Multichannel Energy-Based Noisy Segment Rejection

• The algorithm partitions synchronized multichannel data into non-overlapping frames and uses fixed energy thresholds to identify and reject high-noise segments.
• It streamlines downstream processing by excluding corrupted PCG intervals, thereby improving feature extraction with measurable gains in CAD detection accuracy.
• Its deterministic, channel-agnostic design relies solely on computed frame energies, ensuring consistent application across diverse sensor inputs without adaptive criteria.

A multichannel energy-based noisy-segment rejection algorithm is a deterministic, channel-agnostic procedure for discarding segments of time series exhibiting high nonstationary noise, most commonly applied within biomedical signal processing, multichannel sensing, or distributed detection contexts. By partitioning synchronized multichannel data streams into non-overlapping frames, computing per-channel energies, and applying robust, typically non-adaptive energy thresholds, such an approach identifies and removes corrupted segments prior to feature extraction or model training, thereby increasing robustness to transient noise events. The algorithm has proven utility within phonocardiogram (PCG) analysis for coronary artery disease detection, and its mathematical underpinnings admit connections to statistical signal detection in Gaussian noise settings (Marocchi et al., 26 Jan 2026, &&&1&&&).

1. Mathematical Formulation of the Energy-Based Rejection Criterion

Consider a discrete-time multichannel signal $x^{(c)}[n]$ for $n=0,\dots,L-1$ and channel index $c$ (heart microphones HM$_1$–HM$_4$ and one noise reference, NM$_4$). Define frames of length $F=T_f \cdot f_s$ samples, with $T_f$ the duration (s) and $f_s$ the sampling rate; the $i$-th frame comprises samples $s_i = iF$ to $e_i = (i+1)F-1$. The frame energy for channel $c$ is

$E_i^{(c)} = \sum_{n=s_i}^{e_i} [x^{(c)}[n]]^2.$

To mitigate edge effects, exclude the boundary frames ($i=0,N-1$) and determine the channel-wise robust scale via

$$m^{(c)} = \operatorname{median} \{ E_i^{(c)} : i=1,\ldots,N-2 \}.$$

Frame $i$ in channel $c$ is flagged as noisy if

$E_i^{(c)} > \tau \cdot m^{(c)},$

with $\tau=2.5$ selected empirically to balance sensitivity to extraneous transients against natural signal variability.

Let $\mathcal{I}^{(c)}$ be the set of all such noisy-frame intervals. The set of noise-corrupted indices for a given signal is the union

$$\mathcal{I}_{\mathrm{noisy}} = \left(\bigcup_{c \in \{\mathrm{HM}_1,…,\mathrm{HM}_4,\mathrm{NM}_4\}} \mathcal{I}^{(c)}\right) \cup [0, f_s) \cup (L-f_s, L),$$

adding one-second margins at the signal's boundaries. The complement $$\mathcal{I}_{\mathrm{clean}} = [0, L) \setminus \mathcal{I}_{\mathrm{noisy}}$$ is retained for downstream analysis (Marocchi et al., 26 Jan 2026).

2. Algorithmic Workflow and Channel Integration

The full pipeline entails concatenating all sensor recordings for a subject ($L$ samples), splitting each channel into frames (HM: $T^{\mathrm{HM}}_f=2.5$ s, NM: $T^{\mathrm{NM}}_f=0.25$ s), computing and thresholding energies, and marking noisy intervals as described. All flagged intervals, as well as start/end buffer intervals, are removed wholesale from all channels; no inter-channel ratios or adaptive criteria are used. Only samples unflagged in all channels are “clean” and serve as input for spike removal, bandpass filtering (25–450 Hz), $k$-peak normalization, and segmentation into 4 s fragments.

No feature engineering is conducted during rejection—the sole filtering criterion is instantaneous frame energy, rather than spectral or distributional properties. Notably, the algorithm eschews learned, subject-specific, or adaptive thresholds; its empirical $\tau$ is fixed a priori for all training and test subjects (Marocchi et al., 26 Jan 2026).

Channel Integration Table

Channel Type	Frame Duration	Target Noise Rejection
HM (1–4)	2.5 s	Movement/friction
NM$_4$ (reference)	0.25 s	Impulse/external

Longer frames for HM sensors capture low-frequency, sustained interferences, while shorter NM frame lengths address brief impulsive events.

3. Statistical Detection Context and Relations

Multichannel energy-based segment rejection connects directly to the broader statistical theory of multichannel signal detection in Gaussian noise. In the classical model, an observed $K$-channel vector $Y = (Y_1, ..., Y_K)^\top$ could be either pure noise ($H_0$) or contain a signal present in precisely one channel ($H_1$). Tests—such as the maximum posterior probability (MPP) and the optimal Bayes procedures—are constructed using channel-wise (possibly energy-based) statistics and canonical thresholds. In both the flat amplitude prior and channel-symmetric regimes, the rejection statistics reduce to

$$T_{\mathrm{MPP}}(Y) = \max_{j=1,..,K} \pi_j \exp\left(\frac{Y_j^2}{2\sigma^2}\right), \qquad T_{\mathrm{Bayes}}(Y) = \sum_{j=1}^K \pi_j \exp\left(\frac{Y_j^2}{2\sigma^2}\right),$$

with $\pi_j$ channel priors and $\sigma^2$ the noise variance. Segment rejection is effected by comparing $T_{\mathrm{MPP}}$ or $T_{\mathrm{Bayes}}$ to precomputed thresholds $t_\alpha$ for target false-alarm probability $\alpha$, efficiently filtering noise-dominated intervals (Burnaev et al., 2017). The theory provides limiting distributions and non-detectable regions in $\mathbb{R}^K$; for the Bayes test, the non-detectable parallelepiped is strictly contained within the MPP's, demonstrating higher sensitivity to sub-threshold energies.

4. Pseudocode and Computational Implementation

The core implementation proceeds as follows (all details per (Marocchi et al., 26 Jan 2026)):

INPUT: HM channels hm[1..4][0..L-1], NM channel nm4[0..L-1] Sampling rate f_s Frame durations: T_hm=2.5 s, T_nm=0.25 s Threshold τ = 2.5 PROCEDURE: I_noisy ← empty set of sample indices # Process each HM channel for c in 1..4 do F ← round(T_hm * f_s) N ← floor(L / F) for i in 0..N-1 do s ← i * F; e ← (i+1)*F - 1 E[i] ← sum_{n=s}^e hm[c][n]^2 end for m ← median(E[1..N-2]) for i in 1..N-1 do if E[i] > τ * m then mark interval [i*F, (i+1)*F -1] in I_noisy end if end for end for # Process NM channel 4 F ← round(T_nm * f_s) N ← floor(L / F) for i in 0..N-1 do s ← i * F; e ← (i+1)*F -1 E_nm[i] ← sum_{n=s}^e nm4[n]^2 end for m_nm ← median(E_nm[1..N-2]) for i in 1..N-1 do if E_nm[i] > τ * m_nm then mark interval [i*F, (i+1)*F -1] in I_noisy end if end for # Add 1s at boundaries mark [0, f_s-1] and [L-f_s, L-1] in I_noisy # Compute complement => noise-free indices I_clean ← [0..L-1] \ I_noisy OUTPUT: I_clean

Computational cost is linear in the number of frames across channels ($O(KN)$), and does not require spectral feature computation or complex threshold adaptation.

5. Downstream Processing and MFCC-Conformer Integration

Upon segment rejection, only $\mathcal{I}_{\mathrm{clean}}$ are retained. All harmonized channels are spike-removed, bandpass filtered (25–450 Hz), $k$-peak normalized, and segmented into contiguous intervals. Fragments shorter than 4 s are discarded. From the remainder, fixed 4 s segments are extracted with overlapping windows chosen to balance class representation. MFCCs (128 coefficients, computed with STFT window of 512 and hop size of 160) are extracted for every channel and concatenated along the channel axis; these serve as input to a Conformer encoder for CAD detection. Noisy intervals are excluded entirely at the fragment generation stage—they never enter downstream model training or inference (Marocchi et al., 26 Jan 2026).

6. Quantitative Performance Impact

The inclusion of multichannel energy-based noisy-segment rejection yields measurable gains in both fragment- and subject-level performance metrics in noise-robust CAD detection pipelines. On a dataset comprising 297 subjects, the application of the algorithm prior to MFCC-Conformer classification resulted in:

Metric	Noisy	Denoised	Delta
Fragment Accuracy	71.2%	73.9%	+2.7 pp
Fragment UAR	70.9%	73.7%	+2.8 pp
Subject Accuracy	74.3%	78.4%	+4.1 pp
Subject UAR	73.9%	78.2%	+4.3 pp
MCC	0.490	0.570	+0.08

All metrics are 5 fold × 3 run subject-level averages; Denoised refers to pipelines with noisy-segment rejection (Marocchi et al., 26 Jan 2026).

This demonstrates an absolute improvement of approximately 4 percentage points in both accuracy and balanced accuracy at the subject level by excluding high-energy, nonstationary noise-dominated PCG segments.

7. Broader Signal Detection and Theoretical Properties

The energy-based rejection algorithm, in both practical engineering and theoretical statistical settings, demonstrates robust adaptation to nonstationary, transient noise without sacrificing sensitivity to physiological variability. By referencing the multichannel statistical detection literature, especially frameworks encompassing the MPP and Bayes tests, the mathematical properties of energy rejection—including limiting distributions of test statistics and explicit characterization of non-detectable regions—can inform principled design. For example, the Bayes test's non-detectable parallelepiped is strictly smaller than that for the MPP, independent of $\alpha$, and reflects stronger detection power for low-SNR events (Burnaev et al., 2017). A plausible implication is that extensions of the current empirical approach could leverage channel priors and formal noise models for even finer-grained rejection or confidence calibration in high-noise regimes.