## 5 Conclusions  
The efficiency and performance of a battery is highly dependent on its past and present use. To ensure optimal performance, it is of utmost importance to develop battery grading techniques based on fast methods to determine battery degradation in terms of capacity fade, health status, and remaining useful life. Signal processing plays a leading role in the development of predictive and prognostic methods to determine the state of health and degradation status. This paper has proposed and evaluated the behavior of a new health indicator to estimate the capacity fade of lithium-ion batteries, which is directly related to the state of health. It has been shown that the proposed estimator is highly correlated with the capacity fade since the Pearson correlation coefficient \(R\) between the two indicators is around 0.99 for the analyzed batteries. The proposed health indicator is calculated within a narrow SOC interval where the voltage-SOC has a very linear relationship. This narrow SOC interval is within the normal range of most practical charge-discharge cycles, so it is not necessary to collect data from full charge and discharge cycles. Thus, only a small fraction of the data points of a full charge-discharge cycle are required,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Cell** & **B0034** \\ \hline
**Forecast Start Point (\% of Total Useful Life)** & **RMSE** \\ \hline
25\% & 0.105 \\
50\% & 0.072 \\
80\% & 0.070 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparative RMSE results between the experimental measurements of the terminal voltage for cell B0034 vs. SOC and the Nernst equation adjusted with the proposed degradation model with a discharge current of 4 A (C-rate of 2).  
Figure 10: Prediction of future discharge cycles at different stages of cell aging for cell B0034 with a discharge current of 4 A (C-rate of 2). (**a**) Prediction of future cycle using 25% useful life data. (**b**) Prediction of future cycle using 50% useful life data. (**c**) Prediction of future cycle using 80% useful life data.  
minimizing computational and storage resources while providing accurate results. The results presented in this paper have shown that by using the Nernst battery model, the behavior of future charge-discharge cycles can be accurately predicted. The approach proposed in this paper could be applied in many applications where the batteries are not fully charged or discharged, such as in electric mobility or smartphone applications. Finally, the method proposed in this paper is based on various signal processing methods such as discrete wavelet filtering and prediction methods based on linear fitting and autoregressive integrated moving average algorithms.  
Conceptualization, J.d.I.V., J.-R.R., and J.A.O.-R.; methodology, J.A.O.-R.; software, J.d.I.V. and J.-R.R.; validation, J.d.I.V., J.-R.R., and J.A.O.-R.; formal analysis, J.-R.R. and J.A.O.-R.; investigation, J.d.I.V., J.-R.R., and J.A.O.-R.; resources, J.-R.R. and J.A.O.-R.; data curation, J.d.I.V.; writing--original draft preparation, J.d.I.V. and J.-R.R.; writing--review and editing, J.d.I.V., J.-R.R., and J.A.O.-R.; supervision, J.-R.R. and J.A.O.-R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Ministerio de Ciencia e Innovacion de Espana, grant number TED2021-130007B-I00, and by the Generalitat de Catalunya, grant number 2021 SGR 00392.  
Not applicable.  
The authors declare no conflict of interest.