Table 2 is the IFs1 with different , α β corresponding to SNR data. Table 3 is the IFs2 with different , α β corresponding to SNR data. It can be observed from Table 2 that when 1 26.159 = SNR , the SNR of IFs1 is the largest. When 2 32.633 = SNR , the SNR of IFs2 is the largest in Table 3. Based on the above chart, we can obtain such a conclusion: When the SNR of different signals is the largest, they correspond to different α and β values. Therefore, ACMD uses artificial settings of α and β , and using the same α and β for different signals has drawbacks.
Employing the method proposed in this paper can improve the SNR. The SNR improvement is as follows: where R is the SNR improvement; i SNR denotes the SNR of the method proposed in this article; o SNR denotes the SNR obtained using the ACMD method.
From Equation (17), we can see that the IFs1 and IFs2 obtained by the method proposed in this paper are 9.134% and 11.931% higher than those obtained by the ACMD method, respectively.

Conclusions
The ACMD method has a good effect on processing chirp signals. The parameter α controls the smoothness of the output signal. Too small an α will cause a smooth output signal. The parameter β controls the IFs. If too small a β is used, the output IF will be very smooth. However, rapidly changing IFs require relatively large β . The choice of , α β is artificially set, and there are errors in practical applications. Therefore, it employs STA to adaptively optimize , α β . As the number of species in the STA is an artificial error, this paper proposes the Rastrigin optimization test equation to optimize the number of different species. Finally, the ACMD method optimized in this paper is applied to the chirp signal from marine animals, which verifies the feasibility of the proposed method. The following conclusions can be drawn in this paper: