The Gaussian mother wavelet with frequency resolution $\,\alpha\equiv\omega/\sigma_\omega\,$ is given by
I'm using it to do a continuous wavelet transform (CWT) to a real singal $\,x(t)\,$ to obtain a time-dependent spectrum
In the limit of $\,\alpha\rightarrow\infty$, the transformation approaches the time-independent Fourier transform. Gaussian wavelets have the best time-frequency resolution as they hit the bound of the uncertainty relation. I'd like to know how one can reconstruct the signal $\,x(t)\,$ from the spectrum $X_\alpha(\omega,t)\,$ at a given $\,\alpha$. Is the wavelet basis overcomplete? Is there a reconstruction formula for the inverse CWT, or does the overcompleteness of the basis mean the reconstruction formula is not unique?