NLMS (Normalized LMS) Adaptive Filter for PPG
The Normalized Least Mean Squares (NLMS) algorithm adapts the LMS step size by dividing by the input signal power, eliminating sensitivity to input scaling and providing more consistent convergence across varying motion intensities in PPG wearables.
NLMS modifies the LMS update rule to w(n+1) = w(n) + (μ / (ε + ||x(n)||²)) · e(n) · x(n), where the step size is normalized by the squared Euclidean norm of the input vector plus a small regularization constant ε. This normalization ensures that convergence speed is independent of input signal power, which varies dramatically in PPG applications as motion intensity changes.
For wearable PPG, NLMS is one adaptive-filter option when an accelerometer or another reference correlated with motion artifact is available. Normalization reduces sensitivity to reference-signal power, but the step size, filter order, reference quality, and overlap between motion and cardiac spectra still require validation for the intended activity. TROIKA did not use NLMS: Zhang et al. combined signal decomposition, sparse spectral reconstruction, and spectral peak tracking, reporting 2.34 bpm average absolute error on recordings from 12 subjects during treadmill exercise.
Block NLMS variants process multiple samples simultaneously, reducing per-sample computational overhead on DSP architectures with SIMD support. Affine projection algorithms (APA) generalize NLMS by projecting the weight update onto multiple past input vectors, improving convergence for colored noise at the cost of O(P²N) complexity where P is the projection order.
Frequently Asked Questions
How does NLMS differ from standard LMS for PPG?
NLMS normalizes the step size by input power, making convergence independent of signal level. Standard LMS requires manual step-size tuning for each activity type, while NLMS adapts automatically.
What regularization constant should be used in NLMS for PPG?
Typical values are ε = 0.001–0.01 times the expected input power. Too small values cause numerical instability when the input approaches zero; too large values reduce the normalization benefit.
Is NLMS sufficient for high-intensity exercise PPG?
Not reliably in every case. When motion and pulse components overlap spectrally or acceleration is a poor reference for optical artifact, NLMS may leave residual artifact or remove cardiac content. Published results apply to their specific dataset and complete pipeline; TROIKA is not an NLMS pipeline.