Establishment and Application of Recursive Digital Processing Algorithm with Difference Analysis-Based Parameter Setting and Difference Feedback

1. Introduction

In signal processing, preprocessing and feature extraction are two indispensable steps[1,2]. In methods analyzing signals based on data collected by analytical instruments, data preprocessing is a crucial step[3]. Data preprocessing not only enhances the accuracy of subsequent analysis results but also improves the efficiency of data processing[4]. To achieve effectively signal preprocessing for further data analysis, scholars have employed various methods. The methods for preprocessing mainly include various types of digital filters (high-pass filters[5], low-pass filters[6], band-pass filters[7], notch filters[8], etc.), adaptive filters[9], wavelet transforms[10], Empirical Mode Decomposition[11], methods based on neural networks[12], and machine learning algorithms[13,14]. Feature extraction is a crucial step in the analysis of medical signals[15]. Since biomedical signals typically contain a large amount of raw data, feature extraction can transform the data into representative features, reducing data dimensionality, thus to simplify subsequent analysis processes. It also aids in identifying and extracting key information and patterns from signals, helping reveal underlying physiological or pathological characteristics embedded in the signal. Feature extraction is primarily achieved by extracting features from the time domain[16], frequency domain[17], time-frequency domain[18], decomposition domain[19], and deep features using filters with different functions[20].

However, in most studies, common to establish at least two algorithm models to achieve the aforementioned two goals. Currently, it remains further research and development to use the same algorithm to highlight feature areas as well as smooth non-feature areas that are seemingly contradictory functions.

This study explored the possibility of using the same algorithm to highlight feature areas while smoothing non-feature areas, and established an algorithm model with a sign difference feedback system based on statistical analysis. As shown in Figure 1.

2. Method

2.1. Mathematical Formulation of the Algorithm

For digital signal processing systems, the input of the system is a one-dimensional image, and a one-dimensional image can be abstracted as a binary function, denoted as $f(x,y)$. Here, $(x,y)$ represents the pixel position, and $f(x,y)$ represents the signal intensity at that position. After passing through an image processing system, the output is still a function. Therefore, image processing is essentially an operation on a function. The input to this processing system is a binary function, and the output is also a binary function.

The spatial domain method involves direct operations on pixels in an image. If we denote the output function as $g(x,y)$, at position $(x,y)$, the value of $g(x,y)$ is highly correlated with the pixels at the corresponding position in the input image and its neighborhood. The mathematical model for the spatial domain method is shown in Equation (1).

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$$g\left(x,y\right)=T\left[f\left(x,y\right)\right],$$

(1)

In Equation (1), T: a signal processing system, $(x,y)$: the pixel position, $f(x,y)$: the signal intensity at position (x, y) (the magnitude of the input), $g(x,y)$: the signal output intensity at position $(x,y)$.

For a signal with the following characteristics: continuous gradient ascent and descent segments are considered as valid signals or feature-extracted signals; sporadic or short-duration gradient ascent and descent segments are considered as interference signals or non-feature signals; non-feature extraction signals may exhibit directional interference. It is required that the output signal retains the trend and amplitude of the feature-extracted signal, preserving the shape corresponding to the raw signal. The algorithm model we have established addresses the above issues, as shown in Equation (2).

$$d_n=\left\{\begin{array}{c}{y}_1,n=1\\ \left(\sum _{i=1}^{n-1}\alpha ·{|∆y}_{n-i}|\right)+\left|∆y_n\right|,n\ge 2\\ ∆y_i=\frac{y_i-y_{i-1}}{x_i-x_{i-1}},i\ge 2\\ ∆y_1=y_1\end{array}\right.,$$

(2)

In Equation (2), $y_i$: the amplitude of the $i$th input signal, $x_i$: the index of the $i$th input signal, $d_i$: the amplitude of the $i$th output signal, $\alpha$: a weight power, $\alpha ϵ\left(\mathrm{0.9,1}\right)$.

2.2. Meaning and Purpose

To facilitate the explanation of the meaning and purpose of the algorithm, we present Equation (2) in a logical form, as shown in Equation (3).

$$Y=y_1+y_2,$$

(3)

In the Equation (3), $Y$：$d_n$，$y_1$：$\sum _{i=1}^{n-1}\mathsf{\alpha }·{|∆y}_{n-i}|$，$y_2$：$\left|∆y_n\right|$.

We view the output function of Equation (2) as a superposition of two function results. One is the function $y_1$, and the other is the function$y_2$. The discrete form of function $y_1$ is shown in Equation (4).

$$y_{1i}={\alpha ·d}_{i-1},i\ge 2,$$

(4)

In Equation (4), $y_{1i}$: the value of the $i$th output signal, $\alpha$: the weight power, $\alpha ϵ\left[\mathrm{0,1}\right]$.

In Equation (4), corresponding to function $y_1$, the operation involves self-repetition, indicating a recursive algorithm based on the recursive principle. It influences the current output based on the previous output, and when $0<\alpha <1$, the signal is attenuated. Since the output is a part of the previous output, when the signal undergoes significant changes in adjacent time periods, the output signal remains relatively smooth. When $\alpha$ approaches 1, the smoothing effect of the filter becomes more pronounced. With $\alpha$ closer to 1, the signal changes more smoothly as it depends more on the previous output. A larger $\alpha$ value can better suppress signal fluctuations, reduce the impact of noise, and stabilize the signal. When $\alpha$ is a fixed value, meaning the output is a fixed proportion of the previous output, it implies that the current output is a part of the previous output, and the proportion is fixed. This helps reduce signal fluctuations and noise. Therefore, in Equation (2), we provide a more effective range for $\alpha$.

The discrete form of $y_2$ is shown in Equation (5).

$$y_{2i}={|data}_i-{data}_{i-1}|,$$

(5)

$y_{2i}$：The $i$th predicted value，${data}_i$：The $i$th actual observed value.

Equation (5), which represents function $y_2$, is used to maintain the continuous trend. When the input prediction signal exhibits continuous changes, whether ascending or descending, Equation (5) reflects this change, which is added to the output sequence. This ensures that the output signal maintains consistency with the continuous trend of the input signal. If the input signal is continuously increasing, the output sequence will also continue to rise. Due to the influence of the first part of the function $y_1$, the output sequence smoothly follows the overall trend of the input signal, while the function $y_2$ part retains the trend of the input signal. Therefore, if the input signal is continuously decreasing, the output sequence will also continue to decrease. The waveform shape of the output aligns with the waveform shape of the input signal, which is the result of the combined effects of both $y_1$ and $y_2$ components.

2.3. Parameter Settings based on Difference Analysis

Parameter settings are crucial for the algorithm model. In this study, we further determined the value of the weight constant $\alpha$ based on the empirical range of the smoothing filter algorithm weight constant and the design of each part of this algorithm model, using the method of difference testing analysis.

The purpose of the algorithm model established in this study is to preserve the main features of valid signals while smoothing and attenuating noise and interference signals, thereby highlighting the primary characteristics of the observations and reducing the amplitude of noise signals. Therefore, when setting parameters, we need to consider both aspects, aiming to reduce interference signals without affecting the main feature signals. Based on the above analysis, we provide the train of thought for parameter settings, as shown in Figure 2.

As shown in Figure 2, our algorithm serves two purposes simultaneously: preserving the trend of the raw data and reducing noise, smoothing the data to minimize interference with subsequent signal processing. Therefore, for signals with small amplitude fluctuations, smoothing and noise reduction are necessary, while the shape of the main feature signals needs to be retained. The setting of the parameter α plays a crucial role in achieving these two objectives. In various existing preprocessing algorithms, parameter settings are often based on empirical values. In contrast to previous studies, we determine the parameter $\alpha$ using a method based on statistical difference analysis to enhance the stability and adaptability of the model.

By conducting difference testing between the observed (raw) data and the predicted (output after model processing) data, we obtain the differences between the two sets of data. In statistics, if the p (Significance) of the two datasets is not less than 0.05, it is considered that their difference is not significant. In the context of our algorithm’s functionality, this indicates that the trend of the main data features has been preserved, confirming a successful setting for $\alpha$. For datasets where the differences are significant, it is due to the changes in non-feature signal segments after weakening, leading to a significant difference in the analysis. To avoid the interference of these signal segments on parameter setting, we need to perform a regression setting on these signals. Subsequently, based on difference analysis, we reset the parameter α by setting these signals to the corresponding data values after filtering, reducing their impact on the setting of $\alpha$. This process is iterated by refining the observed data and predicted data through difference testing, adjusting the value of $\alpha$ with an accuracy of 0.01 until the p satisfies the statistical condition (p>=0.05), thus achieving the setting for parameter $\alpha$.

As a decay factor, the parameter $\alpha$ typically ranges between (0,1). When $\alpha \to 0$, the absolute difference between the current data point and the previous data point contributes less to the result, indicating a smoother model that focuses more on past data. Conversely, as $\alpha \to 1$, the model becomes more flexible, placing greater emphasis on recent changes. For algorithm models requiring both fast responsiveness to data and high stability, or when the data itself exhibits a high degree of autocorrelation, higher $\alpha$ values can better capture the dynamic characteristics of the data. Here, we limit $\alpha$ to the range between (0.9, 1), with an accuracy of 0.01. This ensures that the model can react strongly to changes in the most recent data points without overly relying on them, while still retaining some influence from historical data. This approach helps maintain relative stability when processing rapidly changing data, striking a better balance between the weight of historical and recent data.

Based on the analysis above, we need to preliminarily set $\alpha$ within this range. In this study, we set $\alpha$ to 0.94 initially. We then conduct a difference test between the observed and predicted values using this preset $\alpha$. The steps for the difference test are outlined in Figure 3. Since the observed values (raw data) and predicted values (model output data) are paired samples, we need to use a paired sample test to examine the differences between these two sets of data. By calculating the differences between the observed and predicted values and denoting them as d, we assess whether d follows a normal distribution. As d does not exhibit normal distribution characteristics, we use a paired sample Wilcoxon test to compare the two sets of data.

We observe the p of the test result to determine if it is greater than 0.05. If the p is greater than 0.05, it indicates that our parameter setting has been successful. This means that we have effectively smoothed out interference segments with noise and fluctuations while preserving the original feature signals. If the p is not greater than 0.05, it is likely due to changes in the interference signals affecting the test results. In such a case, normalization of the observed data is required.

This involves two steps: Resetting negative values in the observed data. Since the output signal from the algorithm model has already removed sign interference, negative values in the predicted data are set to the corresponding data values in the output signal. Resetting signals near the equilibrium position in the observed data, such as signals close to the zero line, to the corresponding predicted values. This helps reduce the impact of differences caused by the smoothing of signals near the equilibrium position after model processing.

Finally, the reset observed signals are paired with the original predicted signals for a paired sample Wilcoxon test. We iteratively adjust the value of $\alpha$ with an accuracy of 0.01 until the p is greater than 0.05. At this point, the current $\alpha$ value is selected as the coefficient for the algorithm, indicating a successful parameter setting.

Taking a single clinical ECG data sample from Lead II with a sampling frequency of 500Hz and a duration of 10 seconds as the observed data $d_1$, sourced from Zhongshan Hospital affiliated to Fudan University using the Inno-12-U ECG device), we illustrate the difference testing based on the parameter settings. Initially, we set $\alpha =0.94$ as a preset value. Subsequently, after running the algorithm model, we obtain the predicted values $y_1$. We calculate the difference d between $d_1$ and $y_1$, and perform a normality test on data d. The results are shown in Table 1, where the significance Sig. (value p) = 0.000 < 0.05, indicating that data d is non-normally distributed.

Since the data is non-normally distributed, we need to perform a paired sample Wilcoxon test on the two sets of data, $d_1$ and $y_1$. The test results are shown in Table 2. As the significance Sig. (value p) = .000 < 0.05 of the test results for both sets of data, it indicates that $y_1$ has undergone significant changes due to smoothing and noise reduction, resulting in a significant difference between $y_1$ and $d_1$.

At this point, we cannot determine if the value of the parameter $\alpha$ meets the requirement to preserve the main features and trends of $d_1$. Therefore, we need to reset $d_1$. In this example, due to significant differences in the amplitude of the characteristic waveforms in the ECG data, we first set the data points in $d_1$ with amplitudes not exceeding 0.63 times the maximum amplitude in the observed data to the corresponding data values in $y_1$. Let’s designate the reset observed data as $d_2$. Subsequently, we conduct a Wilcoxon test on $d_2$ and $y_1$. If the resulting p-value is greater than 0.05, it indicates that there is no significant difference between the data after model processing and the reset observed values, thereby preserving the main characteristics and trends of the observed data.

To further demonstrate the importance of parameter setting and its impact on the results, we repeat the process with $\alpha$ values of 0.93 and 0.95. The test results are shown in Table 3. Based on the test results in Table 3, when $\alpha =0.94$, p = 0.084 > 0.05, indicating no significant difference between the predicted data and the corresponding reset observed data. For $\alpha =0.93$ and $\alpha =0.95$, p = 0.001 and p = 0.00, both less than 0.01, showing a highly significant difference between the predicted data and the corresponding reset observed data. Therefore, for this type of data, we should set the parameter $\alpha$ to 0.94 when applying this algorithm.

2.4. Absolute Value Differential Feedback——$y_2$

Another part of the algorithm, function $y_2$, represents the magnitude of change between adjacent data points. By calculating the absolute difference between adjacent data points, we consider the size of the differences between data points while ignoring their direction. Disregarding the direction of the data points can make the model smoother and more resilient to potential outliers or noise in the data. This approach allows the model to have a buffering effect on data fluctuations, reducing sensitivity to outliers and improving the stability and robustness of the model. We illustrate the effect of absolute value differential feedback using clinical ECG Lead II signals in Figure 4.

In Figure 4, it is evident that the algorithm without absolute values is more susceptible to the influence of data, especially when there is a higher presence of noise and interference in the data. The output of the algorithm (the bottom plot) deviates significantly from the main trend of the observed data (the top plot), and the smoothing effect on noise is notably reduced.

3. Case Study and Discussion

3.1. Clinical ECG Signal

An ECG is a diagnostic tool that records the heart’s electrical activity over time. Doctors can evaluate the heart’s health status and identify conditions such as arrhythmias, myocardial ischemia, and myocardial infarction by analyzing the characteristic waveforms of the ECG. ECG plays a vital role in clinical diagnosis. The algorithm model established in this study highlights ECG characteristic waveforms while attenuating non-characteristic waveforms, laying a solid foundation for the subsequent localization and extraction of ECG characteristic waves. The results are shown in Figure 5.

We selected 10 typical clinical ECG data from Zhongshan Hospital Fudan University. Each data segment is 10 seconds long, with a sampling frequency of 500 Hz using clinical lead II acquisition. The results are shown in Figure 5. For detailed visualization, we selected samples from the 2000th to the 3500th data points for each data segment.

From Figure 5, it can be observed that the amplitudes of each data segment remain almost unchanged after passing through the model, while the sign of the raw data is removed. The periods of the characteristic waves after the model are generally consistent with the positions in the raw data, and the start and end points of the characteristic waves are more apparent. The low-amplitude non-characteristic waves outside the characteristic waves exhibit lower oscillation frequencies and smoother waveforms after passing the model. Importantly, the model preserves the periodicity of the raw data.

3.2. MIT-BIH ECG Signal

The MIT-BIH Arrhythmia Database is a publicly available ECG database established through collaboration between the Massachusetts Institute of Technology (MIT) and the Beth Israel Hospital (BIH) in Israel. We selected 10 data records with different IDs from the MLII lead in the MIT-BIH database, each with a sampling frequency of 360Hz. As shown in Figure 6, to display the magnified results, we visualized the segment from the 2000th to the 3500th sample points of the selected data in Figure 6. From Figure 6, it can be observed that the periodicity of each data segment remains consistent after model processing compared to the raw data. The characteristic waveforms become more pronounced after model processing, with unchanged amplitudes. The periods of the characteristic waveforms are more distinct after model processing while eliminating directional interference, aiding in the subsequent localization of the characteristic waveforms. The oscillation frequency of non-characteristic waveforms is significantly reduced, enhancing the anti-interference capability for subsequent data processing.

4. Discussion

In this study, we have found that in signal analysis, the preservation or highlighting of characteristic signals and the smoothing of non-feature signals, two seemingly contradictory signal processing processes, can be superimposed to achieve integration. Through the analysis of two algorithms, recursive logic and difference feedback, we have established an algorithm model that accomplishes these two signal processing steps. This algorithm model can simplify the preprocessing process of signals and provide reliable inputs for subsequent feature extraction and classification.

The example results indicate that the output data from the algorithm model maintains the trend and amplitude of the characteristic waveforms while smoothing the waveforms near the equilibrium position. The recursive part $y_1$ in the algorithm (3), determines the next output based on the previous output data. This approach maintains the continuity of signal variations while reducing high-frequency interference, thereby reducing the magnitude of waveform changes near the equilibrium position. The difference feedback part $y_2$ in the algorithm preserves the trend of data slope, allowing the main characteristic waves to maintain their raw trend. In algorithm (2), we removed the sign interference in the difference feedback part, which corresponds to the positive direction shown in Figure 5 and Figure 6. The output of the algorithm model retains the shape of the characteristic wave groups without altering their amplitude, while also attenuating the interference from non-characteristic wave groups. In contrast to general preprocessing and feature extraction algorithms, which can change the shape and amplitude of the signal, this output is more intuitive and reliable.

The example demonstration provides intuitive validation results. Similarly, based on statistical analysis, we present the corresponding comparison results for variance and Pearson similarity in Table 4 and Table 5. The Validation metrics in Table 4 and Table 5 aim to verify the degree of attenuation of non-characteristic waveforms by the model. A decrease in the variance of the output data after model processing indicates reduced fluctuations, making the data smoother than the raw. Since this part pertains to validation near the equilibrium position, we need to ignore the characteristic wave data for comparison. Based on the maximum value position of each characteristic wave period and the time limit of the characteristic wave period, the values of this part of the data are set to the mean of the remaining data after removing this part. After applying the same processing to both the raw and the result data sets, we then calculate and compare their respective variances.

The Correlation coefficient in Table 4 and Table 5 validates the raw data and the result data using the Pearson similarity method. This validation mainly focuses on whether the trend of characteristic waveforms is consistent. We use the Pearson similarity method to analyze the remaining signal after removing the non-characteristic waveforms based on the parameter setting in the “Method” section. The obtained correlation coefficients are all greater than 0.8, indicating a strong correlation between the result data and the raw data for characteristic waveforms, thereby retaining the characteristics of the characteristic waveforms. The analysis results of differences and consistencies demonstrate that the algorithm model has successfully preserved the trend of characteristic waveforms while attenuating the interference of non-characteristic waveforms. Compared to empirical values, this parameter-setting method enhances reliability and applicability.

In the future, we can further research how to more accurately locate feature waveforms based on the research of this study, thereby further simplifying the iterative steps of the signal processing system.

5. Conclusion

This study established a mathematical model based on recursive and differential methods, setting the parameters of the recursive part in the formula through difference testing analysis from statistics. Absolute values were utilized to remove directional interference in the differential feedback. The establishment of this model aims to highlight the periodic positions of the characteristic waveforms while maintaining their amplitude and trend, simultaneously attenuating high-frequency waveforms near the equilibrium position and smoothing non-feature waveforms.