Modified Median Frequency (MMDF) for EMG Signals

• MMDF is a spectral feature extraction method that computes the median frequency using the amplitude spectrum, reducing noise bias in EMG signals.
• The method modifies conventional median frequency calculations by balancing cumulative amplitude sums, ensuring robust performance under additive white Gaussian noise.
• Empirical results demonstrate that MMDF consistently yields lower percentage errors compared to traditional power-based methods, enhancing pattern recognition in noisy conditions.

The Modified Median Frequency (MMDF) is a spectral feature extraction method designed for robust pattern recognition in electromyography (EMG) signals, with particular resilience against additive White Gaussian Noise (WGN). Introduced by Phinyomark et al. (2009), MMDF modifies the standard median frequency calculation by operating on the amplitude spectrum of an EMG segment rather than its power spectrum. This modification stems from the observation that noise-induced bias is reduced in the amplitude spectrum, resulting in more stable feature values for EMG-based recognition tasks under high-noise conditions (0912.3973).

1. Mathematical Formulation

Given a windowed EMG segment $x[n]$ of length $N$, sampled at rate $f_s$ Hz, its discrete Fourier transform (DFT) is

$$X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j\,2\pi\,nk/N}, \quad k=0,\ldots,N-1,$$

and its one-sided amplitude spectrum is given by

$$A[k] = | X[k] |, \quad k = 0, 1, ..., K, \quad K = \lfloor N/2 \rfloor.$$

The Modified Median Frequency $f_\mathrm{MMDF}$ is defined as the frequency bin $k^*$ such that the cumulative sum of amplitudes up to $k^*$ equals the sum from $k^*$ to $K$:

$N$0

In continuous form,

$N$1

2. Computation Pipeline

The calculation of MMDF from raw EMG involves the following systematic steps:

• Acquisition & Preprocessing
  • EMG is recorded at 1 kHz, band-pass filtered to 10–500 Hz, with a gain of approximately 60 dB.
  • Signals are segmented into 256 ms (i.e., $N$2) non-overlapping windows, sliding every 64 ms (75% overlap).
• Spectral Estimation
  • For each window $N$3, compute the DFT $N$4 via the FFT algorithm.
  • Obtain the one-sided amplitude spectrum $N$5 for $N$6 to $N$7.
• Cumulative-sum Search
  • Compute the total amplitude $N$8.
  • Form partial sums $N$9.
  • Find the smallest $f_s$0 for which $f_s$1.
  • Calculate the corresponding frequency as $f_s$2.

This stepwise procedure enables batch extraction of MMDF features for subsequent statistical analysis or classification tasks.

3. Comparison with Standard Median Frequency

Standard Median Frequency (MDF) is defined over the power spectrum $f_s$3:

$f_s$4

The key modification in MMDF is the use of amplitude spectrum ($f_s$5) in place of power spectrum ($f_s$6). This adjustment is motivated by the observation that WGN, when squared in the computation of $f_s$7, introduces greater variance and bias, adversely affecting the MDF estimate. In contrast, the amplitude spectrum’s smaller dynamic range mitigates noise-induced shifts, resulting in a more robust median-frequency measure under noise contamination (0912.3973).

4. Noise Protocols and Error Metrics

Robustness of MMDF is quantitatively evaluated under controlled noise protocols:

• Additive White Gaussian Noise (WGN) is generated and added to EMG such that the signal-to-noise ratio (SNR) is given by

$f_s$8

with SNR levels spanning 20, 15, 10, 5, and 0 dB.

• Signal Strength Conditions:
  • Strong channels (e.g., extensor carpi radialis longus during hand-close/flexion).
  • Weak channels (all other channels/motions).
• Error Metric: For each spectral feature $f_s$9, the relative error under noise is assessed as

$$X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j\,2\pi\,nk/N}, \quad k=0,\ldots,N-1,$$0

Multiple windows (80 total per test condition) and noise realizations are averaged for statistical stability.

5. Empirical Robustness: Quantitative Results

Phinyomark et al. (2009) present percentage error (PE) comparisons for MMDF, MMNF, and competitors at several SNR levels:

SNR (dB)	MMNF (strong)	MMDF (strong)	MMNF (weak)	MMDF (weak)
20	0.4%	0.5%	0.4%	0.6%
10	2.5%	3.0%	4.0%	5.5%
5	3.5%	4.5%	5.0%	7.0%
0	6.0%	6.5%	10.0%	12.5%

Both MMNF and MMDF outperform the standard MDF (PE $$X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j\,2\pi\,nk/N}, \quad k=0,\ldots,N-1,$$120% or more at SNR = 0 dB) and most time-domain features (typically >20–30% error at high noise). MMDF consistently displays higher robustness in noisy conditions compared to power-based metrics, with a slight but consistent margin in favor of MMNF (0912.3973).

6. Classification Performance and Feature Integration

While only MMNF was directly incorporated into multi-feature classification vectors in published experiments, MMDF's comparable robustness permits its substitution or inclusion with similar expected impact. Representative seven-class LDA recognition results (using 256 ms windows, 64 ms steps, and majority-vote post-processing) are as follows:

Features	Clean	20 dB	15 dB	10 dB
HEMG	60.8	49.2	41.8	34.7
WL	79.3	34.2	14.4	12.5
WAMP (10 mV)	86.6	92.3	47.0	21.6
MMNF	41.1	36.4	32.7	17.1
{MAV, WL, ZC, SSC}	95.7	67.4	22.6	8.0
{RMS, AR$$X[k] = \sum_{n=0}^{N-1} x[n]\,e^{-j\,2\pi\,nk/N}, \quad k=0,\ldots,N-1,$$2}	96.5	89.9	64.9	25.3
{HEMG, WAMP, MMNF}	93.1	96.2	64.1	28.1

Under clean conditions, overall accuracy with the best feature set approaches 93.1%; at 15 dB SNR, accuracy remains near 64.1%. Inclusion of MMDF in such vectors is expected to yield similar classification performance, given the structural similarity of the metric (0912.3973).

7. Limitations and Prospective Extensions

MMDF and its related amplitude-domain metric MMNF present several limitations:

• Both rely on direct FFT-based amplitude estimates—spectral leakage and small-window effects can bias the median frequency.
• Only rectangular windowing was evaluated; Hamming or multitaper PSD approaches may further attenuate noise bias.
• Experiments were limited to LDA-based classifiers; broader integration with nonlinear classifiers (e.g., SVM, neural networks) and high-dimensional time–frequency fusion (e.g., wavelet-based features) is suggested.
• Prospective research directions include application to dynamic gestures, adaptive window sizing, and direct comparison to emerging subspace-projection or deep-learning–derived spectral features.

A plausible implication is that further methodological refinement—particularly in windowing, time–frequency resolution, and classifier integration—could optimize MMDF’s practicality for diverse, noisy EMG analysis scenarios (0912.3973).