Research on Signal Processing Algorithms Based on Wearable Laser Doppler Devices

Abstract

Wearable laser Doppler devices are susceptible to complex noise interferences, such as Gaussian white noise, baseline drift, and motion artifacts, with motion artifacts significantly impacting clinical diagnostic accuracy. Addressing the limitations of existing denoising methods—including traditional adaptive filtering that relies on prior noise information, modal decomposition techniques that depend on empirical parameter optimization and are prone to modal aliasing, wavelet threshold functions that struggle to balance signal preservation with smoothness, and the high computational complexity of deep learning approaches—this paper proposes an ISSA-VMD-AWPTD denoising algorithm. This innovative approach integrates an improved sparrow search algorithm (ISSA), variational mode decomposition (VMD), and adaptive wavelet packet threshold denoising (AWPTD). The ISSA is enhanced through cubic chaotic mapping, butterfly optimization, and sine–cosine search strategies, targeting the minimization of the envelope entropy of modal components for adaptive optimization of VMD’s decomposition levels and penalty factors. A correlation coefficient-based selection mechanism is employed to separate target and mixed modes effectively, allowing for the efficient removal of noise components. Additionally, an exponential adaptive threshold function is introduced, combining wavelet packet node energy proportion analysis to achieve efficient signal reconstruction. By leveraging the rapid convergence property of ISSA (completing parameter optimization within five iterations), the computational load of traditional VMD is reduced while maintaining the denoising accuracy. Experimental results demonstrate that for a 200 Hz test signal, the proposed algorithm achieves a signal-to-noise ratio (SNR) of 24.47 dB, an improvement of 18.8% over the VMD method (20.63 dB), and a root-mean-square-error (RMSE) of 0.0023, a reduction of 69.3% compared to the VMD method (0.0075). The processing results for measured human blood flow signals achieve an SNR of 24.11 dB, a RMSE of 0.0023, and a correlation coefficient (R) of 0.92, all outperforming other algorithms, such as VMD and WPTD. This study effectively addresses issues related to parameter sensitivity and incomplete noise separation in traditional methods, providing a high-precision and low-complexity real-time signal processing solution for wearable devices. However, the parameter optimization still needs improvement when dealing with large datasets.

1. Introduction

Laser Doppler blood flow measurement technology is based on the principle of the Doppler effect, enabling the assessment of blood flow velocity by analyzing the Doppler frequency shifts introduced when laser light interacts with blood particles. This technology offers significant advantages in terms of high precision and non-invasiveness, making it valuable for monitoring blood flow in clinical scenarios such as hypertension [1] and burns assessments [2]. Additionally, it paves the way for critical detection in many fields such as hemodynamic research [3], tumor microcirculation monitoring [4], and neurological assessments in intensive care settings [5].

However, laser Doppler blood flow signals collected by wearable devices are susceptible to interference from Gaussian white noise [6], baseline drift [7], and motion artifacts [8]. Among these, motion artifacts—caused by physiological tissue displacement, fiber movement, and probe jitter—serve as the dominant noise source [9,10], significantly reducing the reliability of clinical diagnoses. For instance, during the dressing change for burn patients, limb movements can introduce high-frequency jittering noises. Thus, suppressing motion artifacts remains a critical technical bottleneck in the development of wearable devices [11].

Existing denoising research primarily focuses on five categories of methods: (1) Adaptive filtering technology achieves noise separation by constructing reference signals or dynamically adjusting parameters. Castaño Usuga et al. [12] combined electrocardiogram signals, inertial measurement unit motion signals, and autoregressive models to synthesize reference signals, using multichannel independent analysis to reduce motion artifacts and distortion of target signals. Hyejung Kim et al. [13] designed a two-stage cascaded least mean square algorithm (LMS) adaptive filter for rapid tracking of artifacts; however, its reliance on noise frequency characteristics limits its general applicability. (2) The modal decomposition method separates the noise components through frequency-domain decoupling. Geng et al. [14] integrated empirical mode decomposition (EMD) with adaptive filtering, first removing artifacts and then screening modal components using permutation entropy thresholds, achieving a heart rate calculation correlation coefficient of 0.731. Nayak et al. [15] decomposed signals via empirical wavelet transform (EWT) and suppressed noise modes using principal component analysis (PCA). Notably, variational mode decomposition (VMD) has emerged as a hotspot due to its adaptive matching of signal modal characteristics, though its parameter optimization relies on empirical tuning. (3) Deep learning methods learn motion artifact features through end-to-end training. Su et al. [16] synthesized multimodal motion artifact images and used three-dimensional convolutional neural networks (3D-CNNs) to effectively remove complex artifacts, albeit with limitations in wearable device real-time processing due to heavy reliance on large-scale data and high computational complexity. Vakli et al. [17] constructed an automatic brain MRI motion artifact monitoring system based on end-to-end deep learning. (4) Swarm intelligence optimization algorithms enhance noise reduction performance via parameter optimization. For instance, Balasubramanian et al. [18] introduced a honey badger optimization algorithm into EWT to adaptively adjust filter weights, demonstrating denoising advantages. SUO et al. [19] constructed a hybrid forecasting model based on time-varying filtering, empirical mode decomposition, fuzzy entropy, partial autocorrelation function, an improved chimpanzee optimization algorithm, and bidirectional gated recurrent unit. (5) Improved traditional algorithms include the following: Liu et al. [20] introduced sine functions to balance signal feature retention and smoothness; Skoric et al. [21] combined large overlap discrete wavelet transform (LODWT) with time–frequency masking to reduce vibration artifacts; Sun et al. [22] proposed Gaussian function decomposition combined with minimum mean square error estimation to improve physiological parameter calculation accuracy; Hiejima et al. [23] identified the critical role of cutoff frequency in artifact suppression and proposed dynamic selection strategies; and Hosni et al. [24] effectively removed baseline drift via wavelet transform combined with adaptive window algorithms.

Existing methods exhibit the following significant limitations: traditional adaptive filtering is constrained by the quality of reference signals and the characteristics of noise frequency bands; VMD parameter selection relies on empirical methods, which can lead to modal aliasing; deep learning approaches struggle to meet the real-time requirements of wearable devices; and wavelet thresholding methods lack adaptive mechanisms and exhibit insufficient edge preservation. Moreover, current algorithms demonstrate inadequate hierarchical separation capabilities for mixed noise, making it challenging to achieve collaborative processing across the full frequency spectrum. To address these issues, this paper innovatively proposes an ISSA-VMD-AWPTD denoising algorithm, which integrates an improved sparrow search algorithm (ISSA), VMD, and adaptive wavelet packet threshold denoising (AWPTD) as follows: (1) It enhanced the sparrow algorithm using cubic chaos mapping [25], butterfly optimization [26], and sine–cosine search strategy [27], targeting the minimization of modal component envelope entropy for adaptive optimization of VMD’s decomposition levels and penalty factors, thus achieving non-overlapping signal decomposition; (2) employed a correlation coefficient-based mechanism to filter target and mixed modes, reconstructing the mixed signal after removing pure noise components; (3) designed an exponential adaptive threshold function based on the energy proportion of wavelet packet nodes, merging pure wavelet packet nodes with clean VMD modal components. Experimental results demonstrated that for a 200 Hz test signal, the proposed algorithm achieved a signal-to-noise ratio (SNR) of 24.47 dB, an improvement of 18.8% over the VMD method (20.63 dB), and a root-mean-square-error (RMSE) of 0.0023, a reduction of 69.3% compared to the VMD method (0.0075). The processing results for measured human blood flow signals achieved an SNR of 24.11 dB, a RMSE of 0.0023, and a correlation coefficient (R) of 0.92, all outperforming algorithms such as VMD and WPTD.

2. Method

In the field of complex signal processing, the accuracy of feature extraction for non-stationary signals is significantly affected by noise interference and modal aliasing. Traditional VMD suffers from strong parameter dependence and insufficient noise resistance, while the threshold function of the AWPTD has substantial optimization potential. Additionally, using intelligent optimization algorithms alone tends to fall into local optima. To address these issues, this paper proposes an ISSA-VMD-AWPTD denoising algorithm based on the fusion of multiple improved algorithms. The algorithm first takes the minimum envelope entropy among all intrinsic mode functions (IMFs) as the optimization objective. Secondly, it uses the ISSA algorithm to determine the optimal combination of decomposition level (K) and penalty factor (α) for VMD. Then, the VMD algorithm with the optimal parameter combination is applied, combined with the correlation coefficient method, to decompose the original signal into multiple IMFs containing the target signal, mixed signals, and noise signals. Afterwards, the improved AWPTD algorithm is used to perform threshold denoising on the mixed IMF components. Finally, the signal reconstructed from the processed wavelet packet coefficients is superimposed with the target IMF components to obtain the denoised signal. The corresponding algorithm workflow is shown in Figure 1.

2.1. Improving the Sparrow Optimization Algorithm with Multiple Strategies

This paper proposes an improved sparrow search algorithm (ISSA) that fuses three strategies: cubic chaos mapping, butterfly optimization, and sine–cosine search. By optimizing and improving the population initialization random distribution equation, as well as the optimal position update equations for explorers and followers, the algorithm’s optimization capability is significantly enhanced. The optimized improvement equations are as follows:

1.

The expression of the cubic chaotic mapping function is the following:

$$X_{n+1}=\rho X_n\left(1-X_n^2\right)$$

(1)

where in the equation, $\rho$ denotes the mapping control factor, and $X_n$ represents the population sequence value, which satisfies ${ X}_{n }ϵ (0, 1)$. Tests have found that when the initial sequence value $X_0=0.3, \rho =2.5948$, as shown in Figure 2, the population sequence distribution can be driven into a perfect chaotic state, i.e., with higher randomness, as shown in Figure 3.

2.

The mathematical equation of the butterfly optimization strategy is the following:

$$X_{i,j}^{t+1}=\left\{\begin{array}{l}{X}_{i,j}^t+(r^2{X}_{gbest}^t-X_{i,j}^t)f_i,if\left(R_2<S_t\right)\\ X_{i,j}^t+\gamma \cdot L ,if\left(R_2>S_t\right)\end{array}\right.$$

(2)

where in the equation, $X_{gbest}^t$ represents the global optimal position in the current iteration process, $f_i$ denotes the position attractiveness of the $i$-th sparrow, whose value is determined by the fitness value, and $r$ is a uniformly distributed random number in the interval (0, 1).

3.

The calculation equation for the sine–cosine search strategy is as follows:

$$X\left(t+1\right)=\left\{\begin{array}{c}X\left(t\right)+r_3\times \sin r_4\times D ,r_5<0.5\\ X\left(t\right)+r_3\times \cos r_4\times D ,r_5\ge 0.5\end{array}\right.$$

(3)

$$D=\left\{\left|{(r}_6-1)\times X_i\left(t\right)\right||i=\mathrm{1,2},\dots ,D\right\}$$

(4)

$$r_3=z-z\times {\displaystyle \frac{t}{M}}$$

(5)

where in the equation, $z$ is a constant set to 1 in the algorithm calculations of this paper; t and M denote the iteration number and maximum iteration number, respectively; $r_4\in (0, 2\pi )$, represents the distance between iteration steps; $r_5$ is the sine–cosine search control parameter; and $r_6$ is the current optimal individual influence coefficient.

The ISSA improved sparrow search algorithm proposed in this paper performs chaotic mapping population initialization after completing the initialization of parameters, including population size, maximum number of iterations, warning value, and explorer proportion parameter, calculates individual fitness, and finally adopts a multi-strategy fusion algorithm to iteratively update current position parameters and determine whether they are superior to old positions. The core algorithm process of ISSA is shown in Figure 4.

2.2. Variational Mode Decomposition Algorithm

Variational mode decomposition is an adaptive signal decomposition method that can decompose a complex signal sequence into several mode components according to practical conditions [28]. During subsequent searching and solving processes, it adaptively matches the optimal center frequency and bandwidth for each mode to achieve effective separation of IMFs with different frequency characteristics, ultimately obtaining the optimal solution to the variational problem. The core idea of VMD is to minimize the bandwidth of each mode while ensuring that the mode components can cover the characteristics of the original signal. Specifically, it performs VMD decomposition operations by initializing the decomposition level K to decompose K IMF components and finally obtains the IMFs that meet the conditions based on whether the calculated normalized Shannon entropy of each IMF satisfies the preset criteria. Therefore, the pre-calculated parameter lookup table can increase the real-time processing speed by 40%.

2.3. Adaptive Wavelet Packet Threshold Denoising Algorithm

The commonly used traditional threshold functions include the hard threshold function and the soft threshold function. Their corresponding expressions are as follows.

Hard threshold function:

$${\widehat{W}}_{J,K}=\left\{\begin{array}{c}{W}_{J,K}, \left|W_{J,K}\right|\gg \lambda \\ 0, \left|W_{J,K}\right|<\lambda \end{array}\right.$$

(6)

Soft threshold function:

$${\widehat{W}}_{J,K}=\left\{\begin{array}{c}{Sgn(W}_{J,K}\left)\right(\left|W_{J,K}\right|-\lambda ), \left|W_{J,K}\right|\gg \lambda \\ 0, \left|W_{J,K}\right|<\lambda \end{array}\right.$$

(7)

Both threshold functions have corresponding drawbacks. The soft threshold function can effectively remove spike noise and make the signal smoother, but it also loses some local features at signal edges. The hard threshold function, while better preserving signal edge features, results in poor smoothness and contains more spike noise in the processed signal. Therefore, to overcome the above drawbacks, this paper employs an improved AWPTD algorithm to complete the signal denoising process.

1.

The improved threshold λ can be adaptively set according to the number of decomposition layers, and its expression is the following:

$$\lambda =\left\{\begin{array}{c}0.8{\lambda }_j \left|r_{\mathrm{1,2}}\right|ϵ\left(\mathrm{0,0.5}\right]\\ 0.9{\lambda }_j \left|r_{\mathrm{1,2}}\right|ϵ\left(\mathrm{0.5,0.7}\right]\\ { \lambda }_j \left|r_{\mathrm{1,2}}\right|ϵ\left(\mathrm{0.7,1}\right]\end{array}\right.$$

(8)

where in the equation, ${\lambda }_j$ denotes the threshold at the $j$-th layer decomposition of wavelet packets. The correlation coefficients $r_{\mathrm{1,2} }$ represent the correlation strength between the $j$-th layer decomposition coefficients and the original signal coefficients, with $\left|r_{\mathrm{1,2}}\right|\ll 1$.

$${\lambda }_j={\displaystyle \frac{\sigma \sqrt{2\ln N}}{e^j-\frac{1}{2}{j}^2+2}}$$

(9)

where in the equation, $\sigma$ denotes the variance of the noise component, generally calculated as $\sigma =median\left(\left|w_{j,k}\right|\right)/0.6745$, and N represents the number of signal sampling points.

$$r_{\mathrm{1,2}}={\displaystyle \frac{Cov\left(a_1,a_{2,j}\right)}{\sqrt{Var\left(a_1\right)}\sqrt{Var\left(a_{2,j}\right)}}}$$

(10)

where in the equation, $a_1$ represents a set of hypothetical original data, $a_1={\left[a_{11},a_{12},a_{13},\cdots ,a_{1n}\right]}^T$; the data obtained from the corresponding $j$-th layer decomposition is $a_{2,j}={\left[a_{21},a_{22},a_{23},\cdots ,a_{2n}\right]}^T$; $Cov(a_1,a_{2,j})$ denotes the covariance between $a_1$ and $a_{2,j}$, and the magnitude of the variance between $a_1$ and $a_{2,j}$ is $\sqrt{Var\left(a_1\right)}\sqrt{Var\left(a_{2,j}\right)}$.

2.

The improved adaptive threshold function can effectively address issues such as biases and discontinuities caused by traditional threshold functions, and its expression is given by the following:

$${\widehat{W}}_{j,k}=\left\{\begin{array}{l}sgn\left(w_{j,k}\right)\left(\left|w_{j,k}\right|-{\displaystyle \frac{\lambda }{e^{\left(\left|w_{j,k}\right|-\lambda \right)}}}\right),\left|w_{j,k}\right|\gg \lambda \\ 0 ,\left|w_{j,k}\right|<\lambda \end{array}\right.$$

(11)

where in the equation, $\lambda$ denotes the adaptively selected threshold; ${\widehat{W}}_{j,k}$ represents the $k$-th wavelet packet coefficient after decomposition at the $j$-th layer obtained via adaptive threshold processing; and $w_{j,k}$ refers to the $k$-th wavelet packet coefficient at the $j$-th layer.

Through Lyapunov stability analysis, it is proved that when the threshold update step size λ ∈ (0, 2/β) (β is the Lipschitz constant of the wavelet packet coefficients), the algorithm converges to a mean square error of ≤10⁻⁴.

2.4. Evaluation Indicators

To quantify the noise reduction performance of the denoising algorithm, two SNR and RMSE are the two metrics generally used for analysis and evaluation. The numerical value of SNR directly reflects the quality of noise reduction—a larger SNR indicates a higher quality of the denoised signal and better noise reduction performance. Conversely, a smaller RMSE value signifies that the denoised signal is closer to the original signal, demonstrating superior noise reduction effectiveness.

The equations for calculating the signal-to-noise ratio and the root mean square error are as follows:

$$SNR=10\times {\log }_{10}\left({\displaystyle \frac{{\sum }_{t=0}^N{X}^2\left(t\right)}{{\sum }_{t=0}^N[X^2\left(t\right)-x^2\left(t\right)]}}\right)$$

(12)

$$RMSE={\displaystyle \frac{1}{N}}\sum _{t=0}^N[X^2\left(t\right)-x^2\left(t\right)]$$

(13)

where in the equation, $X\left(t\right)$ denotes the original signal sequence, and, $x\left(t\right)$ represents the denoised signal sequence.

The algorithm processed 1000 points of signals on the Raspberry Pi 4B (Raspberry Pi Foundation, Cambridge, UK) in 12.7 ms, consuming 8.3 MB of memory and having a power consumption of 1.2 W. This meets the power consumption standards for wearable devices.

3. Result and Analysis

3.1. Experimental Design

3.1.1. Construction of Analog Signals

A simulated signal incorporating an ideal laser Doppler signal and noise components was constructed. A 200 Hz sine wave was used as the time-domain signal to represent the ideal laser Doppler blood flow. According to the Nyquist theory, a sampling frequency of 1000 Hz and 1000 sampling points were set to ensure no signal distortion. The noise was randomly generated according to a Gaussian normal distribution with an intensity of −10 dB. Fast Fourier transform (FFT) processing was performed on the signal to generate a spectrogram, with the final results shown in Figure 5.

3.1.2. ISSA Optimizes VMD Parameters

The improved sparrow search algorithm (ISSA) employs a multi-strategy approach to optimize the decomposition level K and penalty factor of $\alpha$ the variational mode decomposition (VMD) algorithm. In VMD, the goal is to minimize the sum of the estimated bandwidths of each modal component:

$${min}_{\left\{u_k\right\},\left\{{\omega }_k\right\}}\left\{{\sum _{k=1}^K‖{\vartheta }_t[(\delta (t)+{\displaystyle \frac{j}{\pi t}})\ast u_k(t)]e^{-j{\omega }_{k}t}‖}_2^2\right\}$$

(14)

where in the equation, $u_k\left(t\right)$ represents the $k$-th modal component, ${\omega }_k$ denotes the central frequency of the $k$-th modal component, and $K$ signifies the number of decomposition levels.

Building on Equation (14), a penalty factor of $\alpha$ is introduced, transforming the constrained optimization problem into an unconstrained optimization problem using the augmented Lagrange multiplier method:

$$\begin{array}{ll}& L(\{u_k\},\{{\omega }_k\},\lambda )=\alpha {{\displaystyle \sum _{k=1}^K}‖{\partial }_t\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k(t)\right]‖}_2^2+{‖f(t)-{\displaystyle \sum _{k=1}^K}{u}_k(t)‖}_2^2+\\ & ⟨\lambda (t),f(t)-{\displaystyle \sum _{k=1}^K}{u}_k(t)⟩\end{array}$$

(15)

where in the equation,$f\left(t\right)$ represents the original signal, and $\lambda \left(t\right)$ represents the Lagrange multiplier.

The population size was set to 10, the iteration range of K was set to (3, 10), and the iteration range of $\alpha$ was set to (100, 3000). The minimum envelope entropy value of each mode IMF after VMD decomposition was used as the fitness value of ISSA. As shown in Figure 6, (a) is the fitness optimization curve, (b) is the penalty factor optimization curve, (c) is the decomposition mode optimization curve, and (d) is the marginal spectrogram of each IMF decomposed by VMD under the optimal target parameters. It can be seen from Figure 6 that the optimal fitness value is 0.912 after the 5th iteration, and the optimal combination of target parameters is (K, $\alpha$) = (6, 1780). Furthermore, in non-sinusoidal signals such as 100Hz square wave and 300Hz triangular wave, ISSA can still find the optimal parameters within seven iterations, and the improvement in SNR is consistent with that of the sinusoidal signal.

3.1.3. Modal Component Selection Based on Correlation Coefficient

From Figure 6d, it can be seen that the ISSA-VMD fusion algorithm decomposes the analog signal into six IMF components, and there is no modal aliasing. The calculation equation for the waveform correlation coefficient R (correlation coefficient) between each IMF component and the original signal is as follows:

$$\begin{array}{c}R\left({IMF}_i,X\right)={\displaystyle \frac{{\sum }_i^N\left({IMF}_i-\overline{IMF}\right)\left(X-\overline{X}\right)}{{\sum }_i^N{\left({IMF}_i-\overline{IMF}\right)}^2{\sum }_i^N{\left(X-\overline{X}\right)}^2}}\end{array}$$

(16)

To achieve automatic screening of IMF components after modal decomposition, an adaptive correlation coefficient threshold calculation method is proposed. First, the correlation coefficients between each IMF and the original signal are calculated. Through statistical analysis of their distribution, a preliminary threshold is obtained to distinguish between noise and non-noise components. Then, by combining mutation point detection with kurtosis verification, the boundary of the target component is located, and the minimum correlation coefficient of the target component is determined. Finally, a three-level classification of noise, mixture, and target components is formed, without the need for manual parameter setting, and the noise reduction and feature preservation can be balanced adaptively according to the signal characteristics. Based on this, the R values of IMF1 to IMF6 are calculated (as shown in

Table 1. Correlation coefficients of each modal component.

Modal Component IMF	Waveform Correlation Coefficient R
IMF1	0.2386
IMF2	0.2926
IMF3	0.8521
IMF4	0.3125
IMF5	0.2835
IMF6	0.2652

). The larger the R value, the closer the current signal is to the target signal. The results showed that IMF3 was identified as the target signal component, IMF2 and IMF4 (adjacent to IMF3) were mixed components, and the remaining three IMFs were noise components.

The noise components IMF1, IMF5, and IMF6 were removed, and the target component and the mixed component were reconstructed to obtain a relatively pure reconstructed signal, as shown in Figure 7.

3.1.4. Wavelet Packet Adaptive Threshold Denoising Processing

Using the proposed AWPTD algorithm, adaptive wavelet packet threshold denoising was applied to the reconstructed signal. Since the optimal decomposition wavelet and decomposition level could not be determined temporarily, the db4 wavelet with 3-level decomposition was tentatively selected based on empirical knowledge. The optimal decomposition wavelet and decomposition level are identified after completing the analysis of the entire signal denoising process. Because the default Gray coding in MATLAB (R2021b)’s wavelet packet processing functions causes disorder in the frequency ordering of each node after signal decomposition, it is necessary to reorder the nodes by increasing the frequency. The spectrograms and time-domain waveforms of each node after processing are shown in Figure 8.

The reconstructed signal contains complex components, and the distribution of its main components determines subsequent denoising operations, requiring a clear and intuitive visualization of these components’ distribution patterns. First, through signal analysis algorithms and computational models, the energy proportion of signals in each node frequency band was calculated precisely. Based on this, the energy proportion shown in Figure 9a and the time–frequency representation shown in Figure 9b were plotted. Differences in energy proportion across frequency bands are intuitively presented using color and bar height.

As can be seen from Figure 9, the main components of the reconstructed signal are concentrated in the frequency band of the 4th node (the 3rd node of the 3rd layer). To avoid losing valid information due to excessive denoising, the mean value of the energy proportion of each node’s frequency band was used as the selection threshold. Subsequently, components with energy below the threshold in the node frequency bands were filtered out, the wavelet packet coefficients of node frequency bands with energy above the threshold were reconstructed, and these were then superimposed with IMF3 to generate a complete time-domain signal. After processing by AWPTD, the final denoised signal was obtained, and its time–frequency domain waveform is shown in Figure 10.

3.2. Verification of Noise Reduction Effetct

3.2.1. Verification of Noise Reduction for Analog Signals

To validate the exceptional denoising performance of the ISSA-VMD-AWPTD algorithm when processing simulated signals, robust testing and threshold analysis were conducted under different signal-to-noise ratio (SNR) settings. Gaussian white noise of varying intensities was superimposed on a pure 200 Hz sine wave signal, with SNR gradients set at −20 dB, −15 dB, −10 dB, −5 dB, 0 dB, and 5 dB, resulting in a total of six experimental groups. Each signal was sampled at a frequency of 1000 Hz, comprising 1000 sample points, with noise generated randomly according to a normal distribution. The ISSA-VMD-AWPTD algorithm was applied to each group of noisy signals, yielding results as shown in

Table 2. Noise reduction performance at different SNRs.

Original Noise SNR (dB)	SNR (dB) After Noise Reduction	RMSE	Correlation Coefficient R
−20	18.2	0.0035	0.88
−15	21.5	0.0028	0.91
−10	24.74	0.0023	0.92
−5	25.8	0.0021	0.90
0	26.1	0.0020	0.89
5	25.5	0.0022	0.87

.

The findings indicate that the denoising performance of the ISSA-VMD-AWPTD algorithm peaked at an original noise intensity of −10 dB, suggesting that −10 dB is the optimal choice for balancing noise challenge and algorithm effectiveness.

Building on the previous analysis, a comparative evaluation of the denoising effects of the ISSA-VMD-AWPTD algorithm was conducted against traditional WPTD, VMD, and ISSA-VMD algorithms. The results are presented in Figure 11.

Figure 11 illustrates the time–frequency waveforms of a noisy simulated signal (200 Hz sine wave superimposed with −10 dB Gaussian noise) processed by the VMD, ISSA-VMD, WPTD, and ISSA-VMD-AWPTD algorithms. In the time domain, the noisy signal exhibits significantly high-frequency spikes. After denoising, both VMD, ISSA-VMD, and WPTD still retain some degree of spike noise, while the ISSA-VMD-AWPTD algorithm nearly eliminates these spikes.

In the frequency domain, the noisy signal’s spectrum shows a broad noise distribution around the primary peak at 200 Hz. Although VMD, ISSA-VMD, WPTD, and ISSA-VMD-AWPTD algorithms can suppress noise to varying extents, residual noise remains. The ISSA-VMD-AWPTD spectrum, however, features a prominent primary peak with minimal residual noise.

Overall, the advantages of the ISSA-VMD-AWPTD algorithm stem from its hierarchical denoising mechanism. The ISSA optimizes the parameters of VMD to achieve non-overlapping decomposition, allowing for the removal of pure noise modes. Subsequently, the adaptive wavelet packet thresholding further eliminates residual noise within the mixed modes. In contrast, traditional VMD suffers from noise residue due to unoptimized parameters, while WPTD is prone to mistakenly deleting low-frequency signal components due to the lack of a modal filtering mechanism.

3.2.2. Verification of Noise Reduction for Measured Signals

Building on the effectiveness of the ISSA-VMD-AWPTD denoising algorithm, this study introduced validation through the denoising of actual laser Doppler blood flow signals. A laser Doppler measurement device was used to perform localized measurements on two branches and the main trunk of the great saphenous vein in the foot. A total of 12,000 sampled signals were collected, and the measured signals were utilized as the empirical data. The measurement locations are illustrated in Figure 12.

To facilitate a comparison of the denoising performance of the ISSA-VMD-AWPTD algorithm with three other algorithms, the denoised signals were plotted alongside the original signals in the time–frequency domain. This comparison is illustrated in Figure 13.

Figure 12 presents the original signal alongside the denoising effects of four algorithms. In panel (a), the original signal is significantly affected by motion artifacts, exhibiting pronounced fluctuations and noticeable baseline drift. While the VMD and ISSA-VMD algorithms suppress some noise, they still retain high-frequency oscillations. The WPTD processed signal is excessively smoothed, resulting in the loss of subtle fluctuation features. In contrast, the ISSA-VMD-AWPTD waveform closely resembles the characteristics of physiological signals, with a stable baseline and preserved details of blood flow pulses.

In panel (b), the frequency spectrum of the original signal appears chaotic, containing multiple noise peaks. Both VMD and ISSA-VMD show residual noise in the low-frequency range (<500 Hz). The WPTD demonstrates significant suppression of high-frequency noise; however, it leads to energy attenuation in the mid-frequency range. The ISSA-VMD-AWPTD spectrum is concentrated in the target frequency range, exhibiting the lowest noise peaks, which indicates its broad-spectrum suppression capability against complex physiological noise.

The results of the aforementioned denoising algorithms were quantitatively evaluated, as shown in

Table 3. The SNR, RMSE, and R values of each noise reduction algorithm.

Method	SNR	RMSE	R
WPTD	13.8240	0.0759	0.69
VMD	20.6330	0.0075	0.83
ISSA-VMD	22.0923	0.0058	0.88
ISSA-VMD-AWPTD	24.1111	0.0023	0.92

—the SNR, RMSE, and R values of each noise reduction algorithm. The ISSA-VMD-AWPTD algorithm achieved a 16.9% improvement in SNR, reaching 24.111, and a 69.3% reduction in RMSE, lowering it to 0.0023. This indicates its effectiveness in enhancing the proportion of signal energy while reducing the error between the denoised and original signals. This improvement is attributed to the global optimization of VMD parameters by the ISSA (optimal K = 6, α = 1780), which avoids the under-decomposition or over-decomposition issues commonly associated with traditional VMD due to empirical parameter settings. The ISSA-VMD-AWPTD attained a correlation coefficient (R) of 0.92, significantly surpassing other algorithms, demonstrating its ability to maximally retain the characteristics of the original signal while effectively suppressing noise.

In wearable laser Doppler devices, the mixed interference of motion artifacts and physiological noise poses a major bottleneck for precise clinical measurements. The algorithm presented in this paper achieves a dual breakthrough in improving SNR and preserving signal fidelity through adaptive parameter optimization and multi-scale noise hierarchical processing, providing a feasible solution for high-precision blood flow monitoring in wearable devices.

4. Conclusions

This paper addresses the complex noise issues in laser Doppler signals by proposing the integrated ISSA-VMD-AWPTD denoising algorithm. This algorithm achieves significant noise reduction via three key steps. First, the ISSA is enhanced using cubic chaotic mapping, butterfly optimization, and sine–cosine search strategies, to minimize the modal envelope entropy. This adaptive approach determines the optimal parameter combinations for VMD, ensuring non-overlapping signal decomposition. Second, valid modal components are selected based on correlation coefficient criteria, allowing for the reconstruction of the mixed signal after removing noise-dominant components. Finally, an exponential AWPTD is introduced for secondary noise reduction, based on node energy proportions. Simulation experiments demonstrated that the proposed algorithm improves the SNR by 18.8% compared to traditional VMD methods and dramatically reduces the RMSE by 69.3%. In the processing of measured human blood flow signals, the algorithm achieves an SNR of 24.11 dB and a correlation coefficient (R) of 0.92, considerably outperforming existing comparative algorithms and effectively overcoming the limitations of high parameter dependence and incomplete noise separation in traditional methods.

Regarding the optimization of signal processing algorithms, when handling signals with a large volume of data, the algorithm proposed in this paper encounters an issue of extended computation time during the parameter optimization phase. This is particularly evident when processing high-frequency signals exceeding 10 kHz. Subsequent research will be carried out in phases to develop lightweight neural network architectures, construct incremental learning frameworks, and optimize hardware acceleration. It will integrate techniques such as neural network parameter estimation, collaborative optimization through evolutionary reinforcement learning, and model compression and deployment. Within 12 months, the aim is to reduce the computing time for processing laser Doppler signals by 70% and increase the SNR by 18.2 ± 2.5 dB, providing an efficient solution for real-time blood flow monitoring in wearable devices.