Empirical Mode Decomposition and EEMD for PPG Signal Analysis

The Mode Mixing Problem

Definition and Impact on PPG

Mode mixing is the most significant practical limitation of standard EMD. It manifests in two ways:

1. Frequency aliasing within an IMF: A single IMF contains oscillations at disparate time scales. For PPG, this commonly occurs when a transient motion artifact (e.g., a hand gesture) creates a burst of high-frequency energy that contaminates the cardiac IMF.

2. Signal splitting across IMFs: The same physical component spreads across multiple IMFs. For PPG, the cardiac pulse may split between $c_3$ and $c_4$ during segments where the heart rate crosses a boundary between the natural frequency scales of those IMFs.

Mode mixing degrades cardiac component extraction because the simple strategy of selecting IMFs by frequency range no longer cleanly isolates the cardiac signal.

Causes of Mode Mixing

Mode mixing arises because EMD identifies oscillatory modes based on local extrema spacing, and any disruption to the extrema pattern can redirect signal energy to the wrong IMF. Common causes in PPG:

• Transient artifacts: Sudden sensor displacement creates a brief high-amplitude disturbance whose extrema disrupt the sifting process for multiple IMFs
• Frequency proximity: When two signal components have similar frequencies (e.g., heart rate at 2.5 Hz and walking cadence at 2.0 Hz), EMD cannot reliably separate them because the combined extrema pattern does not clearly distinguish two scales
• Amplitude modulation: The respiratory modulation of PPG amplitude creates sidebands around the cardiac frequency that can leak into adjacent IMFs

Ensemble EMD (EEMD)

Algorithm

EEMD, introduced by Wu and Huang (2009, doi:10.1142/S1793536909000047), addresses mode mixing by exploiting the statistical properties of white noise:

1. Add white Gaussian noise $w_i(t)$ with standard deviation $\sigma$ to the original signal: $x_i(t) = x(t) + w_i(t)$
2. Perform standard EMD on $x_i(t)$ to obtain IMFs $c_{i,k}(t)$
3. Repeat steps 1-2 for $i = 1$ to $M$ ensemble members (typically $M$ = 100-300)
4. Average the IMFs across all ensembles: $c_k(t) = \frac{1}{M} \sum_{i=1}^{M} c_{i,k}(t)$

The added noise populates all time-frequency scales uniformly, providing a reference distribution that forces EMD to separate similar-frequency components into the correct IMFs. The averaging across ensembles cancels the added noise (which is uncorrelated across realizations) while preserving the signal components (which are identical across realizations).

Noise Amplitude Selection

The noise standard deviation sigma controls the effectiveness of EEMD:

• *Too small (sigma < 0.1 std(x)): Insufficient to resolve mode mixing. The noise does not significantly alter the extrema pattern, and EEMD results approach standard EMD.
• *Optimal (sigma = 0.1-0.4 std(x)): Sufficient to prevent mode mixing while being small enough that averaging effectively cancels the noise.
• *Too large (sigma > 0.5 std(x)): Dominates the signal in some IMFs, creating spurious modes and degrading the decomposition.

For PPG signals, $\sigma = 0.2 \cdot \operatorname{std}(x)$ is the most commonly used value in the literature and works well across rest and moderate exercise conditions. During high-intensity exercise where motion artifacts increase the signal variance, adaptive noise amplitude selection (tied to a running estimate of the clean signal variance) improves results.

Complete EEMD (CEEMDAN)

Torres et al. (2011, doi:10.1109/ICASSP.2011.5947265) introduced CEEMDAN, which adds noise at each stage of the decomposition rather than only at the beginning. This provides more precise control over noise injection and ensures that the residual noise in each IMF converges to zero as the ensemble size increases. CEEMDAN typically requires fewer ensemble members (50-100 vs. 100-300 for EEMD) for equivalent mode-mixing suppression, reducing computational cost.

Motin et al. (2019) compared the EMD family (EMD, EEMD, CEEMDAN, and improved CEEMDAN), each combined with PCA, for extracting breathing rate from PPG on the MIMIC and CapnoBase datasets. EEMD and improved CEEMDAN performed best on both datasets (doi:10.1109/LSP.2019.2900923).

Cardiac Component Extraction Strategies

Frequency-Based IMF Selection

The most straightforward approach selects IMFs whose dominant frequency falls within the expected cardiac range (0.5-4 Hz). The dominant frequency of each IMF is estimated using either:

• The mean instantaneous frequency from the Hilbert transform
• The peak frequency of the IMF's power spectral density
• The average spacing between consecutive zero-crossings

IMFs with dominant frequency in the cardiac range are summed to reconstruct the cardiac signal:

where $K_{\text{cardiac}}$ is the set of cardiac-band IMFs. This approach works well during rest and moderate motion but can include motion artifacts whose frequency falls within the cardiac range.

Correlation-Based Selection

When a reference signal is available (ECG, or a preliminary heart rate estimate from spectral analysis), each IMF is correlated with the reference. IMFs with correlation above a threshold (typically r > 0.3) are included in the cardiac reconstruction.

Energy-Period Ratio Selection

This method computes the ratio of energy in the cardiac frequency band to total energy for each IMF. IMFs with high cardiac-band energy ratio (> 0.6) are selected as cardiac components, while IMFs with low ratio (< 0.3) are classified as noise or motion. This approach does not require an external reference and provides a balance between frequency-based and correlation-based methods.

Computational Considerations

Standard EMD Complexity

EMD's computational cost depends on the number of extrema in the signal and the number of sifting iterations. For a PPG signal of length T with heart rate HR and sample rate fs:

• Number of extrema per cycle: approximately 2 (one max, one min per heartbeat)
• Total extrema: approximately 2 * HR/60 * T/fs
• Cubic spline interpolation: $O(T)$ per envelope
• Sifting iterations per IMF: 5-15
• Number of IMFs: log2(T) approximately 8-12 for typical PPG segments

Total complexity is approximately $O(K \cdot S \cdot T)$ where K is the number of IMFs and S is the average number of sifting iterations. For T=1000 samples (10 s at 100 Hz), K=8, S=10: approximately 80,000 operations.

EEMD Complexity

EEMD multiplies the EMD cost by the ensemble size M:

For M=200, this is approximately 16 million operations per 10-second window. At 100 Hz sample rate with 50% window overlap (new window every 5 seconds), the required throughput is approximately 3.2 million operations per second. This is feasible on ARM Cortex-M7 (200+ MHz) but challenging on lower-power platforms.

Optimization Strategies

Several approaches reduce EEMD computational cost:

Reduced ensemble size. Empirical testing on PPG data shows that M=50-100 provides 90-95% of the mode-mixing suppression of M=200, halving the computation.

Parallel computation. Each ensemble member is independent and can be computed in parallel. On multi-core processors or GPUs, EEMD parallelizes trivially with near-linear speedup.

Selective decomposition. If only the cardiac component is needed, stop the decomposition after extracting the cardiac-band IMFs (typically after 4-5 IMFs out of 8-12 total), reducing cost by approximately 50%.

Sliding-window with warm start. When processing overlapping windows, initialize the sifting process with the previous window's IMFs as starting points, reducing the number of sifting iterations needed for convergence.

Applications Beyond Heart Rate

Respiratory Rate Estimation

The respiratory component of PPG (0.15-0.5 Hz) maps cleanly to one or two IMFs (typically $c_5$ and $c_6$). EMD-based respiratory rate estimation from PPG avoids the need for separate respiratory sensors. In Motin et al.'s (2019) comparison, median absolute error ranged from 0 to 5.03 breaths/min on MIMIC and from 2.47 to 10.55 breaths/min on CapnoBase across the EMD variants, with EEMD and improved CEEMDAN performing best (doi:10.1109/LSP.2019.2900923). Noise-assisted variants help by reducing cardiac harmonic leakage into the respiratory IMFs. For more on respiratory monitoring via PPG, see our conditions overview.

Pulse Wave Analysis

The systolic and diastolic components of the PPG pulse waveform decompose into separate IMFs, enabling automated detection of the dicrotic notch, systolic peak, and diastolic peak. This facilitates computation of augmentation index, stiffness index, and other vascular parameters without explicit waveform morphology algorithms. The data-driven nature of EMD makes it more robust to waveform shape variations across subjects and measurement sites than template-based approaches. See our algorithms reference for more on pulse wave analysis methods.

Signal Quality Assessment

The distribution of energy across IMFs provides a natural signal quality metric. Clean PPG signals concentrate energy in 2-3 cardiac-band IMFs, while motion-corrupted signals spread energy across many IMFs including the high-frequency modes. The ratio of cardiac-band IMF energy to total IMF energy can therefore serve as a signal quality index (SQI) for separating clean from corrupted PPG segments.

Conclusion

Empirical Mode Decomposition and its ensemble variants provide a powerful, data-driven framework for PPG signal analysis that adapts to the signal's own characteristics rather than imposing fixed basis functions. EEMD effectively solves the mode mixing problem that limits standard EMD, at the cost of substantial computational overhead. For researchers and engineers working with PPG signals, EMD/EEMD excels at tasks requiring adaptive signal decomposition: respiratory rate extraction, pulse wave morphology analysis, and cardiac component isolation in moderately noisy conditions.

For motion artifact removal specifically, EEMD is an alternative to adaptive filtering methods that needs no reference signal, but it has a much higher computational cost. The practical recommendation is to use EEMD when reference-free operation is required and computational resources are available, or in hybrid configurations with wavelet denoising or ICA for the best possible signal quality. Explore our complete algorithms reference for implementation guidance across all PPG processing methods.

References

• Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.-C., Tung, C. C., & Liu, H. H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society of London A, 454(1971), 903-995. https://doi.org/10.1098/rspa.1998.0193
• Rilling, G., Flandrin, P., & Gonçalves, P. (2003). On empirical mode decomposition and its algorithms. IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (NSIP-03), Grado, Italy.
• Wu, Z., & Huang, N. E. (2009). Ensemble empirical mode decomposition: a noise-assisted data analysis method. Advances in Adaptive Data Analysis, 1(1), 1-41. https://doi.org/10.1142/S1793536909000047
• Torres, M. E., Colominas, M. A., Schlotthauer, G., & Flandrin, P. (2011). A complete ensemble empirical mode decomposition with adaptive noise. 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 4144-4147. https://doi.org/10.1109/ICASSP.2011.5947265
• Motin, M. A., Karmakar, C. K., & Palaniswami, M. (2019). Selection of empirical mode decomposition techniques for extracting breathing rate from PPG. IEEE Signal Processing Letters, 26(4), 592-596. https://doi.org/10.1109/LSP.2019.2900923