# The Impact of Contact Force on Signal Quality Indices in Photoplethysmography Measurements
## 3 Materials and Methods
### Signal Quality Indexes  
The study evaluated six commonly used SQIs in PPG, which are skewness (\(S_{SQI}\)), kurtosis (\(K_{SQI}\)), relative power (\(R_{SQI}\)), zero-crossing rate (\(Z_{SQI}\)), entropy (\(E_{SQI}\)), and the perfusion index (\(P_{SQI}\)). The selection of these SQIs is based on their ability to provide a comprehensive and multifaceted evaluation of the quality of PPG signals. Below, we review the definitions and mathematical formulations of these SQIs and their relevance to PPG measurements. Figure 3: Variability in the maximum CF reached by each subject, alongside the absolute CF difference between the fingertips and age. For Subject 11 (not shown in the graph), measurements were only taken from the right index finger, recording a maximum CF of 2.16 N. From [35]. Figure 2: As a case study, the raw signals from the right index finger of Subject 8 are displayed, including details of the 940 nm wavelength AC pulses for two distinct CF values. Adapted from [35]. \(S_{SQ}\): Skewness was selected due to its capacity to quantify the asymmetry of a probability distribution of PPG signals, specifically addressing the distribution deviation around its mean [38; 39]. A skewness value near zero suggests a symmetric distribution indicative of balanced physiological rhythms or consistent signal characteristics. This is generally preferable as significant deviations may indicate artifacts or biases in the measurement. Positive skewness indicates a tail extending toward higher amplitudes, while negative skewness points to a tail stretching toward lower amplitudes [58]. This metric offers insights into physiological fluctuations, such as blood volume changes or external influences on signal amplitude. The \(S_{SQ}\) is particularly sensitive to signal morphology, showing behaviors correlated with changes in PPG signal shape. \[S_{SQI}=\frac{1}{N}\sum_{i=1}^{N}\left[x_{i}-\frac{\hat{\mu}_{x}}{\sigma} \right]^{3} \tag{1}\]  
where \(\hat{\mu}\) and \(\sigma\) represent the empirical estimates of the mean and standard deviation (SD) of \(x_{i}\), respectively, and \(N\) is the total number of samples in the PPG signal. 2. \(K_{SQI}\): Kurtosis is used to evaluate the shape of the signal distribution around the mean, indicating whether it is heavy-tailed and peaked or light-tailed and flat, relative to a normal distribution [38; 39]. An optimal value of around zero, mirroring that of a normal distribution, signifies good signal quality without significant outliers [58]. This is crucial to ensure that the analyzed PPG signals do not present extreme variations that could compromise the interpretation of the data. \[K_{SQI}=\frac{1}{N}\sum_{i=1}^{N}\left[x_{i}-\frac{\hat{\mu}_{x}}{\sigma} \right]^{4} \tag{2}\]  
where \(\hat{\mu}\) and \(\sigma\) represent the empirical estimates of the mean and SD of \(x_{i}\), respectively, and \(N\) is the total number of samples in the PPG signal. 3. \(R_{SQI}\): Relative power measures the power of systolic and diastolic waves within the frequency bands 1 to 2.25 Hz, relative to the total signal bandwidth of 0 to 8 Hz [38; 39]. Given that the energy of systolic and diastolic waves predominantly resides within the 1-2.25 Hz frequency band, the ratio of the power spectral density (PSD) within this specific band to the PSD across the entire 0-8 Hz signal bandwidth serves as an indicator of signal quality. A higher value indicates a greater proportion of the useful signal than noise, reflecting better signal quality. This is critical for applications where accurate detection of systolic and diastolic waves is critical. \[R_{SQI}=\frac{\sum_{f=1}^{2.25}PSD}{\sum_{f=0}^{8}PSD} \tag{3}\]  
PSD was estimated using the Welch method [59]. 4. \(Z_{SQI}\): The zero crossing rate was selected for its ability to reflect the oscillation frequency of the signal, indicating its regularity and stability.