Kalman Filtering for Continuous PPG Heart Rate Estimation

Extended Kalman Filter (EKF) for PPG

The standard Kalman filter assumes linear state and observation models, which limits its applicability to PPG. In practice, the relationship between the PPG signal and heart rate involves nonlinear transformations (spectral estimation, peak detection, waveform feature extraction), and the state dynamics may also be nonlinear (heart rate variability is modulated by respiratory and autonomic factors in complex ways).

The Extended Kalman Filter addresses this by linearizing the nonlinear models at each time step using Jacobian matrices. The nonlinear state and observation models are:

where $f(\cdot)$ and $h(\cdot)$ are nonlinear functions. The EKF computes the Jacobians $F = \partial f/\partial x$ and $H = \partial h/\partial x$ at the current state estimate and uses these in the standard Kalman equations.

EKF State Model Design for PPG

One EKF design for PPG heart rate tracking uses a state vector holding heart rate and a signal model parameter describing the expected PPG spectral shape. The observation model h() mapped the state to the expected power spectral density of the PPG signal, and the innovation was computed by comparing the predicted and observed spectra.

A richer EKF formulation includes the heart rate, its rate of change, and the amplitudes of the fundamental frequency and first harmonic in the state vector:

Including harmonic amplitude states lets the filter resolve harmonic ambiguity (distinguishing the fundamental cardiac frequency from its half-frequency or double-frequency harmonics) by tracking the expected harmonic amplitude ratio across time.

Noise Parameter Tuning

The performance of the EKF depends critically on the process noise covariance Q and measurement noise covariance R. These parameters encode prior knowledge about physiological heart rate dynamics and measurement quality.

Process noise Q should reflect the maximum expected heart rate change between update steps. For a 2-second update interval, maximum physiological heart rate change is approximately 5-10 BPM. Setting $Q = \operatorname{diag}([5^2, 2^2])$ for a $[\mathrm{HR}, \mathrm{dHR}]$ state vector provides reasonable tracking bandwidth. During known exercise conditions, Q can be increased to allow faster tracking. Adaptive Q estimation based on accelerometer-derived activity level can help during activity transitions.

Measurement noise R should reflect the reliability of the spectral heart rate observation. At rest with clean signals, R = 1-4 BPM^2 is appropriate. During heavy motion, R should increase to 25-100 BPM^2, effectively telling the filter to distrust the observation and rely on its prediction. Adaptive R estimation based on PPG signal quality indices (SQI) is a powerful technique that enables the filter to automatically adjust its behavior based on signal conditions.

Unscented Kalman Filter (UKF) for PPG

The UKF provides a higher-accuracy alternative to the EKF by avoiding Jacobian computation and instead using deterministic sampling (sigma points) to propagate the state distribution through the nonlinear models. The UKF captures the mean and covariance of the transformed distribution more accurately than the first-order linearization used by the EKF, particularly when the nonlinearities are strong.

For PPG heart rate tracking, the UKF generates ${2n+1}$ sigma points (where $n$ is the state dimension), propagates each through the nonlinear state and observation models, and computes the predicted mean and covariance from the transformed sigma points. With $n$ = 2-4 for typical PPG state models, the computational overhead relative to the EKF is modest.

Particle Filters for PPG

Particle filters (Sequential Monte Carlo methods) represent the most general Bayesian filtering approach, capable of handling arbitrary nonlinear models and non-Gaussian noise distributions. Instead of parametric state estimation (as in the EKF/UKF), particle filters maintain a set of weighted samples (particles) that approximate the full posterior distribution over the state.

For PPG heart rate estimation, particle filters offer two distinct advantages. First, they can represent multimodal distributions, which naturally handles harmonic ambiguity. When the PPG spectrum contains energy at both the fundamental heart rate frequency and a motion harmonic, the particle cloud can maintain hypotheses at both frequencies, automatically resolving the ambiguity as more observations arrive. Second, particle filters can incorporate arbitrary constraints (such as physiological heart rate bounds of 30-220 BPM) directly into the sampling process.

A useful design is a spectral likelihood function that evaluates each particle's predicted heart rate against multiple peaks in the PPG power spectrum, weighted by peak prominence and harmonic consistency. This naturally handles scenarios where the strongest spectral peak is a motion artifact rather than the cardiac fundamental.

The primary disadvantage of particle filters is computational cost. With hundreds of particles, each requiring spectral evaluation, the computation per update step is far greater than the EKF. On modern wearable processors (ARM Cortex-M4/M7), particle filters at low update rates remain feasible, but careful implementation optimization is required.

Multi-Stage Pipeline Integration

In practical wearable heart rate algorithms, the Kalman filter operates as the final stage in a multi-stage signal processing pipeline. The upstream stages handle motion artifact removal and spectral estimation, while the Kalman filter provides temporal smoothing and continuity.

A representative pipeline architecture is:

1. Preprocessing: Bandpass filtering (0.4-4 Hz), baseline removal
2. Motion artifact reduction: Adaptive filtering (NLMS/RLS) with accelerometer reference, or SVD-based denoising
3. Spectral estimation: Short-time FFT or autoregressive spectral estimation on 5-8 second windows
4. Peak selection: Identification of candidate heart rate frequencies from spectral peaks
5. Kalman filtering: Recursive tracking of the heart rate state using spectral peaks as observations

The interface between stages 4 and 5 is critical. Rather than passing a single point estimate to the Kalman filter, passing multiple candidate peaks with their spectral magnitudes enables the filter to resolve ambiguities using its temporal model. This is particularly important during exercise, where motion harmonics often produce spectral peaks that are stronger than the cardiac fundamental.

Zhang (2015) implemented this multi-candidate approach in the TROIKA framework, using a sparse signal reconstruction step to identify candidate frequencies and a spectral peak tracking stage functionally equivalent to Kalman filtering. TROIKA achieved heart rate MAE of 2.34 BPM on the IEEE Signal Processing Cup dataset and established a benchmark that subsequent Kalman-based methods have built upon. For related peak detection methods, see our guide on PPG signal processing algorithms.

Adaptive Kalman Filtering

Fixed-parameter Kalman filters assume that the process and measurement noise characteristics remain constant, which is rarely true for PPG signals recorded during real-world activities. Adaptive Kalman filtering dynamically adjusts Q and/or R based on signal characteristics, achieving significantly better performance during transitions between rest and exercise.

Signal Quality-Based Adaptation

PPG signal quality indices (SQI) provide a direct measure of measurement reliability that can be mapped to the measurement noise covariance R. Common SQI metrics include:

• Perfusion index: The ratio of AC to DC PPG amplitude. Higher perfusion indicates better signal quality.
• Skewness: Clean PPG pulses have characteristic negative skewness. Deviation indicates corruption.
• Kurtosis: Excess kurtosis detects non-Gaussian artifacts.
• Template correlation: Cross-correlation of each pulse with a running average pulse template. Values below 0.8 indicate morphological distortion.

A common design sets R as an increasing function of a composite SQI score (for example, from perfusion index and template correlation). When SQI drops, R rises and the filter coasts on its prediction. When SQI recovers, R falls and the filter re-locks to the observation.

Innovation-Based Adaptation

An alternative approach adapts Q and R based on the Kalman filter's own innovation sequence (the difference between predicted and observed measurements). If the innovation consistently exceeds the predicted innovation covariance, either Q is too small (the model is not allowing enough state variation) or R is too small (the model is trusting observations too much). Innovation-based adaptation algorithms such as those of Mehra (1970, doi:10.1109/TAC.1970.1099422) and Mohamed and Schwarz (1999, doi:10.1007/s001900050236), developed for general and INS/GPS filtering, can be carried over to PPG heart rate tracking.

Sage-Husa adaptive Kalman filtering estimates both Q and R online from windowed innovation statistics, which removes the need for manual parameter tuning.

Performance Benchmarks

Compare any Kalman tracker against established, non-Kalman reference results on public datasets:

Spectral peak tracking with temporal constraints (as in TROIKA) and Viterbi decoding (as in Temko's offline mode) do the same job as a Kalman stage: they exploit the smoothness of heart rate over time.

Computational Requirements and Embedded Implementation

Kalman filters are highly efficient algorithms, particularly in low-dimensional state spaces. The computational requirements for PPG heart rate tracking are dominated by the upstream spectral estimation, not the filter itself.

For a 4-state EKF, each prediction-update cycle requires approximately 100 floating-point multiply-accumulate operations. At a 1 Hz update rate, this is negligible on any modern processor. The UKF with 9 sigma points requires approximately 500 operations per update, still trivial. Even particle filters with 500 particles require only about 50,000 operations per update, feasible on ARM Cortex-M4 at clock speeds above 48 MHz.

Memory requirements are also minimal: the EKF requires storage for the state vector (4 floats), covariance matrix (4x4 = 16 floats), process and measurement noise matrices, and Kalman gain, totaling approximately 200 bytes. Particle filters require storage for particle states and weights, totaling approximately 4 KB for 500 particles with a 4-state model.

These low computational and memory requirements make Kalman filtering an ideal final stage in wearable heart rate estimation pipelines, adding robust temporal tracking with negligible resource overhead.

Conclusion

Kalman filtering provides a principled, efficient, and effective approach to continuous heart rate estimation from PPG signals. By framing heart rate tracking as a recursive Bayesian state estimation problem, Kalman filters exploit the temporal smoothness of heart rate to produce estimates that are robust to transient noise, spectral ambiguity, and brief signal dropouts. The Extended Kalman Filter offers the best balance of accuracy and computational efficiency for most wearable applications, while the Unscented Kalman Filter and particle filters provide additional capability for handling strong nonlinearities and multimodal ambiguity. Adaptive variants that dynamically adjust noise parameters based on signal quality further improve robustness during real-world use. For researchers and engineers building PPG heart rate algorithms, adding a Kalman filtering stage to the signal processing pipeline is one of the highest-value, lowest-cost improvements available.

References

• Reiss, A., Indlekofer, I., Schmidt, P., & Van Laerhoven, K. (2019). Deep PPG: Large-scale heart rate estimation with convolutional neural networks. Sensors, 19(14), 3079. https://doi.org/10.3390/s19143079
• Temko, A. (2017). Accurate heart rate monitoring during physical exercises using PPG. IEEE Transactions on Biomedical Engineering, 64(9), 2016-2024. https://doi.org/10.1109/TBME.2017.2676243
• Zhang, Z., Pi, Z., & Liu, B. (2015). TROIKA: A general framework for heart rate monitoring using wrist-type photoplethysmographic signals during intensive physical exercise. IEEE Transactions on Biomedical Engineering, 62(2), 522-531. https://doi.org/10.1109/TBME.2014.2359372