Modelling a Novel Filtering and Classifier Approach for ECG Signal Processing

Bae et al. [16] provide a detailed analysis of EMD-based filtering approaches. The investigators present diverse approaches for experimental purposes, and ECG signal classification is done effectually. These investigations involve pre-processing, feature extraction, and noise reduction and classification [17]. Researcher discusses three diverse denoising approaches: EMD-based noise reduction approaches, EMD-based partial reconstruction, and extracting inferences for adaptive filtering. Based on the changing noise amplitudes, these novel approaches intend to perform effective noise reduction methods provided ECG signal under a specific frequency ($48 \mathrm{~Hz}-51 \mathrm{~Hz}$). However, these approaches summarize the EMD enhancements method, which adopts the signal dependency property and is an adaptive technique [18]. Subsequently, these investigations show improved performance in diminishing removing noise with wavelets. This system's performance enhancement is generally coupled with two diverse conditions: a) there exist no constraints that the signal magnitude needs to be superior to the noisy signal, and b) lesser SNR [19-21] (Table 1).

Mukherjee and Ghosh [22] anticipate a novel approach for ECG signal improvement. This approach is based on the entire ensemble EMD with adaptive noise and higher-order statistical processing using mixed interval thresholds. Subsequently, the ECG signal is decomposed into the IMF components set with the EMD technique. IMFs attained are partitioned into three diverse groups: noiseless relevant, higher frequency noisy and lower frequency noise IMFs [23]. The novel derived criterion categorizes these approach groups based on the fourth-order cumulant. Therefore, the ECG signal is reconstructed by merging the threshold IMFs and reserved refined lower frequency appropriate IMFs. The author anticipates a new method for ECG denoising: DWT and EMD integration [24]. The anticipated performance of the windowing in Empirical Mode Decomposition to diminish the interference produced by the preliminary IMF is entirely avoided. However, the ECG signal attained varies in the DWT domain with the adaptive soft thresholding approach to diminish the noise [25]. However, the ECG signal is reconstructed in a superior time resolution of essential DWT properties evaluated to the EMD technique for energy conservation during the noise presence. Therefore, this work attempts to preserve the QRS complex to provide a clear ECG signal [26]. Martis et al. [27] utilized the EMD approach to enhance the noise filtering performance by reducing mode-mixing near the IMF's scales. However, the investigator performs experiments to model filtering ECG signals technique that relies focusing on lower IMF scales, including High-frequency noise. Comparing EEMD filtering performance and the existing EMD approach is highlighted in this work. Nonetheless, Wiener filter application is utilized to evaluate the filtering functionality with EEMD, which is another noise filtering approach. But the model fails to fulfil the requirements [28, 29]. This study concentrates on modelling a model for improving ECG signal accuracy by noise reduction using IIR and FIR filter comparison to handle these issues [30]. The ECG signal goes through several stages, including noise reduction, feature extraction, and classification, all aimed at enhancing prediction accuracy. This process allows for a comparison of the latest and most effective methods available.

Table 1. Comparative analysis of state-of-the-art methods for ECG filtering and prediction

Various approaches are discussed for the noise removal process from the ECG signal. This work provides a broader analysis of the filters for low and high frequencies., i.e., FIR/IIR filters. The filtering approaches are commonly applied in signal processing and executed in diverse ECG signal analyses. However, filtering approaches improve the ECG signals, managed and implemented using FIR/IIR filters. Here, both these filters remove the noise over the ECG signals. Moreover, certain intrinsic noise agglutinant of the ECG signals is removed using the proposed approaches. However, ECG signal filtering is determined to be contextual. It is executed when the desired data remains ambiguous and needs further execution. However, the filtering process is considered an essential issue where the data needs to be disposed of and filtered. Later the denoised signals are provided to the feature extraction phase and map the wavelet based on time and frequency. It is determined to be technique for analyzing time-varying signals. The WT is appropriately utilized specifically for denoising as the wavelets show similarity with the energy spectrum, and QRS complexes are focused around low frequencies (See Figure 2). Finally, classification is done for prediction purposes.

2.png

Figure 2. The representation of synthetic ECG signal with interference (Power Line of 50 Hz)

4.1 FIR filter

The digital filters are analog-to-digital converters and perform the vice-versa process (digital-to-analog). The LTI filter produces the filter coefficients specify the designing filters using impulse response. The linear filter coefficients convolution with input sequences provides the output as in Eq. (1):

$Y=f * x$            (1)

Impulse response specified by function f, input signal is denoted by x, and output signal y resulting from convolution. Convolution of input signal with filter as in Eq. (2):

$\begin{gathered}Y[n]=x[n] * f[n] \\ =\sum_k x[k] f[n-k]=\sum_k f[k] x[n-k]\end{gathered}$               (2)

Here, $Y[n]$ specifies the filter output, $x[n]$ represents the digital input (filter), $f[k]$ specifies the impulse response, and * represents convolution operators. The summation is provided by $k$, representing the impulse response (filter). The filter with finite value is known as the FIR filter. The digital filter class with the current and historical input data and none of the prior filter output to attain the present output value is known as the FIR filter. It is termed as FIR filter implementation. FIR filter operation is the time-domain smoothing via moving average. The low-pass (FIR) with the window design approach is quite simple and produces the filter output with superior performance. The passband/stopband deviations are equal approximately. However, it is general to attain pass band deviation is extremely smaller than stop band deviation. These parameters are not controlled independently in the window design. This, it is essential to design the filter in the passband to fulfil the stopband requirements. The ripple is not uniform and diminishes when it moves away from the transition band. The filter shows features like the passband specification $\varphi_p$ frequency, $\omega_s$ stopband frequency and transfer function (desired). The filter class (special) that fulfils the criteria is the equi-ripple FIR filter. This design reduces the maximal deviation from the transfer function. It poses approximation error with weighting among the actual and desired filter response in passband and stopband regions to reduce the maximal error. Therefore, the outcome possesses a stopband and passband ripple specifications $\operatorname{Hd}(\omega)$ specifies the filters' frequency response, $w(\omega)$ defines the weighted frequency response characteristic. The designer can select the relative error size in diverse spectral bands. This is represented by Eq. (3).

$H(\omega)=e^{-j \omega(M-1) / 2} e^{j(p / 2) L} H^*(\omega)$            (3)

Approximating error with weighted approach can be visualized as:

$E(\omega)=W(\omega)\left[H_d(\omega)-H^*(\omega)\right]$             (4)

$E(\omega)=W(\omega)\left[H_d(\omega)-P(\omega) Q(\omega)\right]$             (5)

The $Q(w)$ represents the frequency function,

$E(\omega)=W(\omega) Q(\omega)\left[H_d(\omega) / Q(\omega)-P(\omega)\right]$              (6)

Therefore, the approximation predicts reducing maximal error through coefficient adjustment $E(\omega)$ measure. The approximation formula is provided in Eq. (7):

$|E(\omega)|=\min [\max |E(\omega)|]$              (7)

FIR filter coefficients are produced with SIMULINK tool. The filter response accuracy relies on the filter coefficients. The filter (ideal) needs the filter tap weights is not probable to execute over the hardware. The approach to handle this problem is coefficient round off. It reduces the hardware utilization, therefore diminishing the hardware utilization. The depreciation of filter coefficients influences the filter performance, specifically when the numbers of tabs are significantly higher.

4.2 Low-pass serial and parallel FIR

The architectural view of the serial FIR filter needs one multiplier, one adder, and one delaying unit. Therefore, it is a superior option when considering hardware efficiency. However, the FIR filter is done via the architecture, with reduced device throughput and slower performance. Similarly, the parallel FIR low-pass filter intends to parallel process data, rather than processing it. It provides superior throughput. All adders are associated with the prior adder output. The result is attained from the sum of adders output and output terminal. It provides the valid output, and the delay is affected by key delay factors. It is hugely greater than or equal to the product sum and accumulator delays pretends to handle the high necessary delay. Here, the adders are connected with the bran tree and add data-parallel indeed of serial. It is so advantageous evaluated to the prior design and it diminishes the Critical delay considerations for FIR filter.

4.3 Analysis with IIR filter

Recursive filter implementation where the output is not associated with prior input. However, it is related to the preceding output. Two diverse design approaches are based on the IIR filter: indirect and direct. It is used to model IIR filters by restricting the zero distribution and pole of the transfer function. Similarly, indirect technique is based on analog filter prototype model to calibrate every filter coefficient based on the requirements. Then, Analog filtering techniques are mapped from the $S \rightarrow Z$ analog domain via the Laplace transform via transforming from S-domain to Z-domain yields a digital filter for processing digital signals (Tables 2-4).

Table 2. FIR parameter

Table 3. IIR filter parameters

Table 4. HPF parameters

By the frequency domain analysis relies on the present information gathered during capturing ECG signals, it is identified that noise significantly affects the signal on the attained ECG signal, i.e., Signal baseline shift caused set breathing frequency, as $0 \sim 1 \mathrm{~Hz}$. The successive noise influence over the ECG signals is power line interference due to $220 \mathrm{~V}, 50 / 60 \mathrm{~Hz}$ AC grid infrastructure. Two filters need to be constructed to avoid the noise interference across two frequency ranges. Filters out low-frequency noise 0 to 1 Hz baseline drift, and the filters out high frequencies, it suppresses roughly interference (50 Hz frequency). The digital angular frequency (Z-domain) is expressed as in Eq. (8):

$\Omega=\frac{1}{f_{a d}} * f_s * 2 \pi n, n=0,1,2, \ldots$              (8)

Here, $f_{a d}$ denotes the ADC sampling frequency, while $f_s$ represents the signal frequency. Frequency $f_c=50 / 60$ has to be avoided when $f_{a d}=480, \Omega_c \approx n \pi / 5(n=0,1, \ldots, 9)$. Then, Notches are designed at $\Omega_c \approx \mathrm{n} \pi / 5(\mathrm{n}=0,1, \ldots, 9)$ to reject $50 / 60 \mathrm{~Hz}$ interference. Additionally, the zeros need consideration in 0 Hz and $\frac{50}{60} \mathrm{~Hz}$ at the point $(1,0)$ in the complex plane. A zero and a pole are strategically placed at $(1,0)$ to prevent frequency overlap. Poles at the origin $(0,0)$ or on the real axis like $(1,0)$ can provide a stable transfer function. For stability, poles are typically placed within the unit circle in the transfer function, as shown in Eq. (9).

$\begin{gathered}\left(e^{j * 0}-e^{j \Omega}\right)\left(e^{\frac{1}{5} \pi j-e^{j \pi}}\right) \\ H(j \Omega) \frac{\left(e^{\frac{2}{5} \pi j-e^{j \pi}}\right) \cdots\left(e^{\frac{9}{5} \pi j-e^{j \pi}}\right)}{\left(e^{j 0}-e^{j \Omega}\right) e^{9 j \Omega}}=\frac{e^{10 * j \Omega}-1}{\left(e^{10 * j \Omega}\right) e^{e j \pi}}\end{gathered}$              (9)

Here, $z=e^{j \Omega}$. Then,

$H(z)=\frac{z^{10}-1}{z^{10}-z^9}=\frac{1-z^{-10}}{1-z^{-1}}=\frac{Y(N)}{X(N)}$              (10)

Increasing the filter order improves the low-pass filter's performance. Eq. (11) outlines the transfer function for a second-order low-pass digital filter.

$H(z)=\frac{\left(1-z^{-10}\right)^2}{\left(1-z^{-1}\right)^2}=\frac{1-2 z^{-10}+z^{-20}}{1-2 z^{-1}+z^{-2}}=\frac{Y(N)}{X(N)}$              (11)

Magnitude-frequency characteristics of a low-pass filter is provided based on Eq. (11). Frequency characteristics attained from multiple filters are implemented using adding two filters with linear phase response and matched delay (See Figure 3(a) and Figure 3(b)), i.e., the weighted sum of filter coefficients. The high-frequency filter is modelled based on subtracting low-pass filter output from original signal. The signal filter poses a constant lag filter, expressed in $H a(z)=A z^{-m}$. Under specific ideal conditions, $H a(z), H_{o w}(z)$ poses a similar DC amplification coefficient. A filter that passes low frequencies is designed using threshold frequency of 2 Hz, i.e., $f_c=2 \mathrm{~Hz}$, when sampling rate is $f_{a d}=480$. The related radial frequency $\Omega c \approx n \pi / 120$, where ( $n=0,1,2, \ldots, 9$ ). Based on the integer filter design:

$H_{l p}(z)=\frac{Y(z)}{X(z)}=\frac{1-z^{-240}}{1-z^{-1}}$              (12)

To create a high-pass filter obtained by subtracting low-pass from unity gain. The expression in Eq. (13) is this.

$H_{h p}\left(z^{\prime}\right)=\frac{P\left(z^{\prime}\right)}{X\left(z^{\prime}\right)}=z^{-120}-\frac{H_{l p}\left(z^{\prime}\right)}{240}$              (13)

$H_{h p}\left(z^{\prime}\right)=\frac{-1+240 z^{\prime-121}-240 z^{\prime-120}+z^{\prime-240}}{240-240 z^{\prime-1}}$              (14)

The response of the HPF under the frequency and magnitude curve is obtained from Eq. (14). The transfer function of the desired HPF is obtained using Eq. (15):

$H_{h p}(z)=\frac{P(z)}{X(z)}=z^{-120}-\frac{H_{l p}(z)}{240}$              (15)

3a.png

(a)

3b.png

(b)

Figure 3. (a) Amplitude-frequency response of low-pass filter; (b) Amplitude-frequency response of high-pass filter

Eq. (16) presents the differential form in detail:

$\begin{gathered}y(n)=2 * y(n-1)-y(n-2)+x(n)-2 \\ * x(n-10)+x(n-20)\end{gathered}$             (16)

The high-pass filter's transfer function is given in Eq. (17).

$\begin{gathered}y(n)=y(n=120) \\ -\frac{(y(n-1)+x(n)-y(n-240))}{240}\end{gathered}$             (17)

Generally, ECG signals show slight distortion after the essential process after the IIR filter (integer coefficients). Therefore, the coefficients are modelled to attain an entire ECG signal where the coefficients filter necessary information. An enhanced filter coefficient is proposed to achieve the essential information. Figure 4 depicts the structural design of every a dual-mode IIR filter module supporting both low-pass and high-pass filtering. Then, ECG signals are filtered using IIR filters with integer coefficients, and the original signal subtracts the processed signal and attains difference signals. The filter coefficients need to achieve the compensation signal. The signal needs to avoid the interference and preserves the essential information (features). The ultimate process is the construction of the waveform and the compensation signals are added after the filtering process to attain the final signal outcome (See Figure 4). Extract relevant features from the processed signal where wavelet transform is best suited.

4.png

Figure 4. Improved IIR filter (integer coefficients) block diagram

The neural network (NN) comprises many neurons connected to transfer and receive the data concurrently. Every neuron in the network is allocated with the weight. It represents the network state during the learning process, and the weight of every neuron needs to be updated. The anticipated model of every neuron is completely connected to hidden layers for extracting features and categorizing the ECG signal. It is executed in SIMULINK, and the generalized sparse network model is adopted to diminish the number of features and enhance computational time. The feature extraction shows diverse descriptive parameters and data using the preliminary process from the ECG signals. The network model is trained and classified based on the feature vectors (features extracted using wavelet transform). During the analysis step, and double-check that the model still delivers quick, accurate results. At this point the module lays out its prediction work using a simple neural network pattern.

Cascaded neural network architecture are displayed by the fake neurons. The sources (features) B₁,…,B_n are thought to be unidirectional and produce a sign stream of neurons. Eq. (24) provides the neuron output:

$o=f($network$)=f\left(\sum_{i=1}^n A_i B_i\right)$                   (24)

The capacity is given as f(network), and the weighted vector is indicated by A_iB_i. the network. The variable network is represented as scalar consequences by the weight and information vectors. It is stated in Eq. (25):

network $=A^T B=A_1 B_1+A_2 B_2+\cdots+A_n B_n$                   (25)

where, $T$ specifies the matrix transposition. The value of $O$ is given by Eq. (26).

$O=f($network$)=\left\{\begin{array}{lc}1 & \text { if } A^T x \geq \theta \\ 0 & \text { otherwise }\end{array}\right.$                   (26)

Note that the model sets a fixed range as its ceiling, and a linear threshold unit (LTU) is a type of node. The neurons inside this scheme behave according to the equation shown in Eq. (27):

$v_k=\sum_{i=1}^p A_{k i} B_i$                   (27)

Then, the neuron output $y_k$ is the outcome of the activation function of $v_k$. The error reduction among the evaluated EEG class is essential. The network performance is evaluated using the output (expected) with the raw output value. The proposed system is faster, and the data needs to increase forward like back-propagation NN, where the information is provided to forward and reverse a blunder. The relapse performed by the classifier is inevitable with the desire for $Y$ with the provided $X=a$. It gives the scalar value representing the input vector $a$. Let $f(a, b)$ illustrate vector irregular variable capacity $X$ and scalar arbitrary variable $Y$. The $a$ is provided with the chance to evaluate the stochastic estimation. The relapse variable Y is defined, with a restricted mean given by Eq. (28):

$E\left[\frac{Y}{X}\right]=\int_{\infty}^{\infty} Y, f\left(\frac{b}{a}\right) d y=\frac{\int_{\infty}^{\infty} Y, f(a, Y) d y}{\int_{\infty}^{\infty} f(a, Y) d y}$                  (28)

In this context, X and Y point to the squeezed components, each set with its own chosen settings. The key link between a and b relies on non-parametric estimation done from scratch, so no advance knowledge is used.

6.1 Loss function

Assume N  pair of training sample dataset:

$D=\left\{\left(a_s, b_s\right) \mid s=1, \ldots, N\right\}$            (29)

Here, $s$ indexes a single data point, $X_s$ stands for the input in $R^n$, and $Y_s$ in R records the corresponding output $a_s$. The goal is to estimate the hidden rule so that the prediction for every training pair stays within an accuracy band of size epsilon, while also uncovering how the outputs $a_s$ and $b_s$ relate to one another. That task starts by pushing all samples through a high-dimensional kernel map $\varphi: \mathrm{R}^{\mathrm{n}} \rightarrow \mathrm{R}^{\mathrm{m}}$, after which fit a simple linear model as shown in Eq. (30):

$f(a)=w^T \varphi(a)+c$            (30)

where, $w \in R^m$ the weighted vector, with 'c' indicating the threshold parameter. Here, 'w' stands for the minimal Euclidean distance, similar to described in Eq. (31):

$w=||w||_2^2$             (31)

The $\epsilon$ pair for precision representation is outlined in Eq. (31), which helps to reduce the error between the target output vs predicted output. To optimize the loss function, we express it as shown in Eq. (32):

$L\left(e_s\right)_\epsilon=L\left(b_s-f\left(a_s\right)\right)_\epsilon$             (32)

The proposed neural network optimization problem is shown below:

$J(w, c)=\frac{1}{2}| | w| |_2^2+P \sum_{s=1}^N L\left(b_s-f\left(a_s\right)\right)_\epsilon$             (33)

In this context, $P \in R^{+}$refers to parameters defined by the user. The level of noise present in the training samples is mentioned, but it doesn't factor into the output. As a result, the loss function, which is based on the optimization method, is designed to be sparse in order to find the solution.

$\left|b_s-f\left(a_s\right)\right|_\epsilon=\left\{\begin{array}{cc}0 & \left|b_s-f\left(a_s\right)\right|<\epsilon \\ \left|b_s-f\left(a_s\right)\right|-\epsilon & \text { else }\end{array}\right.$             (34)

From the statistical analysis, we've determined that the loss function is indeed optimal. Taking into account the error distribution, present the insensitive loss function as shown in Eq. (35): Please remember that when crafting responses, it's important to stick to the specified language and avoid using any others.

$L\left(e_s\right)_e=\left(e_s\right)_e^2$             (35)

In this context, $\left(e_s\right)_e^2$ represents a continuous differential function. By integrating Eq. (34) and Eq. (35), express the network model as shown in Eq. (36).

$\begin{aligned} w \in R^m, & b R^{J(w, c)}=\frac{1}{2} w_2^2+P \sum_{s=1}^N\left[\left(b_s-f\left(a_s\right)\right)\right]_\epsilon^2 \\ & =\left\{\begin{array}{c}b_s-w^T \varphi\left(a_s\right)-c \leq \varepsilon+\xi_s \\ -y_s+w^T \varphi\left(a_s\right)+c \leq \varepsilon+\xi_s \\ \xi_s, \xi_s^{\prime} \geq 0, s \in\{1, \ldots, N\}\end{array}\right.\end{aligned}$              (36)

Here, $\xi_s$ and $\xi_s{ }^{\prime}$ are the slack variables used to account for both negative and positive deviations. To calculate the primal objective of Eq. (37), we take the linear regression and multiply it by a non-negative multiplier for each sample set.

$\begin{aligned} w \in R^m, b \in R^{j\left(w, c, \alpha_s, \alpha_{s^{\prime}}^{\prime}, \gamma_s, \gamma_{s^{\prime}}^{\prime}, \xi_{s^{\prime}}, \xi_s^{\prime}\right)} =w, c, \alpha_{s^{\prime}}, \alpha_{s^{\prime}}^{\prime}, \gamma_s, \gamma_{s^{\prime}}^{\prime}, \xi_{s^{\prime}}, \xi_s^{\prime} \geq 0 s \in\{1, \ldots, N\}\end{aligned}$              (37)

$\begin{gathered}\frac{1}{2} w^T w+P \sum_{s=1}^N\left[\left(\xi_s\right)^2+\left(\xi_s^{\prime}\right)^2\right]-\sum_{s=1}^N \alpha_s\left(\varepsilon+\xi_s-\right. \\ \left.b_s+w^T \varphi\left(a_s\right)+c\right)-\sum_{s=1}^N \alpha_s^{\prime}\left(\varepsilon-\xi_s^{\prime}-b_s+\right. \\ \left.w^T \varphi\left(a_s\right)-c\right)-\sum_{s=1}^N\left(\gamma_s \xi_s+\gamma_s^{\prime} \xi_s^{\prime}\right)\end{gathered}$              (38)

Here, $\alpha_s, \alpha\left(s^{\prime}\right), \gamma_s$, and $\gamma_s{ }^{\prime}$ represent the Lagrange multipliers. To pinpoint the best solution, we need to get rid of the primal variable. Consequently, the partial derivative will equal zero.

$\frac{\delta J}{\delta c}=\sum_{s=1}^N\left(\alpha_s^{\prime}-\alpha_s\right)=0$              (39)

$\nabla_w J=w-\sum_{s=1}^N\left(\alpha_s-\alpha_s^{\prime}\right) \varphi\left(a_s\right)=0$              (40)

$\frac{\delta J}{\delta \xi_s}=P\left(2 \xi_s\right)-\alpha_s-\gamma_s=0$              (41)

$\frac{\delta J}{\delta \xi_s}=P\left(2 \xi_s^{\prime}\right)-\alpha_s^{\prime}-\gamma_s^{\prime}=0$              (42)

Substitute Eq. (41) and Eq. (42) in Eq. (43), the model handles optimization problem:

$\begin{gathered}\max _{\propto \in R^N} J\left(\alpha_s \alpha_s^{\prime}\right)=-\frac{1}{2} \sum_{s=1}^N \sum_{\gamma=1}^N\left(\alpha_s-\alpha_s^{\prime}\right)- \\ \left(\alpha_s-\alpha_s^{\prime}\right) \\ \varepsilon \sum_{s=1}^N+\sum_{s=1}^N b_s\left(\alpha_s-\alpha_s^{\prime}\right)-\frac{1}{2 P} \sum_{s=1}^N\left[\left(\alpha_s\right)^2+\left(\alpha_s^{\prime 2}\right)\right]\end{gathered}$              (43)

$\sum_{s=0}^N\left(\alpha_s^{\prime}-\alpha_s\right)=0$ and $\alpha_s^{\prime} \alpha_s \varepsilon[0, \infty]$              (44)

K refers to the kernel matrix, and the kernel function K(a_s, a_r) is essentially the product of two samples, φ(a_s) and φ(a_r).

$\left.K=\left[K\left(a_s, a_r\right)\right]_{s, r}=\left[\varphi^T\left(a_s\right) \cdot \varphi\left(a_r\right)_{s, r}\right]\right]$              (45)

The dual optimization problem tackles quadratic challenges that lead to distinct outcomes and minima, and find the optimal model and decision function for the test set samples detailed in Eq. (46).

$w=\sum_{a_s \in D V}\left(\alpha_s-\alpha_s^{\prime}\right) \varphi\left(a_s\right)$              (46)

$f(a)=\sum_{a_s \in S V}\left(\alpha_s-\alpha_s^{\prime}\right) k\left(a_s, a\right)+c$              (47)

From Eq. (47), evaluate the operation by applying the kernel function to the input space along with the training set samples, transforming them into a high-dimensional space. In this context, SV refers to the training set samples where $\alpha \_\mathrm{s}-\alpha \_\mathrm{s}^{\prime}$ $\neq 0$, and can skip evaluating $f(a)$ and $w$. This approach helps reduce computational time, allowing the model to focus on solving the problem at hand.