Wavelet Analysis

Wavelet analysis provides a time-frequency representation of signals with adaptive resolution: fine frequency resolution at low frequencies and fine time resolution at high frequencies. This makes it superior to the Short-Time Fourier Transform (STFT) for analyzing transient events, non-stationary machinery signals, and impulsive phenomena such as bearing impacts and gear tooth cracks.

Theory

Continuous Wavelet Transform (CWT)

The CWT correlates the input signal with scaled and shifted versions of a mother wavelet ψ(t):

W(a, b) = (1/√a) · ∫ x(t) · ψ*((t − b)/a) dt

where a is the scale parameter (inversely related to frequency), b is the time shift, and ψ* is the complex conjugate of the mother wavelet. The result is a two-dimensional map of wavelet coefficients showing how the signal's frequency content evolves over time.

Time-Frequency Resolution

Unlike the STFT, which uses a fixed window size, the CWT automatically adapts its resolution:

Property	STFT	CWT
Window size	Fixed for all frequencies	Adapts with scale (frequency)
Low-frequency resolution	Limited by window length	Excellent (long effective window)
High-frequency time localization	Limited by window length	Excellent (short effective window)
Heisenberg constraint	Δt · Δf = const	Δt · Δf = const (but optimally distributed)
Best suited for	Stationary or slowly varying signals	Transients, impacts, non-stationary signals

The Heisenberg uncertainty principle prevents simultaneous perfect resolution in both time and frequency. The wavelet transform distributes this uncertainty optimally for most machinery diagnostics applications.

Mother Wavelets

V-Listen supports the following mother wavelets, each with different characteristics:

Wavelet	Type	Properties	Typical Use
Morlet	Complex, continuous	Gaussian-windowed sinusoid. Excellent frequency resolution. Adjustable bandwidth via parameter ω₀ (default 6).	General-purpose time-frequency analysis, vibration monitoring
Daubechies (db4–db20)	Real, orthogonal	Compact support. Higher order = smoother wavelet, longer support. Orthogonality enables energy-preserving decomposition.	Multiresolution analysis, denoising, feature extraction
Symlet (sym4–sym20)	Real, near-symmetric	Modified Daubechies with improved symmetry. Reduced phase distortion.	Signal reconstruction, minimal-distortion filtering
Mexican Hat	Real, continuous	Second derivative of Gaussian. Good time localization, moderate frequency resolution.	Edge and singularity detection, impact identification

Parameters

Parameter	Range / Options	Description
Mother Wavelet	Morlet / Daubechies / Symlet / Mexican Hat	Select the analyzing wavelet.
Wavelet Order	1 – 20 (for Daubechies/Symlet)	Controls smoothness and support length.
Frequency Range	Auto / Manual (Hz)	Minimum and maximum analysis frequencies.
Voices per Octave	4 / 8 / 16 / 32	Number of frequency steps per octave. Higher values give finer frequency resolution at the cost of computation time.
Time Range	Full / Selection	Analyze the entire recording or a selected region.
Display Scale	Linear / Log / dB	Color map amplitude scaling for the scalogram.

Output: The Scalogram

The primary output is a scalogram — a color-coded map with time on the horizontal axis and frequency (or scale) on the vertical axis. The color intensity represents the magnitude of the wavelet coefficients at each time-frequency point.

Scalogram vs Spectrogram

The spectrogram (STFT) uses uniform frequency bins and uniform time steps.
The scalogram (CWT) uses logarithmically spaced frequency bins and time resolution that varies with frequency.
At low frequencies the scalogram shows better frequency detail; at high frequencies it shows better time detail. This matches the physics of most mechanical systems.

Applications in Machinery Diagnostics

Bearing Fault Transients

Impulsive events from bearing defects excite structural resonances as short bursts of high-frequency energy. The CWT localizes these bursts precisely in time while simultaneously identifying their frequency content. This is especially valuable for:

Confirming the repetition period of suspected bearing fault impacts
Separating bearing impacts from other impulsive sources (gear meshing, electrical noise)
Detecting early-stage faults when impacts are too weak for envelope analysis

Gear Tooth Cracks

A cracked gear tooth produces a brief transient disturbance once per revolution. The CWT reveals this as a localized region of elevated coefficients at the mesh frequency band, recurring at the shaft rotation period.

Run-up and Coast-down Analysis

During speed transients, resonances are excited briefly as the operating speed sweeps through them. The CWT's adaptive resolution captures these transient resonance responses more accurately than the fixed-resolution STFT.

CWT computation is significantly more expensive than FFT. For long recordings at high sampling rates, consider analyzing selected time segments rather than the entire file. A 60-second recording at 51.2 kHz with 32 voices per octave may require several minutes to process.

Workflow in V-Listen

Load a time-domain recording.
Open Analysis → Wavelet (CWT).
Select the mother wavelet (Morlet is recommended as a starting point).
Set the frequency range and voices per octave.
Click Calculate.
Examine the scalogram for transient events, time-varying frequency content, and amplitude modulation patterns.

Practical Tips

Start with the Morlet wavelet and ω₀ = 6 for a good balance between time and frequency resolution.
Use Daubechies wavelets when you need multiresolution decomposition for denoising or sub-band energy extraction.
Increase voices per octave (16 or 32) when precise frequency identification is needed; reduce to 4 or 8 for fast overview analysis.
Combine wavelet analysis with envelope spectrum: use the scalogram to identify the optimal demodulation band, then apply envelope analysis for fault frequency identification.