Global EMG Parameter Documentation

This section documents the computation of global HD-sEMG parameters, their theoretical background, scientific relevance, and Python implementation.

Amplitude-Based Features

1. Root Mean Square (RMS)

Description:

  • Quantifies the signal’s energy content.
  • Sensitive to muscle activation level.

Formula: $\text{RMS} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} x_i^2}$

Python Implementation:

import numpy as np

def root_mean_square(signal: np.ndarray) -> float:
    if signal.ndim > 1:
        raise ValueError("Signal must be a 1D array.")
    return np.sqrt(np.mean(signal ** 2))

References:

  • Merletti & Farina (2016), Surface EMG: Physiology, Engineering and Applications
  • Clancy et al. (2023), CEDE recommendations

2. Average Rectified Value (ARV)

Description:

  • Linear estimator of the envelope.
  • Mean absolute value of the signal.

Formula: $\text{ARV} = \frac{1}{N} \sum_{i=1}^{N} |x_i|$

Python Implementation:

import numpy as np

def average_rectified_value(signal: np.ndarray) -> float:
    if signal.ndim > 1:
        raise ValueError("Signal must be a 1D array.")
    return np.mean(np.abs(signal))

References:

  • CEDE Amplitude Normalization Matrix (Dideriksen et al., 2023)
  • Clancy et al. (2023)

3. Integrated EMG (IEMG)

Description:

  • Total electrical activity over a time window.

Formula: $\text{IEMG} = \sum_{i=1}^{N} |x_i|$

Python Implementation:

import numpy as np

def integrated_emg(signal: np.ndarray) -> float:
    if signal.ndim > 1:
        raise ValueError("Signal must be a 1D array.")
    return np.sum(np.abs(signal))

References:

  • Merletti & Farina (2016)
  • CEDE Amplitude Matrix

Frequency-Based Features

4. Median Frequency (MDF)

Description:

  • Frequency dividing the power spectrum into equal halves.
  • Related to muscle fatigue.

Python Implementation:

import numpy as np
from scipy.signal import welch

def compute_mdf(signal: np.ndarray, fs: float) -> float:
    f, pxx = welch(signal, fs=fs)
    cumsum = np.cumsum(pxx)
    mdf_idx = np.searchsorted(cumsum, cumsum[-1] / 2)
    return f[mdf_idx]

References:

  • Farina & Merletti (2003), Biomedical Engineering
  • CEDE Force Estimation Matrix

5. Mean Frequency (MNF)

Description:

  • Center of gravity of the EMG power spectrum.
  • Sensitive to fatigue and conduction velocity.

Python Implementation:

import numpy as np
from scipy.signal import welch

def compute_mnf(signal: np.ndarray, fs: float) -> float:
    f, pxx = welch(signal, fs=fs)
    return np.sum(f * pxx) / np.sum(pxx)

References:

  • Phinyomark et al. (2012)
  • CEDE (Farina et al., 2023)

Complexity-Based Feature

6. Permutation Entropy

Description:

  • Nonlinear metric for signal complexity.
  • Used in motor control and fatigue studies.

Python Implementation:

from antropy import perm_entropy
import numpy as np

def compute_entropy(signal: np.ndarray) -> float:
    return perm_entropy(signal, normalize=True)

References:

  • Bandt & Pompe (2002)
  • CEDE Motor Unit Matrix (Martinez-Valdes et al., 2023)

Envelope Estimation

7. EMG Envelope (ARV, RMS, Median)

Description:

  • Smoothed versions of ARV/RMS/Median using low-pass filtering.

Python Implementation:

import numpy as np
from scipy.signal import butter, filtfilt

def bandpass_filter(signal, fs, lowcut, highcut, order=4):
    b, a = butter(order, [lowcut, highcut], btype='band', fs=fs)
    return filtfilt(b, a, signal)

def lowpass_filter(signal, fs, cutoff, order=4):
    b, a = butter(order, cutoff, btype='low', fs=fs)
    return filtfilt(b, a, signal)

def compute_emg_envelopes(emg: np.ndarray, fs: float, bpf: dict, lpf: dict):
    emg_filtered = np.array([
        bandpass_filter(ch, fs, bpf['fcl'], bpf['fch'], bpf['N']) for ch in emg
    ])
    arv = np.nanmean(np.abs(emg_filtered), axis=0)
    rms = np.sqrt(np.nanmean(emg_filtered ** 2, axis=0))
    mdn = np.nanmedian(np.abs(emg_filtered), axis=0)
    arv_env = lowpass_filter(arv, fs, lpf['fc'], lpf['N'])
    rms_env = lowpass_filter(rms, fs, lpf['fc'], lpf['N'])
    mdn_env = lowpass_filter(mdn, fs, lpf['fc'], lpf['N'])
    return arv_env, rms_env, mdn_env

References:

  • Merletti & Farina (2016)
  • CEDE Best Practice: Amplitude Matrix