Envelope Spectrum

The Envelope Spectrum analysis extracts amplitude modulation patterns from vibration signals using the Hilbert transform. It is the primary tool for detecting localized defects in rolling element bearings, where repetitive impacts produce characteristic modulation frequencies that are often buried beneath higher-energy structural vibration.

Theory

Hilbert Transform Demodulation

The envelope detection process consists of three stages:

Band-pass filtering — isolate the resonance band excited by the bearing impacts (typically 1–20 kHz). The optimal band can be identified from a kurtogram or spectral kurtosis map.
Analytic signal — compute the analytic signal via the Hilbert transform:
z(t) = x(t) + j · H{x(t)}
where H{·} denotes the Hilbert transform.
Envelope extraction — the instantaneous amplitude (envelope) is the modulus of the analytic signal: A(t) = |z(t)|. An FFT of this envelope reveals the repetition frequencies of the impacts.

Bearing Fault Frequencies

Localized bearing defects generate impacts at characteristic frequencies determined by the bearing geometry and shaft speed. The four fundamental fault frequencies are:

Abbreviation	Full Name	Formula	Defect Location
BPFO	Ball Pass Frequency, Outer Race	BPFO = (n/2) · f_r · (1 − d/D · cosα)	Outer race
BPFI	Ball Pass Frequency, Inner Race	BPFI = (n/2) · f_r · (1 + d/D · cosα)	Inner race
BSF	Ball Spin Frequency	BSF = (D/2d) · f_r · (1 − (d/D · cosα)²)	Rolling element
FTF	Fundamental Train Frequency	FTF = (f_r/2) · (1 − d/D · cosα)	Cage

where n = number of rolling elements, f_r = shaft rotation frequency, d = ball diameter, D = pitch diameter, and α = contact angle.

Bearing manufacturers typically publish these ratios normalized to shaft speed. Enter the bearing model number in the V-Listen bearing database to auto-populate the frequency multipliers.

Parameters

Parameter	Range / Options	Description
Band-pass Low	100 Hz – f_s/2	Lower cutoff of the demodulation band-pass filter.
Band-pass High	100 Hz – f_s/2	Upper cutoff. Must be greater than Band-pass Low.
Filter Order	4 / 6 / 8 / 10	Butterworth filter order. Higher orders give sharper roll-off.
Spectrum Size	1024 – 65536	FFT block size for the envelope spectrum.
Window	Hanning / Rectangular	Window function applied before the envelope FFT.
Averaging	1 – unlimited	Number of envelope spectrum blocks to average.
Shaft Speed	Manual / Tacho	Source of rotational speed for cursor overlay.

Interpreting the Envelope Spectrum

Outer Race Fault

Peaks at BPFO and its harmonics (2×BPFO, 3×BPFO, ...) indicate an outer race defect. Because the outer race is stationary, the load zone is fixed and the amplitude modulation is relatively stable, producing clean spectral lines.

Inner Race Fault

Peaks at BPFI with sidebands spaced at f_r (shaft speed). The sidebands arise because the defect on the rotating inner race passes in and out of the load zone once per revolution, causing amplitude modulation of the BPFI impacts.

Rolling Element Fault

Peaks at 2×BSF (each ball contacts both races per spin) with cage frequency (FTF) sidebands. Rolling element defects are harder to detect because the random orientation of the defect as the ball rotates produces irregular modulation.

Cage Fault

Peaks at FTF and harmonics. Cage defects are the rarest and often produce sub-synchronous spectral lines with broadband noise elevation.

The envelope spectrum is only effective after band-pass filtering. Applying it to the raw broadband signal will typically show only shaft speed harmonics and structural resonances, masking the bearing fault frequencies.

Workflow in V-Listen

Load a time-domain vibration recording (accelerometer signal).
Open Analysis → Envelope Spectrum.
Set the band-pass range around a structural resonance excited by the bearing impacts. Use the kurtogram or spectral kurtosis to identify the optimal band.
Enter the shaft speed or assign a tacho channel.
Click Calculate. Fault frequency cursors are overlaid automatically when bearing geometry is provided.
Inspect peaks at BPFO, BPFI, BSF, FTF and their harmonics.

Practical Tips

Mount accelerometers as close to the bearing housing as possible to maximize the signal-to-noise ratio of impact responses.
Use a sampling rate of at least 25.6 kHz to capture the high-frequency resonance bands excited by bearing defects.
For slow-speed bearings (< 100 RPM), increase the spectrum size to obtain sufficient frequency resolution at very low modulation frequencies.
Trend the envelope spectrum over time to track defect progression from incipient fault to functional failure.