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I noticed that there are many ways to relate the scale factor of wavelets to some characteristic frequency, such as the peak frequency, the central instantaneous frequency, and so on(plz see section 2.8 of Aguiar-Conraria2014 for more details). However, I was puzzled to choose which one as the Fourier frequency( or period) equivalent.

For example, if I have an analyzing wavelet whose formula is shown below $$ \psi(x)=\frac{1}{4\sqrt{\pi}}(2-x^2)e^{-x^2/4}, $$ then how to relate the scale factor of its daughter wavelets, i.e. $\frac{1}{\sqrt{a}}\psi(\frac{x-a}{b})$ to the Fourier frequency (period) which is usually used to characterize the spectral properties of signals.

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    $\begingroup$ I'll also write an answer. $\endgroup$ Jul 19 at 14:16
  • $\begingroup$ @OverLordGoldDragon Welcome to post your answer. $\endgroup$
    – Wang Yun
    Jul 19 at 14:30
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The "frequency" in "center frequency" does refer to Fourier frequency - rate of sinusoidal oscillation - but measures (and interpretations) can vary depending on the wavelet.

Said interpretations are often obfuscated in mathematics and jargon; my goal will be to illustrate these intuitively. I'll contrast three notions of "center frequency", and three mappings of scale to frequency - starting with Morlet, then generalizing.

Building a wavelet

Sample by sample; frequency domain zoomed to 32 samples (out of 128), showing real part of resulting ifft:

enter image description here

For Morlet, we have a "bell"; let's make a minimal bell out of three samples. First, visualize what the sides look like:

enter image description here

It looks like a single amplitude-modulated sinusoid (carrier) - and it is:

$$ \cos(f_1t) + \cos(f_2t) = 2 \cos(.5(f_1 - f_2)t)\cos(.5(f_1 + f_2)t) \tag{1} $$

The modulation frequency is $f_m=(f_1 - f_2)/2 = 1$, and carrier frequency $f_c=(f_1 + f_2)/2 = 16$. Note, $f_c$ equals the earlier frequency, which for the symmetric bell is the peak and the center frequency; overlapped:

enter image description here

Now add these two to the earlier peak, and we obtain a minimalistic wavelet (centered (fftshift) for clarity):

enter image description here

So as long as we add frequency-domain samples symmetrically about some center, it's same as adding AM carriers of the same frequency, which meaningfully interprets as a sinusoid of that frequency that's decayed.

Example with actual Morlet at same center frequency:

enter image description here

Measuring "center frequency"

In above example, the center frequency is $16$. How would we compute it? For a symmetric, positive, and real $\hat x$ (freq-domain), the simplest approach is a weighted average of all bins: $f_\psi = \sum k \cdot \psi[k]$. For the "minimalistic wavelet", that's $.5 \cdot 15 + 1 \cdot 16 + .5 \cdot 17 = 32$. Why not 16? What's missing is to normalize by the wavelet's norm: $\cdot \frac{1}{\sum \psi[k]} = \cdot \frac{1}{.5 + 1 + .5} \rightarrow 32 \cdot \frac{1}{2} = 16$.

In the general case we take $|\psi|$ instead, and square it to obtain an energy measure, which can be seen as attenuating small values in favor of dominant ones. In continuous-time, using radian frequencies:

$$ \tilde \omega_\psi = \frac{\int_0^\infty \omega |\hat\psi(\omega)|^2 d\omega} {\int_0^{\infty} |\hat\psi(\omega)|^2 d\omega} \tag{2} $$

Squaring allows 1) interpreting $|\hat\psi|^2$ as a probability distribution, 2) easily translating with related quantities (e.g. time or frequency resolution) via Parseval-Plancherel's theorem. For symmetric $|\hat\psi|$, it coincides with $|\hat\psi|^1$.

Note, we justify taking $|\psi|$ as, otherwise negative frequency values (not bins) would cancel the positives, which isn't valid since the bins are orthogonal:

enter image description here

It's the same center frequency, but phase shifted:

$$ \cos(f_1t) - \cos(f_2t) = - 2 \sin(.5(f_1 - f_2)t)\sin(.5(f_1 + f_2)t) \tag{3} $$

Peak frequency

$\tilde \omega_\psi$ is known as the "energy" or "mean" frequency. Another measure is the peak frequency, $\omega_\psi$, which is the frequency at which $|\hat\psi|$ peaks (is maximum).

To illustrate the difference, consider a strictly-analytic Morlet near Nyquist:

enter image description here

(Image altered - see Note 1) Clearly, mean != peak here. Which is "more accurate"? I'll comment bit later. For now, the key property of the peak center frequency is, it's the sinusoidal oscillation the most resonant with the wavelet - that is, it's the "best-fitting" sinusoid:

enter image description here

This isn't a surprise: the Fourier transform exactly computes similarities of input with sinusoids at different frequencies, so where F.T. peaks is where's most similarity.

For a symmetric $|\hat\psi|$, the two are equal: $\omega_\psi = \tilde\omega_\psi$.

Mapping scale to frequency

In context of CWT, a standard form of the wavelet is called the "mother wavelet", and the "daughter wavelets" are its scaled versions. We can define mappings from scale to frequency as analogues of wavelet center frequency measures.

Consider $x_o = \cos(\omega_o t)$. Its CWT with a strictly analytic wavelet is:

$$ W_o(s, t) = \frac{1}{2} \hat{\psi^{*}} (s\omega_0) e^{j\omega_o t} \tag{4} $$

(Time-domain conv $\Leftrightarrow$ freq-domain mult, $\hat{\psi^{*}}(s \omega) \cdot \delta(\omega - \omega_o)$, negative frequencies vanish per analyticity, and the exp is per time-shift.) The scale at which the maginute of CWT is maximum, obtained by solving

$$ \frac{\partial}{\partial s} |W_o(s, t)|^2 = 0, \tag{5} $$

is $s = s_\psi \equiv \omega_\psi / \omega_o$. This is the peak scale. To better interpret it, note that this is the same as the scale at which the rate of change of CWT's phase is equal to the signal frequency - that is, the scale at which below is satisfied:

$$ \frac{\partial}{\partial t} \Im m \left\{ \ln(W_o(s, t)) \right\} = \frac{\partial}{\partial t} \arg \left\{ W_o(s, t)) \right\} = \omega_o \tag{6} $$

($\ln(W_o(s, t)) = i\omega_o t + \hat{\psi^{*}}(s \omega_o) + \ln(\frac{1}{2})$ -- $\Im m \rightarrow \omega_o t$ -- $\partial/\partial t \rightarrow \omega_o$, set equal to $\omega_o$ and solve for $s$. See Note 2)

The peak frequency $\omega_\psi$ thus controls the location of CWT amplitude maximum, and the rate of phase progression, of an oscillatory feature much broader in time than the wavelet.

Mean scale

Similarly, we define the energy analogue for scale-freq mapping:

$$ \tilde s_\psi = \frac{\int_0^\infty s|W_o (s,t)|^2 ds}{\int_0^{\infty} |W_o(s, t)|^2 ds} = \frac{\int_0^\infty s|\hat\psi (s\omega_o)|^2 ds} {\int_0^{\infty} |\hat\psi (s\omega_o)|^2 ds} \tag{7} $$

With a change of variables, this is simply $\tilde s_\psi = \tilde \omega_\psi / \omega_o$. Thus $\tilde\omega_\psi$ determines the scale at which the mean of CWT energy of a sinusoidal signal occurs.

Disagreement example

To illustrate the mapping distinctions, take CWT of a sinusoid near Nyquist, since that's where a strictly analytic Morlet is asymmetric and has different mappings to frequency:

enter image description here

$s_\psi$ will be at the lowest scale, as expected, and $\tilde s_\psi$ a bit greater. Intuitively, the energy frequency is the first moment of $|\text{CWT}|^2$, or the "center of mass" or "expected value" of time-frequency energy of a sinusoidal oscillation.

Additional measure: instantaneous

Does a wavelet oscillate at same frequency throughout all of its decay? This is the question the next measure answers; the instantaneous center frequency of a wavelet is given by the rate of change of its phase:

$$ \breve \omega_\psi = \frac{d}{dt} \Im m \left\{ \ln(\psi(t)) \right\} = \frac{d}{dt} \left\{ \arg{(\psi(t)} \right\} \tag{8} $$

evaluated at wavelet center: $\breve \omega_\psi = \breve \omega_\psi(0)$. If $\breve \omega_\psi$ differs from $\tilde \omega_\psi$, then the frequency-domain wavelet is asymmetric (look back to "Building a wavelet"; the AM cosines will produce different $(f_1 + f_2)/2$).

More generally, $\breve\omega_\psi \neq \tilde\omega_\psi$ indicates that wavelet frequency content is not uniform in time, and the deviations can be quantified with $|\breve\omega_\psi(t) - \tilde\omega_\psi|$.

Like for peak and energy, there exists a scale analogue: let $W_\delta (s, t)$ be the CWT of a Dirac delta $\delta (t)$. The rate of change of phase is:

$$ \frac{\partial}{\partial t} \Im m\left\{ \ln(W_\delta (s, t)) \right\} = \frac{1}{s} \breve{\omega_\psi} \left(\frac{t}{s}\right) \tag{9} $$

which, at the location of the delta ($t=0$), is $\breve\omega_\psi (0) / s$. The wavelet central (time-center) instantaneous frequency $\breve\omega_\psi$ therefore controls the rate of phase propagation at the center of a feature much narrower in time than the wavelet.

Non-uniform $\breve \omega_\psi$ example

Since the Nyquist-trimmed Morlet is asymmetric, it qualifies, but it's hard to tell per discretization limitations so instead consider the same but at greater sampling frequency:

enter image description here

enter image description here

Note that the frequency at the center is different from that at the sides. However, it appears that decayed frequency stabilizies, and matches peak center frequency. To confirm, take synchrosqueezed CWT:

enter image description here

This behavior won't always manifest for freq-asymmetric wavelets, but it can be shown to do so with a "trimmed symmetric". Note that synchrosqueezing will yield the correct instantaneous frequency of regardless of wavelet's center frequency (limitations apply, but not here), so we don't run into circular reasoning with "use wavelet to measure wavelet".

Summary

Three center frequency measures:

  1. Energy, $\tilde\omega_\psi$: "center of mass", broadset in time frequency
  2. Peak, $\omega_\psi$: "most dominant" frequency
  3. Instantaneous central, $\breve\omega_\psi$: narrowest in time frequency, or "most dominant for narrow features"

and analogue scale-to-frequency mappings:

  1. Energy, $\tilde\omega_s \equiv \tilde \omega_\psi / s$: correctly gives mean of energy scales of the transform
  2. Peak, $\omega_s \equiv \omega_psi / s$: correctly gives frequency of scale at which CWT of a sinusoid obtains maximum
  3. Instantaneous central, $\breve\omega_s \equiv \breve\omega_\psi(0) / s$: correctly gives frequency to be same as rate of phase progression of CWT at the location of an infinitesimally narrow pulse.

That is, we define a scale-to-frequency mapping in context of CWT as the scale at which the CWT satisfies some criterion. Each of these "valid" in different ways, and what's most useful will vary by application (and wavelet).

For more complicated wavelet behaviors, the interpretations are about the same - only the results different.

Mapping nonlinearity

The relations in the "Summary" will not always hold. They imply that scale is always inversely proportional to center frequency, which isn't the case. An example is Morlet, which has a "corrective term" to ensure zero mean, and it becomes pronounced for very low frequencies (see interactive):

enter image description here

The surest mapping is obtained by evaluating the defining expressions directly.

"Fourier" frequency

The first two mappings are defined directly in terms of CWT's interaction with a pure sinusoid, and thus derived interpretations are exact. The same actually holds for the third measure, since delta is the limit of sinc, and we don't care about the $1/x$ since $\Delta x \rightarrow 0$ and the interpretation of "instantaneous sinusoid" remains.

Let us visualize each of the three for the "trimmed Morlet":

enter image description here

Left I show results for the near-Nyquist case as it's easier to interpret in terms of CWT of a pure sinusoid (but same can be done with exact case).

enter image description here

Peak.

enter image description here

Time width of sinusoid of exaggerated for clarity, but its norm rescaled accordingly.

Period

In sense of $T$ such that $\psi(t) = \psi(t + T), \forall t$, wavelets have no such period, as they decay permanently. One can define a pseudo-period as the inverse of center frequency, or a statistical measure as described here.

Multi-peak wavelets

For example, from here:

enter image description here

What's the peak center frequency? It's both; we interpret the wavelet as a multi-component signal, with two modes, and respective peak center frequencies. Since in CWT we seek a single measure, however, this isn't ideal - and thus energy center frequency is preferred (though it too can be interpreted in multi-modal regime).

Alternatively, one can take the dominant peak in time-frequency sense, via instantaneous energy localization (synchrosqueezing):

enter image description here

thus a max scale of $|\text{SSQ_CWT}(\psi(t))|^2$. This has implications for ridge analysis - see 4.5.1 Wigner-Ville Distribution.

"Signal model" approach

As described in the other answer, one can use a reference signal for which we have a good idea of what qualifies as "the frequency". This is sound, but how do we measure its frequency to begin with? And as shown, the center frequency of daughter wavelets will not always be inversely proportional to scale. Thus we require an absolute measure, independent of any reference - and the three described in this answer are such measures.

Notes

1: The time-domain wavelet in that image is actually for the spectrum below, and that image's spectrum gives below time-domain wavelet. Proper analyticity requires halving the Nyquist bin -- see discussion.

enter image description here

2: in this case we observe the quantity is independent of $s$, which states that the CWT has same phase rate of change at every scale. This is accurate: see near top here. The intensity (amplitude envelope) of this phase, however, will only peak at $s_\psi$.

Code

Available at Github.

References

  1. Higher-Order Properties of Analytic Wavelets - J. M. Lilly, S. C. Olhede
  2. Wavelet Tour, Ch1-4 - S. Mallat

Few statements were copied verbatim from 1.

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  • $\begingroup$ You see, even if you post the code, one could appreciate a good answer. $\endgroup$
    – Royi
    Jul 21 at 4:16
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The document “The Continuous Wavelet Transform: A Primer” by Luís Aguiar-Conraria and Maria Joana Soares, which you are referring to, says it well:

  • this [standard] inverse relation between scale and frequency corresponds to a particular interpretation and that there are other meaninful ways of assigning frequencies to scales.
  • from Meyers (1993): "for a general wavelet, the relation between scale and the more common Fourier wavelength is not necessarily straightforward [...] In those cases it is probably a meaningless exercise to find a relation between the two disparate measurements of distance".

Therefore, some people prefer to speak about "pseudo-frequency corresponding to scale". The correspondence is even more complicated due to the different sampling schemes used in the actual implementation of the CWT discretization.

What I use in that situation is more practical than theoretical: I use a "model signal" of the phenomenon that I want to analyze, with some parametric frequency content. A classical model could be a damped sine, some asymmetric lineshape. From this model, one should be able to derive a "frequency content" of interest. Then you can compare the frequency localization $f_F$ in the Fourier to the corresponding feature observed in the wavelet domain, as a pseudo-frequency $\phi_w$. Hopefully, you will find a linear (or an affine) relation behind the "actual frequency" and the corresponding "wavelet pseudo-frequency"

$$\phi_w \simeq \alpha f_F \quad (+\beta) $$

and propagating this rule-of-thumb matching to the scale (by inverse transformation) or to daughter wavelets with the scaling law behind the signal and its spectrum:

$$ s(\alpha t) \mapsto \frac{1}{|\alpha|} S \left( \frac{f}{|\alpha|}\right) $$

could provide you with an application-dependent rule.

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