---
title: Multichannel Energy-Based Noisy Segment Rejection
url: https://www.emergentmind.com/topics/multichannel-energy-based-noisy-segment-rejection-algorithm
type: topic
---

# Multichannel Energy-Based Noisy Segment Rejection

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 [2601.18295], [1712.06335].

## 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 [2601.18295].

## 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 [2601.18295].

**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 [1712.06335]. 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 [2601.18295]):

```text
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 [2601.18295].

## 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 [2601.18295].*

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 [1712.06335]. 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.

Source: https://www.emergentmind.com/topics/multichannel-energy-based-noisy-segment-rejection-algorithm