chirp-speech periodicity entrainment index (cpei)

derived from projections onto ramanujan periodic subspaces (rps), for measuring neural temporal tracking in auditory eeg. for a detailed description on the code base, please refer to the 20260409_Code_Documentation.pdf

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background

when a person listens to speech, their brain tracks the rhythm and pitch of the speaker's voice. this tracking degrades in people with hearing loss or age-related auditory decline. cpei is a scalar metric (0 to 1) that quantifies how strongly a neural signal is entrained to a target modulation frequency. and unlike standard fourier-based methods, it should be robust to the octave error problem that arises when hearing loss preferentially attenuates the fundamental frequency relative to its harmonics.

tools like the welch method treat each frequency bin independently, so when hearing loss weakens the brain's response at the fundamental frequency, the dominant peak migrates to the first harmonic, which causes an octave error. rps avoids this since all harmonics of a fundamental period t collapse into the same ramanujan subspace, so the method accumulates energy from the entire harmonic series regardless of which individual harmonic is strongest.

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algorithms compared

algorithm	description
cpei / rps	proposed method. ramanujan filter bank energy concentration index.
welch	standard power spectral density estimator. nonparametric baseline.
plv	phase locking value. measures phase coherence against a fixed reference frequency.
itpc	inter-trial phase coherence. epoch-based equivalent of plv.
yin	pitch estimator (de cheveigné & kawahara 2002). f0 ground truth reference.

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synthetic signals tested

signal	description	tests
1. clean harmonic ffr	f0 = 120 hz, 8 harmonics, snr swept +20 to −20 db	sensitivity
2. attenuated fundamental	fundamental attenuated from 100% to 5%	octave error immunity
3. drifting f0	pitch drifts 100 → 150 hz over 5 s	quasi-periodic robustness
4. pure noise	white gaussian noise, 10 trials	false positive rate
5. overlapping harmonics	two speakers at 120 hz and 180 hz mixed	multi-source separation

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key results

1. sensitivity

algorithm	detectable down to snr
plv	−10 db
itpc	−10 db
welch	−10 db
cpei	0 db
yin	+20 db only

2. octave error

algorithm	octave errors
welch	3 / 5 conditions
rps	0 / 5 conditions
yin	0 / 5 (but produced no estimates at all)

3. drifting pitch

algorithm	result
cpei	detected (cpei = 0.47)
welch	detected
itpc	partial
plv	near noise floor
yin	0 / 497 valid frames

4. noise floor

algorithm	noise floor
cpei	0.040 ± 0.0003
plv	0.006 ± 0.004
itpc	0.089 ± 0.003
welch snr	0.22 ± 0.77 db
yin	no detections

5. two-speaker separation

algorithm	speaker a (120 hz)	speaker b (180 hz)	separates?
cpei	0.158	0.061	yes: asymmetric
plv	0.478	0.464	yes: symmetric
itpc	0.306	0.306	yes: symmetric
welch snr	29.08 db	29.67 db	yes: symmetric
yin	nan	nan	complete failure

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requirements

- matlab r2019b or later
- signal processing toolbox