Wavelet Transform Denoising for PPG Signal Cleaning

Thresholding Strategies

The Denoising Framework

Wavelet denoising follows a three-step procedure introduced by Donoho and Johnstone (1994, doi:10.1093/biomet/81.3.425):

1. Decompose: Apply the DWT to obtain wavelet coefficients at each level
2. Threshold: Modify the detail coefficients at selected levels according to a thresholding rule
3. Reconstruct: Apply the inverse DWT to the modified coefficients to obtain the denoised signal

The key decisions are which levels to threshold, what threshold value to use, and whether to apply hard or soft thresholding.

Hard vs. Soft Thresholding

Hard thresholding:

This creates a discontinuity at the threshold boundary, which can introduce Gibbs-like ringing artifacts in the reconstructed signal. However, it preserves the amplitude of large coefficients exactly, maintaining the PPG waveform shape for the cardiac component.

Soft thresholding (shrinkage):

This produces a continuous mapping and generally smoother reconstructed signals. The amplitude bias (all coefficients are reduced by T) can attenuate the systolic peak amplitude, potentially affecting waveform-based analyses.

For PPG heart rate estimation, soft thresholding is preferred because the smooth output produces cleaner spectral peaks. For pulse wave morphology analysis (augmentation index, stiffness index, dicrotic notch detection), hard thresholding better preserves the critical waveform features.

Threshold Value Selection

The threshold value T determines the aggressiveness of noise suppression. Several methods are used:

Universal threshold (VisuShrink):

where $\sigma$ is the noise standard deviation (typically estimated from the level-1 detail coefficients using the median absolute deviation: $\sigma = \mathrm{MAD}(d_1) / 0.6745$) and $T$ is the signal length. This provides an upper bound on the threshold that asymptotically ensures complete noise removal with probability approaching 1. For PPG, VisuShrink tends to be too aggressive, removing fine cardiac waveform details.

SureShrink (Stein's Unbiased Risk Estimate): Minimizes Stein's unbiased risk estimate at each decomposition level independently, providing a level-dependent threshold that adapts to the signal-to-noise ratio at each scale. SureShrink is generally recommended for PPG because it automatically balances noise removal against signal preservation at each frequency band (Donoho and Johnstone, 1995, doi:10.1080/01621459.1995.10476626).

BayesShrink:

where $\sigma_j$ is the estimated signal standard deviation at level $j$. This Bayes-optimal threshold adapts to each level's signal content and typically produces less signal distortion than VisuShrink while achieving comparable noise suppression. BayesShrink was introduced by Chang, Yu, and Vetterli (2000, doi:10.1109/83.862633) for image denoising.

Minimax threshold: Uses the minimax principle to minimize the maximum risk over all signals in a given class. For PPG, the minimax threshold is more conservative than VisuShrink and typically preserves more waveform detail at the cost of leaving more noise.

Level-Dependent Thresholding for PPG

Rather than applying a single threshold across all decomposition levels, level-dependent thresholding allows different aggressiveness at each scale:

• Levels 1-2 (high frequency): Aggressive thresholding (high T). These levels contain primarily sensor noise and high-frequency artifacts with minimal cardiac content.
• Levels 3-4 (mid frequency): Moderate thresholding. These levels contain cardiac harmonics mixed with motion artifact. Over-thresholding removes important waveform detail.
• Levels 5-6 (cardiac fundamental): Minimal or no thresholding. These levels contain the cardiac pulse fundamental and should be preserved.
• Approximation coefficients: Generally not thresholded. Baseline wander in the approximation can be removed separately by subtracting a polynomial fit.

This level-dependent strategy is more effective than uniform thresholding for PPG because the signal-to-noise ratio varies dramatically across frequency bands. Levels 1-2 have SNR < 0 dB (noise dominates), while levels 5-6 have SNR > 10 dB (signal dominates at rest).

Stationary Wavelet Transform for PPG

Motivation

The standard DWT is not shift-invariant: shifting the input signal by one sample can change the wavelet coefficients significantly, producing different denoising results. This is problematic for PPG because the cardiac pulse timing is not aligned to the wavelet decomposition grid. Artifacts can appear in the reconstructed signal at positions where the pulse peaks happen to align with downsampling boundaries.

The Stationary Wavelet Transform (SWT), also called the undecimated or redundant wavelet transform, eliminates downsampling and produces shift-invariant coefficients. The cost is increased computation ($O(T \cdot L \cdot 2^L)$ vs. $O(T \cdot L)$) and memory (coefficients at each level have the same length as the original signal rather than being halved).

For applications where shift invariance matters (e.g., beat detection, inter-beat interval estimation), SWT is recommended despite the higher cost. For applications focused on spectral estimation (heart rate from FFT), standard DWT is often sufficient.

Wavelet Packet Decomposition

Enhanced Frequency Resolution

The standard DWT decomposes only the approximation coefficients at each level, producing an octave-band frequency decomposition with decreasing resolution at lower frequencies. Wavelet Packet Decomposition (WPD) decomposes both the approximation and detail coefficients at each level, producing a uniform-resolution frequency decomposition with ${2^L}$ equal-width subbands at level $L$.

For PPG, WPD offers finer frequency resolution in the cardiac band, which can better separate cardiac and motion components when their frequencies are close. At level 6 with 100 Hz sampling, WPD provides 64 subbands of 0.78 Hz width, compared to the DWT's non-uniform bands. This allows isolating a cardiac frequency of 1.5 Hz (90 BPM) from a walking cadence of 2.0 Hz in adjacent subbands.

WPD for PPG Heart Rate Estimation

The trade-off is computational cost: full WPD is $O(T \cdot 2^L)$, compared to $O(T \cdot L)$ for the DWT, so it is considerably more expensive at typical decomposition depths. Best-basis selection can reduce the number of subbands that must be processed.

Benchmark Results and Comparisons

Reference-Free Operation

Unlike adaptive filtering, which needs an accelerometer or other noise reference, wavelet denoising requires no reference signal. This makes it a strong reference-free option for broadband noise types. Reported SNR gains depend heavily on the dataset, noise model, and evaluation protocol, so compare methods on your own data rather than relying on cross-study numbers.

Heart Rate Estimation Accuracy

Wavelet denoising alone (without spectral tracking or adaptive filtering) works well at rest and during light motion, where most noise is well separated from the cardiac band. During vigorous exercise, periodic motion artifacts overlap the cardiac band and resist wavelet separation, so wavelet denoising is insufficient as a standalone method. It performs best when combined with other techniques:

Wavelet + adaptive filter: Apply wavelet denoising as pre-processing to remove broadband noise, then apply NLMS or RLS to cancel periodic motion artifacts using the accelerometer reference. Combining the two addresses both broadband and periodic noise.

Wavelet + EMD: Apply EMD/EEMD for adaptive decomposition, then apply wavelet thresholding to each IMF for fine-grained noise removal. This hybrid is fully reference-free.

Wavelet + ICA: Apply ICA for source separation on multi-channel PPG, then apply wavelet denoising to the selected cardiac component. This two-stage approach removes both structured interference (via ICA) and residual broadband noise (via wavelets).

Real-Time Implementation

Streaming Wavelet Denoising

For real-time PPG processing, wavelet denoising is applied in overlapping windows:

1. Buffer incoming PPG samples into windows of length T (typically 512-2048 samples, 5-20 seconds)
2. Apply DWT decomposition
3. Threshold detail coefficients
4. Reconstruct via inverse DWT
5. Use overlap-add or overlap-save to produce continuous output

The window length determines the latency and the frequency resolution of the lowest decomposition level. A 1024-sample window at 100 Hz provides 10.24 seconds of data with lowest detail band 0.78-1.56 Hz, sufficient for cardiac frequencies above 47 BPM.

Memory and Computation Budget

For an embedded implementation at 100 Hz with db4 (filter length 8) and 6 decomposition levels:

• Memory: Wavelet filter coefficients (16 values) + input buffer (1024 samples) + coefficient storage (approximately 2048 values for DWT, double for SWT) = approximately 12-20 KB
• Computation per window: 6 levels * 2 filters * 1024 convolution operations = approximately 12,288 multiply-accumulate operations for DWT; approximately 50,000 for SWT
• Throughput at 50% overlap: One window every 5.12 seconds, requiring approximately 2,400 MACs/second for DWT

This is trivially feasible on any modern embedded processor, making wavelet denoising one of the most practical PPG signal processing methods for resource-constrained wearable devices.

Comparison with Other Methods for Embedded Deployment

Wavelet denoising occupies a practical middle ground: substantially more effective than simple bandpass filtering, reference-free operation, moderate memory and computation requirements, and latency acceptable for heart rate monitoring (though not for beat-by-beat HRV analysis requiring sub-second resolution).

Advanced Topics

Wavelet Shrinkage with Signal-Dependent Thresholds

Standard thresholding methods (VisuShrink, SureShrink) estimate the noise level from the data and apply a global or level-dependent threshold. For PPG, where the signal amplitude varies with cardiac cycle phase (high during systole, low during diastole), signal-dependent thresholds can improve performance.

Block thresholding groups adjacent wavelet coefficients and thresholds them jointly, exploiting the correlation structure of the signal. For PPG, blocks aligned with the cardiac cycle (one block per beat) can adapt the threshold to the local signal amplitude, preserving weak diastolic features that would be removed by a threshold calibrated to the systolic peak. Cai and Silverman (2001, zbMATH 1192.42020) proposed block thresholding estimators (NeighBlock and NeighCoeff) that use information from neighbouring coefficients when deciding whether to keep each coefficient.

Cross-Scale Coefficient Correlation

Cardiac signal energy propagates across scales in a predictable pattern: large coefficients at one level tend to be accompanied by large coefficients at adjacent levels at the same temporal location. Noise coefficients do not exhibit this cross-scale structure. Cross-scale thresholding exploits this by preserving coefficients that are large at multiple scales and removing those that are large at only one scale.

The cost is extra coefficient storage and comparison operations across levels. For algorithm implementation details, see our algorithms reference.

Matching the Denoiser to the Downstream Task

Not every PPG application values the same output, and wavelet settings should be tuned for the endpoint rather than for how clean the waveform looks.

Heart rate estimation. If the goal is stable heart rate, preserving the dominant periodic energy is what matters. A slightly smoothed waveform can still perform very well as long as peak timing stays stable and the spectrum is cleaner.

Morphology analysis. If the goal is arterial stiffness markers, dicrotic notch timing, augmentation index, or derivative features, smoothing becomes dangerous. Small shape distortions can produce large errors in derived markers, so settings should be validated against morphology-specific metrics, not just output SNR.

Evaluation metrics that matter

Subjective waveform appearance is not a validation method. More useful measures include:

• heart rate MAE against an ECG reference
• beat detection sensitivity and positive predictive value
• normalized RMSE on synthetic benchmarks with a known clean reference
• agreement of morphology landmarks before and after denoising
• SNR improvement under controlled noise injection

Synthetic benchmarks are helpful because the clean signal is known, but real wearable recordings remain essential because synthetic noise rarely reproduces sensor-skin coupling and real motion.

Common implementation mistakes

1. Choosing decomposition levels without mapping them to the device sample rate
2. Using thresholds that are too aggressive for morphology work
3. Assuming the same settings transfer across wrist, finger, and ear PPG
4. Validating only on resting data
5. Judging success by visual smoothness instead of endpoint metrics

A practical starting point is db4 or sym4 with level-dependent soft thresholding, levels mapped explicitly to the sample rate, and separate evaluation on clean and motion-contaminated segments.

Conclusion

Wavelet transform denoising provides a powerful, computationally efficient, and reference-free approach to PPG signal cleaning. The combination of multi-resolution decomposition with intelligent thresholding enables selective noise removal across the full frequency spectrum while preserving the cardiac pulse waveform. The db4 wavelet with SureShrink or BayesShrink thresholding at 5-6 decomposition levels represents a robust default configuration for PPG at typical sampling rates.

For standalone PPG denoising without an accelerometer reference, wavelet methods are the recommended first-line approach. For the highest accuracy during vigorous exercise, wavelet denoising should be combined with adaptive filtering (when an accelerometer is available) or EMD (for reference-free operation). The low computational cost makes wavelet denoising particularly attractive as a pre-processing stage that improves the performance of any downstream algorithm.

Explore our complete algorithms reference for implementation guidance, and visit our conditions pages to understand how clean PPG signals enable accurate monitoring of specific cardiovascular health parameters.

References

• Mallat, S. G. (1989). A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 674-693. https://doi.org/10.1109/34.192463
• Donoho, D. L., & Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425-455. https://doi.org/10.1093/biomet/81.3.425
• Donoho, D. L., & Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association, 90(432), 1200-1224. https://doi.org/10.1080/01621459.1995.10476626
• Chang, S. G., Yu, B., & Vetterli, M. (2000). Adaptive wavelet thresholding for image denoising and compression. IEEE Transactions on Image Processing, 9(9), 1532-1546. https://doi.org/10.1109/83.862633
• Cai, T. T., & Silverman, B. W. (2001). Incorporating information on neighbouring coefficients into wavelet estimation. Sankhya: The Indian Journal of Statistics, Series B, 63(2), 127-148. https://zbmath.org/5769380