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We know that Heisenberg uncertainty Principle states that $$\Delta f \Delta t \geq \frac{1}{4 \pi}.$$
But (in many case for Morlet wavelet) I have seen that they changed the inequality to an equality. Now my question is when are we allowed to change the inequality to an equality:
$$\Delta f \Delta t = \frac{1}{4 \pi} $$
why =
It is important to define the time and frequency widths $\Delta_t$ and $\Delta_{\omega}$ of a signal before discussing any special forms of the uncertainty principle. There is no unique definition of these quantities. With appropriate definitions it can be shown that only the Gaussian signal satisfies the uncertainty principle with equality.
Consider a signal $f(t)$ with Fourier transform $F(\omega)$ satisfying
$$\int_{-\infty}^{\infty}f^2(t)dt=1\quad\textrm{(unit energy)}\\ \int_{-\infty}^{\infty}t|f(t)|^2dt=0\quad\textrm{(centered around }t=0)\\ \int_{-\infty}^{\infty}\omega|F(\omega)|^2d\omega=0\quad\textrm{(centered around }\omega=0)$$
None of these conditions is actually a restriction. They can all be satisfied (for signals with finite energy) by appropriate scaling, translation and modulation.
If we now define time and frequency widths as follows
$$\Delta_t^2=\int_{-\infty}^{\infty}t^2|f(t)|^2dt\\ \Delta_{\omega}^2=\int_{-\infty}^{\infty}\omega^2|F(\omega)|^2d\omega$$
then the uncertainty principle states that
$$\Delta_t^2\Delta_{\omega}^2\ge\frac{\pi}{2}\tag{2.6.2}$$
(if $f(t)$ vanishes faster than $1/\sqrt{t}$ for $t\rightarrow\pm\infty$)
where the inequality is satisfied with equality for the Gaussian signal
$$f(t)=\sqrt{\frac{\alpha}{\pi}}e^{-\alpha t^2}\tag{2.6.3}$$
The equation numbers above correspond to the proof below which is from Wavelets and Subband Coding by Vetterli and Kovacevic (p.80):
I cannot give you all the theory behind this (as it literally fills books), but it turns out that Heisenberg becomes an exact equality for precisely this family of signals:
$$ s_{t_0,\omega_0,\sigma,\phi,\gamma}(t) = \exp\left(-\left(\frac{t-t_0}{\sigma}\right)^2 + i \left(\phi + \omega_0 (t-t_0) + \gamma (t-t_0)^2\right)\right) $$
where all parameters are real numbers. This family is generated by quadratic symplectomorphisms in time-frequency from a single Gabor atom. These symplectomorphisms preserve the Heisenberg uncertainty relation.
Edit: Let me make this more precise and also in fact more correct. The signals I gave above minimise the time-frequency area, but not the time-frequency uncertainty product. If you want minimal $\Delta F \cdot \Delta T$ then $\gamma$ from above must vanish.
The notion of time frequency area can however be generalised to measure the area of shapes that are not aligned with the time and frequency axis. That means instead of the uncertainty product between F and T we measure the minimal uncertainty product of any two conjugate variables spanned by F and T. I'll spare you the details, but for this definition of time-frequency area the family of signals gives you the minimum.
The uncertainty principle sets up a theoretical bound for resolution, so it is never written as an equality.
The equality relationships you are encountering are for for a specific analysis context and analysis implementation. In this case the context is signal analysis so time/frequency are the conjugate variables of interest, and the implementation is the specific wavelet in use.
The equality relationship provides a way of comparing resolutions across different analysis implementations. Care must be taken when interpreting these relationships because the definition of resolution shouldn't, but may vary.
An equality relationship is appropriate once you have defined two things: 1) the mathematical meaning of resolution. 2) the method of analysis (in this case, choice of wavelet).
