# Understanding the units of wavelet time & frequency resolution

While using a continuous wavelet transform for my research project, I came up with some questions.

Having a set of equations for a Morlet wavelet,

$$\psi(t)=\sqrt{\frac{2}{\pi}}e^{-t^2/2}e^{i6t}$$
$$\hat{\psi}(\omega)=2e^{-(\omega-6)^2/2}$$

I have calculated the time and frequency resolution (i.e., $$\Delta t$$ and $$\Delta \omega$$, respecitvely) such that

$$\sigma_\psi^2=\frac{\int_{-\infty}^{\infty}(t-t_0)^2|\psi(t)|^2dt}{\int_{-\infty}^{\infty}|\psi(t)|^2dt}=\frac{1}{2}$$ and $$\sigma_\hat{\psi}^2=\frac{\int_{-\infty}^{\infty}(\omega-\omega_0)^2|\hat{\psi}(\omega)|^2d\omega}{\int_{-\infty}^{\infty}|\hat{\psi}(\omega)|^2d\omega}=\frac{1}{2}$$ where $$t_0=0$$ and $$\omega_0=6=2\pi f_0$$ $$\Delta t=\sigma_t=s\sigma_\psi$$ and $$\Delta \omega=\sigma_\omega=\frac{\sigma_\hat{\psi}}{s}$$ where the scale $$s$$ is approximated with $$\frac{f_0}{f\cdot \text{sampling period}}$$

I have been trying to calculate the Heisenberg box as the intervals $$[t-\frac{\sigma_t}{2},t+\frac{\sigma_t}{2}]$$ and $$[\omega-\frac{\sigma_\omega}{2},\omega+\frac{\sigma_\omega}{2}]$$, but the computation resulted with counterintuitive values, with $$\Delta t$$ being extremely large and $$\Delta \omega$$ being extremely small for almost every $$t$$ and $$\omega$$ values I plugged in.

As I read some previous posts (including here), it seems the problem may be coming from the units of the resolutions. I read that the units of $$\Delta t$$ and $$\Delta \omega$$ are

$$[\Delta t] = \frac{\text{samples}}{\text{cycles}\cdot\text{radians}}$$
$$[\Delta \omega] = \frac{\text{cycles}\cdot\text{radians}}{\text{samples}}$$

If I convert samples to seconds, the resulting intervals for the Heisenberg box seem to make sense. However, even if I change samples to seconds, the units still include cycles in them, and I do not know how I should interpret the values qualitatively.

My main questions therefore are (1) if the equations defining $$\Delta t$$, $$\Delta \omega$$, and $$s$$ are mathematically correct, and (2) how I should change the units of time and frequency resolution so I can interpret them in seconds and Hz.

Thank you very much.

• Hz = cycles / second. (1) are correct. If your question is rather why the values are counterintuitive, it'd help to see a concrete example, and what you expect values instead to be. May 13, 2021 at 14:42
• @OverLordGoldDragon Thank you for your comment. For example, having $f_0=\frac{\pi}{3}$ and sampling period of $\frac{1}{512}$, if I test on $f=40$ Hz, I get $s=13.4041$, $\sigma_t = 9.4782$, and $\sigma_\omega=0.0528$. What I am confused about is if I can interpret $\sigma_t$ as 9.4782 seconds width and $\sigma_\omega$ as 0.0528 Hz width on a time-frequency plane, or if this is not a correct interpretation.
– scho
May 13, 2021 at 15:36
• @OverLordGoldDragon Thank you. And sorry for my confusions, but wouldn't $[\sigma_\omega\cdot f_s]=\frac{\text{cycles}\cdot\text{radians}}{\text{samples}}\frac{samples}{sec}=$rad-Hz? Also, how does $\sigma_t$ end up with the seconds unit? Are the units I wrote above incorrect?
– scho
May 13, 2021 at 15:54
• Ignore my deleted reply, I get where the confusion is - am writing an answer. May 13, 2021 at 15:55
• Oh okay, thank you very much!
– scho
May 13, 2021 at 15:55

Confusion seems rooted in notation:

• In $$s = f_0 / (f \cdot \text{sampling rate})$$, $$f_0$$ is the center frequency of mother $$\psi$$, and $$f$$ is the center frequency of the scaled $$\psi$$, i.e. $$\psi_s$$.
• In $$\sigma_{\hat\psi}=$$ ..., $$\omega_0$$ is the (radian) center frequency of the wavelet of interest, or $$\psi_s$$, rather than $$\omega_0=2\pi f_0$$.
• However, you can compute sigma of scaled from sigma of mother via: $$\sigma_{\hat\psi_s} = \sigma_{\hat\psi} / s$$ (see under 4.54).
• $$\text{sampling rate}$$ is only relevant in discretized computations and must get canceled to yield physical units (Hz, sec, etc), so if you seek only physical use $$s = f_0 / f_s$$ (adjusted $$f\rightarrow f_s$$ to avoid confusion; recall that $$f$$ (or $$\omega$$) is what we integrate over, rather than plug in)
• Do not involve $$\text{sampling rate}$$ in defining $$\psi, \hat\psi$$, scaled, or computing $$\sigma$$, or effectively most formulas you come across unless they explicitly include it, it's only to be used after computation, e.g. you first find $$\sigma_{\hat\psi}$$ without $$\text{sampling rate}$$, and then rescale as in $$(4)$$ here.
• You commented "$$\sigma_\omega$$ Hz", but it's actually "rad-Hz"; to get Hz do $$f=\omega / (2\pi)$$.

Let me know if this makes sense (if not, again with a specific example).

• Thank you for the answer. Now it makes much more sense to me. I have some clarification questions for the third bullet point. First of all, I think $\psi$ should be $\hat{\psi}$ such that $\sigma_{\hat{\psi}_s}=\sigma_{\hat{\psi}}/s$? Please let me know if I am misunderstanding this point. Also, since I have $\sigma_\hat{\psi} = 1/\sqrt{2}$, for the sigma of scaled, I should calculate $\sigma_{\hat{\psi}_s} = \frac{1}{\sqrt{2}s}$. Am I right?
– scho
May 13, 2021 at 16:25
• Right, typo (fixed), also added a bullet. And you're right on scaled sigma. May 13, 2021 at 16:58