Features for Signals and Machines

Why features

A vibration sensor at 4 kHz delivers 4 000 numbers a second, almost all of them redundant. A model fed raw samples needs enormous amounts of data to find the pattern; a model fed a handful of features (numbers that summarize a short stretch of signal and carry the physics) often works with a few hundred examples and a simple, explainable method. Good features are usually worth more than a better algorithm (see Trees, Clustering and Anomaly Detection), and they let you use what you know about the machine.

Signal	Typical features	What they reveal
Vibration	RMS, peak, crest factor, kurtosis, band energies, order amplitudes	Imbalance, misalignment, looseness, bearing and gear faults
Motor current	RMS, harmonics, sidebands around the supply frequency, power factor	Broken rotor bars, eccentricity, load changes, jams
Sound	Spectrum bands, spectral centroid, MFCCs	Leaks, cavitation, wear, a changed running sound
Temperature	Level, rate of rise, difference to ambient or to a neighbor	Friction, blocked cooling, an electrical fault
Pressure, flow, torque	Mean, ripple, slope, stroke time	Valve wear, pump faults, leaks
Process data	Ratios, energy per unit, deviation from set-point	Yield and quality, efficiency loss

Windows

Cut the stream into windows and compute features per window. The window must be long enough to hold several cycles of the slowest thing you care about, and short enough that conditions are steady within it. Overlapping windows (50 %) give more rows but nearly duplicate ones; split by stretch of time, not by window (see Data Preparation, Splitting and Leakage).

Time-domain features

Feature	Formula	Meaning
Mean	x̄ = Σ x ÷ N	The offset; remove it first
RMS	√(Σ x² ÷ N)	The overall energy: a steady measure of how much it shakes
Peak	max |x|	The largest excursion; sensitive to single spikes
Crest factor	peak ÷ RMS	Impulsiveness: 1.41 for a sine, higher with impacts
Kurtosis	Σ (x − x̄)⁴ ÷ N ÷ σ⁴	3 for noise; rises sharply with sparse impacts
Skewness	Σ (x − x̄)³ ÷ N ÷ σ³	Asymmetry: one-sided impacts or clipping
Zero-crossing rate	sign changes per second	A crude frequency measure

Frequency-domain features

The spectrum (from an FFT of a windowed segment) splits the signal into frequencies, where faults live at predictable places: shaft speed and its harmonics (imbalance, misalignment), gear-mesh frequency (gears), bearing frequencies (below), and high-frequency resonances (impacts). Use band energies: sum the squared amplitudes in chosen frequency bands.

• Always window the data (Hann) before the FFT, or energy leaks across the spectrum; average several spectra to cut the scatter.
• Use fixed bands tied to the machine (multiples of shaft speed), not fixed hertz, if speed varies (order analysis).
• A spectrogram (spectrum against time) is a picture: it can be fed to a convolutional network (see Neural Networks and Deep Learning).
• Wavelets and spectral kurtosis pick the best band automatically for impulsive faults.

Bearing and gear frequencies

A rolling bearing defect produces an impact each time a ball passes it. The repetition rate depends only on the geometry and the shaft speed. With n balls of diameter B_d, pitch diameter P_d, contact angle θ and shaft frequency f_r:

BPFO = (n ÷ 2) (1 − B_d/P_d cos θ) f_r BPFI = (n ÷ 2) (1 + B_d/P_d cos θ) f_r BSF = (P_d ÷ 2B_d) (1 − (B_d/P_d cos θ)²) f_r FTF = ½ (1 − B_d/P_d cos θ) f_r

Envelope analysis

The impacts are weak against the machine’s own vibration and ring at a high, unrelated frequency, so the bearing frequency does not appear directly in the ordinary spectrum. Envelope analysis recovers it: band-pass (or high-pass) filter around the ringing, take the envelope (rectify and smooth, or the Hilbert transform), then compute the spectrum of the envelope. The repetition rate of the impacts shows as a line.

Operating conditions

Normal changes in speed and load move every feature. A feature that rises with load is not a fault. Either include the operating conditions as inputs (load, speed, temperature, product) or compare like with like (only windows at the same load).

• Order tracking expresses frequency as multiples of shaft speed, so fault lines stay in the same place when speed changes.
• Record the conditions with every window (see Planning a Logging System): you cannot add context to data afterwards.
• Cover the range in training: a model that has only seen 80 % load has no idea what 40 % looks like.

Trends and remaining life

A single window says how the machine is now; the trend of a feature over days and weeks says how it is changing, and often matters more. Wear usually accelerates, so a feature that is flat for months and then climbs gives a warning.

Current, sound, temperature and images

• Motor current signature analysis (MCSA): spectrum of one phase current; broken rotor bars and eccentricity add sidebands at ± slip frequency around the supply line (see Motion Control and Actuators). It needs no extra sensor on a drive that already measures current.
• Sound: a microphone near the machine; features from the spectrum (band energies, spectral centroid) or mel-frequency cepstral coefficients (MFCCs). It works for leaks and cavitation but is sensitive to background noise.
• Temperature: use the rate of rise and the difference to ambient or a neighbor, not the absolute level.
• Images: classic features (edges, areas, color) or a convolutional network on the picture itself (see Neural Networks and Deep Learning and Robotics Tools).
• Sensors and mounting matter: an accelerometer must be rigidly stud-mounted near the bearing; a loose magnet changes the response above a few kilohertz (see Sensors and Field Devices).

A feature pipeline

import numpy as np
from scipy.signal import welch, hilbert, butter, sosfiltfilt
from scipy.stats import kurtosis

def features(x, fs, band=(700, 1100)):
    x = x - x.mean()
    rms = np.sqrt(np.mean(x ** 2))
    f, p = welch(x, fs=fs, nperseg=len(x) // 2, window="hann")            # power spectrum
    in_band = p[(f >= band[0]) & (f < band[1])].sum() / p.sum()           # fraction of energy in the fault band
    sos = butter(4, [band[0], band[1]], btype="band", fs=fs, output="sos")
    env = np.abs(hilbert(sosfiltfilt(sos, x)))                            # envelope of the ringing
    env = env - env.mean()
    fe = np.fft.rfftfreq(len(env), 1 / fs); pe = np.abs(np.fft.rfft(env * np.hanning(len(env))))
    sel = (fe > 30) & (fe < 300)
    return {"rms": rms, "crest": np.max(np.abs(x)) / rms, "kurt": kurtosis(x, fisher=False),
            "band_frac": in_band, "env_peak_hz": fe[sel][np.argmax(pe[sel])], "env_peak_amp": pe[sel].max()}

• Use exactly the same function in training and in service (and on the microcontroller, if the model runs there; see Deployment, Monitoring and Limits).
• Log the raw windows as well as the features, so features can be redesigned later without new data (see Logging Strategies).
• Validate on a different machine before trusting a feature: a feature that separates faults on one machine may be a property of that machine.

Related

• Machine Learning: the overview of this guide.
• Trees, Clustering and Anomaly Detection: models to feed these features to.
• Deployment, Monitoring and Limits: computing features on a microcontroller.
• Data Logging: sampling, filtering and timing the signals properly.
• Numerical Methods: FFT, filters and windows in code.

FAQ

Which features should I start with?

RMS, crest factor, kurtosis and the energy in a few frequency bands tied to the machine (shaft speed, gear mesh, bearing frequencies), plus the operating conditions. Add an envelope spectrum for bearings.

Why does my model fail when the speed changes?

The features move with speed and load. Include the conditions as inputs, compute features per order of shaft speed, or compare residuals from a healthy baseline at the same operating point.

How fast must I sample vibration?

At least twice the highest frequency of interest, and about 2.5 times in practice, with an anti-alias filter. Bearing resonances can be several kilohertz.

Can a neural network learn the features itself?

Yes, given enough labeled data, for example a convolutional network on spectrograms. With small data, hand-made, physically meaningful features and a simple model are safer and easier to explain.

References

• Randall, R. B., Vibration-based Condition Monitoring, Wiley; Harris, T. A. and Kotzalas, M. N., Rolling Bearing Analysis, CRC Press.
• ISO 20816 (mechanical vibration: measurement and evaluation of machine vibration), ISO 13373 (condition monitoring and diagnostics of machines: vibration).
• Smith, W. A. and Randall, R. B., “Rolling element bearing diagnostics using the Case Western Reserve University data”, Mechanical Systems and Signal Processing, 64–65 (2015).

Last reviewed: October 2026. The signals on this page are synthetic and the thresholds illustrative; use the manufacturer’s bearing data and the standards for real limits.