Wavelet Transform

The wavelet transform is a operation that is not typically taught in school. I wasn't taught this but had to learn the more fundamental transforms of Fourier and Laplace first, which is fair. Still, the wavelet trasnform is much more practical in everyday use.

The Fourier transform doesn't give us any knowledge on how a transient signal (one that changes with time) develops. All those pretty spectrograms can't be created using a Fourier transform because all the time domain information is transformed into the frequency domain. This is because of the uncertainty principal: there can only be so much information in the product of the change in time and frequency $\Delta t \Delta \omega \geq \frac{1}{2}$.

The uncertainty principal quantum mechanics that everyone knows about is Heisenberg's Uncertainty,$\Delta x \Delta p \geq \frac{h}{4 \pi}$ is directly related to the uncertainty principle from signal processing.

But the Fourier Transform doesn't leverage something that we intuitively know about the timescales of signals: low frequency signals take a long time to repeat and high frequency signal repeat much more rapidly. So, with this additional information we can create a new tool that can give us more information about the time dependent frequency distribution

Wavelets: a mathematical microscope