NLMS (Normalized LMS) Adaptive Filter for PPG

NLMS modifies the LMS update rule to

where the step size is normalized by the squared Euclidean norm of the input vector plus a small regularization constant $\varepsilon$. 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 $O(P^2 N)$ complexity where $P$ is the projection order.