The two-sample Kolmogorov–Smirnov (TK–S) test is a non-parametric statistical method used to determine whether two independent datasets originate from the same underlying distribution. Its application requires two fundamental assumptions (Corder & Dale, 2023): (i) observations X₁, …, X_m form a random sample from a continuous population (Population 1), with values that are independent and identically distributed (i.i.d.); and (ii) observations Y₁, …, Y_m form a random sample from another continuous population (Population 2), also i.i.d. The two samples must additionally be independent of each other.

The procedure begins by constructing the empirical distribution functions (EDFs) for both datasets. For each real number t, the EDFs are defined as:

where m and n denote the sample sizes of X and Y, respectively. The maximum absolute divergence between the two EDFs across all possible values of t is then computed as:

This statistic D quantifies the largest observed difference between the two empirical cumulative distribution functions (ECDFs). To account for sample sizes, the test statistic Z is defined as:

Then, use the TK-S test statistic, Z, and the Smirnov (Smirnov, 1948) formula (see Formula 3-8) to find the two-tailed probability estimate p:

where

where

The value of Z is then used to estimate the corresponding probability (p-value), based on the exact distribution of the TK–S statistic (Smirnov, 1948). The p-value reflects the likelihood of observing a divergence equal to or greater than D under the null hypothesis that both samples arise from the same distribution.

It is important to note that while computational formulas exist for approximating the p-value under different ranges of Z, the standard interpretation of the test does not rely on rigid thresholds of Z alone. Instead, the inference is made by comparing the calculated p-value to a predefined significance level α.

This decision-making framework ensures that the TK–S test remains sensitive to differences across the entire distribution rather than localized measures such as mean or variance. In the context of biomedical signal analysis, this feature is particularly valuable, as it allows the detection of subtle yet clinically meaningful differences between signals obtained from prototype devices and calibrated reference instruments.

The categorization of the TK–S test statistic Z into intervals such as Z < 0.7, 0.27 ≤ Z < 1, or Z > 3.1 is not part of the standard interpretation of the TK-S test and does not reflect an established statistical convention. In traditional K-S analysis, the interpretation of Z is made in conjunction with its corresponding p-value, which is calculated based on the sample sizes and the observed value of D, the maximum difference between the empirical cumulative distribution functions (ECDFs).

If such thresholds were mentioned in an earlier version or in informal discussion, it is important to clarify that no universal or formal categorical scale exists for interpreting the magnitude of Z alone. The significance of any observed Z depends on the context: primarily on the sample sizes n₁ and n₂, and the level of significance α. For instance, a small Z (e.g., < 0.7) may correspond to a high p-value in large samples (indicating similarity), while the same Z in small samples might yield a different inference. Conversely, a large Z (e.g., > 1.5 or 2.0) typically leads to rejection of the null hypothesis, provided the sample size is sufficient.

Once we have our p-value, we can compare it against our level of risk α to determine if the two samples are significantly different.

Once the p-value is obtained, it is compared to the predetermined significance level, α, which represents the acceptable level of risk for rejecting the null hypothesis. If the p-value is less than or equal to α, we reject the null hypothesis, concluding that the two samples are significantly different. Conversely, if the p-value exceeds α, we fail to reject the null hypothesis, indicating that there is insufficient evidence to suggest a significant difference between the two samples. This comparison allows researchers to make informed decisions regarding the independence of the data samples.

The interaction of light with biological tissues is inherently complex, as it involves multiple physical phenomena such as reflection, transmission, absorption, and scattering (Anderson & Parrish, 1981). The detected intensity depends not only on the optical properties of the tissue but also on external factors, including the pressure exerted on the sensor, which can alter capillary blood flow and, consequently, the recorded signal (Kyriacou & Chatterjee, 2022). Photoplethysmography (PPG) has been widely adopted as a noninvasive technique for monitoring cardiovascular activity. It is particularly effective for capturing cardiac pulse waveforms, traditionally using a single light source to interrogate tissue (Hertzman, 1938).

From a physiological perspective, PPG signals arise from variations in blood volume within the microvascular bed of tissue. At specific wavelengths, hemoglobin exhibits distinct absorption characteristics depending on its oxygenation state. Red light (660 nm) is preferentially absorbed by deoxygenated hemoglobin (Hb), whereas near-infrared light (940 nm) is absorbed to a greater extent by oxygenated hemoglobin (HbO₂). By alternating illumination between these two wavelengths, it becomes possible to differentiate changes in arterial blood oxygenation and volume. This principle underlies the operation of pulse oximetry and enhances the robustness of PPG signal acquisition in hemodynamic monitoring.

In this study, PPG signals were obtained using a dual-wavelength approach with red (660 nm) and near-infrared (940 nm) light sources, modulated in the form of a pulse train t_p(t) (Equation 9). Signal acquisition was performed in reflectance mode, as illustrated in Figure 1, allowing detection of optical changes as light scattered back from the tissue. The alternating red and infrared pulse trains were generated at a frequency of 122 Hz, while the acquisition system sampled the signal at 1950 Hz, corresponding to a temporal resolution of t=5.12×10^-4 seconds. This configuration ensured high fidelity in capturing both the pulsatile (AC) component, which reflects synchronous arterial expansion during each cardiac cycle, and the baseline (DC) component, which represents static absorption by skin, bone, and venous blood. Together, these components provide the physiological basis for analyzing vascular tone, cardiac rhythm, and tissue oxygenation.

The sampled signal is applied to a low-pass convolution filter (Stearns & Hush, 2011), yielding the following result:

where ω_p and f_p are the angular frequency and spatial frequency of the carrier signal (p), and A and B are constants.

If the cardiac-pulse–derived signal | P(t) | amplitude-modulates a pulse train as defined in Equation (9), and the pulse-train (carrier) frequency f_p greatly exceeds the physiological frequency to be recovered f_r (heart rate), then the modulated signal can be treated as a carrier system satisfying:

Under this separation of timescales, | P(t) | acts as a slowly varying envelope on the high-frequency carrier, facilitating subsequent demodulation and recovery of the heart-rate component f_r.

The following relationship can be considered:

When the signal relationship described in Equation (12) is multiplied by two reference signals—sine and cosine functions with frequencies close to the carrier frequency fp≈fp' —the following expressions are obtained:

By expanding these expressions and recalling that the physiological frequency f_r is much smaller than the carrier frequency (fr≪fp, (Equation 11), it follows that the low-frequency term fr−|fp' remains within the band of interest, while the high-frequency components fr−|fp' and harmonics at 2π fp't are effectively attenuated. These unwanted high-frequency terms are eliminated by applying a low-pass filter. Subsequently, Min–Max normalization yields the following in-phase and quadrature components:

where B'=B2 is a constant term.

Since the interest lies in recovering the magnitude rather than the phase of the cardiac-related signal, the envelope of the demodulated signal is obtained from Equations (15) and (16). The resulting magnitude expression is:

Here, | P(t) | represents the recovered photoplethysmographic signal associated with the cardiac pulse. This recovery is achieved through synchronous detection, as illustrated in Figure 2, ensuring robust isolation of the physiological component from the high-frequency carrier.

The synchronous detection technique is extensively employed in communications and has also been adapted for use in optical interferometry (Servin et al., 1994). Its underlying principle relies on the property that when two sinusoidal signals of different frequencies are multiplied, the mean value of the resulting function approaches zero, thereby suppressing the DC component of the output. By exploiting this effect, synchronous detection enables the extraction of weak signals embedded in noise. Specifically, the method involves multiplying the measured signal by a sinusoidal wave of a defined frequency, followed by a time-averaging process. When the reference sinusoid matches the frequency of the input signal, the desired component is selectively enhanced while other frequency components are attenuated (Ghaderi & Bahreyni, 2021).

A schematic representation of this process is presented in Figure 2.

Two datasets were collected, each comprising 50 measurements obtained from a 37-year-old female subject. The first dataset was acquired using a commercial device, the MD300C23 CHOICEMED oximeter, while the second dataset was recorded with the prototype system illustrated in Figure 1, which incorporates the synchronous demodulation technique described in Figure 2. The recorded signals were subsequently analysed using the Fourier Transform, allowing the identification of dominant spectral peaks corresponding to the cardiac frequency. These peaks were then used to calculate the heart rate for each measurement, with the results summarized in Table 1.

The selection of the carrier frequency and the integration time for the synchronous demodulation stage was guided by fundamental signal processing considerations and by the specific characteristics of PPG signals. The cardiac-related physiological information of interest is concentrated at low frequencies, typically below 3 Hz, corresponding to heart rates within the expected physiological range. To ensure adequate spectral separation between the physiological content and the modulation carrier, the carrier frequency was chosen to be significantly higher than the highest expected heart rate component, while remaining well below the Nyquist frequency defined by the acquisition sampling rate.

In addition, the selected carrier frequency was intentionally placed away from dominant sources of ambient and electrical interference, such as power-line frequencies (50/60 Hz) and their harmonics, as well as from low-frequency illumination fluctuations. This choice minimizes spectral overlap and facilitates efficient rejection of unwanted components during the subsequent low-pass filtering stage following demodulation. The selected carrier frequency therefore represents a compromise between spectral separation, noise immunity, and implementation simplicity for the prototype hardware.

The integration time used in the synchronous demodulator was selected to balance noise reduction and temporal resolution of the recovered PPG signal. A longer integration window improves signal-to-noise ratio by averaging out high-frequency noise components, but excessively long integration times may attenuate physiologically relevant temporal variations in heart rate. Conversely, very short integration times preserve temporal resolution but reduce noise suppression. Based on these trade-offs, the integration time was selected to be sufficiently longer than the carrier period while remaining short relative to the timescale of heart rate variations, ensuring stable recovery of the PPG envelope without distortion of the underlying cardiac rhythm.

Parameter optimization for the prototype was performed through a combination of literature-based design guidelines and preliminary experimental testing. Initial parameter ranges were defined based on prior studies employing synchronous detection in biomedical and optical measurement systems. These ranges were then refined empirically by evaluating the stability of the demodulated signal, the clarity of the recovered cardiac waveform, and the consistency of the resulting heart rate estimates. The final parameter values were selected to maximize signal stability and repeatability while maintaining compatibility with the constraints of the acquisition hardware and real-time processing requirements.

To evaluate the agreement between heart rate (HR) values derived from the proposed system and those obtained from the reference commercial oximeter, a Bland–Altman analysis was conducted. For each paired measurement, the mean heart rate between the two devices was computed and plotted along the horizontal axis, while the difference between the corresponding HR estimates, defined as HR_prototype – HR_reference, was plotted along the vertical axis. The mean of the differences was calculated to quantify systematic bias between the two measurement methods. The 95% limits of agreement were determined as the bias ± 1.96 times the standard deviation of the differences. This graphical representation enables visualization of agreement, identification of potential systematic trends, and assessment of the dispersion of differences across the range of observed heart rate values (Bland & Altman, 1986).

Let X₁, … , X_m and Y₁, … , Y_n denote two independent random samples. The null hypothesis states that there is no difference between the distributions of the two samples X and Y. Conversely, the research hypothesis is two-tailed and non-directional, indicating that a difference exists between the two distributions without specifying its direction (Corder & Dale, 2023). This approach enables the detection of deviations in either direction from the null hypothesis.

Formally, the hypotheses are defined as follows:

The significance level commonly adopted is α = 0.05, which corresponds to a 95% confidence that any observed statistical difference reflects a true underlying difference rather than random variation.

In this study, we compare two random samples, X and Y, where each sample consists of mutually independent and identically distributed observations. Furthermore, the values of X and Y are independent of each other. To assess whether these samples originate from the same distribution or exhibit significant differences, we apply the TK–S, which provides a non-parametric statistical framework for evaluating distributional equality.

To initiate the analysis, the first step involves computing the empirical distribution functions (EDF) for the two samples, X and Y. The EDF provides a non-parametric estimate of the cumulative distribution function (CDF) of the data and serves as the foundation for the TK–S test. The procedure can be summarized as follows:

And

where m = 50 and n = 50.

Using the data presented in Table 1 and applying Equation (1), each divergence value between the empirical distribution functions of the two samples was calculated. These values are summarized in Table 2. This table provides the basis for identifying the maximum divergence between the distributions, which constitutes the TK–S test statistic. By examining these divergences, it becomes possible to determine whether the differences observed between the two datasets are statistically significant, thereby supporting or rejecting the null hypothesis.

Next, we find the largest divergence D_max = 0.10. Now, we use Equation (2) to calculate the K-S test statistic Z:

Now, we find the p-value using Equation (4) since they satisfy the condition that if 0.27 ≤ Z < 1.

We first need Q using Equation (5):

Now, we can use Formula (4):

The two-tailed probability, p = 0.9639, was computed and is now compared with the level of risk specified earlier, α = 0.05. If it is greater than the p-value, we must reject the null hypothesis. If is less than the p-value, we must not reject the null hypothesis. Since it is not greater than the p-value (0.05 > 0. 9639), we do not reject the null hypothesis.

We do not reject the null hypothesis, suggesting that the two signals do not present a significant difference.

When reporting the results from the TK-S test, include such information as the sample sizes for each group, the D statistic, and the p-value’s relation to α.

Figure 3 illustrates the CDF of the estimated spectral frequency data (blue) and the reference CHOICEMED pulse oximeter measurements (orange). The TK–S test quantifies the maximum vertical difference between the two CDFs, representing the point of greatest divergence between the empirical distributions. In this case, the maximum difference is approximately 0.10, as indicated by the black arrow. The red dashed line marks the data value at which this divergence occurs, approximately 68.00.

The progression of the two CDFs demonstrates how cumulative probabilities accumulate across the data range. The close alignment of the two curves across most values suggests a high degree of similarity between the distributions, supporting the assumption that the measurements obtained with the proposed synchronous demodulation method are consistent with those recorded by the commercial pulse oximeter.

From the maximum difference observed in Figure 3 (D = 0.10), the TK–S test statistic was calculated and compared with the corresponding critical value at a significance level of α = 0.05. Since the obtained value of D did not exceed the critical threshold, the null hypothesis could not be rejected. This indicates that there is no statistically significant difference between the distributions of heart rate measurements obtained with the synchronous demodulation method and those recorded with the CHOICEMED oximeter. These findings reinforce the reliability of the proposed method, as it produces results statistically consistent with those of a validated commercial device.

For two independent samples of sizes m and n, the Kolmogorov–Smirnov test statistic is defined as

where F_m(t) and G_n(t) are the empirical distribution functions of the two samples. In this case, the maximum difference observed between the two CDFs was D_m,n = 0.10 (Figure 3).

The corresponding critical value for the test at a significance level of α = 0.05 is given by

where c(0.05) ≈ 1.36. Substituting the sample sizes into this expression yields the threshold for rejection of the null hypothesis. Since the observed value D_m,n = 0.10 is less than the critical value D_α, the null hypothesis cannot be rejected.

Thus, the statistical analysis confirms that the heart rate distributions obtained with the synchronous demodulation method and with the CHOICEMED oximeter do not differ significantly, demonstrating that the proposed method provides results consistent with those of the commercial reference device.

During data acquisition, the subject remained at rest, and no controlled motion was introduced, with the goal of minimizing motion-induced distortions in the recorded PPG signals. It is well known that real-world PPG measurements are prone to motion artifacts, which can alter the morphology and distribution of the acquired signals. Within the context of the proposed TK-S framework, such artifacts would likely manifest as significant divergences between the empirical distributions of the reference and prototype signals, potentially increasing the test statistic and leading to the rejection of distributional equivalence. Although motion artifacts were not explicitly evaluated in the present study, future work will examine the behavior of the method under realistic dynamic conditions to assess its robustness in wearable or ambulatory monitoring scenarios.

To complement the distribution-based comparison provided by the Kolmogorov–Smirnov test, a direct clinical comparison of the derived Heart Rate (HR) values from both devices was performed. The mean absolute error (MAE) between the two measurements was 1.71 bpm, indicating a small average deviation (See Table 3). The mean bias was −0.44 bpm, showing that the proposed device tends to slightly underestimate the heart rate relative to the commercial oximeter. These values fall within clinically acceptable ranges typically reported for photoplethysmography-based heart-rate monitors. Although a complete Bland–Altman analysis is recommended for future studies with larger populations, the present results suggest that the distributional equivalence observed with the TK-S test is consistent with a clinically reasonable agreement in HR estimation.

Figure 4 presents the Bland–Altman plot comparing the heart rate (HR) values derived from the proposed system and those obtained from the reference commercial oximeter. The analysis shows a small positive mean difference (bias = 0.44 bpm), indicating minimal systematic overestimation of HR by the proposed method relative to the reference device. The 95% limits of agreement, defined as −2.83 to 3.71 bpm, demonstrate that the majority of the paired measurements fall within a narrow and clinically reasonable range.

The dispersion of the differences does not exhibit a clear dependence on the magnitude of the mean heart rate, suggesting the absence of proportional bias across the observed HR range. Furthermore, no evident clustering of extreme deviations is observed, and the differences remain relatively uniformly distributed across low and moderate heart rate values. These findings indicate good agreement between the two measurement methods under the controlled experimental conditions considered in this study.

When interpreted in conjunction with the signal-level distributional equivalence demonstrated by the TK-S test, the Bland–Altman results support the conclusion that statistical equivalence at the PPG signal level translates into practically consistent heart rate estimates in this proof-of-concept validation. Nevertheless, given the single-subject and motion-free nature of the dataset, these results should be interpreted as preliminary evidence of methodological consistency rather than definitive clinical validation.

Data acquisition was performed under controlled experimental conditions in which the subject remained at rest, with no intentional body or limb movement during signal recording. This protocol was adopted to minimize motion-induced artifacts and to ensure stable optical coupling between the sensor and the measurement site, thereby allowing the proposed TK-S based validation framework to be evaluated under motion-free conditions. Therefore, the results reported in this study correspond to a controlled scenario and should be interpreted as a methodological validation of the signal processing and statistical comparison approach, rather than as an evaluation under ambulatory or real-world monitoring conditions.

Motion artifacts in photoplethysmography measurements typically manifest as non-physiological amplitude fluctuations, baseline shifts, and spectral distortions, which alter the empirical distribution of the recorded signal. In the presence of such artifacts, these effects would be reflected as increased divergence between empirical distributions and would likely lead to rejection of the null hypothesis of equivalence in the TK-S test. Accordingly, while the proposed framework does not aim to suppress or compensate for motion artifacts, it can serve as a diagnostic tool to assess their impact on signal integrity and agreement between measurement systems. Extension of the present analysis to datasets acquired under controlled or semi-controlled motion conditions constitutes an important direction for future work.

An important limitation of the present study is that the experimental validation was conducted using data acquired from a single subject. As a result, the findings reported here should not be interpreted as a definitive clinical validation of the proposed photoplethysmography-based system. Instead, this work is intended as a methodological and statistical proof of concept, demonstrating the feasibility and sensitivity of the TK–S test for assessing distributional equivalence between PPG signals obtained from a prototype device and a calibrated commercial reference under controlled conditions.

The use of a single subject was a deliberate design choice aimed at minimizing inter-individual physiological variability, such as differences in vascular compliance, skin tone, tissue composition, or body mass index, in order to isolate and evaluate the statistical behaviour of the validation methodology itself. By analysing multiple repeated measurements (n = 50 per device) obtained under comparable conditions, the proposed framework enables a robust within-subject distributional comparison, allowing subtle differences in signal behaviour to be detected without confounding population-level effects.

Consequently, the statistical equivalence demonstrated in this study should be understood as conditional on the specific experimental context and subject considered. While this approach provides strong internal validity for the proposed TK–S based validation strategy, it does not address inter-subject variability and therefore does not support population-wide generalization at this stage.

Future studies will extend the proposed validation framework to a heterogeneous cohort of subjects (N > 10), encompassing variability in age, skin tone, and body mass index, to evaluate the robustness and generalizability of the method at the population level. Applying the same TK–S–based distributional comparison across multiple subjects will enable assessment of inter-subject consistency and will constitute a necessary step toward clinical validation of the proposed system.