Wavelet Time-Frequency Analysis

Layout

From top to bottom: the time waveform of the current analysis frame in engineering units; a 3D waterfall on a black background (time × frequency × amplitude, rendered with hidden lines for a real 3D impression); and a false-color 2D map with a nine-step color scale. The parameter panel on the left covers set-up, plotting, reading and export on one screen.

Parameters

Interaction

• 3D waterfall: drag to rotate (yaw and pitch), right-drag to pan, zoom buttons and one-step reset to the isometric view, double-click to maximize. Three cursor lines for time, frequency and amplitude show the 3D coordinates.
• 2D color map: click for crossed cursor lines that stay visible and follow the mouse; color is mapped to amplitude with a nine-step scale.
• Rendering: while dragging the plot is redrawn at reduced resolution and at full resolution on release; computation and display are cached separately, so interaction does not repeat the calculation.

Export

• Copy the 3D image to the clipboard for Word or presentation reports.
• Export the coefficients as CSV: rows are the scale center frequencies (Hz), columns are time (s, reduced to at most 1024 columns); the file opens directly in Excel.
• Built-in help, stay on top and double-click to maximize.

An FFT tells you which frequencies a signal contains; a wavelet transform tells you when they occur and how long they last. Non-stationary signals, whose frequency changes with time or whose components appear with events, are its main field.

Early bearing and gear faults

Spalling in a rolling bearing produces periodic impacts; gear pitting produces sideband modulation of the mesh frequency. Their energy is spread out and often buried in an FFT spectrum. In the time-frequency plane they form clear patterns: an impact lights up a wide band at one instant (a vertical streak), and modulation brightens and dims periodically along the mesh frequency (a horizontal band).

Run-up, run-down and non-stationary processes

During variable-speed start-up, shutdown and passage through critical speeds, characteristic frequencies move with speed. One FFT frame smears the whole process; the wavelet waterfall draws the frequency path against time, so critical resonance and beating regions are clear.

Weak transient features

The color map helps locate weak transient components in time and frequency. Visibility depends on signal-to-noise ratio, wavelet settings and display range. The source gives inconsistent gain-normalization descriptions, so absolute ridge amplitude and a fixed weak-signal detection range require confirmation.

Acoustic transients and impacts

For abnormal noises, knocks and bangs, the time waveform does not show the frequency content and the FFT does not show the timing. The color map shows when the sound occurs and in which band on one chart.

Training and algorithm verification

Test mode generates a signal with known components (three stepped sine segments, a decaying impact and noise), so the result can be judged at a glance.

The processing has three steps: build a family of Morlet frequency-domain filters, multiply with the FFT, and take the magnitude of the inverse FFT.

Why wavelets

An FFT decomposes the whole frame into global sine functions with a constant resolution Δf = Fs/N and no information about when something happens. A short-time FFT uses a sliding window as a compromise, but with only one window width: a short window blurs frequency, a long one blurs time. Wavelets are scalable, shiftable wave packets — narrow at high frequencies for precise timing and wide at low frequencies for fine frequency resolution — giving an optimal set of compromises under the Δt·Δf uncertainty principle.

Complex Morlet wavelet and scale-to-frequency mapping

ψ(t) = exp(−t² / 2) · exp(jω₀t) f = ω₀ / (2π · s)

The Morlet wavelet is a complex sine in a Gaussian envelope and is Gaussian in the frequency domain, a classical compromise between time-frequency concentration and amplitude fidelity; ω₀ = 6 is used. Scale s maps exactly to center frequency. With the band [Fmin, Fmax] and the number of scales J, the scales are distributed to give the center frequency of each band.

Fast FFT implementation and analytic signal

The source describes a frequency-domain implementation: transform the waveform, multiply by a Morlet filter for each scale, and inverse-transform to obtain time-varying coefficients. It uses positive-frequency filters for an analytic representation. The supplied gain formula conflicts with its stated center gain; the numerical normalization is withheld pending algorithm and software validation.

Physical amplitude scaling

The manual reports a NumPy cross-check, but no reproducible validation package was supplied for this website. Do not interpret a coefficient ridge as an independently verified physical peak amplitude until normalization, scale dependence and the reference signal have been checked.

• The FFT shows which frequencies are present; the wavelet transform shows when they occur.
• Events smeared in a single spectrum become clear in the waterfall.
• Coefficient ridges locate components; absolute peak-amplitude interpretation requires confirmation.
• Demonstration without hardware in test mode.
• Large transforms are accelerated by a compiled DSP library, so the 3D view rotates smoothly.
• The 3D image copies to the clipboard and the coefficient matrix exports for further analysis.

1. Select the analysis channel, or enable test mode for a demonstration without hardware.
2. Set the lower and upper frequency, the number of scales and ω₀; the defaults suit first use.
3. Click Refresh to generate the three views.
4. Rotate or pan the 3D view to find the best angle, and use the color-map cursor to locate features.
5. Copy the 3D image into the report, or export the coefficient matrix as CSV.

Recommended settings