Hilbert Envelope

The Hilbert Envelope analysis extracts the instantaneous amplitude, frequency, and phase of a signal using the analytic signal concept. It is the foundational technique for envelope-based diagnostics, demodulation, and characterization of amplitude- and frequency-modulated signals.

Analytic Signal

The analytic signal z(t) is a complex-valued extension of the real signal x(t), formed by adding its Hilbert transform as the imaginary part:

z(t) = x(t) + j · H{x(t)}

The Hilbert transform H{x(t)} is defined as:

H{x(t)} = (1/π) P.V. ∫_−∞^∞ x(τ) / (t − τ) dτ

In practice, the Hilbert transform is computed efficiently in the frequency domain by zeroing the negative-frequency components of the FFT and taking the inverse transform. This produces the one-sided analytic signal whose spectrum contains only positive frequencies.

Instantaneous Parameters

Instantaneous Amplitude (Envelope)

a(t) = |z(t)| = √(x²(t) + H{x(t)}²)

The envelope traces the peak amplitude of the signal as a function of time, removing the carrier oscillations. It is the primary output for bearing fault detection, where periodic impacts produce characteristic envelope patterns.

Instantaneous Phase

φ(t) = arctan(H{x(t)} / x(t))

The instantaneous phase represents the total accumulated phase of the signal. Phase unwrapping is applied to remove 2π discontinuities, producing a continuous monotonically increasing function for a signal with positive frequency content.

Instantaneous Frequency

f(t) = (1/2π) · dφ(t)/dt

The time derivative of the unwrapped instantaneous phase yields the instantaneous frequency. For a pure sinusoid, f(t) is constant. For a frequency-modulated signal, f(t) varies according to the modulation pattern, enabling direct tracking of speed fluctuations and FM content.

Relationship to Envelope Spectrum

The envelope spectrum is the FFT of the instantaneous amplitude a(t):

E(f) = FFT{ a(t) }

This two-stage process (Hilbert envelope extraction followed by spectral analysis) is the standard approach for bearing diagnostics. The envelope spectrum reveals the repetition rates of impulsive events that are buried in broadband vibration, making it far more sensitive than direct spectral analysis for early fault detection.

The Hilbert transform assumes the signal is narrowband (mono-component). For broadband signals, bandpass filter to the frequency range of interest before applying the Hilbert transform. Without filtering, the envelope may contain artifacts from spectral components beating against each other.

Parameters

Parameter	Options	Description
Input Channel	Any signal channel	Signal for envelope extraction.
Bandpass Filter	Center + Bandwidth, or f_low–f_high	Isolates the frequency range before Hilbert transform. Critical for meaningful results.
Output	Envelope, Phase, Inst. Frequency	Which instantaneous parameter to display.
Envelope FFT Size	256 – 65536	Block size for the envelope spectrum computation.
DC Removal	On / Off	Removes the mean from the envelope before spectral analysis.

Applications

Application	Carrier Band	What to Look For
Rolling element bearing faults	Resonance band excited by impacts (typically 2–10 kHz)	Envelope spectrum peaks at BPFO, BPFI, BSF, FTF and their harmonics
Gear fault detection	Gear mesh frequency band	Modulation sidebands at shaft speed in the envelope spectrum
Electrical motor diagnostics	Stator slot frequency band	Envelope peaks at slip frequency, 2× line frequency
Acoustic emission analysis	High-frequency band (> 50 kHz)	Burst patterns correlated with mechanical events

Practical Tips

The bandpass filter selection is the most critical step. Use a spectral kurtosis analysis or manual spectrum inspection to identify the band with the highest impulsiveness before applying the Hilbert transform.
Always remove the DC component from the envelope before computing the envelope spectrum; otherwise, the DC peak will dominate and obscure low-frequency modulation components.
For very early fault detection, combine the envelope spectrum with kurtosis trending. A rising kurtosis alerts you to increasing impulsiveness even before clear spectral peaks emerge.
Compare the instantaneous frequency trace with the tachometer signal to verify speed tracking accuracy and detect torsional vibration.

The Hilbert transform produces edge effects at the beginning and end of finite-length signals. Discard the first and last few milliseconds of the envelope when analyzing short data segments to avoid artifacts.