Wavelet Transform Scale-to-Frequency Calculator
Convert a continuous wavelet transform scale into an equivalent pseudo-frequency, for reading scalograms in familiar Hz terms.
🌊 What is Wavelet Scale-to-Frequency Conversion?
Wavelet scale-to-frequency conversion translates the abstract "scale" axis of a continuous wavelet transform (CWT) into an equivalent frequency in Hz, so scalograms can be read and compared using the same familiar frequency units as an FFT spectrogram. A wavelet transform naturally organizes its output by scale, not frequency, since scale is what actually controls how the mother wavelet is stretched or compressed, but frequency is what most engineers actually want to see on an axis label.
Neuroscientists convert EEG and MEG wavelet scales to frequency to identify standard brain rhythm bands (delta, theta, alpha, beta, gamma) directly on a scalogram. Structural health monitoring engineers convert vibration-signal wavelet scales to frequency to match detected transients against known resonance or fault frequencies. Geophysicists analyzing seismic wavelets use the same conversion to relate CWT scale ridges to physically meaningful frequency content in earthquake or exploration data.
A common misconception is that wavelet "frequency" works exactly like an FFT bin frequency. It does not: because a wavelet is a localized, oscillating waveform rather than a pure sinusoid, its pseudo-frequency is an approximation of its dominant oscillation rate, useful and widely adopted, but not an exact one-to-one correspondence the way FFT bins are.
This calculator applies the standard pseudo-frequency formula used by wavelet toolboxes, letting you convert any scale value into Hz for the Morlet wavelet, the Mexican Hat wavelet, or any other mother wavelet whose center frequency you supply.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Morlet Wavelet at Scale 10
Morlet (Fc = 0.8125), a = 10, fs = 1,000 Hz
Example 2 — Mexican Hat Wavelet at Scale 4
Mexican Hat (Fc = 0.25), a = 4, fs = 500 Hz
Example 3 — Custom Wavelet at Scale 50
Custom (Fc = 1.0), a = 50, fs = 44,100 Hz
❓ Frequently Asked Questions
🔗 Related Calculators
What is a wavelet transform scale?
In a continuous wavelet transform (CWT), the scale parameter a stretches or compresses the mother wavelet in time before it is correlated against the signal. Small scales use a compressed (narrow) wavelet that responds to fast, high-frequency features; large scales use a stretched (wide) wavelet that responds to slow, low-frequency features.
How do you convert wavelet scale to frequency?
The standard pseudo-frequency formula is Fa = Fc * fs / a, where Fc is the mother wavelet's own center frequency (a fixed property of its shape), fs is the sampling rate, and a is the scale. This lets a scalogram's scale axis be relabeled in familiar Hz terms.
What is the center frequency Fc of a wavelet?
Fc is the dominant oscillation frequency built into the mother wavelet's own shape, at scale a = 1 and a sampling rate of 1 Hz. It depends only on the wavelet family, not on any input signal, and is typically published in wavelet toolbox documentation (the Morlet wavelet's Fc is about 0.8125; the Mexican Hat's is about 0.25).
Why is a small scale associated with a high frequency?
A small scale compresses the mother wavelet in time, packing its characteristic oscillations into a shorter window, which corresponds to a higher-frequency wiggle. Since the pseudo-frequency formula divides by scale, a smaller a directly produces a larger Fa, matching this intuition.
Why is wavelet frequency called a 'pseudo-frequency' instead of an exact frequency?
Unlike a Fourier basis function, a wavelet is not a pure sinusoid, it is a localized, oscillating waveform with its own bandwidth around a dominant frequency. The pseudo-frequency is a practical, widely used approximation of that dominant frequency, useful for labeling axes and comparing to Fourier-based analysis, but not a bin-exact equivalence.
Why do wavelet scales get chosen logarithmically instead of linearly?
Logarithmically spaced scales give roughly constant relative frequency resolution across octaves (similar to how musical notes and human pitch perception work), letting a single scalogram efficiently resolve both fast, high-frequency transients and slow, low-frequency trends without needing an impractically large number of scales.
How does wavelet analysis differ from STFT for the same purpose?
Both trade time resolution against frequency resolution, but a fixed-window STFT uses the same time/frequency trade-off at every frequency, while a wavelet transform automatically uses short (fine time, coarse frequency) windows at high frequencies and long (coarse time, fine frequency) windows at low frequencies, adapting the trade-off per octave.
What is the Morlet wavelet used for?
The Morlet wavelet, a sinusoid modulated by a Gaussian envelope, is the most common choice for time-frequency analysis of oscillatory signals like EEG rhythms, audio, and vibration data, since its smooth, symmetric shape gives a clean, easily interpreted scalogram.
What is the Mexican Hat (Ricker) wavelet used for?
The Mexican Hat wavelet, the negative normalized second derivative of a Gaussian, is well suited to detecting sharp, symmetric peaks and edges in a signal (seismic wavelets, blob detection in images), since its shape closely matches an isolated spike or localized feature rather than a sustained oscillation.
Can I use this calculator for any wavelet family, not just Morlet or Mexican Hat?
Yes. Selecting Custom lets you enter any center frequency Fc, so this calculator works for any mother wavelet as long as you know (or look up) its own Fc value, since the pseudo-frequency formula itself is the same regardless of which wavelet family it came from.