A 1/f Noise Detection Method for IGBT Devices Based on PSO-VMD

Abstract

The generation of 1/f noise is closely related to the quality defects of IGBT devices. In the process of detecting single-tube noise of IGBT, thermal noise and shot noise show obvious white noise characteristics in the low-frequency range. This paper investigates how to accurately detect the 1/f noise under strong white noise, and thus proposes a particle swarm optimization method known as variational mode decomposition. First, the particle swarm optimization algorithm was used twice to search the optimal parameter combination between the penalty parameter and the decomposition modulus of the VMD model. Then, the parameters of the variational mode decomposition algorithm were set in optimal parameter combination. The frequency center and bandwidth of each IMF component were determined by continuous iteration in the variational framework. Finally, the 1/f noise signal was adaptively separated from background noise. Extensive experimental investigations carried out under different signal-to-noise ratios, compared with the optimal wavelet denoising algorithm, revealed that the PSO-VMD algorithm improved the signal-to-noise ratio by 6.6%, 16.82%, and 42.48%, whereas the mean square error is reduced by 7.12%, 19.80%, and 33.76%.

1. Introduction

IGBT (Insulated Gate Bipolar Transistor) is a power semiconductor device composed of BJT (Bipolar Junction Transistor) and MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) equipped with the most advanced technology in power electronics, and the former is widely used in high-precision fields such as high-speed rail, aviation, electric vehicles, and wind power. Traditional IGBT (Insulated Gate Bipolar Transistor) operational reliability studies evaluate the health of the device in order to predict the remaining life by observing changes in the parameters of the aging or failure process of the IGBT device [1,2]. M. Fukuda et al. [3,4] found that for semiconductor devices with PN junctions, the low-frequency noise is mainly dominated by 1/f noise. The generation of 1/f noise is closely related to the defects and impurities of the device. Its power spectral density is obviously easy to observe in the low-frequency band. Therefore, the reliability of the device can be determined by examining the low-band noise [5,6].

In recent years, the gradually developed wavelet analysis theory has provided a powerful tool for analyzing 1/f noise due to its multi-resolution, good local characteristics, time-varying analysis, and scale analysis functions [7]. In 1992, Wornwell et al., proposed a wavelet-based 1/f noise signal model [7], which opened up a new way for subsequent studies. In the same year, Mallat [8] developed a signal modulus maximum value detection method, with which a quick decomposition of signals into components at different scales can be achieved by the orthogonal wavelet base. On that basis, Donoho et al. [9] presented a threshold detection method based on wavelet transform, which was widely used because of its easy implementation. Early methods of soft and hard thresholding, such as heuristic thresholding [10] and other similar methods for determining thresholds, are mostly speculative. Aiming at this characteristic, Zhang Rongbiao et al. [11] detected a weak signal based on wavelet entropy. This method used signals with different wavelet entropy at different decomposition scales to adaptively determine the high-frequency coefficient component threshold. However, similar to traditional wavelet threshold denoising, the drawbacks in these algorithms are fixed wavelet base and poor flexibility. In addition to wavelet analysis theory, EMD (Empirical Mode Decomposition) algorithms are also widely used in weak signal detection. Lin Zhujun et al. [12] introduced a joint denoising method based on empirical mode decomposition and wavelet soft threshold, but there are usually obvious modal aliasing and port effect in EMD algorithms.

Variational mode decomposition is a method of adaptive signal processing, proposed by Dragomiretskiy et al., in 2014 [13], which is essentially an adaptive Wiener filter group. When the model parameters are set properly, the modal aliasing of the traditional EMD algorithm can be effectively avoided. Applications of the VMD (Variational Modal Decomposition) method are studied in [14,15,16,17,18], which show that VMD performs better than EMD in terms of pitch detection, separation, and noise robustness. However, the VMD method is not adaptive in different applications, and its performance depends largely on internal parameter settings. The penalty parameter α and the number of components $K$ in VMD affect the result of the signal processing. Therefore, in order to achieve excellent performance, it is necessary to first find an optimal combination of the penalty parameter $\alpha$ and the component number $K$. This paper proposes the application of the PSO (Particle Swarm Optimization) algorithm to the selection of VMD model parameters and introduces the signal-to-noise ratio of the output signal as the fitness function. In this way, the original VMD method is modified to be an adaptive algorithm. To verify the detection performance of PSO-VMD, an abbreviated Brownian model is utilized to simulate 1/f noise, and mixed additive white noise is deployed as background noise. Based on the wavelet analysis, the 1/f noise detection is performed by using the aforementioned multiple algorithms, and, thus, the improved EMD algorithm and the PSO-VMD algorithm are proposed in this paper. According to the experimental results obtained, the proposed PSO-VMD method performs better than other algorithms under the two indicators of signal-to-noise ratio and mean square error.

2. Basic Theory of 1/f Noise in IGBT

For IGBT modules, material failure or an imperfect manufacturing process causes 1/f noise, which is a main low-frequency noise found in the performance of IGBT. The internal noise of IGBT can be divided into white noise and low-frequency noise. While the former is mainly composed of thermal noise and shot noise, the latter essentially includes 1/f noise, RTN noise (Random Telegraph Noise), and G-R noise (Generation-Recombination Noise) [5]. Since low-frequency noise is often caused by device defects and/or impurities, it is necessary to study how to improve the quality and reliability of devices. 1/f noise, the main low-frequency noise in BJT and MOSFET devices, has obvious power spectral density in the low-frequency range. Therefore, some studies showed that structural defects of BJT and MOSFET (such as microtubules, dislocations, doping, carrier lifetime, and stacking) affect their reliability, and these defects cause 1/f noise [19]. Therefore, for the low-frequency noise research of IGBT devices, this paper mainly detects the 1/f noise with white noise as the background.

There are two main characteristics of 1/f noise: statistical self-similarity and long-range correlation. Among them, statistical self-similarity is a time domain feature, which means that the 1/f noise time domain waveforms are approximately the same on different observation scales. Suppose the input signal is $x(t)$, $\alpha$ is the scale factor, and $H^{\prime }$ the similarity index. Their relation can be expressed as:

$$x{(t)}^P={\alpha }^{-H^{\prime }}x(\alpha t)$$

(1)

According to Equation (1), the 1/f noise power spectral density expression can be obtained as follows:

$$S_x(\omega )={\left|\alpha \right|}^{\gamma }{S}_x(\alpha \omega )$$

(2)

In $\gamma =2H^{\prime }+1$, it is observable that the power spectral densities of 1/f noise on different scales differ only by the fixed amplitude factor ${\left|\alpha \right|}^{\gamma }$.

The long-range correlation is characterized by long-term persistence. Using the generalized inverse Fourier transform of Equation (2), the autocorrelation function of the 1/f noise is obtained as follows:

$$R_x(\tau )=\frac{{\tau }^{\gamma -1}}{2\Gamma (\gamma )\cos (\gamma \pi /2)}~{\left|\tau \right|}^{\gamma -1}$$

(3)

where Γ is a Gamma function and τ is a time delay. Equation (3) shows that $R_x(\tau )$ decreases slowly with $\tau$. Also, there is a long-range correlation of 1/f noise in the time domain.

The fractional Brownian motion model has fractal features, long-range correlation, and power-law spectral features, which can simulate the 1/f noise model well [7,19], as shown in Formula (4).

$$B_H(t)-B_H(0)=\frac{1}{\Gamma (H+1/2)}{\displaystyle {\int }_{-\infty }^{t}K(t-t^{\prime })dB(t^{\prime })}$$

(4)

where the kernel function is:

$$K(t-t^{\prime })=\{\begin{array}{c}{(t-t^{\prime })}^{H-1/2},0\le t^{\prime }\le t\\ {(t-t^{\prime })}^{H-1/2}-{(-t)}^{H-1/2},t^{\prime }<0\end{array}$$

(5)

where the range of the parameter $H$ is between 0 and 1. $B(t)$ is a normal Gaussian random process with a mean of 0, and $\langle B^2(t)\rangle =t$ is set.

3. Principle of PSO-VMD Algorithm

3.1. Variational Mode Decomposition Algorithm (VMD)

Variational mode decomposition is an adaptive signal processing method proposed by Dragomiretskiy et al., in 2014 [13]. The VMD algorithm implements the acquisition of IMF (Intrinsic Mode Function) components in the variational framework, and adaptively decomposes the signals by searching for the optimal constrained variational model. Provided that the original signal is decomposed into IMF components, the corresponding constrained variational model is as follows:

$$\begin{array}{c}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{{\displaystyle \sum _k\Vert {\partial }_t\left[\left(\sigma \left(t\right)+\frac{j}{\pi t}\right)u_k\left(t\right)\right]e^{-j{\omega }_{k}t}\Vert }}_2^2\right\}\\ {\displaystyle \sum _k{u}_k=f}\end{array}\}$$

(6)

where: $\left\{u_k\right\}=\left\{u_1,\cdots ,u_k\right\}$ represents the $K$ modal components resulting from the decomposition; $\left\{{\omega }_k\right\}=\left\{{\omega }_1,\cdots ,{\omega }_k\right\}$ represents the frequency center of each component; and ${\displaystyle \sum _k:}={\displaystyle \sum {}_{k=1}^K}$ represents the sum of all components.

In order to obtain the optimal solution of the above model, we combined the two penalty function terms $\alpha$ and introduced the augmented Lagrange function [13]. When solved, Formula (6) can be expressed as:

$$\begin{array}{l}L\left(\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda \right)=\alpha {\displaystyle \sum _k{\Vert {\partial }_t\left[\left(\sigma \left(t\right)+\frac{j}{\pi t}\right)•u_k\left(t\right)\right]e^{-j{\omega }_{k}t}\Vert }_2^2}\\ +{\Vert f\left(t\right)-{\displaystyle \sum _k{u}_k}\left(t\right)\Vert }_2^2+\langle \lambda \left(t\right),f\left(t\right)-{\displaystyle \sum _k{u}_k\left(t\right)}\rangle \end{array}$$

(7)

where $\alpha$ is the penalty parameter, and $\lambda$ is the Lagrangian multiplier.

The saddle point of the augmented Lagrange function (the optimal solution of the constrained variational model) is obtained by the alternating direction multiplier algorithm. The specific algorithm steps are as follows:

• Initialize $\left\{u_k^1\right\}$, $\left\{{\omega }_k^1\right\}$, ${\lambda }^1$, and n. Set the initial value to 0;
• Define $n=n+1$ and execute the entire loop;
• Define the number of cycles as $k$. Perform the first and second loops of the inner layer according to the step size of 1. Update $u_k$, Formulas (8) and (9) until $k=K$.

  $$u_k^{n+1}={\arg }_{u_k}\min L\left(\left\{u_{i<k}^{n+1}\right\},\left\{u_{i\ge k}^n\right\},\left\{{\omega }_i^n\right\},{\lambda }^n\right)$$

  (8)

  $${\omega }_k^{n+1}={\arg }_{{\omega }_k}\min L\left(\left\{u_i^{n+1}\right\},\left\{{\omega }_{i<k}^{n+1}\right\},\left\{{\omega }_{i\ge k}^n\right\},{\lambda }^n\right)$$

  (9)
• Update $\lambda$ according to ${\lambda }^{n+1}={\lambda }^n+\tau \left(f-{\displaystyle \sum _k{u}_k^{n+1}}\right)$ in the outer loop.

Repeat steps 2~4 until the iteration stop condition ${\displaystyle \sum _k\left({\Vert u_k^{n+1}-u_k^n\Vert }_2^2/{\Vert u_k^n\Vert }_2^2\right)}<\epsilon$ is met, before ending the entire inner loop and outputting the result.

3.2. Particle Swarm Optimization Algorithm (PSO)

The decomposition result of the VMD algorithm depends on the preset number of components $K$. Variance in the value of $K$ determines the different detection effects of the algorithm. Research shows that the size of the penalty parameter $\alpha$ of the VMD algorithm also has a great influence on the result. The magnitude of the $\alpha$ value is inversely proportional to the bandwidth of the decomposed modal component [20]. Since the actual input detection signal has large differences, the selection of the component number $K$ and the penalty parameter $\alpha$ is key to the detection effect of the VMD algorithm.

After consulting the literature, we compare the advantages and disadvantages of genetic algorithm, simulated annealing, hill climbing algorithm, ant colony algorithm, and particle swarm optimization. Since the parameter combinations $\left[\alpha ,K\right]$ of the required solutions are real numbers, and the population range to be solved is large and requires a fast convergence speed, this paper selects the particle swarm optimization algorithm to optimize the two parameters of the VMD model. The PSO algorithm is used twice to reduce the risk of falling into local optimum.

Assuming that the number of particles is $N$, the population in the D-dimensional space is $X=\left(X_1,X_2,\cdots ,X_N\right)$, the position of the i-th particle in space is $X_i=\left(x_{i1},x_{i2},\cdots ,x_{iD}\right)$, and the update step (i.e., the update speed) is $V_i=\left(v_{i1},v_{i2},\cdots ,v_{iD}\right)$ for each iteration process. The speed update formula can be expressed as:

$$\begin{array}{ll}{V}_{id}\left(t+1\right)=& V_{id}t+c_1\times randperm\times \left(g_{id}-x_{id}\left(t\right)\right)+\\ & c_2\times randperm\left(z_d-x_{id}\left(t\right)\right)\end{array}$$

(10)

and the update of the argument is:

$$x_i\left(t+1\right)=x_i\left(t\right)+V_i\left(t\right)$$

(11)

In the formula, $d=1,2,\cdots ,D$; $i=1,2,\cdots ,M$; $c_1,c_2$ represent the learning factor, which determines the ability to follow the historical excellent solution. As depicted in Figure 1, $c_1$ and $c_2$, respectively, move towards the individual optimal solution and the global optimal solution. In this paper, randperm was adopted instead of rand to generate integers.

In order to improve the 1/f signal detection, the maximum output of the modal component, the signal-to-noise ratio function is adopted as a fitness function, as shown in Equation (12). Each time the particle is updated, the fitness value is calculated, and the optimal fitness is obtained through continuous iteration.

$$SNR=10\times {\log }_{10}\left(\frac{u_i{}^2}{{(s-u_i)}^2}\right)$$

(12)

where $u_i$ is the decomposed $i-th$ modal component and $s$ is the original signal.

3.3. PSO-VMD Algorithm

Since the actual detection of 1/f noise is often submerged in white noise, an attempt is made to separate the 1/f noise from the background noise by VMD with the particle swarm optimization algorithm for parameter searching. Two PSO were used in order to avoid being trapped into local optimal solution. Also, the output of the first PSO algorithm optimization was taken as the initial value of $\left[\alpha ,K\right]$ (i.e., the initial particle position) for the second PSO. Finally, a 1/f noise detection method based on motivational mode decomposition with optimized parameters is proposed. The specific implementation steps are shown in Figure 2.

• First, the parameters of the particle swarm algorithm are set to complete an optimal $\left[\alpha ,K\right]$ parameter combination search;
• After optimization of the first particle swarm, a set of parameter combinations $\left[{\alpha }_0,K_0\right]$ is obtained. The initial position of the next PSO algorithm particle is set by $\left[{\alpha }_0,K_0\right]$, and the quadratic particle swarm optimization is performed. Generally, the parameter combination $\left[{\alpha }_1,K_1\right]$ is considered to be the global optimal solution;
• Consequently, $\left[{\alpha }_1,K_1\right]$ is adopted to set the value of the VMD model penalty parameter and the number of the decomposition modulus. Thereafter, use is made of the VMD model of the optimized parameter to realize the 1/f signal detection;
• Lastly, the measured signal is processed by the VMD algorithm, and several modal components are obtained. The component IMFk with the largest output signal-to-noise ratio is selected; that is, IMFk is regarded as the 1/f signal. This completes the detection of the 1/f signal.

4. PSO-VMD Simulation Experiment

4.1. Experimental Condition

In order to study the reliability of high-power IGBT devices, IGBT components that meet the test conditions are selected first. Two control groups are set up in this paper for comparison and analysis of the influence of basic high-temperature reverse bias conditions on low-frequency 1/f noise [20,21]. One group of components is the IGBT devices of the jt020n after aging test, and the other group is the qualified IGBT devices of the same model and batch produced by Huawei Company. Given that 1/f noise is a low-frequency signal noise, the experimental frequency range is set as 0.1 Hz–100 kHz. The detection scheme is shown in Figure 3, and the experimental hardware platform established based on the IGBT low-frequency noise test environment is shown in Figure 4.

It is known that under the rated working condition of IGBT, the starting voltage V_GS = 2~4 V; however, in order to ensure the normal starting of IGBT devices during the test, it is necessary to set the bias voltage to 6 V or higher.

In order to study the low-frequency noise data of IGBT, before the test the default noise of the device is measured after an aging test is conducted. Normal devices are qualified products of the same batch. 100 normal devices are selected as the normal device group for accuracy, and the aging test devices are divided into two groups, with 100 aging test devices in each group fitted with a cluster of curves. Figure 5 shows the comparison of low-frequency power spectrum data of IGBT devices under normal and high-temperature aging test conditions.

According to the analysis of the above results, for conventional devices, the 1/f noise trend is mainly shown in the frequency range of 0.1–100 Hz. The power spectral density decreases with the increase in frequency, and there is an obvious Lorentz spectral trend in the frequency range of 100–100,000 Hz. After the high-temperature test, the low-frequency noise power spectral density of IGBT devices increases significantly in the overall noise variation range; its value decreases when the frequency increases, and fluctuates greatly when it is less than 100 Hz. The noise power in the broadband and the narrowband noise power corresponding to each frequency point are shown in

Table 1. Low-frequency noise data of IGBT.

Test Number	Broadband Voltage (5 Hz–100 Hz) (V)	Narrowband Noise (V2/Hz)
		1 Hz	10	100	1000
Normal device group	1.16 × 10−7	5.76 × 10−16	1.28 × 10−16	7.29 × 10−17	8.91 × 10−17
Aging test device group 1	2.56 × 10−6	4.95 × 10−13	8.62 × 10−14	2.23 × 10−14	6.65 × 10−15
Aging test device group 2	5.32 × 10−6	6.28 × 10−13	9.79 × 10−14	2.25 × 10−14	6.33 × 10−15

.

It can be seen from Table 1 that at 1 Hz, the low-frequency noise change rate of aging device group 1 is 828.4% compared with that of the normal device group, and the low-frequency noise change rate of aging device group 2 is 1089.3% compared with that of the normal device group. The change rates at 10 Hz were 6724% and 763.8%, respectively. The change rates at 100 Hz were 304.9% and 304.6%, respectively. The change rates at 1000 Hz were 73.6% and 70.0%, respectively. By analyzing the 1/f noise data of IGBT in the low-frequency band, it is obvious that the noise spectrum of the aging device is inversely proportional to the frequency. Compared with the normal device, the aging device is a faulty device, and its spectrum is significantly different from the normal device. Therefore, the reliability of IGBT devices can be reflected by 1/f noise.

In order to prove that the PSO-VMD decomposition algorithm is better than the improved EMD decomposition [12], we conducted experimental simulations of the traditional wavelet threshold denoising [11], the wavelet entropy soft threshold denoising [10], and the wavelet entropy soft threshold denoising algorithm based on lifting wavelet decomposition. The experiment was carried out with the signal-to-noise ratio of 1.44 dB, −3.56 dB, and −8 dB, respectively. The measured 1/f signal $s(t)$ in the IGBT device was simulated by the fractional Brownian model [7]. The 1/f noise power spectral density was obtained by setting $H=0.3$, as shown in Figure 6.

Figure 7 shows a computer-generated 1/f signal containing a large amount of white noise, where $L=1000$, with the signal-to-noise ratio being: $R_{SNR}=20\lg \left(\frac{ps}{pn}\right)=1.44 \mathrm{dB}$.

The parameters set by the PSO algorithm in this paper are shown in

Table 2. Parameters of the PSO algorithm.

Genmax	Sizepop	c1	c2	Vmax	Vmin	Popmax	Popmin
10	80	1.49445	1.49445	1	−1	40	2

, where gen-max is the largest evolutionary algebra, and size-pop is the population size.

4.2. Experimental Process

Using the particle swarm optimization algorithm, the output value can be obtained from Figure 8, where the penalty parameter $\alpha =37$, the decomposition modulus number $K=4$, and the optimal output signal-to-noise ratio peaks at 6.3891 dB with the evolution algebra.

Using the results of the algorithm to set the VMD model parameters, the four components which are depicted in Figure 9 are decomposed after measuring the input of the signal. IMF1, the 1/f noise extracted from the white noise background, is also included among them. The extracted 1/f noise power spectral density is shown in Figure 10, and it can be seen that the amplitude of the signal decreased significantly with increasing frequency, which shows significant 1/f noise characteristics. Consequently, the proposed particle swarm optimization algorithm, which is based on the variational mode decomposition, proves to be an effective method for detecting the 1/f noise.

Table 3. VMD algorithm for optimizing parameters to detect 1/f noise.

VMD Algorithm for Optimizing Parameters to Detect 1/f Noise
Signal to Noise Ratio	SNR	MSE	Penalty Parameterα	Decomposition Modulus K
1.44 dB	6.3891	0.6648	37	4
−3.56 dB	4.5942	0.8043	28	4
−8 dB	2.9838	1.0025	4	8

shows the values of signal-to-noise ratio, mean square error, optimal penalty parameter, and decomposition.

4.3. Simulation Algorithm Performance Analysis

It can be seen from the simulation experiments of the seven algorithms that the power spectral density of the detected 1/f noise signal decreases with increasing frequency, showing obvious spectral characteristics of 1/f noise. The signal extracted under background noise needs to meet two requirements: first, it has to ensure the smoothness of the extracted signal and the original signal; second, the greatest possible similarity between the extracted signal and the original signal needs to be achieved. Figure 10 shows the comparison of the signal power spectra of the seven denoising algorithms. It is observed in the figure that the power spectral density curve obtained after the maximal minimum threshold and the heuristic threshold denoising is obviously mixed with other noise. However, there is a distorting phenomenon in this method. The 1/f noise reduction range detected by the improved EMD algorithm is shorter and its denoising effect is poor. The performance of the other algorithms is indistinct in Figure 11.

In order to accurately measure the detection effect of the algorithm, the signal-to-noise ratio and the mean square error of the de-noising signal are selected as the measurement parameters. The expressions of the signal-to-noise ratio and the mean square error are shown by Equations (13) and (14). The original signal-to-noise ratio is set to 1.44 dB, −3.56 dB, and −8 dB, respectively.

$$SNR=10\times {\log }_{10}\left({\displaystyle {\sum }_i^N\frac{y_i{}^2}{{(x_i-y_i)}^2}}\right)$$

(13)

$$MSE=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{(x_i-y_i)}^2}$$

(14)

where $x_i$ is the denoised signal, $y_i$ is the original signal, and $N$ is the signal length.

It is also evident from

Table 4. Comparison of noise reduction effects for different SNR cases.

Signal to Noise Ratio	Parameter	Improved EMD Algorithm	Heuristic Threshold	Minimaxi Threshold	Lifting Wavelet Threshold	Wavelet Entropy Soft Threshold	Lifting Wavelet Decomposition Wavelet Entropy Soft Threshold	PSO-VMD
1.44 dB	SNR	3.8410	3.5747	5.6157	5.8679	5.8219	5.9898	6.3891
	MSE	0.8099	1.1892	0.7777	0.7353	0.724	0.7159	0.6649
−3.56 dB	SNR	3.0834	1.0235	2.0647	3.7093	3.8175	3.9327	4.5942
	MSE	0.9235	2.6333	1.6485	1.0566	1.0077	1.0029	0.8043
−8 dB	SNR	0.6377	0.3380	0.5647	1.8885	2.0183	2.0942	2.9838
	MSE	1.1294	4.6011	3.3930	1.6247	1.5141	1.5134	1.0025

that under different noise-to-noise ratios, the PSO-VMD algorithm can obtain a larger signal-to-noise ratio and a smaller noise error in the same noise background. In other words, the detection advantage of the PSO-VMD algorithm becomes more apparent when the signal to noise ratio is low.

Table 4 compares the detection effects of the PSO-VMD algorithm with respect to: the improved EMD algorithm, the heuristic threshold, the minimax threshold, the lifting wavelet threshold, the wavelet entropy soft threshold, and the lifting wavelet decomposition wavelet entropy soft threshold in the case of calculating different signal-to-noise ratios; a positive number indicates a percentage increase in the signal-to-noise ratio, and a negative number indicates a percentage decrease in the mean square error. It can be seen from

Table 5. Comparison of denoising effects compared with the PSO-VMD algorithm under different SNR.

Signal to Noise Ratio	Parameter	Improved EMD Algorithm	Heuristic Threshold	Minimaxi Threshold	Lifting Wavelet Threshold	Wavelet Entropy Soft Threshold	Lifting Wavelet Decomposition Wavelet Entropy Soft Threshold	PSO-VMD
1.44 dB	SNR	3.8410	3.5747	5.6157	5.8679	5.8219	5.9898	6.3891
	MSE	0.8099	1.1892	0.7777	0.7353	0.724	0.7159	0.6649
−3.56 dB	SNR	3.0834	1.0235	2.0647	3.7093	3.8175	3.9327	4.5942
	MSE	0.9235	2.6333	1.6485	1.0566	1.0077	1.0029	0.8043
−8 dB	SNR	0.6377	0.3380	0.5647	1.8885	2.0183	2.0942	2.9838
	MSE	1.1294	4.6011	3.3930	1.6247	1.5141	1.5134	1.0025

that the wavelet entropy soft threshold denoising algorithm based on lifting wavelet decomposition is the optimal 1/f among all noise detection algorithms that are based on wavelet theory. Compared with this algorithm, the signal-to-noise ratio detected by the PSO-VMD algorithm proposed in this paper was increased by 6.6%, 16.82%, and 42.48%, whereas the mean square error was reduced by 7.12%, 19.80%, and 33.76%, which greatly improves the detection effect.

5. Results

This paper proposes a 1/f noise signal detection method based on VMD (Variable Modal Decomposition) for optimizing parameters with the PSO algorithm. Through a series of experimental simulations, the detection method of PSO-VMD adopts suitable VMD model parameters and adaptively realizes the effective separation of detection signal components. The modal aliasing problem of the traditional EMD algorithm is also surmounted. Under various SNR conditions, the PSO-VMD algorithm achieves the best signal detection performance. Its advantages become more obvious especially when the signal-to-noise ratio is low. Compared with the optimal wavelet-based denoising algorithm, the signal-to-noise ratio increases by 6.6%, 16.82%, and 42.48%, whereas the mean square error reduces by 7.14%, 19.80%, and 33.76%. The signal power spectral density has obvious characteristics of the 1/f noise signal, and its power spectral density decreases with the increase in frequency. When the PSO-VMD algorithm is used for processing the 1/f noise signal, the extracted signal has the same smoothness as the original signal, and its power spectrum is more accurate. In this sense, this method improves the accuracy of detection.